Brain Struct Funct DOI 10.1007/s00429-014-0936-0
ORIGINAL ARTICLE
The connectome of the basal ganglia Oliver Schmitt • Peter Eipert • Richard Kettlitz Felix Leßmann • Andreas Wree
•
Received: 27 February 2014 / Accepted: 30 October 2014 Ó Springer-Verlag Berlin Heidelberg 2014
Abstract The basal ganglia of the laboratory rat consist of a few core regions that are specifically interconnected by efferents and afferents of the central nervous system. In nearly 800 reports of tract-tracing investigations the connectivity of the basal ganglia is documented. The readout of connectivity data and the collation of all the connections of these reports in a database allows to generate a connectome. The collation, curation and analysis of such a huge amount of connectivity data is a great challenge and has not been performed before (Bohland et al. PloS One 4:e7200, 2009) in large connectomics projects based on meta-analysis of tract-tracing studies. Here, the basal ganglia connectome of the rat has been generated and analyzed using the consistent cross-platform and generic framework neuroVIISAS. Several advances of this connectome meta-study have been made: the collation of laterality data, the network-analysis of connectivity strengths and the assignment of regions to a hierarchically organized terminology. The basal ganglia connectome offers differences in contralateral connectivity of motoric regions in contrast to other regions. A modularity analysis of the weighted and directed connectome produced a specific grouping of regions. This result indicates a correlation of structural and functional subsystems. As a new finding, significant reciprocal connections of specific network motifs in this connectome were detected. All three principal basal ganglia pathways (direct, indirect, hyperdirect) could be determined in the connectome. By identifying these pathways it was found that there exist many further equivalent pathways possessing the same length and mean O. Schmitt (&) P. Eipert R. Kettlitz F. Leßmann A. Wree Department of Anatomy, University of Rostock, Rostock, Germany e-mail: [email protected]
connectivity weight as the principal pathways. Based on the connectome data it is unknown why an excitation pattern may prefer principal rather than other equivalent pathways. In addition to these new findings the local graphtheoretical features of regions of the connectome have been determined. By performing graph theoretical analyses it turns out that beside the caudate putamen further regions like the mesencephalic reticular formation, amygdaloid complex and ventral tegmental area are important nodes in the basal ganglia connectome. The connectome data of this meta-study of tract-tracing reports of the basal ganglia are available for further network studies, the integration into neocortical connectomes and further extensive investigations of the basal ganglia dynamics in population simulations. Keywords Connectome Connectomics Basal ganglia Caudate putamen Striatum Substantia nigra Neuroontology Digital atlasing Tract tracing Multiscale Network analysis Graph analysis Abbreviations A All (all inputs and outputs) AD Average degree Ac Accumbens nucleus AC Amygdaloid complex AGl Lateral agranular prefrontal cortex AGm Medial agranular prefrontal cortex Aut Authoritativeness AW Average weight BC Betweenness centrality BG Basal ganglia C Circle CC Cluster-coefficient CE Closeness centrality
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chain CL CM CNS CPu DG Dic Dii Dis DNN Doc Doi Dos DR EC Ent HIPP Hub I in INN L LGP LHb MDL MDM MDS MGP MRF O out PC PCA PF Pir PL PRC Pub RADin RADout Rec Rel Sic Sii Sis SG SNC SNR Soc Soi Sos
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Chain pattern of a motif Centrolateral thalamic nucleus Central medial thalamic nucleus Central nervous system Caudate putamen Degree Direct input from contralateral Direct input from ipsilateral Direct input from ipsi- and contralateral Direct neighbor network Direct output to contralateral Direct output to ipsilateral Direct output to ipsi- and contralateral Dorsal raphe nucleus Eigenvector centrality Entorhinal cortex Hippocampus Hubness In (input to a region; used in tables) Symmetric input connection to a central node of a motif Indirect neighbor network Laterality Lateral globus pallidus Lateral habenular nucleus Mediodorsal thalamic nucleus lateral part Mediodorsal thalamic nucleus medial part Metric multidimensional scaling Medial globus pallidus Mesencephalic reticular formation Out (Output of region; used in tables only) Symmetric output connection from a central node of a motif Paracentral thalamic nucleus Principal component analysis Parafascicular thalamic nucleus Piriform cortex Path length Page rank centrality Number of articles Radiality of the input Radiality of the output Reciprocal Reliability Subtree input from contralateral Subtree input from ipsilateral Subtree input from ipsi- and contralateral Subgraph centrality Substantia nigra compact part Substantia nigra reticular part Subtree output to contralateral Subtree output to ipsilateral Subtree output to ipsi- and contralateral
SP SPN STh VA VL VM VTA
Length of shortest path Spiny neurons of the CPu Subthalamic nucleus Ventro anterior thalamic nucleus Ventrolateral thalamic nucleus Ventromedial thalamic nucleus Ventral tegmental area A10
Introduction The basal ganglia (BG) are composed of the four major nuclei neostriatum, globus pallidus (GP), substantia nigra (SN) and subthalamic nucleus (STh). The canonical basal ganglia system consists of the BG and of those cortical areas that are connected with BG (Gerfen and Bolam 2010). In rodents the neostriatum is a single nucleus, the caudate putamen (CPu) which is divided by the internal capsule into the caudate nucleus and putamen in higher vertebrates. The GP consists of an external segment or lateral globus pallidus (LGP) and the internal segment or medial globus pallidus (MGP). The MGP in rodents is also called the entopeduncular nucleus. The SN consists of the pars compacta (SNc) and the pars reticulata (SNr). These nuclear complexes are related to the dorsal aspects of the BG. The ventral aspects contain the accumbens nucleus (Ac), the ventral pallidum and the medial aspects of the STh and SN. The dorsal aspect of the basal ganglia has motor and associative functions, whereas the ventral aspect is related to limbic functions. Additional motor components that are strongly associated with the dorsal aspect of the BG are the frontal cortex especially the primary motor cortex (lateral agranular prefrontal cortex) and subnuclei of the thalamus (Gerfen 2004). Principal afferents of the BG arise from the ipsi- and contralateral cerebral cortex (layer 5 glutamatergic neurons), intralaminar thalamic nuclei such as the centromedian and parafascicular nucleus, the dorsal raphe nucleus (DR) and the amygdaloid complex (AC). The most extensive input to the BG is glutamatergic from the cerebral cortex and thalamus. These projections terminate in the heads of spines of the GABAergic spiny neurons (SPN) of the CPu. The latter are the principal projection neurons innervating the SN and GP. These SPN possess a dense local axon collateral system that innervates other SPN and interneurons of the CPu. At the level of macroconnectomes and macrocircuits all three major pathways of the BG are represented by the connectome data in this study. The direct pathway is an excitatory pathway which has the same source region (CPu) and BGoutput region (STh) as the inhibitory indirect pathway. The macrocircuitry of the BG is dominated by these two
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principal pathways (direct, indirect) (Alexander and Crutcher 1990; DeLong 1990; Smith et al. 1998) and a hyperdirect pathway (Kita and Kitai 1994) which will be investigated in the introductory part of the ‘‘Results’’. These pathways along with the cerebral cortex and thalamus form three loops (cortical-basal ganglia loops). It is known that the GABAergic neurons in the LGP influence the activity at every level of the BG by their extensive axon collaterals (Tepper et al. 2007). Single cell labeling studies (Kita and Kitai 1994; Bevan et al. 1998; Sato et al. 2000) have indicated that LGP projection neurons have axonal collaterals to the STh and BG output nuclei. Additionally, up to one-third of LGP neurons project to the CPu (Kita and Kitai 1994; Bevan et al. 1998) and to its GABAergic interneurons which innervate the spiny projection neurons (Kita 1993; Koos and Tepper 1999; Bennett and Bolam 1994). Besides the major inputs to and outputs from the BG nearly all areas of the neocortex are interconnected with regions of the BG (Gerfen and Bolam 2010). In terms of functional segregation, frontal cortical areas and intralaminar thalamic nuclei are involved in planning and execution of movement behavior. Outputs of the BG to the intermediate layers of the superior colliculus are involved in the generation of eye and head movements and output to the pedunculopontine nucleus is necessary for orienting movements of the body. A functional and hodological grouping of subregions the BG into four categories has been proposed (Parent 1986; Alexander and Crutcher 1990; DeLong 1990; Smith et al. 1998; Gerfen and Bolam 2010). The input nuclei are the CPu and STh which receive cortical inputs. The output nuclei are the MGP and SNR which project to the thalamus, midbrain and brainstem. The LGP is a relay nucleus connecting the input and output nuclei. The modulator nucleus is the SNC. The major objectives of this study are 1. 2. 3. 4.
the collation of connectivity data, a weighted and directed analysis of the BG connectome, the analysis at two levels of detail and the intrinsic, extrinsic, and differential analysis
which are specified in the following. In addition to this general organization of the BG abundant information of interconnections and densities of connections between topographic, functional, cytoarchitectonic and chemoarchitectonic subdivisions of BG regions (intrinsic BG connections) is available. More specifically, afferent, efferent intrinsic and extrinsic connections of the BG regions are described in 789 reports (Lanciego and Wouterlood 2011) (objective 1). These connectivity data can be collated for a meta-study as shown by others (Bakker et al. 2012; Sporns et al. 2000b; Stephan et al. 2001b; Press et al. 2001) to
analyze and characterize the resulting network in terms of graph theory (Sporns 2002, 2011; Rubinov and Sporns 2010; Young et al. 1994, 1996; Young 1992b). One particular objective of this study is a weighted and directed analysis of the BG connectome with regard to strengths of connections in combination with ipsi- and contralateral connectivity (objective 2). Connectivity data sets of parts of nervous systems have been developed and analyzed by several groups (Felleman and Essen 1991; Young 1992a, 1993; Sporns et al. 2000a, 2002, 2004; Sporns and Ko¨tter 2004; Honey et al. 2007; Modha and Singh 2010; Sugar et al. 2011), however, these data do not provide a complete collection of BG connections. Most of the connections in these data sets are compiled into neocortical connectomes within the framework of meta-studies of tract-tracing publications. The sources of information in this study originate exclusively from peerreviewed tract-tracing publications where anterograde and/ or retrograde tracers were applied (lesion studies were not evaluated). Only those publications were considered that describe connectivity in juvenile or adult rats. Connections between neuronal regions are handled with the help of a consistent neuroontology (Schmitt and Eipert 2012) that can be updated frequently in parallel to the fast progress of the identification of connections in tract-tracing publications. In most tract-tracing studies that describe new details of afferents and efferents, modifications of nomenclatures and overlap of regions occur which must be interpreted and integrated into connectome projects (Schmitt and Eipert 2012). Therefore, it is necessary to propose refinements of subdivisions of regions that are documented in stereotaxic atlases (Paxinos and Watson 2004, 2009; Swanson 2003). Such subdivisions can be presented in acyclic graphs or hierarchies. This approach allows to expand and aggregate regions and to apply such a selection of super- and subregions for defining a connectome and partial connectomes, respectively, without loosing non-standard terminologies of publications. This technique was applied here to define a core connectome of the basal ganglia (BG1) and a more detailed connectome that includes especially limbic connections (BG2) (objective 3) (the composition of the BG1and BG2-networks is described in the ‘‘Materials and methods’’ as well as in the ‘‘Conditional connectomics’’). The BG-connectome was analyzed in three contexts (objective 4): (1) embedded in regions that directly connect to it (Vlachos et al. 2012) (connectome embedding), to measure extrinsic connectivity; (2) stand-alone, to measure intrinsic connectivity (3) differential, to compare the role of local regions in the other two contexts (differential connectomics). A further aim of this investigation was to collate the connectivity data (Burns and Cheng 2006) of all peer-
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reviewed non-viral tract tracing publications (Bohland et al. 2009a) in which connections of regions of the BG were described. Viral tract tracing was not considered in this work because some neurotrophic viruses propagate transsynaptically in a time-dependent manner. The density and reliability of collated connections should be as large as possible. Such a BG data source has been proposed recently (Looi et al. 2013), but was not available, so far. Now, this BG connectome is available within a database (http://neuroviisas.med.uni-rostock.de/connectome/index. php) and can be downloaded for sharing (Van Horn and Gazzaniga 2013; Koslow 1997, 2005; Ko¨tter 2003), (https://neuinfo.org-/datasharing/index.shtm). Based on these tract-tracing data, the intrinsic, extrinsic, bilateral and weighted connectivity of the BG connectome was characterized in terms of global and local parameters, as well as multivariate statistics to determine regions of specific importance of intrinsic and extrinsic BG connectomes of the rat brain. By applying modularity analysis it was intended to investigate a possible correlation of functionally similar regions and their connectivity patterns. A population-based large-scale simulation of a thalamocortical network of the whole human cortex (Izhikevich and Edelman 2008) has provided evidence for the propagation of an excitatory wave from the visual cortex to frontal areas following a spike injection into the occipital areas. Because macroscopic connectivity in this model was primarily based on tractographic data of DTI-measurements, one assumption of this model was a complete reciprocity of all macroscopic interconnections of cortical regions. The hypothesis that reciprocal connections in tract tracing studies are highly specifically distributed has been investigated. The result of this analysis of reciprocal connections is of particular interest for modeling connectivity at macroscopic levels based on DTI-measurements and tractographic analysis.
Materials and methods In a meta-analysis of 4,513 peer reviewed tract-tracing publications describing non-viral tract-tracing experiments in juvenile and adult rats we found 3,265 articles that provide connectivity data that are useful for a mesoscale connectome analysis (Schmitt et al. 2012b; Burns and Cheng 2006; Burns et al. 2008a, b; Bota and Swanson 2007b, 2010). Nearly 800 of these 3,265 publications are considered as the data core because they contain connectivity data of particular regions of the basal ganglia. The inquiry of these publications was performed through PubMed (http://www.ncbi.nlm.nih.gov/pubmed) using search expressions (Schmitt et al. 2012b). Among the 3,265 articles numerous studies were found that applied tract tracing
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techniques for verifying and confirming experiments in adult laboratory rats as controls. Also these studies were taken into consideration, however, only those connections of them were collated which were observed in control rats. The sources, targets, semiquantitative weights (strengths of connections), type of tracer, directions of transport and features of somata (transmitters, enzymes, receptors) were transferred to tables. The raw data were verified by at least two experts in neuroanatomy. The verified data tables were imported into neuroVIISAS. This software is a platformindependent generic framework (Schmitt and Eipert 2012) developed in JAVATM for advanced connectome analysis (Bullmore and Sporns 2009; Stam and Reijneveld 2007; Albert and Barabasi 2002; Sporns et al. 2000a, 2004), visual analytics (Ware 2013; Archambault et al. 2008; Wong et al. 2006a, b) and nervous system specific 3Dmultiscale visualizations using stereotaxic coordinate systems (Hjornevik et al. 2007; Gustafson et al. 2004, 2007; Ju et al. 2006; Bjaalie 2002). The advantage of neuroVIISAS is that different modalities like ontologies, digital atlasing, connectomics and neuron population simulation (Feng et al. 2005; Burns et al. 2006; Gewaltig and Diesmann 2007; Moore et al. 2007; Niggemann et al. 2008; Lee et al. 2008; Gouws et al. 2009) are integrated into one flexible framework (Martone et al. 2004). The neuroVIISAS installation package can be downloaded from (http://neu roviisas.med.uni-rostock.de). All computations and visualizations were realized exclusively with the neuroVIISAS framework (chord diagrams were generated using the circular layout engine CIRCOS (http://circos.ca) through the CIRCOS–neuroVIISAS-interface). In most reports, connections are described in the form of ordinal categories (1, 2, 3, 4), symbols (*, **, ***) or expressions like: sparse, moderate, dense. Occasionally, authors have performed a stereologic analysis of retrogradely marked perikarya or a densitometric quantification of axonal terminals. There exist no standardization or normalization between these ordinal categories and estimations of the density of connections between independent studies or different reports. Obviously, connection weights are not discrete categories (Hilgetag and Grant 2000) and extend over several orders of dimensions (Markov et al. 2011, 2014). The latter determined a lognormal distribution of the fraction of neurons following injections of different tract tracers. Lognormal distributions of corticocortical connectivity was also found by Oh et al. (2014), ErcseyRavasz et al. (2013) and Wang et al. (2012). Therefore, ordinal categories were transformed to a logarithmic scale. This was realized in neuroVIISAS by a transformation table to approximate the distribution of the connection strengths logarithmically (Appendix Fig. 10). We would like to point out that these estimated connections strengths should be considered as a hypothetical estimate.
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In general, connections are described in tract tracing studies in terms of ipsi- and contralateral courses rather than by providing exact stereotaxic coordinates (The sagittal axes in the rat atlas of Paxinos and Watson (2007) have the same values for the left and right side. In the rat atlas of Swanson (2003) one hemisphere is considered, only.) or an explicit indication of a left and/or right side injection into a specific region. In neuroVIISAS it is possible to manage injections and labelled sites either with explicit laterality, or with a symmetric representation where the labelled sites are either ipsi- or contralateral. To judge the reliability R of the data of connections, a reliability estimation of connectivity data is introduced. Because the connection data and not the measurement of connections itself constitute R it should be considered as an observation score. If this observation score is large then the probability that the connection really exists is large. An important assumption for calculating R is, that at least most of the tract tracing reports of a set of regions of interest have been evaluated and connectivity data are available for computation. The observation score of a connection is estimated by adding reliability weights. Reliability weights (Table 1) are defined for Table 1 Reliability weights used for estimating the reliability parameter R Variable
Case
Value
t
a/r
0.25
t
r
0.5
t
a
0.5
t
r?a/r
0.7
t
a?a/r
0.7
t
a?r
1.0
t w
a?r?a/r 3:0 unknown
1.0 0.7
w
2:0 fibers of passage
0.0
w
1:0 not clear
0.8
w
0:5 exists
w
0.0 not present
w
0.5 very light
1.0
w
1.0 light/sparse
1.0
w
1.5 light/moderate
1.0
w
2.0 moderate/dense
1.0
w
2.5 moderate/strong
1.0
w
3.0 strong
1.0
w
4.0 very strong
1.0
0.9 -1.0
Value reliability weight of connection strength, t variable of reliability weight for transport directions of tracers, w variable of reliability weight for strengths of connections, a anterograde tracer transport, r retrograde tracer transport, a/r bidirectional tracer transport, a?r?a/ r means that a connection has been proved by an anterograde, a retrograde and a bidirectional transported tracer
–
–
types of tracer transport directions (anterograde, retrograde) (t weight of the transport direction of a tracer) and the weight or strength of a connection (w weight of the connection strength).
The reliability R is defined as follows: X X R¼ wþ þ tþ þ w t
ð1Þ
Identical connections (connections between identical regions) may be described severalfold in a particular report in different animals or ‘‘cases’’ and/or in different reports. Since all these data are available in our database we are able to calculate sums of w of such case-based and noncase-based experimental observations. However, if a connection has been proved only by a retrograde or anterograde method then a smaller t ¼ 0:5 is assigned instead of a prove by a anterograde and retrograde method in two independent experiments (t ¼ 1). A bilateral transport (a/r) of a tracer is considered also in this weighting scheme (Table 1). The variables have subscripts and þ which indicate an observation of an existing connection (wþ , tþ ) or an explicit description that a connection does not exist (w , t ). Hence, explicitly not existent connections are weighted by negative reliability weights. The number of P observations of a specific connections is added ( w). However, the value of t is determined by identifying different directions of tracer transports within all connections that were added. For example, a specific connection that was observed in 10 different tract tracing experiments by applying anterogradely transported tracers obtains a w ¼ 10 and a t ¼ 0:5 (R ¼ 10:5). If the connection that was observed 10 times was found using 7 anterograde and 3 retrograde tracers then t ¼ 1 (R ¼ 11). And if the connection is observed 9 times (wþ ¼ 9) with 7 retrograde and 2 anterograde tracers (tþ ¼ 1) and in addition this specific connection has not been found in one experiment (w ¼ 1) with an anterograde tracer (t ¼ 0:5) then R ¼ 8:5. If there are many descriptions in publications that a specific connection does not exist, it gets a strong negative observation score. Finally, these scores are presented in a reliability matrix. We must emphasize that the observation score is defined for the data of a connection between a specific source or efferent region and a target or afferent region. So, it is not a score for connections rather than for the underlying data of the connections. Some connections and regions may receive more attention in tract tracing research so that frequently investigated or popular regions may have higher observation scores. This aspect is explicated in ‘‘Discussion’’. Regions of the nervous system of this particular project of the rat connectome are organized hierarchically (Schmitt and Eipert 2012), in graph theory such a hierarchy is called
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a tree. An example of a hierarchy is shown in the Appendix Fig. 11. The root region of this hierarchy can be the nervous system of the rat. If a region is divided into smaller parts, these subregions are the childs of this parent region, which could be further subdivided. A region and all its descendents give rise to a so called subtree. Regions without descendents are called leafs. With regard to this concept the terms direct input or direct output of a region sum up all those connections that were described to end at (afferents), respectively, start from (efferents) this specific region. The terms subtree input or subtree output specify all those connections that end in, respectively, start from any region of the subtree of a given region including the region itself. The BG1 network consists of 10 regions (SNC, SNR, VA, VL, LGP, MGP, CPu, STh, AGl, AGm). These core regions of the BG are involved in the processing of motor functions. There exist several additional regions which are strongly interconnected with the BG1 regions, however, they are merely indirectly connected with regard to the principal BG pathways (direct, indirect, hyperdirect). To obtain a more comprehensive network, conditions for the selection of further regions can be applied (conditional connectomics). These selected regions belong to specific thalamic nuclei (lateral habenula among other regions), mescencephalic nuclei like the ventral tegmental area as well as to the limbic system like the ventral striatum (ventral CPu, accumbens nucleus). Further regions like the piriform cortex, hippocampus, entorhinal cortex and mesencephalic reticular formation (MRF) were selected in combination with their thalamic relay nuclei. In the following we will refer to this network as the BG2 network. The BG2 network is more extensive and contains with regard to the BG1 regions 15 further regions (VTA, MRF, CM, CL, PC, PF, MDL, MDM, VM, LHb, AC, Ac, HIPP, Ent, Pir). These two network configurations of the BG can be considered in a wider sense as a motoric BG connectome (BG1) and an associative BG connectome (BG2). The 10 regions of the BG1 connectome are localized on levels 9–14 (most on level 12) and the regions of the BG2 connectome on levels 8–14 of the hierarchy. The complex spatial arrangement of regions (Fig. 1) and their connections (Fig. 2) becomes visible by applying interactive 3Dvisualization of neuroVIISAS. The BG1 network was embedded in a surrounding network of regions that all have at least one input and one output connection to a region of the BG1 network. All regions of the surrounding network are located at a comparable hierarchical level like the BG1 region. The regions or parts of the selected regions of the BG1 or BG2 network do not overlap with or are not included in other regions (Bohland et al. 2009b). A further hypothesis to be tested in this study is, that specific distributions of reciprocal connections could
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characterize the three principal pathways of the BG. Such specific distributions were analyzed by a recently developed pathway analysis of the BG-connectome data. Moreover, it was tested if additional routes from a source to a target region may exist in the directed BG connectome, a phenomenon which may entail a routing problem in structural connectomics. To determine pathways from source to target regions a sequential search of interconnected regions (pathway analysis) was applied in neuroVIISAS. The search-technique allows to add individual regions that should be passed in pathways from the provided source to the target regions. The result is shown in a table that contains weights, reliability and reciprocal connections within the detected pathways. To share the connectivity data of the BG, the rat connectome database can directly be queried through a web interface (http://neuroviisas.med.uni-rostock.de/connectome/ index.php).
Results Principal pathways The well-known principal pathways of the BG can be identified in the connectome. The adjacency matrix of the core regions of the BG is shown in Fig. 3a (BG1 network). Each of the following pathways can be derived from this matrix. The direct excitatory pathway [AGl, AGm] ! [CPu] ! [SNR, MGP] ! [VA, VL] ! [AGl, AGm] of the BG can be reconstructed using the adjacency matrix of principal regions (DeLong 1990; Alexander and Crutcher 1990; Albin et al. 1989) (Fig. 4a). The indirect inhibitory pathway [AGl, AGm] ! [CPu] ! [LGP] ! [STh] ! [SNR, MGP] ! [VA, VL] ! [AGl, AGm] of the BG can also be built using the same selection of regions (DeLong 1990; Alexander and Crutcher 1990; Albin et al. 1989) (Fig. 4b). The hyperdirect inhibitory pathway [AGl, AGm] ! [STh] ! [SNR, MGP] ! [VA, VL] ! [AGl, AGm] of the BG can be built by means of the same selection of regions (Bosch et al. 2012; Nambu et al. 2002) (Fig. 4c), too. The number of tract-tracing experiments that describe a certain connection is indicated by the number above each edge. Hence, the connection of the SNC to the CPu is described in 44 tract-tracing publications (Fig. 4a–c). The adjacency matrix of the ipsi- and contralateral BG network suggests that there could be additional pathways from AGl over the CPu back to AGl. Therefore, a pathway analysis was applied to detect additional routes or parallel pathways in the BG network (Table 2). If the maximum pathlength is
Brain Struct Funct Fig. 1 The regions of the BG2 network in stereotaxic space. a The stereotaxic space of the regions of the BG2 network. This view shows the region from dorsal, respectively, from top. Cortical regions are shown in red, subcortical parts in light red, thalamic regions are light magenta and SN are presented in dark magenta. The same color scheme was used for the coloring of abbreviations of regions in matrices. b Abbreviations of the regions of one hemisphere. c View from rostral directly onto the accumbens nucleus and ventral striatum. d View from caudal where midbrain structures and the turquoise MRF appear
restricted to 4 (because the minimum length of the principal pathways is 4), 653 ipsilateral pathways exist from AGl to AGl (circular pathway) in the BG2 network. If the CPu is determined as the first region in a pathway from the AGl back to the AGl, then 74 ipsilateral pathways (8 of these pathways have less than 4 edges) are found. The latter were sorted with regard to the reliability value (see ‘‘Reliability of connections’’). 6 pathways have reliability values that are larger than 10 and all of them have the SNR as the third region within the pathway. Most variability was found for thalamic regions (VM, PF, VL, PC, CM, CL) as output region to AGl. The direct pathway of this group has a reliability of 12. The indirect pathway may use MDL and PF as thalamic output regions to AGl or the VTA as an efferent region to AGl (reliability values are lying between 4 to 5). The hyperdirect pathway has a reliability of 7.5 and may use different output regions (MDL, PF, VM, VTA) to the AGl, too. Besides variations of BG-output nuclei of the thalamus, there are principal changes of input nuclei within these pathways, e.g. AGl has projections to AGm, CL, MDL, MGP, MRF, PC, PF, SNR, VL and VM. The sequence of connections of directly connected regions within a pathway were examined with regard to reciprocal connections. Then the patterns of reciprocity of connections within all principal pathways of the BG (only those regions were considered that were selected in the
BG2-network) were analyzed. In the first part, MGP was selected as a connecting region from the BG to thalamic and mesencephalic regions that have connections to AGl. In the second part SNR was selected as a connecting region from the BG to thalamic and mesencephalic regions that have connections to AGl. Interestingly, it was found in the first part of the analysis that PF is the sole region with reciprocal connections to MGP and AGl within the output of the BG2 network in all three principal BG-pathways. There were no reciprocal connections reported from VA, VL, VM and MDL to MGP. However, MGP seems to have a reciprocal connection with VTA, however, VTA is nonreciprocally connected with AGl. In the second part of this reciprocal pattern analysis with regard to SNR it was found that in addition to PF also VM has reciprocal connections to SNR and to AGl in all three principal BG pathways. Again VTA has a reciprocal connection to SNR and the non-reciprocal connection to VTA in this SNR BG-output pathway. Contralateral connectivity The core regions of the basal ganglia show larger connection strengths in ipsi- than in contralateral connections. The AGl and AGm possess more contralateral connections than the core regions of the BG (Fig. 3b). Reciprocal
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Brain Struct Funct Fig. 2 The weighted connections and color-coded weighted DGAll parameter of the bilateral BG2 network. a The ipsilateral connections of the BG2 network are shown using a 3-axis expansion technique of neuroVIISAS. b A unilateral view with labels of regions. c Bilateral regions in a comparable view like in (a) with labels
connections of the BG core regions occur exclusively ipsilaterally, however, two reciprocal connections of the contralateral AGl with VL and CPu were identified. Ipsiand contralateral connections are visualized in 3D in Fig. 2. Stronger and weaker ipsi- and contralateral connectivity of AGl is clearly visible in the low-resolution BG1-network visualization using the chord diagrams of CIRCOS (Appendix Fig. 12). However, the CIRCOS visualization is not suitable for the high-resolution BG2 network (Appendix Fig. 13). Reliability of connections The observation scores of the BG network are presented in Fig. 3c. The most reliable connection in the BG network is the ipsilateral SNC ! CPu projection (43.6) followed by
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the ipsilateral SNR ! CPu (24.1) projection. The largest reliability of a contralateral connection is observed for SNC ! CPu (5). Further, large reliabilities exist for ipsilateral efferences of STh to SNR (14) and MGP (10). Global and local connectome analysis of the BG2 network The ipsilateral BG2 network consists of 25 regions and 235 connections and the adjacency matrix has a line density of 39.167 %. The connections of thalamic regions to mesencephalic and diencephalic regions of the adjacency matrix are sparse, a phenomenon that is well presented by the adjacency matrix (Fig. 3d) as well as in the distance matrix that show relatively larger distances from these regions to other regions of the BG network (Appendix Fig. 14). The
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a
b
c
d
Fig. 3 Matrix representations of the BG connectome. a Adjacency matrix of the core regions of the motoric basal ganglia. Numbers and gray values are indicating the number of experiments that describe connections. Source regions are shown in the rows and target regions in the columns. b The bilateral connectivity of the core regions of the motoric basal ganglia. The weighted adjacency matrix displays the mean weights of connections. Weights are ranked as shown in the color table. Altogether, the ipsilateral connectivity is denser than the contralateral. Relative strong connections exist between contralateral cortical regions, only. c The reliability of the weighted connectivity of the core regions of the motoric basal ganglia. The reliability of the weighted bilateral connections is calculated using the number of
experiments that describe connections (if different experiments within a particular publication occur than all experiments are considered) and variability of weights. The maximum is 43.6 for SNC ! CPu and the minimum is 0.1. d The adjacency matrix of the ipsilateral BG2 network with regions that are not primarily involved in motoric processing (BG2). The intrinsic connectivity of thalamic regions (magenta) is very weak. The regions of the midbrain and MRF have abundant connections to most other part of the BG2 network. Especially SNR has strong connections to nearly all thalamic regions in this selection. The amygdalar complex and the CPu have comparable number of output connections. The variability of weights of the bilateral BG2 network is shown in the Appendix Fig. 17
central connectional position of the CPu and the substantial contralateral connections have been visualized using an organic planar graph visualization (Fig. 4d). The parafascicular thalamic region has a conspicuously dense connectivity to other regions. This is confirmed by computing the communicability (Estrada and Hatano 2008) (Appendix Fig. 15). Communicability is a measure for the number of paths from a source to a target region. Here, PF has significantly larger values than other thalamic regions. Furthermore, the LGP has relatively small communicability values. A related matrix is the generalized topological overlap matrix GTOM (Yip and Horvath 2007). It is a
measure of pairwise interconnectedness that is proportional to the number of neighbors that a pair of nodes share in common. Interestingly, VTA and MRF possess a noticeable amount of common neighbors resulting in relative large GTOM values (Appendix Fig. 16). The global analysis of the BG2 network reveals a network architecture close to a small-world network because the small worldness is 1.258. The average cluster coefficient of 0.661 is larger than in the Erdo¨s Re´nyi, Watts– Strogatz or Barabasi–Albert randomizations. The scale-free property (power law distribution of edges) of the BG2 network can be asserted because with 2.2, the deviation D
123
Brain Struct Funct Fig. 4 The three principal pathways of the motoric BG. a Direct pathway. b Indirect pathway. c Hyperdirect pathway. The levels of the regions within the hierarchy (the hierarchical organization of regions is shown in Appendix Fig. 11) are shown at the left side of the graphs. The number of publications of specific parts of a pathway are displayed by the numbers on the connections and the maximal weight by the same colors as shown in Fig. 3b. Thick lines indicate reciprocal connections. d The bilateral BG2 network with weighted edges in a planar graph visualization with an organic edge layout. The CPu is localized around the center of regions of each hemisphere. _L region of the ipsilateral hemisphere, _R region of the contralateral hemisphere
a
b
c
d
from the power function is relatively small. The mean scale-free property as well as mean global values of the rewiring randomization resemble the real BG2 network values more closely, than those of other randomizations (Appendix Fig. 18). The weighted modularity analysis (Hagmann et al. 2008) distinguishes 5 modules (Fig. 5). In module 1 most regions are strongly interconnected with the CPu. Module 2 contains the primary motor cortex and important core regions of the BG (SNR, STh, LGP) as well as specific thalamic regions and MRF. Module 3 and 5 consist of regions that have memory functions and are involved in emotional processing. Module 4 contains MGP and VL (VA has light connections only). Based on the
123
weighted modularity analysis modules 1, 2 and 4 contain regions that are associated with motoric function and modules 3 and 5 those regions involved in memory and emotional processing. Hence, the computed modules partly reflects the functional separation of regions. Some parameters of the local-network analysis of the BG2 network are summarized in Table 6 (a complete list of all local parameters of this analysis is shown in Appendix Table 15). After sorting the regions by DGall , it turns out that AC has most inputs and outputs (42), followed by the CPu (41) and VTA (40). If the analysis is done on the weighted rather than binarized data, the top three regions are CPu, VTA and SNR (Appendix Table 7). The
Brain Struct Funct Table 2 Pathway analysis starting with projections from the AGl to the CPu Path
PL
Rec
AW
Pub
Rel
AGl ! CPu ! SNR ! VM ! AGl
4
1
3
13
13.13
AGl ! CPu ! SNR ! PF ! AGl
4
1
3
12
12.95
AGl ! CPu ! SNR ! VL ! AGl
4
0
3
11.75
12.02
AGl ! CPu ! SNR ! PC ! AGl
4
0
3
11.25
11.73
AGl ! CPu ! SNR ! CM ! AGl
4
0
3
11
11.48
AGl ! CPu ! SNR ! CL ! AGl
4
0
3
10.75
10.98
AGl ! CPu ! SNR ! MDL ! AGl
4
0
2.75
10
9.75
AGl ! CPu ! SNR ! VTA ! AGl
4
0
2.25
9.25
9.75
AGl ! CPu ! SNR ! VA ! AGl
4
0
2.25
9
9.30
AGl ! CPu ! SNC ! CM ! AGl
4
0
3
7.25
7.63
AGl ! CPu ! SNC ! CL ! AGl AGl ! CPu ! MGP ! VL ! AGl
4 4
0 0
3 3
7.5 8.5
7.53 7.50
AGl ! CPu ! SNC ! VTA ! AGl
4
0
2.75
6.75
7.13
AGl ! CPu ! SNC ! AGm ! AGl
4
0
2.38
7
7.03
AGl ! CPu ! SNC ! PC ! AGl
4
0
2.75
7
7.03
AGl ! CPu ! SNC ! PF ! AGl
4
1
2.38
6.75
7.03
AGl ! CPu ! MGP ! PF ! AGl
4
1
2.75
7.75
6.97
AGl ! CPu ! SNC ! VM ! AGl
4
0
2.38
7.75
6.97
AGl ! CPu ! SNC ! MDL ! AGl
4
0
2
6.25
6.32
AGl ! CPu ! Ac ! VTA ! AGl
4
0
2.25
5.5
6.05
AGl ! CPu ! VL ! AGl
3
1
2.67
5.33
5.93
AGl ! CPu ! MGP ! VM ! AGl
4
0
2.5
7
5.85
AGl ! CPu ! STh ! PF ! AGl
4
1
2.25
4.75
5.78
AGl ! CPu ! AGl
2
1
2.5
5
5.55
AGl ! CPu ! MRF ! CL ! AGl
4
0
2
4.75
5.55
AGl ! CPu ! AGm ! VL ! AGl
4
1
2.5
4.75
5.32
AGl ! CPu ! VTA ! Ent ! AGl AGl ! CPu ! VM ! AGl
4 3
0 1
2.38 2.67
5.25 4.67
5.32 5.17
AGl ! CPu ! MRF ! PF ! AGl
4
1
2
4.25
5.07
AGl ! CPu ! MGP ! MDL ! AGl
4
0
2.5
6.25
4.88
AGl ! CPu ! MGP ! VTA ! AGl
4
0
2
5.75
4.88
AGl ! CPu ! AGm ! AGl
3
1
2.33
4.33
4.87
AGl ! CPu ! PC ! AGl
3
1
2.33
4.33
4.87
AGl ! CPu ! PF ! AGl
3
1
2.33
4.33
4.87
AGl ! CPu ! VTA ! AGl
3
0
2.33
5
4.83
AGl ! CPu ! MGP ! VA ! AGl
4
0
2
5.75
4.80
AGl ! CPu ! MRF ! VL ! AGl
4
0
1.25
4.25
4.78
AGl ! CPu ! VTA ! CL ! AGl
4
1
2.25
4.75
4.75
AGl ! CPu ! VTA ! AGm ! AGl
4
1
2.25
5
4.72
AGl ! CPu ! VTA ! VM ! AGl
4
1
2.5
4.75
4.72
AGl ! CPu ! AC ! VL ! AGl
4
0
2.25
4.25
4.70
AGl ! CPu ! STh ! MDL ! AGl AGl ! CPu ! STh ! VTA ! AGl
4 4
0 0
1.75 1.75
4 4.25
4.70 4.61
AGl ! CPu ! MRF ! PC ! AGl
4
0
2
3.75
4.57
AGl ! CPu ! MRF ! VM ! AGl
4
1
1.75
3.75
4.57
AGl ! CPu ! CM ! AGl
3
0
2.33
4
4.53
AGl ! CPu ! AGm ! CL ! AGl
4
1
2.5
4
4.53
AGl ! CPu ! AGm ! VM ! AGl
4
1
2.5
4
4.50
123
Brain Struct Funct Table 2 continued Path
PL
Rec
AW
Pub
Rel
AGl ! CPu ! AGm ! PC ! AGl
4
1
2.25
3.75
4.28
AGl ! CPu ! PF ! AGm ! AGl
4
1
2
3.75
4.28
AGl ! CPu ! CM ! AGm ! AGl AGl ! CPu ! PC ! AGm ! AGl
4 4
1 1
2 2.25
4 4
4.25 4.25
AGl ! CPu ! PF ! VM ! AGl
4
0
2.25
3.75
4.25
AGl ! CPu ! VTA ! CM ! AGl
4
0
2.25
4.75
4.15
AGl ! CPu ! MRF ! CM ! AGl
4
0
2
3.5
4.13
AGl ! CPu ! VTA ! PF ! AGl
4
1
2.5
4.75
4.13
AGl ! CPu ! MDL ! AGm ! AGl
4
0
1.75
4
4.05
AGl ! CPu ! AGm ! CM ! AGl
4
0
2.25
3.5
4.03
AGl ! CPu ! AGm ! PF ! AGl
4
1
2.25
3.5
4.03
AGl ! CPu ! VL ! AGm ! AGl
4
1
2.5
3.5
4.03
AGl ! CPu ! VM ! AGm ! AGl
4
1
2.5
3.75
4.03
AGl ! CPu ! PF ! VTA ! AGl
4
0
2
3.5
4.00
AGl ! CPu ! MDL ! AGl
3
0
1.33
3.67
3.93
AGl ! CPu ! MRF ! AGm ! AGl
4
0
1.5
3.5
3.88
AGl ! CPu ! MRF ! MDL ! AGl
4
0
1.75
3
3.88
AGl ! CPu ! VTA ! MDL ! AGl AGl ! CPu ! AGm ! MDL ! AGl
4 4
1 1
1.75 2.25
4 3.5
3.80 3.78
AGl ! CPu ! AGm ! VTA ! AGl
4
0
2
3.25
3.72
AGl ! CPu ! MRF ! VTA ! AGl
4
0
1.5
3.25
3.72
AGl ! CPu ! AC ! CM ! AGl
4
0
2
3.25
3.71
AGl ! CPu ! AC ! VTA ! AGl
4
0
1.5
3.75
3.61
AGl ! CPu ! VM ! VTA ! AGl
4
0
2.13
3
3.53
AGl ! CPu ! PC ! VTA ! AGl
4
0
1.75
3
3.50
AGl ! CPu ! MDL ! VTA ! AGl
4
0
1.25
3
3.33
PL path length, Rec reciprocal, AW average weight, Pub number of articles, Rel reliability
importance of a region in this network is estimated by the Shapley-value (Keinan et al. 2004; Ko¨tter et al. 2007). The lowest Shapley-value (largest importance) was calculated for CPu (0:309), then the AGl (0:277) and AGm (0:243) also have smaller values than other regions. The betweenness centrality (BC) (Freeman 1977) is a coefficient that divides the number of shortest paths between regions passing a particular region by the number of all shortest paths. It provides a useful measure for the interconnectedness and embedding of a particular region in the network (Vlachos et al. 2012). Again the CPu is most important and has the largest BC of 0.092, followed by the AC (0.074) and the AGm (0.069). The BC has been visualized in 3D in the bilateral BG network by assigning the BC values with a color code to each region (Fig. 6). The same ranking of CPu (1), AC (0.997) and AGm (0.817) holds for the authoritativeness as well (Kleinberg 1999). The local parameters have been visualized in parallel coordinates (Appendix Fig. 19) after sorting by their similarities. The presentation of the high-dimensional parameter space clearly indicates the central role of the CPu in
123
the BG2 network. Also the cortical input areas AGl and AGm are lying close to the CPu-curve in this visualization. A further finding concerns the laterality (LA ) of connections (Appendix Tables 8, 9). The AGm and AGl have a laterality of 0.62 and 0.607 (ipsilateral connections divided by DGall ). A smaller laterality value was found for MRF (0.552). Interestingly, the core regions of the BG all have lateralities larger than 0.7. This phenomenon is also found when considering reciprocal laterality (LR ). Rich-club analysis and knotty-centers A rich club is a set of high-degree nodes (rich nodes) which are connected with each other even more frequently than it would be expected by their high degree. Nodes with numerous links are much more likely to form well interconnected subgraphs (clubs) than low degree nodes. The rich-club phenomenon can be computed by the rich-club coefficient / (Zhou and Mondragn 2004). /ðkÞ measures the fraction of edges actually connecting nodes with degree [ k out of the maximum number of possible connections
Brain Struct Funct
(Colizza et al. 2006). Rich-club analysis has been applied successfully to detect structural linkages of hubs of the cerebral cortex (Shanahan et al. 2013; Harriger et al. 2012; van den Heuvel and Sporns 2011). The diagram of richclub coefficients in dependence on node degrees /ðkÞ of the bilateral BG2 network allows to detect a rich-club
phenomenon of nodes with degrees k 44 (Fig. 7). These nodes have a / 0:8. The feeder nodes are those nodes of the BG2-network that have direct connections to the richclub nodes. This connectivity is shown in Appendix Fig. 20. It visualizes the dense connectivity of rich nodes and a substantial connectivity between rich-club nodes and feeder nodes. CPu, AGl, AGm and SNR are members of the rich club. Interestingly, the non-canonical BG regions AC, VTA and MRF are also rich-club nodes. The knottycentrality (KC) quantifies the connectivity within a topologically central connective core (Shanahan and Wildie 2012). The knotty-center consists of the following regions found in the ipsi- and contralateral hemispheres (Fig. 21): VTA, MRF, CPu, AGl, AGm, AC. The thalamic output regions do not belong to the knotty-set. However, knottyset members MRF and VTA are massively connected with cortical as well as subcortical hub regions. Motif analysis
Fig. 5 The weighted-modularity analysis of the ipsilateral BG2 network. Five modules have more weighted connections within than among the modules. Thick lines indicate large weights
Motif analysis has been applied by cortical and subcortical connectomes in different species to determine such subgraphs that are significantly more abundant or considerably less frequent than in suitable randomizations (Sporns and Ko¨tter 2004; Sporns 2006; Schreiber and Schwo¨bbermeyer 2004, 2005). The motifs of the ipsi- (Fig. 8a) and bilateral (Fig. 8b) BG2 networks show characteristic distribution patterns. The frequency of 13 3-node motifs and 3 complex 5-node motifs was determined. The 5-node motifs consist of 2 reciprocal edges and two non-reciprocal edges which
Fig. 6 The weighted connections and color-coded weighted betweenness (BC) parameter of the bilateral BG2network. The color code of connections weights is the same as in Fig. 3d. The weighted betweenness centrality BC of the regions has been integrated into this 3D-view. The CPu belongs to the class of maximalweighted BC values indicating its connectional importance in the BG2-network AGm and Pir have smaller values. The large amount of contralateral projections is also visible
123
Brain Struct Funct Fig. 7 The rich-club coefficients / of the ipsi and contralateral BG2 network in dependence of the ranks k of the regions. 1,000 rewiring randomizations were applied and mean rich-club coefficients were computed. A secondary yaxis in green shows the coefficient /ðkÞ=rand/ðkÞ
are oriented in 3 ways: chain-like (chain), centripetal (in) to and centrifugal (out) from the center vertex to the vertices with reciprocal edges. The frequency of motifs in the real ipsilateral BG2 network can be compared with the frequencies of motifs in 1,000 rewiring randomizations with the same number of nodes and edges. It turns out that the motifs 3-09 and 3-13 are significantly more abundant in the real BG2 network than in randomizations (Fig. 8a). In the bilateral BG2 network 3-04, 3-06, 3-09, 3-13 motifs are more abundant than in 1,000 rewiring simulations. All these 3-node motifs have at least one reciprocal edge. In contrast to the ipsilateral network, the 5-node motifs are very abundant (Fig. 8b). The contribution of regions to motifs is shown in Table 3 and Appendix Table 10. AC, CPu and VTA are most often elements of the fully reciprocal motif 3-13 (Table 3). The 5-node motifs only appear sporadically. The MGP is most often (326) an element of the centripetal 5-node motif and 333 times an element in the chain 5-node motif. The centrifugal 5-node motif is built by AGl in the majority of cases. In the bilateral BG2 network MRF participates most often in the full reciprocal motif 3-13. Multivariate analysis Modularity analysis provides a classification of regions based on their number of connections in modules and in between modules. The result of a weighted modularity analysis has been described before (Fig. 5). Metric multidimensional scaling (MDS) arranges regions that have similar connectivity patterns closer together than to regions that have more different connections. The result of a MDS in a normalized and quadratic diagram, respectively, is shown in Appendix Fig. 24 for the ipsilateral and in Appendix Fig. 25 for the bilateral BG2 network. Most of the thalamic regions are located within the lower left quadrant whereas LHb and MDM are located in the upper left quadrant. Cortical motor regions, MGP, CPu, SNR and
123
MRF are in the lower right quadrant. SNC and VTA are located close together in the upper right quadrant. Ent, Pir and HIPP which also share functional features are relatively closely arranged in the upper left quadrant. In the bilateral BG2-network cortical motor, SNC, SNR, VTA, CPu and AC are located close together in the lower quadrants whereas MRF appears as an outlier in the lower quadrants. All other regions are located in the upper quadrants. The principal component analysis was applied to the DGAll , CluCAll , CluC2 , AvgDGnb , VCDG , Loc as proposed by (Echtermeyer et al. 2011) (Appendix Table 13, the definitions of these parameters are summarized in the Appendix) to determine those two dimensions that have the largest influence in the bilateral BG2 network (Fig. 9a, b). It turns out that 5 parameters of the first component (first row of Appendix Table 13) have similar absolute values and determine the distribution of regions in the first dimension. The second dimension (second row of Appendix Table 13) is determined by the largest absolute value by CluC2 . This means, that the variance within the y-axis of the PCA-plane is dominated by the variance of CluC2 . The gray values of the PCA-plane diagram (Fig. 9a) indicate the densities of regions within a ring. The lightest gray value in the middle covers most regions. The STh was selected and the connectional relations of STh in the BG2 network are shown in the circular layout in Fig. 9b. In the inner ring 24 directly connected neighbors are located and in the outer ring 25 indirect neighbors are shown. The number of indirect and direct neighbors is well balanced and typical for all regions within the center of the PCA plane. This result is important since the target of deep brain stimulation is the STh. The CPu shows another pattern of direct–indirect-neighbor connectivity. The number of direct neighbors is much larger than the number of indirect neighbors (Fig. 9c). Interestingly, the number of indirect neighbors of VL (smaller first component than within the CPu) is larger than the number of direct neighbors (Fig. 9d).
Brain Struct Funct
a
b Fig. 8 Overview of the distributions of motifs in the ipsi and contralateral BG2 network. a The frequencies of motifs in the ipsilateral BG2 network. 13 directed 3-node motifs and 3 directed 5-node motifs have been searched: syOut (‘‘Out’’ with regard to the central node of the motif), syIn and syCh. The frequency of a motif in the real network is indicated by the blue point. The BG2 network was randomized using a rewiring procedure. Each randomization contains exactly the same number of regions and connections. The rewiring randomization was repeated 1,000 times, after each randomization the frequencies of motifs were determined, indicated by a small black dot. The red bar indicates the variation of frequency within the standard
syOut
syIn
syCh
syOut
syCh
syIn
deviation. The motifs are sorted by their z values. Hence, motifs that have larger frequencies in the real network appear on the right side of the diagram. Most motifs that have reciprocal edges are more abundant in the real network than in the 1,000 rewiring simulations. b The frequency of motifs in the contralateral BG2 network. The complex motifs syOut, syCh and syIn are much more often represented in the contralateral BG2 network than in the ipsilateral case. The differences of frequencies of motifs in the real BG2 network and in rewiring BG2 networks are stronger in the contralateral than in the ipsilateral case
123
Brain Struct Funct Table 3 Frequency of motifs of the bilateral BG2 network Region
3-01
3-02
3-03
3-04
3-05
MRF_L
121
MRF_R
109
27
2
228
26
28
3
220
23
AC_L
32
17
15
184
AC_R
32
16
15
VTA_L
41
28
11
AGm_L
22
36
VTA_R
41
31
AGm_R
23
CPu_R
3-06
3-07
3-08
3-09
3-10
3-11
3-12
3-13
chain
in
out
36
3
115
98
33
20
164
125
7
0
0
42
2
112
103
31
18
157
125
2
0
0
16
56
2
54
134
19
16
154
114
72
130
8
191
16
56
2
53
162
21
16
157
111
68
136
8
166
10
48
0
54
110
25
15
139
95
78
54
25
24
145
16
112
0
25
151
21
26
105
93
188
273
82
13
162
10
51
0
51
104
24
15
131
90
113
60
25
42
28
143
18
120
1
25
140
22
27
102
88
212
254
77
30
29
30
103
33
176
2
44
94
35
41
127
80
135
42
45
CPu_L
31
28
33
103
33
172
2
41
93
37
39
131
80
114
51
58
AGl_L AGl_R
8 8
7 8
5 5
76 73
1 1
135 136
1 1
2 2
198 184
15 15
4 4
80 72
78 76
249 294
104 108
131 173
PF_L
25
24
15
150
9
65
0
31
100
8
7
67
62
145
160
33
PF_R
22
20
15
145
8
64
0
29
102
7
5
63
62
136
184
30
STh_L
20
45
14
144
6
60
3
23
81
18
13
60
60
265
161
61
STh_R
17
42
12
141
6
63
2
21
83
16
11
55
60
243
173
69
SNR_L
77
55
39
142
13
91
5
63
54
33
13
58
56
193
191
63
SNR_R
78
56
39
141
13
90
5
63
53
33
13
58
56
190
204
51
SNC_R
58
31
20
140
29
77
1
55
41
23
25
68
55
168
153
96
SNC_L
58
30
20
141
29
78
1
55
42
23
25
68
55
151
135
107
MGP_R
34
50
23
105
12
67
2
34
62
17
6
52
49
333
326
105
MGP_L
34
48
23
108
12
67
2
34
63
17
6
52
49
329
299
100
Ac_L
32
45
20
140
19
51
2
17
44
28
16
68
33
201
165
75
LHb_R
33
57
14
124
20
67
6
30
48
23
7
60
31
251
155
111
LHb_L
33
57
14
127
20
66
6
30
48
23
7
60
31
242
149
93
Ac_R
35
52
24
137
21
53
2
20
40
28
15
63
30
192
163
87
VM_R VM_L
51 51
30 29
13 13
99 101
26 26
72 73
2 2
31 31
83 83
11 11
6 6
56 56
27 27
149 141
68 68
34 34
CM_R
70
39
17
110
24
66
4
27
88
16
10
49
21
108
54
20
CM_L
67
37
19
104
24
64
4
28
85
21
12
51
21
101
55
19
Ent_L
25
43
29
76
10
79
3
16
67
14
12
51
19
281
158
112
PC_R
52
20
12
77
2
54
0
6
98
7
5
29
14
120
74
51
PC_L
52
20
12
77
2
53
0
6
99
7
5
29
14
103
73
56
HIPP_L
37
58
33
82
24
117
2
23
35
21
22
62
13
258
120
135
HIPP_R
37
53
32
91
27
112
4
21
39
23
22
59
10
277
189
143
Pir_L
56
47
18
88
29
71
3
44
37
22
16
46
10
249
110
100
CL_R
81
43
18
75
8
53
3
19
59
21
9
30
8
96
51
63
CL_L
82
44
18
75
8
52
3
19
61
21
9
30
8
91
48
67
MDL_R
76
37
8
103
18
24
0
29
62
4
2
35
7
84
145
39
MDL_L
76
38
8
103
18
23
0
29
63
4
2
35
7
71
136
39
Pir_R
59
55
21
85
32
71
4
42
37
24
13
44
6
250
91
96
LGP_L LGP_R
46 44
30 31
7 7
89 87
6 6
24 24
2 2
12 12
19 19
8 8
3 3
25 25
6 6
194 183
193 204
29 26
VL_R
105
28
23
113
22
49
0
46
45
7
8
26
6
62
191
30
VL_L
105
29
23
113
22
48
0
46
46
7
8
26
6
59
181
35
MDM_L
77
19
2
108
6
18
0
20
31
1
8
19
5
159
140
48
MDM_R
78
20
2
105
6
12
0
20
32
1
7
15
2
91
116
17
123
Brain Struct Funct Table 3 continued Region
3-01
3-02
3-03
3-04
3-05
3-06
3-07
3-08
3-09
3-10
3-11
3-12
3-13
chain
in
out
Ent_R
4
19
4
28
0
13
0
0
28
1
0
2
1
57
63
34
VA_L
30
22
17
58
1
36
1
2
0
6
1
0
0
0
67
0
VA_R
30
22
17
58
1
35
1
2
0
6
1
0
0
0
70
0
Regions are sorted using motif 3-13 frequencies chain symmetricCHAIN, in symmetricIN, out symmetricOUT (see legend of Fig. 8)
a
b
c
d
Fig. 9 Principal component analysis of the ipsi and contralateral BG network. a PCA plane of the ipsi- and contralateral BG2 network. The regions of the BG2 network are distributed in the PCA-plane (defined by the first (abscissa) and second component (ordinate) of the PCA) as strongly overlapping pairs. The gray values within a ring present the density of regions in the ring. Most regions are located in the innermost ring. This presentation of regions in the PCA plane need to be considered together with the values of the local network parameters (Appendix Table 13) and their contribution to the first (abscissa) and second component (ordinate). b STh was selected from the bilateral BG2 network. The direct (first) neighbors of STh are located on a ring around STh. The connections of these regions are
dark blue. The indirect (second) neighbors of STh are on the external ring. Connections between second neighbors are light blue and connections between second and first neighbors are yellow. The number of direct and indirect neighbors is similar as well as the number of connections inside the sets of neighbors. c CPu was selected from the bilateral BG2 network. The number of first neighbors of the CPu is much larger than the number of second neighbors. Also the density of connections between second neighbors is much more different from the knotty-center presentation of STh (Appendix Fig. 21). d VL was selected from the bilateral BG2 network. The number of indirect neighbors is much larger than the number of direct neighbors of the VL
123
Brain Struct Funct Table 4 Vulnerability of regions in the ipsilateral BG2 network Region Caudate putamen
Vulnerability 2.2792
Amygdaloid complex
2.1273
Ventral tegmental area A10
1.7475
Mesencephalic reticular formation
1.5196
Medial agranular prefrontal cortex
1.4057
Substantia nigra compact part
0.8360
Lateral agranular prefrontal cortex
0.7600
Substantia nigra reticular part
0.7220
Subthalamic nucleus
0.6461
Medial globus pallidus
0.6081
Accumbens nucleus Parafascicular thalamic nucleus
0.3802 0.2663
Lateral habenular nucleus
-0.3034
Piriform cortex
-0.3414
Entorhinal cortex
-0.4173
Ventromedial thalamic nucleus
-0.5313
Hippocampus
-0.6072
Central medial thalamic nucleus
-0.6452
Mediodorsal thalamic nucleus lateral part
-0.9870
Centrolateral thalamic nucleus
-1.1010
Lateral globus pallidus
-1.1010
Ventrolateral thalamic nucleus
-1.2529
Paracentral thalamic nucleus
-1.3289
Mediodorsal thalamic nucleus medial part
-1.3668
Ventro anterior thalamic nucleus
-2.5062
Differential connectomics: vulnerability Vulnerability analysis provides an estimate of the effect of removing a region from a network. If the CPu is removed, then the mean distance between all resting regions increases by 2.279 % and if AC is removed then the mean distance increases by 2.127 %. A sorted list of vulnerabilities of regions is shown in Table 4. The number of connections or the degree of regions is positively correlated with most local network measures especially the centrality parameters (Appendix Table 12). The number of connections of the ipsilateral BG2 network was visualized using the K-core analysis where regions are sorted with regard to their number of connections and those regions which have the smallest number of connections are removed until the regions with the largest number of connections remain in the center of the K-core visualization (Appendix Fig. 22). This has been performed for the ipsilateral BG2 and the bilateral BG2 network (Appendix Fig. 23). It turns out that CPu, AGl, AGm, VTA, MRF and PF are the core regions of the BG2 network. The MDS of the bilateral BG2 network is relatively symmetric. In this case, all thalamic regions of the ipsilateral hemisphere are
123
arranged in the upper left quadrant. Ent, Pir and HIPP are in between these thalamic regions. SNR, SNC, CPu, AGl and AGm are located in the lower left and lower right quadrant, respectively, for the contralateral regions. If MDS is performed on the weighted connectivity matrix, then the regions are moving closer together in direction of the midline of the MDS diagram (not shown). Connectomics of an embedded BG-network An embedding of the BG1 network into a network of regions which have direct input and output connections to the BG1 regions was performed to determine the importance and ranks of BG1 regions in the context of extrinsic direct and indirect neighbors. The results of the analysis are shown in Appendix Table 14. Changes of ranks of regions were found when comparing the BG1 network with the network of all regions that are directly connected to the BG1 network. In terms of BC, the CPu, AGl and STh are covering the first three ranks. In the direct neighbor network (DNN) the AGl, AGm and LGP have ranks 1, 2 and 3. In the indirect neighbor network (INN) CPu, SNC and AGl have ranks 1, 2 and 3. With regard to EC, the SNC, CPu and MGP are on the first three ranks in the BG1 network, in the DNN and INN these are SNC, SNR and MGP. Hence, with regard to EC there is not so much change in the ranks as compared with the BC parameter. Interestingly, the ranking of regions by means of PRC, SC, Hub, Aut, RADout and RADin in the BG1 network, DNN and INN is similar: CPu, AGl, STh, MGP, SNC or SNR are covering the first three ranks. A further interesting finding is that VL is considered as an important output region of the BG, however, the region rankings of the embedded BG1-network indicate variable ranks of 4 or 7, only. Furthermore, the CPu changes its rank in the DNN when considering the RADin parameter from 1 in the BG1-network to 7. Global extrinsic connectivity of the BG2 network The global extrinsic connectivity of the BG2 network has been determined ipsilaterally (Table 5) and bilaterally (Appendix Table 11). Extrinsic connections are those where at least the source or target is not a member of the set of regions of BG2. The CL has most direct ipsi- and contralateral inputs (284) in the BG2 network. The AGl has most direct outputs followed by MRF and AGm. However, AGm has the most ipsilateral outputs (516). When considering all connections of subregions of a region of BG2, then the MRF has most inputs (78,691) followed by AC (5,424) and CPu (1,536). The ranks of regions changes when regarding their subtree outputs: AC is 4,903, for MRF it is 1,893 and for AGl it is 1,375. In the bilateral
Brain Struct Funct Table 5 Global extrinsic connections of regions of the ipsilateral BG2 network Region
Dic
Dii
Dis
Doc
Doi
Dos
Sic
Sii
Sis
Soc
Soi
Sos
Centrolateral_thalamic_nucleus_L
398
446
844
10
68
78
413
470
883
10
103
Accumbens_nucleus_L
277
378
655
5
74
79
373
829
1,202
17
304
321
Central_medial_thalamic_nucleus_L
278
322
600
18
70
88
288
340
628
18
184
202
Paracentral_thalamic_nucleus_L
242
309
551
10
81
91
245
413
658
10
89
99
Mesencephalic_reticular_formation_L
226
311
537
235
334
569
3,418
4,273
7,691
1,383
3,510
4,893
Amygdaloid_complex_L
148
375
523
15
71
86
951
4,473
5,424
933
3,970
4,903
Ventrolateral_thalamic_nucleus_L
154
344
498
12
52
64
175
384
559
13
56
69
82
311
393
31
79
110
479
1,057
1,536
35
172
207
Parafascicular_thalamic_nucleus_L
124
213
337
42
136
178
160
377
537
127
291
418
Ventromedial_thalamic_nucleus_L
91
244
335
37
54
91
128
356
484
44
161
205
Ventral_tegmental_area_A10_L Medial_agranular_prefrontal_cortex_L
78 70
238 131
316 201
138 130
231 416
369 546
127 112
476 212
603 324
188 203
440 584
628 787
Hippocampus_L
37
137
174
27
46
73
263
1,030
1,293
228
631
859
Entorhinal_cortex_L
35
137
172
36
63
99
166
900
1,066
80
546
626
Lateral_habenular_nucleus_L
30
128
158
44
99
143
70
226
296
52
129
181
Subthalamic_nucleus_L
25
124
149
31
83
114
38
181
219
42
127
169
Substantia_nigra_compact_part_L
32
101
133
48
126
174
47
136
183
58
181
239
Lateral_agranular_prefrontal_cortex_L
30
95
125
284
381
665
89
436
525
574
801
1,375
Piriform_cortex_L
17
103
120
27
53
80
86
300
386
73
191
264
Mediodorsal_thalamic_nucleus_medial_part_L
15
91
106
1
14
15
15
91
106
1
14
15
Mediodorsal_thalamic_nucleus_lateral_part_L
22
82
104
2
18
20
22
83
105
2
18
20
Substantia_nigra_reticular_part_L
20
73
93
50
133
183
20
86
106
74
200
274 91
Caudate_putamen_L
113
Medial_globus_pallidus_L
15
53
68
17
59
76
16
58
74
18
73
Lateral_globus_pallidus_L
10
22
32
0
4
4
10
22
32
0
5
5
Ventro_anterior_thalamic_nucleus_L
12
12
24
0
10
10
15
14
29
0
10
10
The regions were sorted by the direct input sum (Dis) Dic direct input from contralateral, Dii direct input from ipsilateral, Dis direct input from ipsi- and contralateral, Doc direct output to contralateral, Doi direct output to ipsilateral, Dos direct output to ipsi- and contralateral, Sic subtree input from contralateral, Sii subtree input from ipsilateral, Sis subtree input from ipsi- and contralateral, Soc subtree output to contralateral, Soi subtree output to ipsilateral, Sos subtree output to ipsi- and contralateral
BG2 network (Appendix Table 11) CL has most inputs (838) followed by Ac (645) and CM (587). The AGl has most outputs (640) followed by MRF (561) and AGm (524). Most subtree (for details see ‘‘Materials and methods’’) inputs are going to MRF (7,589), AC (5,291) and HIPP (1,358). The output from subtrees is dominated by AC (4,821) followed by MRF (4,700) and AGl (1,307). Local extrinsic connectivity of the BG2 network In this section extrinsic regions are determined that have abundant connections to the intrinsic regions of the BG2 network. Furthermore, extrinsic regions were determined that receive many connections from intrinsic regions. To identify these extrinsic regions which are densely interconnected with intrinsic regions, a search through the whole connectome was performed to filter out those regions which are located around hierarchical level 9 and
possess direct ipsilateral inputs or outputs to the intrinsic regions. Those regions that have no input or no output were removed from this selection. After this conditional expansion of the network, the interactive hierarchical table method of neuroVIISAS was used to determine extrinsic regions that have abundant inputs to or get outputs from intrinsic regions. It turns out that the medulla oblongata provides most inputs to the CPu, followed by the lateral thalamic nuclear group, medial geniculate nucleus, ventral thalamus and intralaminar nuclei of the thalamus. Most outputs of the CPu are going to intrinsic nuclei. However, the lateral thalamic group and the substantia innominata nucleus basalis complex receive a few outputs of the CPu. Abundant inputs to the SNC derive from the medulla oblongata, tuberomammillary nucleus and the pedunculopontine tegmental nucleus. Most external outputs from SNC are going to the ventral thalamus. The medial global pallidus does not have many extrinsic outputs nor does it
123
Brain Struct Funct
receive many extrinsic inputs. The subthalamic nucleus receives abundant extrinsic inputs from the pedunculopontine tegmental nucleus and the ventral thalamus. The sole outputs of MGP are going to the olfactory tubercle. The accumbens nucleus receives many inputs from mesocortical regions, the medulla oblongata as well as from hypothalamic zones. Most extrinsic outputs are going to hypothalamic zones, the olfactory tubercle and the medulla oblongata. The VL receives abundant connections from the cerebellum and hypothalamic zone. VL projects extrinsically to the medulla oblongata and mesocortical regions.
Discussion The two levels (BG1, BG2) of the mesoscale connectome of the BG were investigated with regard to intrinsic, extrinsic, weighted, ipsi- and contralateral configurations. Connectome embedding and differential connectomics were applied to determine the importance of regions (ranking) in terms of their connectivity in the network. The routing problem and reciprocity patterns The three principal pathways (direct, indirect, hyperdirect) were revealed in the BG connectome. By applying pathway analysis, many further possible pathways from the AGl over the CPu back to AGl were detected in the ipsilateral BG2 network. So far, alternative pathways in the orofacial system (Mascaro et al. 2005) and the pedunculopontine nucleus of the interface between the BG and brainstem nuclei (Martinez-Gonzalez et al. 2013) have been described. Which pathway or set of pathways are used predominantly in terms of biosignal processing and function cannot be worked out by network analytical methods. Maybe this routing problem can be resolved by considering the coding features of spike patterns (intrinsically bursting, regularspiking, fast-spiking) or inhibitory and excitatory functions of local circuits (Tiesinga et al. 2008; Steriade 2004). In an ongoing meta-study, the transsynaptic pathways after virus (especially pseudorabies) injections are being collated and will be compared with the pathway results of monosynaptic tracing studies. Certain reciprocal and non-reciprocal patterns between MGP and SNR to thalamic and further on to the AGl were detected. Continuous reciprocal links of VM and PF between BG output nuclei and AGl were determined. However, VTA is reciprocally connected with SNR and MGP but non-reciprocally with AGl. Rat connectome studies There are not so many publications available which report graph theoretic analyzes of partial connectomes of the rat
123
nervous system. A major connectional analysis of some parts of the rat central nervous system was made (Burns 1997) and a sophisticated web-page (BAMS2 ) for connectivity retrieval of the rat CNS is available (http://bran cusi.usc.edu). Connectome studies focusing on specific parts or functional systems of the rat CNS were performed in terms of the nucleus of the solitary tract (Palombi et al. 2006), the hippocampus (Burns and Young 2000), the thalamocortical connectivity (Costa and Sporns 2006), the brainstem reticular formation (Humphries et al. 2006) and the retrosplenial cortex (Sugar et al. 2011). Moreover, there exist an increasing amount of meta-studies of connectomes of many other organisms from Caenorhabditis elegans (Towlson et al. 2013) up to primates (Ko¨tter 2004; Stephan et al. 2001a; Passingham et al. 2002). The unique feature of the rat project within neuroVIISAS is (1) the preservation of hierarchical terminologies in the form of an advanced neuroontology (Bota and Swanson 2007a), (2) the explicit collation of ipsi- and contralateral connections, (3) the extraction of collateral projections, (4) transneuronal and transsynaptic connectivity, (5) the multimodal integration of digital atlases, (6) and the integration of populationbased simulations using NEST (Gewaltig and Diesmann 2007). Moreover, the rat connectome project encompasses about 90 % of the available tract-tracing literature of the rat nervous system indexed in PubMed (http://www.ncbi. nlm.nih.gov/pubmed). A list of all reports is available on the webpage http://neuroviisas.med.uni-rostock.de/refer ences.html. In the future, the databases Web of Science, Scopus and Google Scholar will be queried for searching those reports that are not indexed in PubMed. Nevertheless, it is unknown how many connections at the scale of nuclei exist (ground truth of connections) and how dynamic changes of connectivity strengths influences variability of the estimates of connectivity strengths. In particular, the question of completeness becomes evident with regard to contralateral connectivity because in most tract-tracing studies contralateral connections are not systematically investigated. This means that especially contralateral connectivity should be considered as underestimated. In the macaque Barbas et al. (2005) provide evidence that ipsiand contralateral connectivity depend on topological proximity and the structural type of linked areas. Furthermore, they found that projections of both hemispheres are highly correlated with the structural architecture of cortical areas and that the contralateral connectivity mirrors ipsilateral connectivity while being weaker. This suggests that contralateral connectivity brings out a specific laterality pattern which should be investigated further. Hence, the successive accumulation of contralateral connections in the rat connectome project should elucidate the many aspects of contralateral connectivity at the cortical and subcortical level of the central nervous system.
Brain Struct Funct
Reliability of connections
Network granularity
It should be emphasized that some connections and regions may receive more attention in tract-tracing research so that frequently investigated or popular regions may have higher observation scores. This means, that the observation score introduced here should be taken with caution. There could be different reasons like disease (Parkinson syndrome, Huntington-disease and so forth), function (whisker cortex) or accessibility (size of source and target regions in tract tracing experiments, e.g., CPu) why connections or regions are investigated more often than others. Furthermore, most experimental tract-tracing studies typically have been hypothesis driven and restricted to the analysis of one or a few brain regions. The number of tract tracing studies that aim to map connections of a larger functional or structural system of the CNS is rather limited. However, the mouse connectome project at the Allen Institute (http://www. mouseconnectome.org) performs a high-throughput mapping of the interconnectivity of the whole mouse brain based on the tract-tracing technique (Oh et al. 2014). Nevertheless it is useful to have the reliability score available, so that one can filter the connectivity matrix to only contain results that have been observed multiple times. It is reasonable to consider specific reliability weights of connections between two regions in a particular report and in independent reports. In such a way, it could be possible to judge inherent observer biases of connections. The analysis of observation scores shows large values for most connections of the BG network data. Especially, the ipsilateral connections have larger reliability values than the contralateral connections. Obviously, the ipsilateral connectivity of the bilateral BG1 network is stronger than the contralateral connectivity. Furthermore, the laterality of cortical regions is larger than those of the subcortical regions. In the chord diagram of the BG1 network the strong connectivity of SNC to CPu attracts attention. However, in the vulnerability analysis the removal of the SNC, comparable with its neuro-degenerative loss of dopaminergic neurons in the SNC in Parkinson disease, shows an unexpected 6th position in the vulnerability ranking, only. This is in agreement with the modularity role of the SNC, indicating that the SNC is not a complex integrator or hub-region (hub = 0.672) (Sporns et al. 2007). This is also confirmed by the relatively large Shapley-value of 0.008 in the bilateral BG2 network (Small or negative Shapley-values indicate an increase of importance of nodes in a network. Because the Shapley-value is a sum of the differences of connected components, negative values occur. The formal definition can be found in the Appendix.).
The BG2 network contains functionally important limbic regions and specific thalamic, mesencephalic and brainstem regions. In particular, the brainstem and mesencephalic regions show a very dense connectivity to most BG2 regions. However, diencephalic regions of the thalamic nuclei have dense connections to cortical areas and specific subcortical regions of the BG. The larger granularity of the BG2 network emphasizes the impressive contralateral connectivity as shown in the planar network diagram (Fig. 4). This means that the laterality should be considered in dependence of the network granularity. The global connectivity of the BG2 network indicates that it has a small world (Sporns 2006; Bassett and Bullmore 2006) and scale-free structure (Stam and Reijneveld 2007). However, it remains to be seen whether the conclusion will hold when the complete rat connectome is considered (LimaMendez and van Helden 2009). Connectome visualization and thalamic regions The BG2 network was also visualized in 3D-stereotaxic coordinate space (Paxinos and Watson 2007) in neuroVIISAS (Fig. 1) in combination with local-network parameters and weighted connections (Fig. 2). Before computing the surfaces of regions, the contours of all regions of the stereotaxic atlas were manually edited and related to the ontology (Schmitt and Eipert 2012). The surfaces were determined by the marching-cube algorithm of VTK (http://www.vtk.org/) in neuroVIISAS. In the BG2network the CPu has a strong ipsilateral connectivity whereas cortical regions can be distinguished by many contralateral connections. The thalamic PF region is a particular region which has many direct (adjacency matrix) and indirect (communicability matrix) connections to most other regions of the BG1- and BG2-network. In contrast, VA appears to be a region with marginal connectivity only. Modularity analysis and MDS The modular grouping that was found by modularity analysis and MDS indicates a relation of the connectional architecture with functional entities. The regions of the ipsilateral and contralateral network were also grouped by applying metric multidimensional scaling (MDS). Those regions that have similar connections are located close together in the MDS-diagram (Appendix Fig. 24) and those which have different connections are located far away from each other. In the ipsilateral BG2 network nearly all thalamic regions are located in one quadrant of the MDS-
123
Brain Struct Funct
diagram and the motor cortical regions, CPu, MGP and SNR are located together with MRF in another quadrant. A trend towards such a thalamic—basal ganglia—grouping was also found in the bilateral BG2 network (Appendix Fig. 25). Local parameters of the basal ganglia connectome Most local parameters of the ipsilateral BG2 network show relatively high correlations with the DGAll parameter. Especially the centrality parameters build a group with similar numeric trends in the parallel coordinate visualization (Appendix Fig. 19). Comparable patterns of correlations of local network parameters were also described by Bounova and de Weck (2012). However, not for each region does the correlation of DGAll (Appendix Table 12) and another parameter like the Shapley-value and EC-value show comparable trends. The CPu has rank 2 with regard to DGAll , however, for EC it has rank 3 (0.92) and for the Shapley-value it has rank 1 (0:31) (Table 6). AGl has a Shapley-value of 0:28 and rank 2, whereas, the EC-value is 0.685 and has a rank of 11, only. The DGAll of AGl is 31 which equals rank 8. AGm shows comparable values: its Table 6 Local parameters of the ipsilateral BG2 network
The regions are sorted using the degree all. Abbreviations and parameter definitions are described in the Appendix. A complete list of all local parameters is shown in the Appendix Table 15 A all, DG degree, I in, O out
123
Shapley-value is 0:24 (rank 3) and the EC-value is 0.693 (rank 10), however, its DGAll is 34 with rank 5, only. This means that the total number of connections of a particular region does not fully determine its importance. Furthermore, connectional aspects like connections to other important regions (authoritativeness) and local clustering influence its importance on the connectional level. The weighted-ipsilateral BG2 network provides other ranks, yet the CPu still belongs to the group of regions with highest ranks. The VTA and SNR have ranks 2 and 3 with regard to DGAll . The CPu and VTA stay on ranks 1 and 2 when ECvalues are considered. However, SNR has only rank 7 and MRF gets up to rank 3. The Shapley-value cannot be computed for weighted networks. Hence, some numeric similarities can be observed when connectional roles of particular regions are considered in a non-weighted and weighted BG2 network. A complex region at midbrain and brainstem levels like the MRF with many large weights may strongly change the ranks of importance of regions in a particular functional system like the BG. This means important regions of an isolated functional system or subnetwork like the BG2 that is embedded in a significantly more complex network may change their significances of
Region
REC
DGA
DGO
DGI
BC
Shapley
Hub
Aut
AC
20
42
22
20
0.0736
-0.16
0.96
1
CPu
18
41
20
21
0.0923
-0.31
0.89
1
VTA
17
40
23
17
0.051
-0.1
1
0.89
MRF
15
38
23
15
0.0287
-0.01
1
0.87
AGm
15
34
17
17
0.0694
-0.24
0.72
0.82
SNC
11
32
21
11
0.0177
0.03
0.93
0.67
SNR
11
32
19
13
0.0279
-0.04
0.81
0.77
AGl
14
31
15
16
0.0487
-0.28
0.67
0.76
MGP
11
30
17
13
0.0239
-0.02
0.74
0.77
STh
12
30
17
13
0.0175
0.02
0.78
0.78
PF
11
28
17
11
0.0073
0.06
0.78
0.73
Ac
10
28
13
15
0.0214
0.01
0.58
0.82
LHb
8
22
11
11
0.0057
0.07
0.51
0.71
Pir
5
22
8
14
0.0105
0.07
0.36
0.8
VM Ent
7 8
21 21
9 9
12 12
0.0034 0.0086
0.11 0.05
0.45 0.42
0.77 0.65
CM
5
20
10
10
0.0042
0.12
0.48
0.66
HIPP
6
20
11
9
0.0048
0.11
0.52
0.55
MDL
4
17
4
13
0.0025
0.19
0.2
0.8
CL
4
16
8
8
0.0017
0.14
0.38
0.55
LGP
4
16
5
11
0.0017
0.16
0.23
0.69
PC
5
14
6
8
0.0009
0.15
0.31
0.56
MDM
4
14
4
10
0.0013
0.2
0.19
0.59
VL
3
14
3
11
0.0008
0.25
0.15
0.7
VA
0
5
2
3
0.0001
0.42
0.1
0.18
Brain Struct Funct
regions heavily. This was proved by embedding the lesscomplex BG1 network (contains all BG-core regions) into (1) the extrinsic direct neighborhood network and (2) into the indirect neighborhood network. It was found that some regions-ranks of the direct and indirect networks are constant in terms of EC whereby SNC has rank 1 in this comparison. For the parameters PRC, SC, Hub, Aut, RADout and RADin region-ranks change a little, however, the most important regions stay in the upper ranks. The low ranking of VL does not reflect its significance with regard to its BG output role. This finding is in line with the high variability of regions which have direct input to the AGl in the pathway analysis and supports the hypothesis that the projections from BG-core regions are relatively divergent to different thalamic nuclei. In the non-weighted bilateral BG2 network the MRF, AGm, AGl and VL have the largest DGAll values. However, the Shapley-value of the CPu is conspicuously small (0:37) as well as the value for the AGl (0:37), followed by the AC (0:35). Again, another context like the bilateral selection of BG2 regions changes the importance and ranks of particular regions. With this direct confirmation of the change of importance of brain regions in different contexts of networks it is emphasized that connectomeanalysis depends strongly on the selection of intrinsic regions and extrinsic regions of particular functional subsystems of nervous systems. Motif-analysis A further local feature is the frequency by which regions contribute to particular motifs (Fig. 8a, b; Table 3 and Appendix Table 10). This was investigated in the ipsilateral and contralateral BG2 network. The most impressing result of this analysis was the significantly higher frequency of some motifs (3-12, 3-13) with reciprocal edges. A significantly higher number of complex 5-node motifs was found in the BG2 networks. Those regions, that have high degrees are more frequently found to build reciprocal motifs such as AC, CPu, VTA and MRF in the ipsilateral BG2 network and MRF, AC, VTA and AGm in the bilateral BG2 network. The frequencies in Table 3 of the bilateral BG2-network were sorted with regard to the fully reciprocal motif 3-13. However, regions with high abundance in motifs 3-12 and 3-13 do not have high abundance in the complex 5-node motifs. It is striking that regions with many contralateral connections like AGl and AGm contribute significantly more often to the symmetricINmotif. Surprisingly, the frequency of the MGP-contribution to the symmetricIN-motif is maximal which means that MGP is most often involved in building the symmetricINmotif.
Principal component analysis Principal component analysis (PCA) was performed to determine significant parameters out of a set of 6 local parameters that reflect local connectivity (Echtermeyer et al. 2011). In the PCA-plane the thalamic regions have a relatively small component 1 and a larger component 2, while limbic regions tend to be localized in the center of the PCA-plane and BG-core regions appear to have a relatively larger component 1 (Fig. 9). Again, this distribution of regions can be interpreted as a grouping with functional meaning. Within the PCA-analysis it was found that the CPu has a very large amount of direct connections to most of the BG2 regions (Fig. 9c). Knotty centers and rich-club analysis The knotty-center in the bilateral BG2 network contains the cortical regions AGl and AGm, followed by the CPu and AC as well as MRF and VTA (Appendix Fig. 21). By comparing the knotty-center with the rich-club regions (Appendix Fig. 20) of the bilateral BG2 network, the SNR is the sole region which is a rich-club member, however, not in the knotty-center. Some of these regions were also found in the K-core analysis. This analysis has assigned in the ipsilateral BG2 network the CPu, STh, MGP, AC, PF, MRF, VTA, SNR and SNC to the inner core and in the bilateral BG2 network AGm was assigned to the interior core, however, AGl, SNR, SNC and STh to the first shell. Partly, these groupings provide comparable results for CPu, AC, MRF and VTA, however, some differences for SNC, SNR, AGl, AGm, MGP and STh were found as well. Representation and interpretation of ipsi- and contralateral connections So far, no evidence has been reported of an asymmetry of structural connectivity in the rat brain. However, Chida and Toyosawa (1994) reported a functional asymmetry of the rat hippocampus using small animal EEG. Only, a few studies exist that describe asymmetries of olfactoric processing (Riddle and Purves 1995) and the volume of whisker representing fields of the primary somatosensory cortex in rats (Parthasarathy and Bhalla 2013). In the peripheral nervous system of the rat a significant asymmetry of connections exists, especially for the innervation of impar organs of the abdomen. To allow the representation of peripheral and central connections for extensive connectivity analysis of the complete nervous system of an organism it is mandatory to realize side-dependent connectivity mapping. Finally, the well-known structural as well as functional asymmetry of connections of the human
123
Brain Struct Funct
CNS support the concept of a generic framework allowing left- and right-side representations of connectivity. Organization of connections in a hierarchy of regions A further important challenge of the approach described before by Schmitt and Eipert (2012) is the hierarchical organization of subdivisions of regions in the rat connectivity project in neuroVIISAS. The concept of hierarchy is based on spatial scales and semantic relations. The concept is extended by defining variants of the hierarchy to map connections in terms of functional subdivisions. This approach allows the consistent representation of connections of very fine details such as terminal fields as well as connections of functional subsystems like, e.g., the sensory and motoric barrel cortices. Such a scale-based organization of connectivity data has several advantages with regard to multiscale connectivity analysis and connectome based population simulations. By controlling the granularity of a connectome through spatial scales it is also possible to filter and compare connectomes at coarse scales where most connectivity data have larger reliability values. This scaling-technique can be used in combination with functional and chemoarchitectonic data (data are not presented here) which were collated along with the connection data. Such an exceptional database of experimental raw data could be the basis for more realistic large scale simulations of the rat physiom [http://neuroviisas.med.unirostock.de/ratphysiome.html (Bai et al. 2006)]. One important perspective is to use these tract tracing based connectome data of the BG and to combine it with functional attributes at the microcircuit level (Diesmann et al. 1999) to define more realistic and population-based models for simulation studies (Gewaltig and Diesmann 2007). As shown before by (Schmitt and Eipert 2012) this step can be performed also in neuroVIISAS using a hybrid neuron model (Thibeault and Srinivasa 2013) in the NEST simulation engine (Gewaltig and Diesmann 2007).
connectivity reflects, at last partly, functionality. We have worked out a problem by pathway analysis termed the routing problem because multiple pathways from a particular source to a target were found in the data. So far, it had not been possible by network analytical methods to determine a pathway between a source and a target node that is the principal route or a restricted set of functional routes in vivo. Now a comprehensive data set is available and can be shared by the neuroscience community to be used for simulation studies in computational neuroscience as well as in connectomics and neuroinformatics. Acknowledgments The authors thank Klaus-Peter Schmitz (Department of Biomedical Engineering, University of Rostock) for the support of the neuroVIISAS project. We thank Frauke Winzer, Susanne Lehmann, Antje Schu¨mann, Jennifer Meinhardt, Ann-Christin Klu¨nker for their faithful work on extending the database and mappings. All work was supported by the Faculty of Mathematics and Natural Sciences and of the Faculty of Medicine of the University of Rostock.
Appendix A The appendix contains in the first part figures, in the second part tables and the third part formal definitions of matrices, graph-theoretical parameters and randomization models. See Figs. 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26.
Summary The connections of the basal ganglia and important regions directly connected with the basal ganglia were collated in a standardized and systematic way from all available articles which have documented results of tract-tracing experiments in normal juvenile and adult rats. The connectome was analyzed with regard to laterality in a ipsilateral and contralateral way as well as with regard to connection weights in a weighted and non-weighted configuration. Functional groups like thalamic, limbic, basal ganglia core regions and a motor associated region can be deduced from the connectional patterns by multivariate methods. Hence,
123
Fig. 10 Ordinal categories of connection strengths are plotted on the abscissa and the hypothetical estimates of connection densities shown on the left ordinate which scales the blue straight line. The right ordinate is linearly scaled for the red graph. The largest slope lies between the semiquantitative values 3.5 and 4. This logarithmic transformation has been used for adapting the ordinal categories to a distribution of connection densities that was observed by others as well [Hilgetag and Grant (2000); Markov et al. (2011, 2014); Oh et al. (2014); Ercsey-Ravasz et al. (2013); Wang et al. (2012)]. The function for transforming ordinal categories of connection strengths to the logarithmic distributed connection densities is f ðxÞ ¼ elnð10Þx4lnð10Þ
Brain Struct Funct
Fig. 11 The regions of the BG2 network are organized in a hierarchy. It is presented as a triangle hierarchy (top) and a region hierarchy (left). The sequence of regions of the hierarchy (from left to right) is
the same as that in the adjacency matrix. All regions of the connectomes that were investigated are organized in a hierarchy
Appendix B
disjunctive region that contains neuron perikarya (sources of physiological action potential) and/or axonal terminals (targets of physiological action potentials). Set of indexed nodes: The set of all indices of nodes is
Tables 7, 8, 9, 10, 11, 12, 13, 14, 15.
Appendix C: Formal definitions of parameters, matrices and concepts The definitions of expressions, parameters, matrices and simulation models (random graph models) used in this article are summarized in the following. More detailed descriptions, algorithms and proofs are provided elsewhere (Newman 2010; Rubinov and Sporns 2010; Newman et al. 2006; Ko¨tter 2007; Brandes and Erlebach 2005; Ko¨tter 2003). Basic definitions Node, vertex: The smallest subunit of a network. With regard to connectomes a node is a circumscribed or
N ¼ f1; 2; 3; . . .; ng
ð2Þ
Number of nodes: The number of nodes (regions, vertices) is n ¼ jNj
ð3Þ
Edge: A directed edge ði; jÞ 2 N N is the line that connects vertices i and j with source i and target j. The set of directed edges E is E NN
ð4Þ
Edges: The number of edges (connections, links) is ¼ jEj
ð5Þ
Set of edges:
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Brain Struct Funct
Fig. 12 Chord diagram visualization of the bilateral BG1 network using CIRCOS. The diagram consists of 4 nested rings. The outer ring of arcs indicates the relative frequency of inputs and outputs to a particular region. The second ring is the input arc ring and the third ring is the output ring of arcs. The inner ring indicates the absolute
numbers of connections followed by a thin layer of arcs with an input part and output part for each region. The extensive connectivity of the CPu with SNC and SNR is visualized prominently. Stronger contralateral connections can be easily found by the thicker red connections between the CPu and contralateral AGl
L ¼ fði; jÞ 2 Eji 6¼ jg
Weighted matrix: The weighted matrix W is
ð6Þ
W ¼ ðwij Þni;j¼1
The set of all not self-referencing edges is ‘ ¼ jLj
ð7Þ
Graph: G ¼ ðN; EÞ
ð8Þ
Adjacency matrix: The adjacency matrix (connectivity matrix) A is A¼
ðaij Þni;j¼1
123
where aij ¼
1
if ði; jÞ 2 E
0
else
ð9Þ
ð10Þ
whereas wij is the weight of the edge ði; jÞ that connects i and j. 0 wij 1. Path: A sequence of vertices (v1 ; . . .vk ) is a path from (v1 to vk ) if 8i 2 f1; . . .; k 1g : ðvi ; viþ1 Þ 2 E. The length of a path v1 ; . . .; vk is k 1. Distance matrix: The distance matrix D is D ¼ ðdij Þni;j¼1 where
ð11Þ
Brain Struct Funct
Fig. 13 Chord diagram visualization of the bilateral BG2 network using CIRCOS. The amount of connections is relatively large. The strong connections of the CPu and AC are diverging into many different regions
dij ¼ dði; jÞ length of the shortest path from i to j; ¼ 1;
if such a path exists else
ð12Þ Generalized topology matrices (GTOM): Let Nm ðiÞ be the m step neighborhood of node i: Nm ðiÞ ¼ fj 6¼ ij minfdði; jÞ; dðj; iÞg mg
ð13Þ
Nmout ðiÞ ¼ fj 6¼ ijdði; jÞ mg
ð14Þ
Nmin ðiÞ ¼ fj 6¼ ijdðj; iÞ mg
ð15Þ
then the GTOM-matrix of step m is defined as
n m GTOMðmÞ ¼ ðgm ij Þi;j¼1 with gij 8 jNm ðiÞ \ Nm ðjÞj þ aij aji > > > > < minfjNm ðiÞj; jNm ðjÞjg þ 1 aij aji ¼ > > > > : 1
if i 6¼ j if i ¼ j
ð16Þ out The definitions of GTOMin ðmÞ and GTOMðmÞ with the directed in out m step neighborhoods Nm ðiÞ and Nm ðiÞ are analog. Degree all (degall , DGAll ): Self-references of nodes are not considered for all three degree measures. degðiÞ ¼ degall ðiÞ
123
Brain Struct Funct
Fig. 14 The distance matrix of the ipsilateral BG2 network. The distances between regions correspond to the smallest number of connections that are necessary to connect pairs of regions. The thalamic paths are in most cases longer (lighter gray values) than paths of midbrain regions or MRF
Fig. 15 The communicability matrix of the ipsilateral BG2 network. The number of shortest paths through a pair of nodes is presented by the communicability matrix (Estrada and Hatano 2008). Light gray values are indicating large communicability values. The CPu, AC, STh, PF, MRF and VTA are contributing to shortest paths of the BG2network
degðiÞ ¼
n X j¼1 j6¼i
Degree out:
123
aij þ aji
Fig. 16 The generalized topology overlap matrix (GTOM) of the ipsilateral BG2 network. The pairwise interconnectedness in relation to the number of neighbors that a pair of nodes share in common is presented by the GTOM. Large values indicate many similarities of neighbors and connections of a pair of nodes. E.g., the pair CL and CPu has a relatively large GTOM value like the pair STh and VTA
Fig. 17 The variability of connection strengths of the bilateral BG2 network. Large-standard deviations of connection weights are indicated by light gray values. The variability of connection weights of the SNC has relatively smaller values in comparison to VTA, HIPP and AGl
ð17Þ degout ðiÞ ¼
n X j¼1 j6¼i
aij
ð18Þ
Brain Struct Funct
Fig. 18 The global parameters of the ipsilateral BG2 network. The first row (beginning with ‘‘Nodes’’) presents the general parameters of the BG2 network. The line density is 52.333 %. If each regions would be connected with each other region then the line density is 100 %. Cy number of shortest cycles and CyC is the cycle coefficient, AvgHD is the average hierarchical level of the regions in the BG2 network. Six-different randomizations have been performed. Each
Degree in: degin ðiÞ ¼
n X
Latout ðiÞ ¼ aji
ð19Þ
j¼1 j6¼i
Reciprocal edge count RecðiÞ: RecðiÞ is the number of reciprocal edges adjacent to a node i. X RecðiÞ ¼ aij aji ð20Þ j2N i6¼j
degIPSI out ðiÞ degout ðiÞ
ð23Þ
Laterality of the reciprocal edge count LatRec ðiÞ: The laterality of the reciprocal edge count is the fraction of ipsilateral reciprocal edges. For a node i 2 N IPSI N. X 1 LatRec ðiÞ ¼ aij aji ð24Þ RecðiÞ IPSI j2N i6¼j
Neighborhoods: Out-neighbors of i: IPSI
N be a subset of nodes and Laterality: Let N degIPSI ðiÞ the degree of the node i in the subset N IPSI . Then the lateralities are defined as follows: degIPSI all ðiÞ degall ðiÞ
ð21Þ
degIPSI in ðiÞ Latin ðiÞ ¼ degin ðiÞ
ð22Þ
Latall ðiÞ ¼
randomization was repeated 1,000 times using exactly the same number of regions and connections as in the real BG2 network. The global network parameters of the real BG network are shown in the first column followed by mean values of the randomizations. The rewiring randomization is showing the smallest differences with regard to global parameters when comparing with the real network
Niout ¼ fj 2 Nnfigjaij ¼ 1g
ð25Þ
In-neighbors of i: Niin ¼ fj 2 Nnfigjaji ¼ 1g
ð26Þ
All neighbors of i: Ni ¼ Niout [ Niin
ð27Þ
123
Fig. 19 The local parameters of the ipsilateral BG2 network in parallel coordinates. Parallel coordinate visualization provides an overview of a high-dimensional parameter space. Parameters are sorted by similarity. The CPu, AGl and AGm are highlighted by dashed lines. The CPu has almost largest or smallest values with regard to centrality measures. For abbreviations of parameters see Appendix
Brain Struct Funct
123
Niþ ¼ Ni [ fig
ð28Þ
Network parameters Communicability matrix G: Gpq ¼
1 ðAk Þ X pq
k!
k¼0
¼ ðeA Þpq
ð29Þ
Modularity measure: Let M ¼ fM1 ; . . .; Mm g be a partition of N. Mi is a group, module or cluster of vertices. With 1 X ðajk þ akj Þ; ei ¼ ð30Þ ‘ j;k2M i
j\k
the fraction of edges that fall within group Mi N and 1X X ðajk þ akj Þ; ai ¼ ð31Þ 2‘ j2M k2Nnfjg the fraction of ends of edges that are attached to vertices in group Mi , the Modularity Q¼
m X
ðei a2i Þ;
ð32Þ
i¼1
whereas a2i is the fraction of edges that would connect vertices within group Mi if they were connected at random. A large modularity implies that the fraction of edges that fall within groups is larger than expected in the random case. The partition is generated by a ‘‘Greedy’’ optimization algorithm. Starting with a partition where every single node has its own group, stepwise those two groups are joined that increase Q most. The algorithm ends if there are no more such groups. The weighted case is similar, only the aij are replaced by wij and ‘ is replaced by the sum of the edge weights X ! wij ‘w ¼ ð33Þ i;j2N i6¼j
The method of Newman and Girvan (2004) was used. Global efficiency GE: X 1 1 GE ¼ ð34Þ nðn 1Þ i;j2N dðijÞ i6¼j
! ! GE and GE w analog with d! ði; jÞ and d w ði; jÞ Directed global efficiency: X 1 1 GE! ¼ nðn 1Þ i;j2N d! ðijÞ !
i6¼j
Harmonic mean HM:
ð35Þ
Brain Struct Funct
Fig. 20 The rich-club regions of the ipsi and contralateral BG2 network have a rich-club coefficient / [ 0:8 respectively a rank of 44: The direct neighbors or feeders are shown in the upper arc. Their interconnections are presented by blue edges. The connection of
HM ¼
1 GE
ð36Þ
The directed and weighted versions use the directed and weighted global efficiencies. Local efficiency: The local efficiency indicates how strong neighbors of nodes are interconnected. For each node i the inverse lengths of the shortest paths of the neighbors of i that are passing i are added. The local efficiency is this sum divided by the maximal possible sum of paths between neighbors that are connected with i. The efficiency of the network (global efficiency) is the average local efficiency of all nodes. Directed local efficiency: P 1 j;k2Ni djk ðNi Þ X 1 j6¼k ! LE ¼ ð37Þ n i2N ni ðni 1Þ
rich nodes with feeder nodes is shown by yellow edges and the interconnectivity of rich nodes by black edges. The indices _L and _R are indicating the sides of the hemispheres
Weighted directed local efficiency: ! P 1 w j;k2Ni djk ðNi Þ X ! 1 j6 ¼ k LE w ¼ n i2N ni ðni 1Þ
ð38Þ
ni [ 1
! whereby ni ¼ jNi j and djk ðNi Þ, respectively, djkw ðNi Þ is the length of the shortest path between j and k that contains only neighbors of i. Directed assortativity coefficient r ! : P r
!
¼ 1 2
P
ði;jÞ2L
degout ðiÞ degin ðjÞ 4‘1
2 ði;jÞ2L ðdegout ðiÞ
hP
þ degin ðjÞ2 Þ 4‘1
ði;jÞ2L ðdegout ðiÞ
hP
þ degin ðjÞÞ
ði;jÞ2L ðdegout ðiÞ
i2
þ degin ðjÞÞ
i2
ð39Þ ! Directed and weighted assortativity coefficient r w :
ni [ 1
i2 w w w ðdeg ðiÞ þ deg ðjÞÞ ij ! out in ði;jÞ2L rw ¼ P hP i2 2 2 w w w w 1 1 w ðdeg ðiÞ þ deg ðjÞ Þ w ðdeg ðiÞ þ deg ðjÞÞ ij ij out in out in ði;jÞ2L ði;jÞ2L 2 4‘ P
w w 1 ði;jÞ2L wij ðdegout ðiÞ degin ðjÞÞ 4‘
hP
ð40Þ
123
Brain Struct Funct
Fig. 21 The regions of the ipsi- and contralateral BG2 network that belong to the knotty-center in an arc diagram visualization. The knotty-center regions are on the lower arc. Direct neighbors of knotty centers are shown in the upper arc. Their interconnections are
presented by blue edges. The connections of the knotty-center with their direct neighbors are shown by yellow edges and the interconnectivity of rich-nodes by black edges. The indices _L and _R are indicating the sides of the hemispheres
Fig. 22 The K-core analysis of the ipsilateral BG2 network. The connections of CPu are highlighted. In the outer circles are those regions which have the smallest number of connections and in the center are regions that have the largest number of connections
Fig. 23 The K-core analysis of the ipsi- and contralateral BG2 network. The connections of the CPu are highlighted
123
Brain Struct Funct
The correlation of the degrees of nodes that are connected: 1 r 1. Large positive values imply that nodes are mainly connected to nodes with similar degrees. Large negative values imply that nodes with a large degree are mainly to nodes that have a small degree. If r 0 there is no relation detectable. Average path length = characteristic path length ðdÞ: With P ¼ fði; jÞ 2 N Njdði; jÞ\1g, the set of paths. 8 X > < 1 dði; jÞ; P 6¼ £ ð41Þ d ¼ jPj ði;jÞ2P > : 0; P¼£ In the weighted case the distances dði; jÞ are replaced by the weighted distances. Average directed degree deg: deg ¼
Fig. 24 Metric multidimensional scaling (MDS) applied to the ipsilateral BG2 network. Regions that are adjacent to each other have more similarities with regard to their connectivity than with regions that are further away from each other. The 8 thalamic regions that are located within the lower-left quadrant are relatively similar with regard to connectivity. AGm and AGl in the lower-right quadrant have also a large similarity. Interestingly, SNR is adjacent to MRF, however, the second major output node of the BG is MGP which is not to far from SNR. Limbic regions like Ac, Ent, HIPP and Pir are located in the upper left quadrant
2‘ n
ð42Þ
Heterogeneity VC: Coefficient of variation (VC) of the degreeall parameter. X 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdegall ðiÞ degÞ2 HVC ¼ ð43Þ deg i2N If HVC ¼ 0, all nodes have the same degree. The larger HVC the more diverse are the node degrees. In the weighted case the versions of the degrees are used. The heterogeneity measure of Estrada (2010) was not implemented because it is not defined for directed and weighted graphs. Line density Ld: Ld ¼
‘ n ðn 1Þ
ð44Þ
Without self-referencing edges. Rich-club coefficient /ðkÞ: With Nk ¼ fi 2 NjdegðiÞ [ kg and Ek ¼ fði; jÞ 2 Nk Nk jði; jÞ 2 Eg Gk ¼ ðNk ; Ek Þ is the subgraph of G ¼ ðN; EÞ containing all vertices with a degree greater than k. The rich-club coefficient /ðkÞ of a graph G is defined as the line density of the subgraph Gk : /ðkÞ ¼ LdðGk Þ
ð45Þ
Diameter Diam: Diam ¼ maxfdði; jÞjdði; jÞ\1g
ð46Þ
Katz index: The Katz index (Katz status index, Katz centrality) is a measure for the direct and indirect input of a node (Foster et al. 2001). CKatz ðiÞ ¼ Fig. 25 MDS applied to the bilateral BG2 network. The major output regions SNR and MGP are located in the upper- and lower- left quadrant, however, not too far from each other. Most limbic and all thalamic regions are located in the upper-left quadrant (left hemisphere) or in the upper-right quadrant (right hemisphere)
1 X X
ak ðAk Þji
ð47Þ
k¼1 j¼1
The attenuation factor a has to be smaller than the reciprocal of the absolute value of the largest eigenvalue of A. For a better readability and comparability of the results, in
123
Brain Struct Funct Fig. 26 Counts of cyclic connections or loops in the bilateral BG2 network. The logarithm (Log) of the frequency of contributions of regions to a cycle of size 2–7 is presented by the y-axis. Every second region (all regions of the left hemisphere) are labeled. For a cycle from MRF back to MRF passing 4 regions within the BG2 network there exist 8,255 possible paths or 41,191,973 paths when 7 regions are allowed
neuroVIISAS the Katz centrality is multiplied by the mean in ðiÞ of all nodes with CKatz ðiÞ [ 0. Hence, of the quotient Cdeg Katz ðiÞ
the values lie in the same range as the indegrees. Number of triangles: X ðaij þ aji Þðaik þ aki Þðajk þ akj Þ t! ðiÞ ¼ j;k2Nnfig j\k
ð48Þ
The maximum number of possible triangles that can be deviated from a complete reciprocal triangle is 8. Weighted number of triangles: X 1 1 1 1 1 1 ! t w ðiÞ ¼ ðw3ij þ w3ji Þðw3ik þ w3ki Þðw3jk þ w3kj Þ ð49Þ j;k2Nnfig j\k
Instead of the sum of triangles (t! ðiÞ) the sum of geometric means of edge weights of each triangle is calculated. The 1
following example provides ðwij wjk wik Þ3 as the summand:
Directed transitivity: The general definition of transitivity (T) is the sum of number of triangles around all nodes divided by the maximum possible sum of triangles around all nodes. P t! ðiÞ ! T ¼ P i2N ð50Þ i2N tmax ðiÞ Directed and weighted transitivity: ! P ! t w ðiÞ w i2N T ¼P i2N tmax ðiÞ
ð51Þ
whereby tmax ðiÞ ¼ degðiÞ ðdegðiÞ 1Þ 2 recðiÞ with degðiÞ ¼ number of adjacent edges of i and recðiÞ ¼ number of reciprocal edges of i (the two directions of one reciprocal edge are considered as one reciprocal edge). The degree deg and the reciprocity rec are defined as: X degðiÞ ¼ aij þ aji ð52Þ j2Nnfig
recðiÞ ¼
X
aij aji
j2Nnfig
For the directed and weighted case: 1 wij [ 0 aij ¼ 0 else
123
ð53Þ
ð54Þ
Brain Struct Funct Table 7 Local parameters of the weighted-ipsilateral BG2 network Region
DGA
DGO
DGI
CDC
Katz
SPC
Triag
CyclC
EccO
CPu
26.25
12.25
14.0
0.53333
12.17519
0.16667
503.6196
0.66462
375
VTA
20.875
12.5
8.375
0.4012
8.04604
0.25
436.14384
0.55527
0.41667
SNR
19.125
11.0
8.125
0.42484
8.51141
0.29167
323.26377
0.66477
0.41667
MRF
18.875
12.375
AC
18.0
9.625
6.5
0.34437
6.58671
0.25
399.3736
0.56189
0.5
8.375
0.46528
7.82869
0.25
402.70699
0.49854
0.45
SNC
17.375
11.125
6.25
0.35971
6.90902
0.25
317.85196
0.59338
0.5
STh
16.875
9.0
7.875
0.46667
7.93142
0.25
313.32162
0.64057
0.41667
AGm
16.25
9.75
6.5
0.4
6.16135
375
246.8733
0.51577
0.41667
MGP
15.5
8.75
6.75
0.43548
6.93033
0.33333
270.12611
0.59842
0.41667
AGl
14.5
7.625
6.875
0.47414
6.36241
0.16667
237.70384
0.56222
0.5
PF
13.875
8.5
5.375
0.38739
5.68001
375
263.27964
0.5863
0.5
Ac
12.75
6.25
6.5
0.5098
6.29047
0.25
227.32308
0.55463
0.5
Pir
11.0
4.5
6.5
0.59091
6.51555
0.25
157.8558
0.56038
0.54167
VM
10.375
4.375
6.0
0.57831
6.2192
0.33333
154.06266
0.5957
0.5
HIPP CM
10.25 10.0
5.25 4.75
5.0 5.25
0.4878 525
4.75053 5.51932
0.25 375
133.31931 131.01033
0.57456 0.59741
0.5 0.5
LHb
10.0
4.875
5.125
0.5125
5.42794
0.25
164.73628
0.50748
0.5
VL
9.0
1.5
7.5
0.83333
7.67315
0.2
81.79831
0.73374
625
MDL
8.75
1.5
7.25
0.82857
7.35711
0.41667
101.08281
0.58321
625
LGP
8.75
2.625
6.125
0.7
6.23351
0.41667
106.87098
0.65648
0.54167
CL
8.5
4.0
4.5
0.52941
4.73061
0.41667
96.16541
0.62436
0.5
Ent
8.375
3.25
5.125
0.61194
4.82634
0.25
113.68525
0.43629
0.66667
PC
7.0
2.5
4.5
0.64286
4.89828
0.41667
71.94521
0.60614
0.54167
MDM
6.0
1.5
4.5
0.75
4.34144
0.58333
56.44901
0.52978
625
VA
2.5
1.0
1.5
0.6
1.51929
625
7.97699
0.70278
625
Lev
Loc
Region
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG
CPu
0.41667
0.29079
0.26399
0.25321
0.31398
0
12.54348
366
0.3729
0.96524
VTA
0.5
0.27075
0.30556
0.27075
0.27191
0
12.92935
0.37537
0.2589
0.98554
SNR
0.5
0.25804
0.41506
0.25952
0.33326
0.4375
12.70238
0.43298
0.23739
0.8122
MRF
0.5
0.2747
0.37619
0.2747
0.29024
0
13.0163
0.38224
0.21062
0.98553
AC
0.5
0.28436
0.30362
0.28436
0.23942
0
13.26136
0.37947
0.17976
0.93529
SNC
0.5
0.28185
0.4625
0.28185
0.32768
125
13.2381
0.39005
0.1655
0.86018
STh
0.45
0.33686
0.40545
0.32557
0.37036
0.03333
14.38889
0.34253
0.10538
0.74536
AGm
0.5
0.23989
0.27574
0.25
0.22607
375
12.54605
0.45852
0.17302
0.68563
MGP
0.54167
0.3102
0.42468
0.29496
0.31854
0.23125
13.50658
0.40979
0.1091
0.76064
AGl
0.5
0.3125
0.27552
0.29412
0.26353
0.24405
13.53676
0.43892
0.08385
0.64739
PF
0.5
0.34835
0.45455
0.34835
0.35869
0.04167
14.90441
0.32782
-0.00814
0.68479
Ac Pir
0.5 0.5
0.37099 0.32639
0.34286 0.36882
0.31127 0.33869
0.30886 0.31955
0.3 0.20238
13.70139 14.35294
0.4002 0.36632
0.00466 -0.09458
0.72482 0.68825
VM
0.5
0.41667
0.42898
0.38805
0.37946
0.11528
15.75893
0.29619
-0.18057
0.54
HIPP
625
0.31439
0.41319
0.33333
0.32676
0.20139
14.39167
0.37484
-0.12721
0.56576
CM
0.5
0.31389
0.42778
0.34345
0.35408
0.0625
15.2
0.32213
-0.17637
0.52686
LHb
625
375
0.46023
0.3875
0.3362
0.16493
15.325
0.31079
-0.18284
0.62755
VL
0.41667
625
0.40568
0.40568
0.46476
0.11298
17.03409
0.25543
-0.28858
0.38215
MDL
0.5
0.34375
0.40064
0.40064
0.38289
0.07273
16.52885
0.26345
-0.28663
0.4676
LGP
0.54167
0.64375
0.43523
0.45265
0.46065
0.16761
16.5
0.28107
-0.28201
0.49548
CL
0.54167
0.41295
0.49107
0.45265
0.41451
0.14489
16.75
0.25151
-0.30743
0.48319
123
Brain Struct Funct Table 7 continued Region
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG
Lev
Loc
Ent
0.58333
0.34201
0.32576
0.35897
0.2814
0.18977
15.03846
0.34071
-0.24859
0.47373
PC
0.5
0.40417
0.47098
0.4566
0.41829
0.13929
18.22222
0.20137
-0.43369
0.3041
MDM VA
0.58333 0.66667
375 625
0.38611 0.16667
0.38611 0.4625
0.32442 0.39885
0.23214 0.22807
15.2875 18,325
0.26926 0.23199
-0.41216 -0.75126
0.34497 0.14264
Region
CEO
CEI
BC
EC
SC
PRC
FC
Stress
CPu
4.20438
4.4376
0.23583
1
243.76129
1
0.50716
182
1.41218
VTA
3.94521
3.25424
0.05577
0.91751
154.20322
0.58706
0.39402
45
1.86591
SNR
3.86577
3.65482
0.0939
0.69998
130.14345
0.53169
0.33929
72
0.64769
MRF
3.86577
2.88
0.01208
897
129.7987
0.41633
0.31682
10
0.78248
AC SNC
3.37632 3.91837
3.13043 3.40828
0.04026 0.02159
0.68099 0.80884
107.60925 116.2092
0.64706 0.38099
0.38934 0.2869
30 26
0.65307 0.74325
STh
3.81457
3.42043
0.0237
0.77207
125.52561
0.4873
0.38685
23
0.26545
AGm
3.72574
2.83744
0.07555
0.67544
88.77745
0.70423
0.38213
64
1.49691
MGP
3.34884
3.18232
0.0226
0.59287
86.04644
0.43381
0.31835
22
-0.02124
AGl
3.40828
2.83744
0.01736
0.66608
85.19876
0.64031
0.38948
17
1.02062
PF
3.40828
2.8263
0.00645
0.72779
99.67888
0.32626
0.31893
10
0.45657
Ac
3.21788
2.83744
0.00543
0.56331
77.21747
0.43082
0.27614
7
-0.18659
Pir
2.96907
2.97214
0.01585
0.35111
45.73839
0.45033
0.21232
14
0.74636
VM
2.89447
3.22148
0.00218
0.3923
55.50867
0.36658
0.27266
5
-0.40825
HIPP
2.66667
2.67907
0.01479
0.35601
34.50464
0.35927
0.24167
10
0.37318
CM
3.16484
2.86567
0.00494
0.39959
45.3308
0.30888
0.22619
7
-0.64639
LHb
3
2.71698
0.00453
0.42068
54.47565
0.29981
0.25625
5
-0.65307
VL
2.26772
3.64557
0.00595
0.17041
24.99282
0.46127
0.16023
5
-1.12268
MDL
2.16867
3.23595
0.00177
0.14331
22.56384
0.43209
0.14423
4
-1.49273
LGP CL
2.67907 2.78261
3.08021 2.8263
0 0.00203
0.26767 0.35679
38.75035 35.96419
0.35154 0.27445
0.20786 0.24242
0 6
-0.5946 -0.69016
Ent
2.37037
2.65438
0.00242
0.26855
26.31024
0.36863
0.25
2
PC
2.65438
2.75598
0.00112
0.26166
28.23121
0.28222
0.2934
4
-1.12268
MDM
2.18347
2.75862
0.00192
0.10016
9.94491
0.28595
0.16944
4
-1.30614
VA
2.19847
2.20015
0
0.11114
5.36512
0.1381
0.15
0
-2.21914
Region
ZI
ZA
PCO
PCI
PCA
RadO
RadI
ZO
0
CenO
CenI
CPu
1.63169
1.63651
0.64973
0.64541
0.64907
1.42882
1.44132
3
10
VTA SNR
1.45048 0.46465
1.89849 0.59929
0.6392 0.66219
0.63711 0.61917
0.63839 0.6512
1.41319 1.40799
1.35938 1.39306
-5 -4
-16 -10
MRF
-0.42875
0.49136
0.65177
0.65385
0.65471
1.40799
1.31944
-5
-19
0.72524
0.73019
0.66352
0.66251
0.66397
1.37049
1.34722
-11
-17
AC SNC
-0.31337
0.23626
0.64083
0.5952
0.62771
1.41146
1.37326
-3
-11
STh
0.65916
0.49556
0.63735
0.59965
0.63473
1.40451
1.37431
-8
-15
AGm
0.77174
1.44677
0.65976
0.66198
0.66225
1.39826
1.31424
-9
-17
MGP
0.46465
0.23626
0.66245
0.55967
0.63541
1.36806
1.35243
-11
-16
AGl
1.37199
1.25569
0.59446
0.62281
0.6088
1.37326
1.31424
-11
-18
PF
-0.79964
-0.17863
Ac
0.48349
Pir
-0.96699
VM
0.1715
HIPP
0.48349
123
0.63149
0.64467
0.64216
1.37326
1.31285
-12
-18
0
0.5632
0.63018
609
1.3559
1.31424
-18
-17
0.29208
0.27778
0.64793
0.58264
1.32986
1.33021
-20
-16
-0.27298
0.65959
0.63281
0.6538
1.32118
1.35625
-18
-19
0.43811
0.52608
0.54
0.55205
1.29167
1.2934
-16
-15
Brain Struct Funct Table 7 continued Region
ZI
ZA
PCO
PCI
PCA
RadO
RadI
CenO
CenI
CM
-1.62924
-1.03731
0.65374
0.63492
0.66375
1.35069
1.31771
-16
-18
LHb
-1.69223
-1.02226
0.65746
0.56871
0.62625
1.33333
1.29861
-16
-18
VL MDL
0.77174 -0.48349
-0.65514 -1.31434
0.44444 0.5
0.65778 0.65161
0.64815 0.66286
1.22569 1.20556
1.39236 1.35764
-23 -22
-12 -14 -19
0.65916
0.02881
0
0.46647
0.35714
1.2934
1.34201
-25
CL
LGP
-0.89689
-0.85283
0.53125
0.54938
0.55536
1.30729
1.31285
-21
-22
Ent
0.96699
0.29208
0.27219
0.51636
0.44108
1.24479
1.28993
-21
-16
PC
-1.02899
-1.22839
0.66
0.64815
0.65561
1.28993
1.30382
-22
-21
MDM
-0.96699
-1.31434
0.44444
0.62346
0.60069
1.20868
1.30417
-24
-19
VA
-1.86942
-2.20123
0.5
0.44444
0.48
1.21181
1.21215
-25
-25
The regions are sorted using the weighted degree all. Abbreviations and parameter definitions are described in the Supporting Information A all, AD average degree, C circle, CC cluster-coefficient, CE closeness centrality, DG degree, I in, O out, SP length of shortest path
Table 8 Local parameters of the bilateral BG2 network Region
HD
REC
DGA
DGO
DGI
CDC
LA
LO
LI
LR
Katz
SPC
MRF_R
12
24
67
41
26
0.39
0.55
0.54
0.58
0.63
25.16
1
MRF_L
12
24
68
42
26
0.38
0.56
0.55
0.58
0.63
25.2
1
AGm_R
14
21
55
27
28
0.51
0.58
0.59
0.57
0.67
26.34
1
AGl_R
14
22
48
24
24
0.5
0.6
0.58
0.63
0.59
23.25
1
AGm_L
14
22
56
28
28
0.5
0.61
0.61
0.61
0.68
26.39
1
AGl_L
14
23
50
25
25
0.5
0.62
0.6
0.64
0.61
23.96
1
VL_L
12
4
22
5
17
0.77
0.64
0.6
0.65
0.75
18.12
1
VL_R
12
4
22
5
17
0.77
0.64
0.6
0.65
0.75
18.08
1
HIPP_L
10
8
31
18
13
0.42
0.65
0.61
0.69
0.75
11.15
1
HIPP_R
10
8
31
18
13
0.42
0.65
0.61
0.69
0.75
11.11
1
AC_R
8
26
64
37
27
0.42
0.66
0.59
0.74
0.77
24.82
1
CM_L
11
8
30
13
17
0.57
0.67
0.77
0.59
0.63
19.12
2
LHb_L
10
10
33
16
17
0.52
0.67
0.69
0.65
0.8
16.61
1
CM_R
11
8
30
13
17
0.57
0.67
0.77
0.59
0.63
19.08
2
LHb_R AC_L
10 8
10 25
33 62
16 36
17 26
0.52 0.42
0.67 0.68
0.69 0.61
0.65 0.77
0.8 0.8
16.56 24.76
1 1
CPu_R
11
21
58
23
35
0.6
0.69
0.87
0.57
0.86
31.66
1
PC_L
11
7
20
9
11
0.55
0.7
0.67
0.73
0.71
13.3
2
VM_L
11
9
30
12
18
0.6
0.7
0.75
0.67
0.78
19.37
2
PC_R
11
7
20
9
11
0.55
0.7
0.67
0.73
0.71
13.27
2
VM_R
11
9
30
12
18
0.6
0.7
0.75
0.67
0.78
19.33
2
VTA_R
11
21
54
33
21
0.39
0.7
0.67
0.76
0.76
20.72
1
CPu_L
11
21
58
23
35
0.6
0.71
0.87
0.6
0.86
31.67
1
PF_R
11
15
38
21
17
0.45
0.71
0.76
0.65
0.73
17.29
2
VTA_L
11
22
56
34
22
0.39
0.71
0.68
0.77
0.77
21.41
1
Pir_R
10
6
28
12
16
0.57
0.71
0.58
0.81
0.67
16.27
1
PF_L
11
15
39
22
17
0.44
0.72
0.77
0.65
0.73
17.33
2
SNR_L
12
13
44
26
18
0.41
0.73
0.73
0.72
0.85
17.68
2
CL_L
11
5
22
10
12
0.55
0.73
0.8
0.67
0.8
13.02
2
SNR_R CL_R
12 11
13 5
44 22
26 10
18 12
0.41 0.55
0.73 0.73
0.73 0.8
0.72 0.67
0.85 0.8
17.64 12.98
2 2
123
Brain Struct Funct Table 8 continued Region
HD
REC
Ent_L
11
Pir_L
10
SNC_L SNC_R
12 12
Ac_R
11
10
34
14
20
0.59
0.76
0.86
0.7
0.9
18.73
1
STh_R
9
14
37
19
18
0.49
0.78
0.84
0.72
0.86
18.09
1
STh_L
9
14
38
20
18
0.47
0.79
0.85
0.72
0.86
18.14
1
Ac_L
11
11
35
15
20
0.57
0.8
0.87
0.75
0.91
18.75
1
MDL_L
12
5
21
5
16
0.76
0.81
0.8
0.81
0.8
16.92
2
MDL_R
12
5
21
5
16
0.76
0.81
0.8
0.81
0.8
16.84
2
VA_L
12
0
6
2
4
0.67
0.83
1
0.75
0
4.23
3
MGP_L
12
12
36
22
14
0.39
0.83
0.77
0.93
0.92
VA_R
12
0
6
2
4
0.67
0.83
1
0.75
0
MGP_R
12
12
36
22
14
0.39
0.83
0.77
0.93
0.92
13.95
1
Ent_R
12
2
6
3
3
0.5
0.83
0.67
1
1
2.76
1
MDM_R
12
3
14
3
11
0.79
0.86
1
0.82
1
11.32
2
MDM_L
12
4
16
4
12
0.75
0.88
1
0.83
1
12.1
2
LGP_L LGP_R
12 12
4 4
16 16
5 5
11 11
0.69 0.69
1 1
1 1
1 1
1 1
10.96 10.91
2 2
Region
Triag
CyclC
EccO
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG
MRF_R
1,975
0.07
2
3
0.39
0.56
0.39
0.45
0.1
36.4
0.39
MRF_L
2,021
0.07
2
3
0.39
0.56
0.38
0.45
0.05
36.23
0.39
AGm_R
1,279
0.07
2
2
0.43
0.45
0.39
0.44
0.34
36.29
0.48
940
0.07
2
2
0.43
0.42
0.42
0.42
0.38
36.58
0.44
AGl_R AGm_L AGl_L VL_L
DGA
DGO
DGI
CDC
LA
LO
LI
LR
Katz
SPC
9
28
14
14
0.5
0.75
0.64
0.86
0.89
13.85
1
7
29
12
17
0.59
0.76
0.67
0.82
0.71
17.02
1
13 13
42 42
27 27
15 15
0.36 0.36
0.76 0.76
0.78 0.78
0.73 0.73
0.85 0.85
15.25 15.24
1 1
14.03
1
4.22
3
1,324
0.07
2
2
0.42
0.45
0.39
0.44
0.34
36.41
0.47
988 296
0.07 0.09
2 3
3 2
0.42 0.85
0.41 0.64
0.41 0.64
0.41 0.65
0.39 0.18
36.26 50.28
0.44 0.23
VL_R
296
0.09
3
2
0.85
0.64
0.64
0.65
0.18
50.17
0.23
HIPP_L
510
0.08
2
3
0.58
0.55
0.52
0.56
0.25
42.3
0.41
HIPP_R
479
0.08
2
3
0.52
0.54
0.52
0.52
0.25
42.17
0.41
AC_R
1,714
0.07
2
3
0.4
0.48
0.4
0.43
0.31
36.79
0.41
CM_L
522
0.09
3
2
0.61
0.65
0.57
0.61
0.18
46
0.31
LHb_L
634
0.1
3
3
0.59
0.64
0.55
0.61
0.26
42.52
0.32
CM_R
498
0.09
2
2
0.53
0.65
0.54
0.58
0.19
44.82
0.36
LHb_R
634
0.1
3
3
0.59
0.64
0.55
0.61
0.27
42.39
0.32 0.39
AC_L
1,724
0.08
2
3
0.42
0.53
0.42
0.46
0.26
37.76
CPu_R
1,423
0.07
2
2
0.5
0.42
0.41
0.44
0.28
37.43
0.4
PC_L
266
0.12
3
3
0.71
0.75
0.72
0.73
0.24
53.77
0.17
VM_L
564
0.1
3
2
0.69
0.67
0.62
0.66
0.18
47.33
0.28
PC_R
266
0.12
2
3
0.71
0.75
0.72
0.73
0.23
53.69
0.18
VM_R VTA_R
564 1,434
0.1 0.08
2 2
2 3
0.69 0.46
0.67 0.59
0.62 0.46
0.66 0.51
0.19 0.2
47.14 39.82
0.28 0.38
CPu_L
1,433
0.07
2
2
0.5
0.42
0.41
0.44
0.28
37.49
0.4 0.32
PF_R
0.1
2
3
0.58
0.65
0.56
0.61
0.22
43.91
1,514
0.08
2
3
0.46
0.56
0.46
0.5
0.2
39.59
0.37
Pir_R
418
0.08
3
3
0.49
0.66
0.56
0.56
0.28
42.36
0.36
PF_L
865
0.09
3
3
0.56
0.65
0.54
0.6
0.22
43.38
0.32
VTA_L
123
838
Brain Struct Funct Table 8 continued Region
Triag
CyclC
EccO
SNR_L
916
0.08
3
3
0.39
0.64
0.39
0.49
0.37
36.23
0.44
CL_L
293
0.1
3
3
0.64
0.68
0.63
0.65
0.26
46.71
0.27
SNR_R CL_R
916 293
0.08 0.1
3 3
3 3
0.39 0.64
0.64 0.68
0.39 0.63
0.49 0.65
0.39 0.27
36.1 46.41
0.44 0.27
Ent_L
453
0.09
2
3
0.6
0.63
0.58
0.61
0.27
44.11
0.34
Pir_L
460
0.08
3
3
0.51
0.65
0.58
0.58
0.29
42.64
0.36
SNC_L
948
0.09
3
3
0.47
0.75
0.48
0.56
0.29
39.76
0.41
SNC_R
948
0.09
2
3
0.47
0.75
0.48
0.56
0.3
39.59
0.41
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG
Ac_R
641
0.1
3
3
0.63
0.56
0.51
0.58
0.37
39.21
0.41
STh_R
804
0.1
2
3
0.61
0.63
0.55
0.62
0.25
42.83
0.34
STh_L
837
0.1
3
3
0.6
0.63
0.54
0.61
0.25
42.42
0.35
Ac_L
679
0.1
3
3
0.6
0.57
0.51
0.58
0.37
39.38
0.4
MDL_L
284
0.12
3
3
0.75
0.69
0.69
0.69
0.25
48.31
0.25
MDL_R
284
0.12
3
3
0.75
0.69
0.69
0.69
0.25
48.19
0.26
20
0.15
3
3
1
0.33
0.67
0.67
0.34
49.83
0.15
VA_L MGP_L
728
0.1
3
3
0.49
0.78
0.48
0.59
0.39
37.79
0.42
20
0.15
3
3
1
0.33
0.67
0.67
0.35
49.5
0.16
MGP_R Ent_R
728 18
0.1 0.12
2 3
3 3
0.49 0.83
0.78 0.67
0.48 0.58
0.59 0.69
0.39 0.42
37.63 39
0.42 0.37
MDM_R
138
0.11
3
3
0.83
0.79
0.79
0.78
0.29
49.91
0.28
MDM_L
180
0.11
3
3
0.92
0.77
0.77
0.78
0.29
48.5
0.3
LGP_L
202
0.15
3
3
1
0.82
0.83
0.87
0.38
44.17
0.3
LGP_R
202
0.15
3
3
1
0.82
0.83
0.87
0.36
43.83
0.31
VA_R
The regions are sorted using the degree all. Abbreviations and parameter definitions are described in the Supporting Information A all, AD average degree, C circle, CC cluster-coefficient, DG degree, I input, L laterality, O out, O output, R reciprocal, SP length of shortest path
Table 9 Local parameters of the bilateral BG2 network Region
Lev
Loc
CCO
CCI
BC
EC
SC
PRC
FC
Stress
Shapley
MRF_R
0.32
0.88
0.86
0.67
0.04
0.98
25,442,898.34
0.58
0.58
735
-0.21
MRF_L
0.33
0.91
0.88
0.67
0.04
1
26,026,701.64
0.58
0.56
763
-0.21
AGm_R
0.25
0.62
0.69
0.7
0.04
0.64
17,358,891.41
0.85
0.66
649
-0.22
AGl_R
0.18
0.49
0.66
0.66
0.04
0.55
13,243,875.48
0.89
0.85
516
-0.37
AGm_L
0.26
0.63
0.7
0.7
0.05
0.66
18,011,872.88
0.83
0.68
675
-0.22 -0.4
AGl_L
0.2
0.5
0.67
0.66
0.04
0.57
14,180,107.63
0.92
0.86
552
VL_L
-0.37
0.31
0.51
0.6
0
0.15
2,727,904.72
0.41
0.26
26
0.21
VL_R
-0.37
0.32
0.51
0.6
0
0.15
2,700,497.46
0.41
0.26
26
0.21
HIPP_L
-0.1
0.43
0.61
0.54
0.01
0.51
5,787,286.52
0.34
0.45
181
0.07
HIPP_R
-0.09
0.43
0.61
0.54
0.01
0.49
5,588,129.56
0.34
0.45
217
0.03
AC_R
0.3
0.76
0.8
0.67
0.07
0.89
22,779,869.63
0.78
0.69
942
-0.35
CM_L
-0.18
0.39
0.57
0.6
0.01
0.34
6,998,306.99
0.43
0.46
129
0.09
LHb_L
-0.1
0.45
0.59
0.6
0.01
0.48
8,298,279.27
0.39
0.52
141
0.03
CM_R
-0.15
0.38
0.58
0.6
0.01
0.32
6,632,895.27
0.43
0.46
230
0.09
LHb_R AC_L
-0.1 0.27
0.46 0.75
0.59 0.79
0.6 0.66
0.01 0.04
0.48 0.89
8,215,504.87 22,732,362.73
0.4 0.73
0.52 0.68
138 698
0.03 -0.25
CPu_R
0.25
0.74
0.65
0.78
0.05
0.55
18,102,852.8
1
0.59
726
-0.37
123
Brain Struct Funct Table 9 continued Region
Lev
Loc
CCO
CCI
BC
EC
SC
PRC
FC
Stress
Shapley
PC_L
-0.45
0.23
0.54
0.55
0
0.28
3,976,306.75
0.31
0.59
34
0.16
VM_L
-0.2
0.4
0.56
0.61
0
0.35
7,402,875.66
0.43
0.49
83
0.1
PC_R VM_R
-0.45 -0.2
0.23 0.4
0.55 0.57
0.56 0.61
0 0
0.28 0.35
3,930,742.69 733,2016.86
0.32 0.44
0.59 0.49
35 82
0.16 0.1
VTA_R
0.18
0.68
0.75
0.62
0.03
0.88
18,725,331.26
0.55
0.64
429
-0.18
CPu_L
0.25
0.74
0.65
0.78
0.05
0.56
18,415,083.57
0.99
0.59
714
-0.37
-0.04
0.44
0.64
0.59
0.01
0.59
10,995,770.08
0.38
0.68
195
-0.01
0.2
0.7
0.77
0.63
0.03
0.9
19,880,271.57
0.57
0.65
485
-0.2
Pir_R
-0.16
0.44
0.56
0.58
0.01
0.29
4,746,984.35
0.45
0.4
123
0.09
PF_L
-0.02
0.46
0.64
0.58
0.01
0.61
11,479,685.57
0.38
0.65
205
-0.02 -0.03
PF_R VTA_L
SNR_L CL_L SNR_R
0.14
0.54
0.67
0.59
0.02
0.58
11,064,009.13
0.45
0.49
336
-0.33
0.31
0.54
0.56
0
0.27
3,797,240.62
0.3
0.42
67
0.17
0.14
0.54
0.67
0.6
0.02
0.57
10,923,968.28
0.45
0.49
347
-0.03
CL_R
-0.33
0.31
0.54
0.56
0
0.27
3,721,042.61
0.3
0.42
67
0.17
Ent_L
-0.19
0.37
0.58
0.57
0.01
0.4
5,650,397.26
0.42
0.55
125
0.06
Pir_L
-0.15
0.45
0.56
0.59
0.01
0.29
5,000,332.71
0.46
0.43
116
0.09
SNC_L
0.07
0.58
0.68
0.58
0.01
0.68
10,703,603.42
0.37
0.48
260
0
SNC_R Ac_R
0.07 -0.03
0.59 0.48
0.69 0.57
0.58 0.61
0.01 0.01
0.67 0.4
10,568,914.69 7,650,218.92
0.38 0.48
0.48 0.49
266 150
0 0.02
STh_R
-0.04
0.46
0.62
0.6
0.01
0.56
10,461,940.53
0.46
0.65
252
-0.02
STh_L
-0.02
0.48
0.62
0.6
0.01
0.59
10,996,004.71
0.46
0.63
263
-0.03
Ac_L
-0.02
0.49
0.58
0.61
0.01
0.43
8,161,457.86
0.48
0.52
166
MDL_L
-0.37
0.31
0.52
0.59
0
0.17
3,135,070.85
0.41
0.31
31
0.2
MDL_R
-0.37
0.31
0.52
0.59
0
0.17
3,078,873.46
0.41
0.31
32
0.2
VA_L
-0.78
0.09
0.43
0.48
0
0.05
249,113.28
0.15
0.27
1
0.47
0.02
0.49
0.64
0.56
0.01
0.57
8,355,735.62
0.38
0.54
217
0.02
-0.78
0.09
0.43
0.48
0
0.05
241,958.15
0.15
0.27
1
0.47
MGP_L VA_R MGP_R
0
0.03
0.49
0.64
0.56
0.01
0.57
8,210,767.81
0.38
0.54
229
0.02
Ent_R
-0.71
0.11
0.47
0.44
0
0.1
268,736.61
0.13
0.58
4
0.31
MDM_R
-0.54
0.22
0.48
0.54
0
0.09
1,061,603.37
0.27
0.27
8
0.27
MDM_L
-0.48
0.25
0.49
0.54
0
0.11
1,396,820.87
0.3
0.33
11
0.23
LGP_L
-0.44
0.3
0.46
0.52
0
0.14
1,710,007.94
0.29
0.39
6
0.21
LGP_R
-0.44
0.3
0.46
0.53
0
0.14
1,670,074.84
0.3
0.39
6
0.21
Region
ZO
ZI
ZA
PCO
PCI
PCA
RadO
RadI
CenO
CenI
Hub
Aut
MRF_R
1.3
0.64
1.15
0.65
0.64
0.65
2.84
2.51
-1
-10
0.98
0.87
MRF_L
1.4
0.71
1.25
0.66
0.65
0.66
2.86
2.51
1
-10
1
0.88
AGm_R
0.53
1
0.75
0.62
0.64
0.63
2.55
2.57
-15
-7
0.67
0.81
AGl_R
0.53
1.36
0.88
0.53
0.52
0.53
2.49
2.49
-18
-11
0.61
0.59
AGm_L
0.59
1.09
0.82
0.64
0.65
0.65
2.57
2.57
-14
-7
0.69
0.81
AGl_L
0.79
1.47
1.11
0.56
0.59
0.58
2.51
2.49
-17
-11
0.63
0.61
-1.24 -1.38
-0.42 -0.43
-1.03 -1.12
0.48 0.48
0.64 0.6
0.62 0.58
2.02 2.02
2.35 2.35
-39 -39
-18 -18
0.15 0.15
0.65 0.65
HIPP_L
0.72
0.52
0.72
0.62
0.52
0.59
2.37
2.14
-24
-25
0.49
0.4
HIPP_R
0.38
0.52
0.49
0.64
0.47
0.59
2.37
2.16
-24
-24
0.47
0.4
AC_R
1.42
1.5
1.64
0.64
0.55
0.61
2.76
2.51
-5
-10
0.89
0.78
CM_L
-0.63
-0.42
-0.6
0.64
0.62
0.64
2.24
2.35
-31
-19
0.35
0.65
LHb_L
-0.23
-0.8
-0.46
0.62
0.65
0.64
2.31
2.33
-27
-19
0.41
0.59
VL_L VL_R
123
Brain Struct Funct Table 9 continued Region
ZO
ZI
ZA
PCO
PCI
PCA
RadO
RadI
CenO
CenI
Hub
Aut
CM_R
-0.8
-0.43
-0.72
0.64
0.58
0.62
2.27
2.35
-30
-19
0.34
0.65
LHb_R
-0.23
-0.79
-0.46
0.59
0.62
0.62
2.31
2.33
-27
-19
0.41
0.59
AC_L CPu_R
1.07 1.3
1.01 2.08
1.18 1.68
0.65 0.42
0.58 0.64
0.63 0.58
2.73 2.47
2.49 2.71
-6 -19
-11 0
0.88 0.54
0.78 1
PC_L
-1.24
-1.18
-1.31
0.67
0.6
0.65
2.16
2.18
-35
-27
0.27
0.45
VM_L
-0.63
0.34
-0.32
0.63
0.59
0.6
2.22
2.37
-32
-18
0.35
0.67
PC_R
-1.19
-1.15
-1.26
0.64
0.51
0.59
2.18
2.2
-34
-26
0.27
0.45
VM_R
-0.61
0.28
-0.32
0.57
0.54
0.55
2.24
2.37
-31
-18
0.35
0.67
VTA_R
1.3
1
1.28
0.62
0.54
0.59
2.67
2.39
-9
-16
0.81
0.67
CPu_L
1.4
2.23
1.82
0.47
0.65
0.61
2.47
2.71
-19
0
0.54
1
PF_R
0.53
-0.08
0.34
0.54
0.55
0.55
2.43
2.31
-22
-21
0.54
0.64
VTA_L
0.72
-0.94
0.04
0.62
0.57
0.6
2.69
2.41
-8
-15
0.83
0.7
Pir_R
0.38
-0.94
-0.19
0.49
0.51
0.56
2.2
2.29
-32
-21
0.3
0.54
PF_L
0.59
-0.04
0.39
0.6
0.58
0.6
2.43
2.29
-22
-22
0.56
0.65
SNR_L
0.99
0.34
0.82
0.6
0.59
0.6
2.51
2.31
-18
-21
0.64
0.61
-0.74
0.56
0.54
0.56
2.16
2.2
-35
-26
0.26
0.48
CL_L SNR_R
-0.63
-0.8
0.91
0.28
0.75
0.57
0.55
0.56
2.51
2.33
-18
-20
0.63
0.6
CL_R Ent_L
-0.61 0.72
-0.79 0.52
-0.72 0.72
0.46 0.52
0.44 0.49
0.46 0.52
2.16 2.29
2.2 2.24
-34 -28
-26 -23
0.25 0.37
0.47 0.44
Pir_L
0.38
0.04
0.26
0.49
0.55
0.56
2.2
2.31
-32
-20
0.3
0.57
SNC_L
1.2
-0.42
0.68
0.59
0.6
0.6
2.53
2.27
-16
-22
0.67
0.54
SNC_R
1.11
-0.43
0.61
0.57
0.55
0.56
2.55
2.27
-15
-22
0.67
0.54
-1.02
0.52
-0.42
0.5
0.54
0.53
2.24
2.37
-29
-17
0.32
0.65
0.34
0.28
0.34
0.53
0.55
0.54
2.39
2.35
-23
-18
0.47
0.63
Ac_R STh_R STh_L
0.38
0.34
0.39
0.59
0.59
0.59
2.39
2.35
-23
-18
0.49
0.63
Ac_L
-0.32
0.52
0.04
0.55
0.58
0.57
2.27
2.37
-29
-17
0.34
0.65
MDL_L
-1.45
-0.04
-1.03
0.64
0.59
0.61
2.06
2.31
-40
-21
0.13
0.59
MDL_R
-1.38
-0.08
-0.99
0.56
0.54
0.54
2.08
2.31
-38
-21
0.13
0.58
VA_L
-1.45
-2.31
-1.88
0
0.38
0.28
1.69
1.92
-47
-37
0.06
0.14
0.52
0.44
0.5
2.43
2.2
-21
-25
0.53
0.47
0
0.38
0.28
1.67
1.92
-47
-37
0.06
0.14
MGP_L
0.99
0.34
0.82
-1.57
-2.59
-2.06
0.91
0.28
0.75
0.48
0.36
0.44
2.45
2.22
-20
-24
0.53
0.47
Ent_R MDM_R
-1.37 -1.72
-2.4 -0.94
-2.03 -1.57
0 0.44
0.44 0.58
0.28 0.56
1.86 1.92
1.71 2.14
-44 -42
-39 -27
0.06 0.08
0.1 0.43
MDM_L
-1.37
0.04
-0.88
0.38
0.54
0.51
1.94
2.16
-41
-26
0.1
0.46
LGP_L
-0.84
-0.42
-0.74
0
0.4
0.3
1.84
2.08
-42
-32
0.13
0.37
LGP_R
-0.99
-0.43
-0.86
0
0.3
0.22
1.82
2.1
-42
-31
0.12
0.36
VA_R MGP_R
The regions are sorted using the degree all. Abbreviations and parameter definitions are described in the Supporting Information A all, AD average degree, C circle, CC cluster-coefficient, DG degree, I in, L laterality, O out, SP length of shortest path
123
Brain Struct Funct Table 10 Frequency of motifs of the ipsilateral BG2 network Region
3-01
3-02
AC&1
0
1
27
0
CPu&1
5
2
20
2
VTA&10
0
1
40
2
MRF&23
0
0
44
2
STh&3
9
3
41
AGm&1
5
3
37
SNR&25
9
6
42
PF&8
3
6
45
SNC&26
2
0
43
MGP&12
17
2
42
5
AGl&3 Ac&5
2 14
2 7
26 34
0 7
LHb&4
10
1
47
5
12
2
13
21
8
1
27
26
2
5
VM&17
9
3
24
8
13
1
14
36
3
1
34
18
2
1
Ent&12
7
3
26
0
24
0
5
34
5
8
32
15
1
6
HIPP&12
8
6
28
8
29
0
12
22
6
8
29
11
2
2
CM&24
11
10
26
10
16
3
15
27
12
4
30
10
0
0
PC&22
4
4
30
0
11
0
3
47
4
1
15
9
3
2
LGP&18
9
4
52
6
6
2
12
11
8
3
25
6
3
5
13
7
27
15
15
2
28
11
14
11
32
6
3
2
MDL&28
9
3
36
11
8
0
22
28
3
2
25
5
0
3
MDM&26
8
1
38
1
11
0
13
22
1
7
13
5
1
3
CL&24
12
5
25
5
14
1
7
28
13
5
23
5
0
0
VL&31
7
2
43
4
7
0
22
23
1
4
15
3
0
1
VA&13
7
5
30
0
25
1
1
0
5
1
0
0
0
7
Pir&20
3-03
3-04
3-05
3-06
3-07
3-08
3-09
3-10
3-11
3-12
3-13
chain
5
0
12
30
0
9
53
1
38
12
0
65
83
0
0
3
73
68
0
1
0
28
34
0
8
3
66
65
0
0
2
0
40
17
8
3
54
65
0
0
4 2
8
0
22
0
16
29
8
6
38
48
0
0
6
63
5
11
36
46
3
2
16
7
3
32
16
20
1
33
45
0
3
5 10
9
0
17
27
6
2
34
45
0
0
14
0
34
10
10
9
39
44
0
0
11
1
21
21
10
3
38
42
5
8
30 15
1 1
0 8
57 24
8 19
3 5
41 41
39 32
0 0
0 0
in
out
Regions are sorted using motif 3-13 frequencies chain symmetricCHAIN, in symmetricIN, out symmetricOUT (see legend of Fig. 8)
Table 11 Global extrinsic connections of regions of the bilateral BG2 network Region
Dic
Dii
Dis
Doc
Doi
Dos
Centrolateral_thalamic_nucleus_L
392
446
838
7
68
75
403
470
873
7
103
Centrolateral_thalamic_nucleus_R
392
446
838
7
68
75
403
470
873
7
103
110
Accumbens_nucleus_R
265
380
645
4
74
78
308
853
1,161
11
309
320
Accumbens_nucleus_L
265
378
643
4
74
78
318
829
1,147
11
304
315
Central_medial_thalamic_nucleus_L
265
322
587
12
70
82
270
340
610
12
184
196
Central_medial_thalamic_nucleus_R
265
322
587
12
72
84
270
340
610
12
188
200
Paracentral_thalamic_nucleus_L
235
309
544
6
81
87
238
413
651
6
89
95
Paracentral_thalamic_nucleus_R
235
309
544
6
81
87
238
413
651
6
89
95
Mesencephalic_reticular_formation_L
222
311
533
227
334
561
3,316
4,273
7,589
1,188
3,510
4,698
Mesencephalic_reticular_formation_R
222
311
533
227
334
561
3,316
4,273
7,589
1,188
3,512
4,700
Amygdaloid_complex_R Amygdaloid_complex_L
144 144
378 375
522 519
4 4
74 71
78 75
632 633
4,659 4,473
5,291 5,106
525 528
4,296 3,970
4,821 4,498
Ventrolateral_thalamic_nucleus_L
144
344
488
8
52
60
160
384
544
9
56
65
Ventrolateral_thalamic_nucleus_R
144
344
488
8
52
60
160
384
544
9
56
65
Caudate_putamen_L
54
311
365
21
79
100
279
1,057
1,336
22
172
194
Caudate_putamen_R
54
311
365
21
79
100
273
1,064
1,337
22
172
194
123
Sic
Sii
Sis
Soc
Soi
Sos 110
Brain Struct Funct Table 11 continued Region
Dic
Dii
Dis
Doc
Doi
Dos
Sic
Sii
Sis
Soc
Soi
Sos
Ventromedial_thalamic_nucleus_L
82
244
326
33
54
87
105
356
461
40
161
201
Ventromedial_thalamic_nucleus_R
82
244
326
33
54
87
105
356
461
40
161
201
Parafascicular_thalamic_nucleus_L Parafascicular_thalamic_nucleus_R
111 111
213 213
324 324
35 35
136 137
171 172
140 140
377 377
517 517
112 112
291 292
403 404
Ventral_tegmental_area_A10_L
60
238
298
99
231
330
96
476
572
124
440
564
Ventral_tegmental_area_A10_R
60
238
298
99
235
334
96
476
572
124
446
570
Medial_agranular_prefrontal_cortex_R
50
138
188
106
418
524
85
219
304
159
586
745
Hippocampus_R
20
162
182
8
47
55
138
1,220
1,358
130
649
779
Medial_agranular_prefrontal_cortex_L
50
131
181
106
416
522
85
212
297
159
584
743
Entorhinal_cortex_L
27
137
164
14
63
77
154
900
1,054
21
546
567
Hippocampus_L
25
137
162
8
46
54
154
1,030
1,184
130
631
761
Lateral_habenular_nucleus_L
24
128
152
37
99
136
55
226
281
40
129
169
Lateral_habenular_nucleus_R
24
128
152
37
99
136
55
226
281
40
129
169
Subthalamic_nucleus_L
19
124
143
23
83
106
29
181
210
34
127
161
Subthalamic_nucleus_R
19
124
143
23
83
106
29
181
210
34
129
163
Substantia_nigra_compact_part_L
24
101
125
30
126
156
37
136
173
33
181
214
Substantia_nigra_compact_part_R
24
101
125
30
126
156
37
136
173
33
181
214
Piriform_cortex_R Lateral_agranular_prefrontal_cortex_R
11 22
112 99
123 121
11 257
55 383
66 640
34 71
309 440
343 511
37 504
227 803
264 1,307
Lateral_agranular_prefrontal_cortex_L
22
95
117
257
381
638
71
436
507
504
801
1,305
Piriform_cortex_L
11
103
114
11
53
64
34
300
334
38
191
229
Mediodorsal_thalamic_nucleus_medial_part_L
11
91
102
0
14
14
11
91
102
0
14
14
Mediodorsal_thalamic_nucleus_medial_part_R
11
91
102
0
16
16
11
91
102
0
16
16
Mediodorsal_thalamic_nucleus_lateral_part_L
17
82
99
0
18
18
17
83
100
0
18
18
Mediodorsal_thalamic_nucleus_lateral_part_R
17
82
99
0
18
18
17
83
100
0
18
18
Substantia_nigra_reticular_part_L
14
73
87
34
133
167
14
86
100
45
200
245
Substantia_nigra_reticular_part_R
14
73
87
34
133
167
14
86
100
45
200
245
Medial_globus_pallidus_L
12
53
65
11
59
70
13
58
71
11
73
84
Medial_globus_pallidus_R
12
53
65
11
59
70
13
58
71
11
73
84
Entorhinal_cortex_R
3
40
43
0
5
5
9
68
77
0
22
22
Lateral_globus_pallidus_L
4
22
26
0
4
4
4
22
26
0
5
5
Lateral_globus_pallidus_R
4
22
26
0
4
4
4
22
26
0
5
5
12 12
12 12
24 24
0 0
10 10
10 10
14 14
14 14
28 28
0 0
10 10
10 10
Ventro_anterior_thalamic_nucleus_L Ventro_anterior_thalamic_nucleus_R
The regions were sorted by the direct input sum (Dis) Dic direct input from contralateral, Dii direct input from ipsilateral, Dis direct input from ipsi- and contralateral, Doc direct output to contralateral, Doi direct output to ipsilateral, Dos direct output to ipsi- and contralateral, Sic subtree input from contralateral, Sii subtree input from ipsilateral, Sis subtree input from ipsi- and contralateral, Soc subtree output to contralateral, Soi subtree output to ipsilateral, Sos subtree output to ipsi- and contralateral
123
123 0.971
0.829 -0.558 0.957
-0.062
-0.090
-0.006
0.181
-0.009
0.030
-0.043
-0.152
0.002 0.147
-0.090
0.080
DGA
DGO
DGI
CDC
LA
LO
LI
LR
Katz LC
Triag
CyclC
0.015 0.024
CCI BC
0.114
-0.017
-0.012
-0.139
-0.139
PRC
FC
Stress
ZO
ZI
-0.172
-0.110
CCO
-0.084
-0.127
Loc
EC
-0.059
Lev
SC
0.106
VCDG
-0.138
0.285
-0.018
CCT
-0.093
-0.012
CCA
CC2
-0.086
CCI
ADnb
-0.833
0.094
CCO
0.822
0.822
0.944
0.814
0.834
0.969
0.915
0.801 0.904
0.922
0.907
0.926
0.686
-0.760
-0.827
-0.445
-0.808
-0.146
EccI
-0.672
0.019
-0.289
EccO
-0.741
0.047
-0.433
-0.487
-0.573
-0.624
0.855
0.930
1.000
-0.018
1.000
-0.018
REC
HD
HD
REC
Region
0.863
0.863
0.941
0.697
0.811
0.978
0.936
0.833 0.878
0.953
0.973
0.969
0.719
-0.780
-0.219
-0.863
-0.867
-0.402
-0.859
-0.166
-0.631
-0.810
0.981
0.860 -0.589
0.076
-0.486
-0.511
-0.615
-0.636
0.884
0.955
1.000
0.971
-0.062
DGA
0.896
0.896
0.881
0.693
0.629
0.950
0.988
0.636 0.786
0.989
0.935
0.926
0.709
-0.787
-0.199
-0.824
-0.862
-0.320
-0.867
0.031
-0.616
-0.748
0.957
0.678 -0.547
0.036
-0.375
-0.529
-0.560
-0.803
0.705
1.000
0.955
0.930
-0.090
DGO
0.648
0.648
0.858
0.573
0.945
0.839
0.679
0.987 0.860
0.718
0.851
0.856
0.600
-0.622
-0.209
-0.763
-0.711
-0.456
-0.685
-0.446
-0.535
-0.756
0.834
0.987 -0.545
0.125
-0.570
-0.386
-0.586
-0.253
1.000
0.705
0.884
0.855
-0.006
DGI
Table 12 Correlation coefficients of local parameters of the bilateral BG2 network
-0.807
-0.807
-0.532
-0.657
-0.217
-0.620
-0.821
-0.179 -0.418
-0.767
-0.616
-0.677
-0.625
0.673
0.009
0.622
0.729
0.107
0.776
-0.278
0.476
0.555
-0.627
-0.221 0.442
-0.048
0.148
0.477
0.377
1.000
-0.253
-0.803
-0.636
-0.624
0.181
CDC
-0.423
-0.423
-0.577
-0.450
-0.548
-0.593
-0.545
-0.603 -0.540
-0.616
-0.536
-0.576
-0.305
0.285
0.503
0.735
0.653
0.491
0.649
0.416
0.493
0.779
-0.565
-0.612 0.412
0.281
0.830
0.814
1.000
0.377
-0.586
-0.560
-0.615
-0.573
-0.009
LA
-0.448
-0.448
-0.452
-0.469
-0.388
-0.480
-0.520
-0.356 -0.413
-0.587
-0.460
-0.505
-0.373
0.340
0.339
0.603
0.565
0.213
0.634
0.156
0.433
0.700
-0.464
-0.405 0.589
0.052
0.373
1.000
0.814
0.477
-0.386
-0.529
-0.511
-0.487
0.030
LO
-0.246
-0.246
-0.446
-0.280
-0.493
-0.464
-0.365
-0.630 -0.417
-0.408
-0.412
-0.447
-0.121
0.114
0.506
0.576
0.488
0.523
0.440
0.526
0.330
0.578
-0.436
-0.606 0.106
0.365
1.000
0.373
0.830
0.148
-0.570
-0.375
-0.486
-0.433
-0.043
LI
0.118
0.118
-0.044
0.094
0.038
-0.004
0.071
0.076 -0.054
0.052
0.151
0.182
0.347
-0.195
0.131
0.193
0.078
0.652
-0.103
0.110
0.064
-0.106
0.033
0.117 -0.404
1.000
0.365
0.052
0.281
-0.048
0.125
0.036
0.076
0.047
-0.152
LR
0.598
0.598
0.812
0.559
0.920
0.816
0.655
0.988 0.807
0.703
0.823
0.831
0.529
-0.541
-0.292
-0.721
-0.667
-0.389
-0.675
-0.494
-0.511
-0.747
0.808
1.000 -0.496
0.117
-0.606
-0.405
-0.612
-0.221
0.987
0.678
0.860
0.829
0.002
Katz
Brain Struct Funct
-0.211
-0.043
-0.194
-0.135
-0.019
-0.103
-0.016
-0.096
LC
PCI
PCA
RadO
RadI
CenO
CenI
Hub
Aut
Region
0.589
0.106
-0.404
-0.496
1.000 -0.537
0.680
0.401
0.071
0.595
0.114
0.596
0.564
0.026
LO
LI
LR
Katz
LC Triag
CyclC
EccO
EccI
CCO
CCI
CCA
CCT
CC2
-0.682
0.412
LA
VCDG
0.442
CDC
0.623
0.808
-0.545
DGI
ADnb
0.033
-0.547
DGO
0.957
0.642
-0.747
-0.273
-0.810
-0.822
-0.401
-0.786
-0.107
-0.617
-0.725
-0.537 1.000
-0.436
-0.464
-0.565
-0.627
0.834
0.957
0.981
-0.558
-0.589
REC
-0.090
Triag
0.925
0.820
0.934
0.790
0.912
0.521
0.481
0.517
DGA
0.147
-0.109
HD
0.747
-0.013
ZA
PCO 0.861
REC
HD
Region
Table 12 continued
-0.722
0.647
0.228
0.884
0.853
0.341
0.906
0.284
0.514
1.000
0.680 -0.725
-0.747
-0.106
0.578
0.700
0.779
0.555
-0.756
-0.748
-0.810
-0.741
0.080
CyclC
0.949
0.857
0.962
0.833
0.949
0.601
0.545
0.584
0.897
0.765
DGA
-0.446
0.422
0.216
0.580
0.534
0.382
0.538
0.210
1.000
0.514
0.401 -0.617
-0.511
0.064
0.330
0.433
0.493
0.476
-0.535
-0.616
-0.631
-0.672
0.019
EccO
0.997
0.673
0.995
0.648
0.967
0.518
0.409
0.554
0.863
0.618
DGO
-0.074
-0.010
0.204
0.224
0.160
0.286
0.051
1.000
0.210
0.284
0.071 -0.107
-0.494
0.110
0.526
0.156
0.416
-0.278
-0.446
0.031
-0.166
-0.146
-0.289
EccI
0.695
0.987
0.731
0.969
0.743
0.619
0.657
0.521
0.783
0.855
DGI
-0.779
0.757
0.105
0.877
0.898
0.260
1.000
0.051
0.538
0.906
0.595 -0.786
-0.675
-0.103
0.440
0.634
0.649
0.776
-0.685
-0.867
-0.859
-0.808
0.094
CCO
-0.813
-0.231
-0.795
-0.200
-0.788
-0.290
-0.094
-0.433
-0.666
-0.278
CDC
-0.208
0.345
-0.057
0.658
0.542
1.000
0.260
0.286
0.382
0.341
0.114 -0.401
-0.389
0.652
0.523
0.213
0.491
0.107
-0.456
-0.320
-0.402
-0.445
-0.086
CCI
-0.590
-0.656
-0.594
-0.619
-0.642
-0.747
-0.684
-0.693
-0.449
-0.402
LA
-0.796
0.856
-0.008
0.966
1.000
0.542
0.898
0.160
0.534
0.853
0.596 -0.822
-0.667
0.078
0.488
0.565
0.653
0.729
-0.711
-0.862
-0.867
-0.827
-0.012
CCA
-0.559
-0.438
-0.562
-0.389
-0.610
-0.586
-0.429
-0.605
-0.419
-0.299
LO
-0.721
0.769
0.057
1.000
0.966
0.658
0.877
0.224
0.580
0.884
0.564 -0.810
-0.721
0.193
0.576
0.603
0.735
0.622
-0.763
-0.824
-0.863
-0.833
-0.018
CCT
-0.398
-0.637
-0.400
-0.639
-0.446
-0.666
-0.722
-0.565
-0.314
-0.352
LI
0.293
-0.297
1.000
0.057
-0.008
-0.057
0.105
0.204
0.216
0.228
0.026 -0.273
-0.292
0.131
0.506
0.339
0.503
0.009
-0.209
-0.199
-0.219
-0.138
0.285
CC2
0.021
0.109
0.076
0.108
0.134
0.134
0.101
0.167
0.198
0.309
LR
-0.930
1.000
-0.297
0.769
0.856
0.345
0.757
-0.010
0.422
0.647
0.623 -0.747
-0.541
-0.195
0.114
0.340
0.285
0.673
-0.622
-0.787
-0.780
-0.760
-0.093
ADnb
0.674
0.987
0.708
0.986
0.735
0.673
0.700
0.581
0.735
0.822
Katz
Brain Struct Funct
123
123
0.709
0.600
-0.625
-0.305
-0.373
-0.121
0.347
0.529
-0.682
CDC
LA
LO
LI
LR
Katz
LC
VCDG
Region
DGI
-0.538
Aut
DGO
-0.569
Hub
0.719
-0.590
CenI
DGA
-0.487
CenO
0.686
-0.605
RadI
0.106
-0.426
RadO
REC
-0.397 -0.373
PCI PCA
HD
-0.618
-0.480
FC
PCO
-0.527
PRC
-0.537
-0.516
SC
-0.598
-0.570
EC
ZA
-0.489
BC
ZI
-0.486
CCI
-0.511
-0.556
CCO
-0.598
-0.622
Loc
ZO
-0.645
Lev
Stress
LC
Region
Table 12 continued
-0.645
0.831
0.182
-0.447
-0.505
-0.576
-0.677
0.856
0.926
0.969
0.926
-0.059
Lev
0.947
0.797
0.953
0.767
0.919
0.531
0.502 0.504
0.840
0.682
0.828
0.828
0.940
0.652
0.742
0.996
0.942
0.864
0.779
0.959
0.968
0.913
Triag
-0.622
0.823
0.151
-0.412
-0.460
-0.536
-0.616
0.851
0.935
0.973
0.907
-0.127
Loc
-0.757
-0.794
-0.786
-0.749
-0.839
-0.762
-0.732 -0.654
-0.783
-0.679
-0.753
-0.753
-0.723
-0.558
-0.711
-0.720
-0.722
-0.679
-0.728
-0.771
-0.774
-0.849
CyclC
-0.556
0.703
0.052
-0.408
-0.587
-0.616
-0.767
0.718
0.989
0.953
0.922
-0.110
CCO
-0.630
-0.536
-0.648
-0.494
-0.633
-0.303
-0.359 -0.161
-0.615
-0.544
-0.583
-0.583
-0.658
-0.604
-0.572
-0.634
-0.619
-0.639
-0.516
-0.634
-0.564
-0.588
EccO
-0.486
0.988
0.076
-0.630
-0.356
-0.603
-0.179
0.987
0.636
0.833
0.801
0.015
CCI
0.013
-0.493
0.001
-0.480
-0.034
-0.265
-0.136 -0.392
-0.088
-0.322
0.046
0.046
-0.204
-0.062
-0.470
-0.125
0.067
-0.267
-0.519
0.015
-0.107
-0.153
EccI
-0.489
0.807
-0.054
-0.417
-0.413
-0.540
-0.418
0.860
0.786
0.878
0.904
0.024
BC
-0.872
-0.693
-0.890
-0.673
-0.932
-0.669
-0.731 -0.493
-0.840
-0.632
-0.860
-0.860
-0.753
-0.684
-0.644
-0.783
-0.849
-0.682
-0.641
-0.869
-0.823
-0.912
CCO
-0.570
0.655
0.071
-0.365
-0.520
-0.545
-0.821
0.679
0.988
0.936
0.915
-0.172
EC
-0.313
-0.442
-0.300
-0.368
-0.242
-0.081
0.010 -0.197
-0.307
-0.299
-0.272
-0.272
-0.557
-0.330
-0.563
-0.440
-0.280
-0.616
-0.456
-0.289
-0.300
-0.310
CCI
-0.516
0.816
-0.004
-0.464
-0.480
-0.593
-0.620
0.839
0.950
0.978
0.969
-0.084
SC
-0.855
-0.701
-0.865
-0.639
-0.862
-0.496
-0.506 -0.448
-0.817
-0.613
-0.837
-0.837
-0.830
-0.653
-0.692
-0.820
-0.830
-0.776
-0.659
-0.839
-0.818
-0.882
CCA
-0.527
0.920
0.038
-0.493
-0.388
-0.548
-0.217
0.945
0.629
0.811
0.834
0.114
PRC
-0.822
-0.763
-0.832
-0.703
-0.832
-0.547
-0.530 -0.490
-0.803
-0.644
-0.799
-0.799
-0.858
-0.634
-0.774
-0.818
-0.783
-0.830
-0.726
-0.813
-0.797
-0.857
CCT
-0.480
0.559
0.094
-0.280
-0.469
-0.450
-0.657
0.573
0.693
0.697
0.814
-0.017
FC
-0.228
-0.279
-0.226
-0.308
-0.267
-0.549
-0.483 -0.495
-0.043
-0.041
-0.039
-0.039
-0.102
0.081
-0.005
-0.274
-0.248
0.005
-0.254
-0.297
-0.267
-0.121
CC2
-0.511
0.812
-0.044
-0.446
-0.452
-0.577
-0.532
0.858
0.881
0.941
0.944
-0.012
Stress
-0.759
-0.555
-0.774
-0.488
-0.731
-0.178
-0.203 -0.180
-0.800
-0.617
-0.814
-0.814
-0.765
-0.621
-0.626
-0.731
-0.745
-0.727
-0.525
-0.731
-0.777
-0.824
ADnb
Brain Struct Funct
-0.121
-0.824
-0.446
-0.074
-0.779
-0.208
-0.796
-0.721
0.293
-0.930
1.000 0.816
0.713
0.665
0.509
0.649
0.673
0.623
0.610
0.580
0.675
0.768
0.768
0.680
0.794
0.346
0.244 0.307
0.714
0.500
0.721
0.563
0.692
EccO
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG Lev
Loc
CCO
CCI
BC
EC
SC
PRC
FC
Stress
ZO
ZI
ZA
PCO
PCI
PCA RadO
RadI
CenO
CenI
Hub
Aut
0.913
0.923
0.837
0.944
0.823
0.952
0.517 0.618
0.635
0.933
0.804
0.892
0.892
0.866
0.700
0.786
0.903
0.913
0.804
0.801
0.915
0.949
0.816 1.000
-0.857
-0.882
-0.310
-0.912
-0.153
-0.588
-0.849
0.642
-0.722
Lev
CyclC
VCDG
Triag
Region
Table 12 continued
0.923
0.821
0.941
0.801
0.923
0.514 0.572
0.545
0.885
0.735
0.862
0.862
0.890
0.570
0.733
0.949
0.922
0.809
0.795
0.936
1.000
0.713 0.949
-0.777
-0.267
-0.797
-0.818
-0.300
-0.823
-0.107
-0.564
-0.774
0.968
Loc
0.989
0.699
0.992
0.675
0.977
0.467 0.590
0.624
0.839
0.621
0.860
0.860
0.874
0.671
0.627
0.952
0.980
0.774
0.656
1.000
0.936
0.665 0.915
-0.731
-0.297
-0.813
-0.839
-0.289
-0.869
0.015
-0.634
-0.771
0.959
CCO
0.629
0.989
0.667
0.982
0.689
0.685 0.639
0.532
0.725
0.821
0.584
0.584
0.809
0.524
0.935
0.786
0.610
0.817
1.000
0.656
0.795
0.509 0.801
-0.525
-0.254
-0.726
-0.659
-0.456
-0.641
-0.519
-0.516
-0.728
0.779
CCI
0.770
0.812
0.783
0.757
0.741
0.412 0.395
0.330
0.777
0.736
0.708
0.708
0.978
0.668
0.901
0.880
0.734
1.000
0.817
0.774
0.809
0.649 0.804
-0.727
0.005
-0.830
-0.776
-0.616
-0.682
-0.267
-0.639
-0.679
0.864
BC
0.989
0.654
0.989
0.632
0.968
0.413 0.541
0.589
0.848
0.600
0.884
0.884
0.837
0.705
0.586
0.935
1.000
0.734
0.610
0.980
0.922
0.673 0.913
-0.745
-0.248
-0.783
-0.830
-0.280
-0.849
0.067
-0.619
-0.722
0.942
EC
0.941
0.803
0.946
0.771
0.913
0.511 0.534
0.506
0.823
0.676
0.808
0.808
0.948
0.695
0.762
1.000
0.935
0.880
0.786
0.952
0.949
0.623 0.903
-0.731
-0.274
-0.818
-0.820
-0.440
-0.783
-0.125
-0.634
-0.720
0.996
SC
0.622
0.927
0.654
0.895
0.655
0.525 0.486
0.403
0.747
0.837
0.610
0.610
0.850
0.652
1.000
0.762
0.586
0.901
0.935
0.627
0.733
0.610 0.786
-0.626
-0.005
-0.774
-0.692
-0.563
-0.644
-0.470
-0.572
-0.711
0.742
PRC
0.701
0.548
0.706
0.510
0.714
0.281 0.336
0.407
0.655
0.547
0.645
0.645
0.673
1.000
0.652
0.695
0.705
0.668
0.524
0.671
0.570
0.580 0.700
-0.621
0.081
-0.634
-0.653
-0.330
-0.684
-0.062
-0.604
-0.558
0.652
FC
0.867
0.813
0.877
0.763
0.835
0.449 0.451
0.402
0.826
0.725
0.784
0.784
1.000
0.673
0.850
0.948
0.837
0.978
0.809
0.874
0.890
0.675 0.866
-0.765
-0.102
-0.858
-0.830
-0.557
-0.753
-0.204
-0.658
-0.723
0.940
Stress
Brain Struct Funct
123
123
-0.583
0.046
-0.860
-0.272
-0.837
-0.799
-0.039
-0.814
0.768
0.892
0.862
0.860
0.584 0.708
0.884
0.808
0.610
0.645
0.784
0.000
1.000
EccO
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG
Lev
Loc
CCO
CCI BC
EC
SC
PRC
FC
Stress
ZO
ZI
0.598 -0.598
0.828
0.118
LR
Katz LC
-0.753
-0.246
LI
CyclC
0.309
-0.448
Triag
-0.352
-0.423
LO
0.671
1.000
0.725
0.547
0.837
0.676
0.600
0.821 0.736
0.621
0.735
0.804
0.680
-0.617
-0.041
-0.644
-0.613
-0.299
-0.632
-0.322
-0.544
-0.679
0.682
0.822 -0.537
-0.299
-0.402
-0.278
0.855
0.618
LA
0.896
DGO
0.765
0.648
0.863
DGA
0.747
-0.013
-0.807
0.822
REC
CDC
-0.139
HD
ZI
DGI
ZO
Region
Table 12 continued
0.953
0.671
0.826
0.655
0.747
0.823
0.848
0.725 0.777
0.839
0.885
0.933
0.794
-0.800
-0.043
-0.803
-0.817
-0.307
-0.840
-0.088
-0.615
-0.783
0.840
0.735 -0.618
0.198
-0.314
-0.419
-0.449
-0.666
0.783
0.863
0.897
0.861
-0.109
ZA
0.445
0.953
0.402
0.407
0.403
0.506
0.589
0.532 0.330
0.624
0.545
0.635
0.346
-0.203
-0.483
-0.530
-0.506
0.010
-0.731
-0.136
-0.359
-0.732
0.502
0.581 -0.397
0.167
-0.565
-0.605
-0.693
-0.433
0.521
0.554
0.584
0.517
-0.211
PCO
0.276
0.445
0.449
0.281
0.525
0.511
0.413
0.685 0.412
0.467
0.514
0.517
0.244
-0.180
-0.495
-0.490
-0.448
-0.197
-0.493
-0.392
-0.161
-0.654
0.504
0.700 -0.373
0.101
-0.722
-0.429
-0.684
-0.094
0.657
0.409
0.545
0.481
-0.043
PCI
0.406
0.276
0.451
0.336
0.486
0.534
0.541
0.639 0.395
0.590
0.572
0.618
0.307
-0.178
-0.549
-0.547
-0.496
-0.081
-0.669
-0.265
-0.303
-0.762
0.531
0.673 -0.426
0.134
-0.666
-0.586
-0.747
-0.290
0.619
0.518
0.601
0.521
-0.194
PCA
0.883
0.406
0.835
0.714
0.655
0.913
0.968
0.689 0.741
0.977
0.923
0.952
0.714
-0.731
-0.267
-0.832
-0.862
-0.242
-0.932
-0.034
-0.633
-0.839
0.919
0.735 -0.605
0.134
-0.446
-0.610
-0.642
-0.788
0.743
0.967
0.949
0.912
-0.135
RadO
0.584
0.883
0.763
0.510
0.895
0.771
0.632
0.982 0.757
0.675
0.801
0.823
0.500
-0.488
-0.308
-0.703
-0.639
-0.368
-0.673
-0.480
-0.494
-0.749
0.767
0.986 -0.487
0.108
-0.639
-0.389
-0.619
-0.200
0.969
0.648
0.833
0.790
-0.019
RadI
0.899
0.584
0.877
0.706
0.654
0.946
0.989
0.667 0.783
0.992
0.941
0.944
0.721
-0.774
-0.226
-0.832
-0.865
-0.300
-0.890
0.001
-0.648
-0.786
0.953
0.708 -0.590
0.076
-0.400
-0.562
-0.594
-0.795
0.731
0.995
0.962
0.934
-0.103
CenO
0.610
0.899
0.813
0.548
0.927
0.803
0.654
0.989 0.812
0.699
0.821
0.837
0.563
-0.555
-0.279
-0.763
-0.701
-0.442
-0.693
-0.493
-0.536
-0.794
0.797
0.987 -0.569
0.109
-0.637
-0.438
-0.656
-0.231
0.987
0.673
0.857
0.820
-0.016
CenI
0.894
0.610
0.867
0.701
0.622
0.941
0.989
0.629 0.770
0.989
0.923
0.923
0.692
-0.759
-0.228
-0.822
-0.855
-0.313
-0.872
0.013
-0.630
-0.757
0.947
0.674 -0.538
0.021
-0.398
-0.559
-0.590
-0.813
0.695
0.997
0.949
0.925
-0.096
Hub
0.560
0.894
0.736
0.441
0.803
0.785
0.646
0.950 0.700
0.690
0.822
0.798
0.459
-0.460
-0.430
-0.643
-0.610
-0.268
-0.642
-0.460
-0.434
-0.727
0.788
0.969 -0.462
0.179
-0.630
-0.368
-0.597
-0.204
0.944
0.653
0.826
0.759
-0.094
Aut
Brain Struct Funct
0.954
0.649
0.669
1.000
Cluster coefficient (triangle based): The triangle based cluster coefficient (Fagiolo 2007) of a node n is the number of triangles around n divided by the maximum possible number. In this version of the cluster coefficient reciprocal edges to a neighbor of a node n can affect the cluster coefficient of node n. In the other version only edges between neighbors of n have an influence to the cluster coefficient of node n. t! ðiÞ tmax ðiÞ ! ! t w ðiÞ w CT ¼ tmax ðiÞ
0.710
0.995
0.984
0.648
0.669
0.685 0.995 1.000 0.686
1.000
0.964 0.648 0.686 1.000
0.710
0.724 0.972 0.985 0.723
0.984
0.736 0.545 0.572 0.715
0.739
0.760 0.421 0.445 0.706
0.699
0.635 0.586 0.612 0.635
0.720
0.688 0.859
0.603
0.765 0.613
0.879 0.730
0.749
0.655 0.825
0.836
Aut CenO RadI
CenI
Hub
Brain Struct Funct
CT! ¼
ð55Þ ð56Þ
Cluster coefficient: Number of edges between the neighbors of a node divided by the maximum possible number. C ! ðiÞ refers to all neighbors of i. X 1 ajk C ! ðiÞ ¼ ð57Þ jNi j ðjNi j 1Þ j;k2N i
! ðiÞ refers to the out-neighbors of i. Cout X 1 ! Cout ðiÞ ¼ out ajk out jNi j ðjNi j 1Þ j;k2N out
0.972
0.739
0.985
0.723
1.000
0.673
0.519
0.724
0.876
0.677
RadO
j6¼k
ð58Þ
i
0.545
! Cin ðiÞ refers to the in-neighbors of i. X 1 ! Cin ðiÞ ¼ in ajk jNi j ðjNiin j 1Þ in j;k2Ni j6¼k
ð59Þ
In the weighted case the aij are replaced by the wij . Average cluster coefficient: n 1X C! ¼ C! ð60Þ n i¼1 i Most local parameters show high correlations with DGA
0.421 Aut
0.894
0.613
0.859
0.586
0.699 0.720 Hub
0.610
0.836
0.749
0.603
0.572 0.445 CenI
0.899
0.655
0.879
0.612
0.715 0.706 CenO
0.584
0.825
0.730
0.635
0.673 0.519
1.000 0.880
0.876
0.724
0.496
0.677 0.883
0.406 RadO
RadI
0.479
0.942
0.880 1.000 PCA
0.276
0.421
0.355
0.704
0.942 0.704 PCI
0.445
0.491
0.505
1.000
0.479 0.355
0.496 0.421
0.505 1.000
0.491 1.000
0.862 0.953
0.671 ZA
PCO
0.862
PCI ZO Region
Table 12 continued
ZI
ZA
PCO
PCA
j6¼k
and n ! 1X ! Cw ¼ Ciw n i¼1
ð61Þ
Small worldness S: C Crand
S¼
d
ð62Þ
drand
Centrality: Pn degmax degðiÞ n degmax 2 ‘ ¼ CD ¼ i¼1 ðn 1Þ ðn 2Þ ðn 1Þ ðn 2Þ
ð63Þ
This centrality (degree centrality) is defined for an undirected network based on undirected degrees. A directed or
123
Brain Struct Funct Table 13 The local parameters of the bilateral BG2 network were applied to the PCA Component
DGAll
CluCAll
CluC2
AvgDGnb
VCDG
Loc
Share (%)
1
0.440
-0.439
0.154
-0.457
0.438
0.434
74.978
2
-0.280
0.101
0.860
-0.166
0.216
-0.312
19.582
3
-0.040
0.818
-0.148
-0.182
0.434
0.292
2.705
4
-0.425
-0.296
-0.417
-0.049
0.626
-0.404
1.875
5
-0.521
-0.056
-0.133
-0.713
-0.408
0.180
0.621
6
-0.523
-0.191
0.151
0.468
0.117
0.659
0.24
The first two components determine the axes of the PCA-plane Share The share of variance of the data represented by the component
weighted version is not available yet. For the calculation the directed network is transferred to an undirected one. Circle length LC: dði; iÞ; dði; iÞ\1 LCðiÞ ¼ ð64Þ 0; dði; iÞ ¼ 1
! 1 X degwall ðjÞ degNB w ðiÞ ¼ jNi j j2Ni
ð70Þ
Variation coefficient of neighbor degree: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P 2 1 ðdeg ðjÞ degNBðiÞÞ all j2N jNi j i
ð71Þ
Eccentricity out: Eccentricity out, the output eccentricity of the vertex i is the maximum distance from i to any vertex.
VCðiÞ ¼
Eccout ðiÞ ¼ maxfdði; jÞjj 2 Ng
The weighted case is analogue. Locality index of node i ðLocðiÞÞ: The locality index of node i is the fraction of edges adjacent to nodes in Niþ whose source and target lie in Niþ . P P k2Niþ ajk j2N þ
ð65Þ
Eccentricity in: Eccentricity in, the input eccentricity of the vertex i is the maximum distance from i to any vertex. Eccin ðiÞ ¼ maxfdðj; iÞjj 2 Ng
ð66Þ
Cluster-coefficient of second neighbors: The cluster-coefficient of second neighbors (Hierarchical directed cluster coefficient of second (indirect) neighbors) C2 ðiÞ is the number of edges between the 2nd neighbors of node i, divided by the maximum possible number of edges. In the weighted case it is the sum of weights of the edges between the 2nd neighbors of node i, divided by the maximum possible sum. With ![ N2 ðiÞ ¼ Nj Niþ ; ð67Þ
degNBðiÞ
i
LocðiÞ ¼ P
j2Niþ
CCOUT ðiÞ ¼ P ð68Þ
In the weighted case the aij are replaced by wij . Average neighbor degree: The non-weighted average neighbor degree NBðiÞ of node i is 1 X degNBðiÞ ¼ degall ðjÞ ð69Þ j2Ni jNi j Weighted average neighbor degree: The weighted average neighbor degree NBðiÞ of node i is
123
k2N k6¼j
ð72Þ
ajk
The weighted case is analogue. A value of 0 means that the node is isolated. The larger the value, the less edges connect the neighborhood of i to outside node. The maximum of one is reached if the neighborhood of i is not connected to outside nodes. Closeness centrality out CCout ðiÞ: The closeness centrality out with indices of nodes from which node i can be reached (RNOUT ðiÞ)RNOUT ðiÞ ¼ fj 2 Nnfigjdði; jÞ\1g
j2Ni
the set of second neighbors of node i is: 8 1 < ; if jN2 ðiÞj [ 1 C2 ðiÞ ¼ jN2 ðiÞj ðjN2 ðiÞj 1Þ : 0; otherwise
P k6¼j
jRNOUT ðiÞj j2RNOUT ðiÞ dði; jÞ
ð73Þ
Closeness centrality in CCin ðiÞ: The closeness centrality in with indices of nodes which can be reached from node i (RNIN ðiÞ)RNIN ðiÞ ¼ fj 2 Nnfigjdðj; iÞ\1g CCIN ðiÞ ¼ P
jRNIN ðiÞj j2RNIN ðiÞ dðj; iÞ
ð74Þ
Betweenness centrality BC: BCðiÞ ¼
1 ðn 1Þðn 2Þ
X j;k2Nnfig
qj;k ðiÞ qj;k
ð75Þ
0.60000
1.00000
0.69500
0.71600
0.62300
MGP
CPu
STh
AGl
AGm
PRC
0.41700
0.04282
AGm
LGP
0.06481
AGl
0.62400
0.06366
STh
0.35400
0.17824
CPu
VL
0.04282
MGP
VA
0.01736 0.00000
VL LGP
0.51000 0.59000
PRC
0.00694
VA
SNC SNR
0.02778
0.03472
SNR
0.35100
0.44400
0.59000
0.58900
0.28800
0.20600
0.34100
0.11000
0.46800 0.32300
0.03206
0.02035
0.02062
0.00850
0.00697 0.02064
0.00114
0.00011
0.00002
0.02083
SNC
Direct BC
BG1 BC
Region
0.17400
0.21900
0.27900
0.29300
0.14200
0.10300
0.17000
0.05400
0.23000 0.16000
PRC
0.00290
0.00610
0.00590
0.00880
0.00200
0.00030 0.00002
0.00001
0.00510
0.00710
Indirect BC
VA
LGP
SNC
SNR
MGP
AGm
VL
STh
CPu AGl
PRC
LGP
VA
VL
SNC
SNR
MGP AGm
STh
AGl
CPu
BG1 BC
Table 14 The local parameters of the embedded BG1 network
VA
LGP
MGP
SNR
VL
AGm
AGl
SNC
STh CPu
PRC
SNC
SNR
VA
VL
MGP
CPu STh
LGP
AGm
AGl
Direct BC
VA
LGP
MGP
SNR
VL
AGm
AGl
SNC
CPu STh
PRC
VA
LGP
VL
MGP
AGm
STh SNR
AGl
SNC
CPu
Indirect BC
0.39200
0.47700
0.79600
1.00000
0.63900
0.42600
0.31000
0.11800
0.59200 0.55700
SC
0.64900
0.86500
0.92800
0.98200
0.94400
0.39400 0.72100
0.29200
0.71700
1.00000
BG1 EC
0.54100
0.65600
0.82500
0.50400
0.46700
0.14000
0.33100
0.02600
1.00000 0.70500
SC
0.43700
0.40000
0.39900
0.48000
0.48200
0.23700 0.16300
0.06400
0.57900
0.68000
Direct EC
0.55020
0.46600
0.64800
0.81660
0.50320
0.13510
0.33170
0.02520
1.00000 0.70280
SC
0.39000
0.35300
0.35300
0.42500
0.42800
0.21100 0.13900
0.05500
0.51400
0.60200
Indirect EC
VA
VL
AGm
LGP
AGl
SNR
SNC
MGP
CPu STh
SC
VA
VL
AGm
SNR
LGP
STh AGl
MGP
CPu
SNC
BG1 EC
VA
LGP
VL
MGP
CPu
AGm
AGl
SNR
SNC STh
SC
VA
LGP
VL
STh
AGl
CPu AGm
MGP
SNR
SNC
Direct EC
VA
LGP
VL
AGl
MGP
AGm
STh
SNR
SNC CPu
SC
VA
LGP
VL
STh
AGl
CPu AGm
MGP
SNR
SNC
Indirect EC
Brain Struct Funct
123
123
0.62000
2.66667
2.22222
2.33333
2.55556
2.77778
2.88889
2.77778
2.66667
2.55550
SNR
VA
VL
LGP
MGP
CPu
STh
AGl
AGm
3.04255
3.00000
3.08511
3.09574
3.04255
2.79787
2.85106
2.61702
3.24468
3.24468
RADout
0.45600
0.41500
0.38500
0.50200
0.50800
0.17000
0.23100
0.06700
0.67600
0.74700
Hub
3.07580
3.01080
3.00720
3.07580
3.09390
2.72560
2.86280
2.54870
3.14800
3.15520
RADout
0.37400
0.33500
0.31000
0.40600
0.40900
0.13300
0.18700
0.05300
0.54300
0.60400
Hub
VA
VL
AGm
LGP
AGl
SNR
STh
MGP
SNC
CPu
RADout
VA
VL
AGm
LGP
SNR
AGl
STh
MGP
SNC
CPu
Hub
VA
LGP
VL
AGl
AGm
MGP
STh
CPu
SNR
SNC
RADout
VA
LGP
VL
STh
AGl
AGm
CPu
MGP
SNR
SNC
Hub
VA
LGP
VL
STh
AGl
AGm
CPu
MGP
SNR
SNC
RADout
VA
LGP
VL
STh
AGl
AGm
CPu
MGP
SNR
SNC
Hub
2.55600
2.55600
2.77800
3.00000
2.66700
2.44400
2.66700
2.33300
2.66700
2.55600
RADin
0.63900
0.48800
0.92200
1.00000
0.82100
0.58600
0.80600
0.36200
0.83700
0.67300
Aut
3.01100
3.20200
3.19100
2.97900
3.01100
2.89400
3.04300
2.74500
3.01100
3.11700
RADin
0.49300
0.42900
0.66400
0.68800
0.42900
0.33300
0.58000
0.18600
0.48800
0.63700
Aut
3.03250
3.02890
3.09390
3.11550
3.00000
2.95670
3.05050
2.80140
3.01440
3.07580
RADin
0.43900
0.38300
0.57600
0.59700
0.37600
0.29600
0.50900
0.16200
0.42600
0.55600
Aut
VA
LGP
AGm
AGl
SNC
MGP
VL
SNR
STh
CPu
RADin
VA
AGl
LGP
AGm
SNC
VL
MGP
SNR
STh
CPu
Aut
VA
LGP
CPu
AGm
MGP
SNR
VL
SNC
STh
AGl
RADin
VA
LGP
AGl
MGP
SNR
AGm
VL
SNC
STh
CPu
Aut
VA
LGP
MGP
SNR
AGl
AGm
VL
SNC
STh
CPu
RADin
VA
LGP
MGP
AGl
SNR
AGm
VL
SNC
STh
CPu
Aut
EC eigenvector centrality, PRC page rank centrality, Hub hubness, Aut authoritativeness, RADout radiality of the output, RADin radiality of the input
The BG1-network was embedded in a network that contains all extrinsic regions that have direct input and output (direct) or indirect input and output (indirect) connections to the regions of the BG1-network. The italic short names are presenting the rank of a particular region with regard to a descending sorted specific local parameter (maximum rank 1 for a region; smallest value: rank 10). For example the CPu has the rank 1 concerning the betweenness centrality (BC) in the BG1-network. In the extrinsic direct neighbor network it has the rank 4 and in the indirect neighbor network it has rank 1. The subgraph centralities (SC) were normed
2.77778
SNC
RADout
0.87100
AGm
0.89800
MGP
AGl
0.73700
LGP
1.00000
0.36900
VL
0.87400
0.25800
VA
STh
0.79400
SNR
CPu
0.97200
SNC
Hub
Table 14 continued
Brain Struct Funct
11
11
11
10
12
11
12
11
12
12
12
VM
Ent
CM
HIPP
MDL
CL
LGF
PC
MDM
VL
VA
MGP
10
12
AGI
10
14
SNR
Pir
12
SNC
LHb
12
AGm
9
14
MRF
11 11
12
VTA
PF Ac
11
CPu
STh
8
11
AC
HD
Region
0
3
4
6
4
4
4
6
5
8
7
5
8
11 10
12
11
14
11
11
15
15
17
18
20
REC
5
14
14
14
16
16
17
20
20
21
21
22
22
28 28
30
30
31
32
32
34
38
40
41
42
DGA
Table 15 Local parameters of the ipsilateral BG2 network
2
3
4
6
5
8
4
11
10
9
9
8
11
17 13
17
17
15
19
21
17
23
23
20
22
DGC
3
11
10
8
11
8
13
9
10
12
12
14
11
11 15
13
13
16
13
11
17
15
17
21
20
DGI
0.6
0.79
0.71
0.57
0.69
0.5
0.76
0.45
0.5
0.57
0.57
0.64
0.5
0.39 0.54
0.43
0.43
0.52
0.41
0.34
0.5
3.39
0.43
0.51
0.48
CDC
3.04
11.47
10.08
8.94
10.97
8.45
13.37
9.2
10.73
11.85
12.37
13.67
11.51
11.69 14.02
13.19
12.92
14.76
13.17
11.33
15.75
14.86
16.05
18.79
18.82
Katz
3
1
2
2
2
2
2
1
2
1
2
1
1
2 1
1
1
1
2
1
1
1
1
1
1
SPC
15
142
135
148
202
188
205
264
275
284
325
299
367
551 492
600
562
499
603
624
558
840
864
886
950
Triag
3.17
0.12
0.11
0.14
0.15
0.13
0.13
0.1
0.11
0.11
0.12
0.09
0.13
0.12 0.11
0.11
0.11
0.09
0.1
0.1
0.08
0.1
0.09
0.09
0.09
CyclC
3
3
2
2
2
2
2
2
2
2
2
3
2
2 2
2
2
2
2
2
2
2
2
2
2
EccO
Brain Struct Funct
123
EccI
2
2
2
2
2
2
2
2
2
2
2 2
2
2
2
2
2
3
2
2
2
2
3
2
3
Region
AC
CPu
VTA
MRF
AGm
SNC
SNR
AGI
MGP
STh
PF Ac
LHb
Pir
VM
Ent
CM
HIPP
MDL
CL
LGF
PC
MDM
VL
VA
Table 15 continued
123
1
1
0.92
0.83
1
0.75
0.92
0.68
0.68
0.71
0.78
0.61
0.75
0.68 0.71
0.67
0.58
0.57
0.5
0.54
0.48
0.54
0.53
0.58
0.54
CCO
0.33
0.81
0.78
0.88
0.82
0.88
0.76
0.81
0.84
0.69
0.84
0.71
0.9
0.86 0.64
0.78
0.79
0.54
0.81
0.89
0.54
0.73
0.61
0.53
0.59
CCI
0.75
0.81
0.78
0.85
0.83
0.82
0.76
0.67
0.69
0.72
0.74
0.67
0.75
0.68 0.59
0.65
0.56
0.56
0.5
0.54
0.49
0.54
0.53
0.51
0.54
CCA
0.75
0.81
0.78
0.86
0.87
0.81
0.78
0.72
0.74
0.7
0.8
0.66
0.82
0.75 0.67
0.71
0.66
0.55
0.62
0.64
0.51
0.61
0.57
0.55
0.56
CCT
0.47
0.27
0.41
0.27
0.36
0.32
0.17
0.37
0.13
0.35
0.26
0.36
0.33
0.07 0.57
0.07
0.65
0.52
0.83
0.5
0.75
0
0
0
0
CC2
33.6
32.82
30.9
35.33
31.42
31.67
31.85
28.79
30
29.85
30.43
28.24
30.07
29.18 26.56
28.33
25.95
26.06
24.81
25.67
24.53
25.43
25.35
24.91
25.68
ADnb
0.12
0.22
0.24
0.13
0.25
0.2
0.22
0.33
0.26
0.29
0.27
0.3
0.26
0.28 0.35
0.3
0.39
0.41
0.41
0.36
0.42
0.34
0.34
0.37
0.32
VCDG
-0.74
-0.39
-0.36
-0.43
-0.3
-0.31
-0.29
-0.15
-0.18
-0.15
-0.16
-0.1
-0.14
0 0.06
0.05
0.11
0.13
0.16
0.14
0.2
0.22
0.24
0.27
0.26
Lev
0.13
0.39
0.35
0.29
0.47
0.46
0.46
0.52
0.54
0.5
0.54
0.69
0.58
0.68 0.72
0.74
0.75
0.64
0.79
0.85
0.69
0.99
0.99
0.96
0.94
Loc
0.5
0.52
0.55
0.57
0.56
0.6
0.55
0.65
0.63
0.62
0.62
0.59
0.65
0.77 0.69
0.77
0.77
0.73
0.83
0.89
0.77
0.96
0.96
0.86
0.92
CEO
0.52
0.65
0.62
0.6
0.65
0.6
0.69
0.59
0.63
0.67
0.67
0.71
0.65
0.65 0.73
0.69
0.69
0.75
0.69
0.65
0.77
0.73
0.77
0.89
0.86
CEI
Brain Struct Funct
1.35
0.95
SNC
AGI
1.35
AGm
0.32
0.32
MRF
SNR
0.55
VTA
ZA
Region
1.35
0.0001
VA
1.43
0.0008
VL
CPu
0.0013
MDM
AC
0.0017
0.0009
0.0017
CL
LGP
0.0025
MDL
PC
0.0042
0.0048
CM
HIPP
0.63
0.66
0.67
0.62
0.67
0.67
0.6
0.67
PCO
0.11
0.17
0.17
0.34
0.29
0.41
0.23
0.49
0.33
0.6
0.68
0.45
0.0214
Ac
0.82
0.86
0.45 0.44
0.0073
PF
0.0034 0.0086
3.0175
STh
0.8
0.69
VM Ent
0.0239
MGP
0.0057
0.0487
AGI
0.77
0.0105
0.0279
SNR
0.92
0.69
LHb
0.0177
SNC
1
Pir
0.0694
AGm
1
0.0510
0.0287
VTA
MRF
0.92
0.0923
CPu
0.97
0.0739
AC
EC
BC
Region
Table 15 continued
0.63
0.6
0.51
0.65
0.66
0.66
0.66
0.67
PCI
1,668.26
8,992.33
8,278.07
15,064.76
16,085.3
17,062.22
15,199.19
20,626.62
23,714.09
27,703.34 23,826.42
20,854.48
32,238.35
44,068.22
47,202.32
52,625.08
47,865.52
47,501.37
50,872.73
48,687.55
50,691.03
69,052.81
74,470.26
80,557.24
84,751.47
SC
0.63
0.66
0.65
0.64
0.67
0.66
0.66
0.67
PCA
0.18
0.45
0.44
0.35
0.41
0.33
0.53
0.4
0.41
0.46 0.57
0.6
0.43
0.6
0.43
0.54
0.53
0.86
0.54
0.45
0.9
0.58
0.74
1
0.94
PRC
2.63
2.79
2.88
2.71
2.96
2.96
2.83
2.92
RadO
0.3
0.27
0.4
0.6
0.39
0.45
0.31
0.51
0.45
0.55 0.64
0.39
0.62
0.6
0.65
0.68
0.61
0.83
0.56
0.52
0.8
0.65
0.74
0.79
0.91
FC
2.67
2.54
2.46
2.71
2.63
2.71
2.88
2.83
RadI
1
5
7
6
6
10
12
20
22
17 30
35
19
67
35
61
71
118
82
62
141
113
154
208
187
Stress
-8
-5
-2
-6
0
0
-3
-1
CenO
0.42
0.25
0.2
0.15
0.16
0.14
0.19
0.11
0.12
0.11 0.05
0.07
0.07
0.01
0.06
0.02
-0.02
-0.28
-0.04
0.03
-0.24
-0.01
-0.1
-0.31
-0.16
Shapley
-5
-9
-10
-4
-6
-4
1
-1
CenI
-1.24
-0.75
-1.32
-0.75
-2.06
-0.75
-1.88
0.9
-0.27
-0.27 0.35
0.35
-1.15
-0.21
0.69
0.69
0.69
1.19
-0.23
0.89
1.67
0.69
0.9
1.19
0.9
ZO
0.67
0.81
0.93
0.72
1
1
0.89
0.96
Hub
-1.51
-0.24
0.73
-0.66
-1.89
-0.66
-1.6
-0.44
-0.66
-0.24 0.73
-0.44
-0.38
-0.44
-0.38
-0.38
1.13
1.46
1.13
1.13
1.03
-0.38
-0.44
1.46
1.9
ZI
0.76
0.77
0.67
0.82
0.87
0.89
1
1
Aut
Brain Struct Funct
123
123
0.55
-2.09
-0.71
-2.21
-0.71
MDL
CL
LGP
PC
0
0
0.38
0.5
0.32
0.66
0.5
0.51
0.64
0.41
0.63
0.66 0.56
0.44
0.66
0.54
0.66
0.64
0.59
0.64
0.49
0.62
0.63
0.61
0.66
0.56
0.6 0.66
0.64
0.6
PCI
0.48
0.62
0.52
0.64
0.59
0.63
0.66
0.56
0.67
0.57
0.65
0.62
0.6
0.64 0.63
0.65
0.64
PCA
2
2.08
2.17
2.25
2.21
2.33
2.17
2.46
2.42
2.38
2.38
2.29
2.46
2.71 2.54
2.71
2.71
RadO
2.08
2.46
2.38
2.33
2.46
2.33
2.54
2.29
2.42
2.5
2.5
2.58
2.46
2.46 2.63
2.54
2.54
RadI
-23
-20
-20
-18
-19
-16
-20
-12
-14
-14
-15
-16
-12
-7 -10
-6
-6
CenO
-19
-10
-12
-14
-11
-14
-9
-13
-12
-9
-10
-7
-10
-11 -6
-8
-8
CenI
A all, AD average degree, C circle, CC cluster-coefficient, CE closeness centrality, DG degree, I in, O out, SP length of shortest path
The regions are sorted using the degree all. Abbreviations and parameter definitions are described in the Supporting Information
-1.4
-0.48
CM
HIPP
VA
0.55
-0.77
-0.25
VM
Ent
-0.48
0.11
MDM
0.64
-0.95
LHb
Pir
VL
0.44
0.32 -0.33
PF Ac
0.65
0.32
0.63
0.95
PCO
STh
ZA
MGP
Region
Table 15 continued
0.1
0.15
0.19
0.31
0.23
0.38
0.2
0.52
0.48
0.32
0.45
0.36
0.51
0.78 0.58
0.78
0.74
Hub
0.18
0.7
0.59
0.56
0.69
0.55
0.8
0.55
0.66
0.65
0.77
0.8
0.71
0.73 0.82
0.78
0.77
Aut
Brain Struct Funct
Brain Struct Funct
where qj;k is the number of shortest paths from j to k and qj;k ðiÞ is the number of shortest paths from j to k that pass through i. The directed and weighted definitions are the same. ^ [ 1 be a subset of Knotty centrality KC: Let N^ N, jNj N. Then P ^ ¼ LdðNÞ ^ Pi2N^ BCðiÞ KCðNÞ ð76Þ i2N BCðiÞ ^ of the subgraph N. ^ The knotty with the line density LdðNÞ center of a graph G is a subset NKC of nodes with ^ KCðNKC Þ ¼ min fKCðNÞg ^ NN ^ [1 jNj
ð77Þ
KCðNKC Þ is called the knotty-centerdness of the graph G. The knotty-centrality of a node i is defined as KCðiÞ ¼
1;
i 2 NKC
0;
else
ð78Þ
Stress S: SðiÞ ¼
X
qj;k ðiÞ
j;k2Nnfig
ð79Þ
The directed and weighted definitions are the same. Central point distance CPD: n 1 X BCmax BCðiÞ CPD ¼ ð80Þ n 1 i¼1 BCmax where BCmax ¼ maxi2N fBCðiÞg is the maximum Betweenness centrality. The directed and weighted versions use the directed and weighted Betweenness centralities. Participation coefficient: The partition M¼ fM1 ; . . .Mm g is generated as described in the definition of modularity. X deg ði; Mj Þ2 x ! PCx ðiÞ ¼ 1 ð81Þ degx ðiÞ Mj 2M with x 2 fin; out; allg and X degin ði; Mj Þ ¼ aki k2Mj nfig
(Number of edges from vertices of Mj to i). X aik degout ði; Mj Þ ¼ k2Mj nfig
(Number of edges from i to vertices of Mj ). X ðaik þ aki Þ degall ði; Mj Þ ¼ k2Mj nfig
ð82Þ
(Number of edges between i and vertices of Mj ). X degw ði; Mj Þ2 ! w x PCx ðiÞ ¼ 1 degwx ðiÞ Mj 2M
ð85Þ
with the same x and weighted definitions of degrees. One has 0 PCðiÞ 1. If PCðiÞ ¼ 1, the node i has no edges (in, out, all). If PCðiÞ ¼ 0 all edges (in, out all) come from, go to or stay in the same cluster. The larger PCðiÞ the more clusters are involved in the edges of node i. Z score/within module degree: Let Mi be the module containing node i. degx ði; Mi Þ x 2 fin; out; allg is defined in the participation coefficient. 1 X degx ðMi Þ ¼ degx ðj; Mi Þ ð86Þ jM j i j2Mi
is the mean and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u u 1 X rdegx ðMi Þ ¼ t degx ðj; Mi Þ degx ðMi Þ jMi j j2Mi
ð87Þ
the standard deviation of the within module Mi degree distribution. Then the Z score is defined as Zx! ðiÞ ¼
degx ði; Mi Þ degx ðMi Þ rdegx ðMi Þ
ð88Þ
and analogous ! degwx ði; Mi Þ degw x ðMi Þ Zxw ðiÞ ¼ rdegwx ðMi Þ
ð89Þ
with the weighted versions of the mean and standard deviation. A value above one or below minus one implies that a node has significantly more or less edges from, to or from and to nodes in its cluster than the average node in its cluster has. Eigenvector centrality: The eigenvector centrality ECðiÞ is the i-th component of the eigenvector with the largest corresponding eigenvalue of the adjacency matrix resp. weight matrix. Shapley rating /: The Shapley rating is a measure that provides information about the loss of connectivity following the removal of a node. X ^ SRðiÞ ¼ ðjSCCðN^ [ figÞj jSCCðNÞjÞ ^ NNnfig
ð83Þ
ð84Þ
ð90Þ ^ 1Þ! jNj! ^ ðn jNj n! ^ where SCCðNÞ is the set of strongly connected components ^ The smaller the value is, the more important is the of N. node in the sense of connectivity of the graph. Because of the exponential number of subsets, this parameter can be approximated for large networks, only.
123
Brain Struct Funct
Radiality: The radiality of a node Rad is a measure of the distance of a node to all other nodes. Nodes that have a small radiality have larger distances to other nodes than those with a greater radiality. Input radiality Radin : The input radiality of a node Radin is X 1 Diam þ 1 dðj; iÞ Radin ðiÞ ¼ ð91Þ n 1 j2N dðj;iÞ\1
In the weighted case the weighted distances are used. Output radiality Radout : The output radiality of a node Radout is X 1 Radout ðiÞ ¼ Diam þ 1 dði; jÞ ð92Þ n 1 j2N dði;jÞ\1
In the weighted case the weighted distances are used. Centroid value Cen: With gout ði; jÞ ¼ jfk 2 Njdði; kÞ\dðj; kÞ\1gj and gin ði; jÞ ¼ jfk 2 Njdðk; iÞ\dðk; jÞ\1gj which are the number of nodes closer to node i than to node j with regard to In- and Outdistance, the centroid value is defined in the following. Output centroid value Cenout : Cenout ðiÞ ¼ minfgout ði; jÞ gout ðj; iÞjj 2 Nnf igg
ð93Þ
FCðiÞ ¼
X 1 aji aik jNi j ðjNi j 1Þ j;k2N
In the weighted case we define the flow coefficient as the sum of weights of paths of length 2 between neighbors of a node i that pass node i divided by the maximum possible sum. X X ! 1 FC w ðiÞ ¼ ðwji þ wik Þ 2 jNi j ðjNi j 1Þ j2Ni k2N nfjg
Cenin ðiÞ ¼ minfgin ði; jÞ gin ðj; iÞjj 2 Nnf igg
ð94Þ
In the weighted case the weighted distances are used. A value \0 implies, that there exists a node that is closer to most other nodes. A value 0 implies, that this node is most central in the network. A value ¼0 implies, that there are more than one most central nodes. Page rank centrality PRC: PRCðiÞ ¼ ri where r is the solution of the linear system 0 1 1 1 B .. C T ð95Þ ðI a A BÞ r ¼ ð1 aÞ @ . A n 1 with the damping factor a ¼ 0:85, the identity matrix I and the diagonal matrix B, whereby 8 1 < ; degout ðiÞ [ 0 bii ¼ degout ðiÞ ð96Þ : 0; otherwise In the weighted case the weight matrix W is used instead of ! A and the weighted version deg w ðiÞ of the outdegree. out
Flow coefficient FC: Number of paths of length 2 between neighbors of a node i that pass node i divided by the maximum possible numbers of sub paths.
123
i
wji [ 0
wik [ 0
ð98Þ Average flow coefficient FC: Number of paths of length 2 between neighbors of a node i that pass node i divided by the maximum possible numbers of sub paths. FC ¼
n 1X FCðiÞ n i¼1
ð99Þ
n ! ! 1X FC w ðiÞ FC w ¼ n i¼1
ð100Þ
Subgraph centrality SC: SCðiÞ ¼
1 X ðAk Þ k¼0
Input centroid value Cenin :
ð97Þ
i
j6¼k
ii
ð101Þ
k!
1 X ! ðW k Þii SC w ðiÞ ¼ k! k¼0
ð102Þ
The subgraph centrality of the network is the average subgraph centrality of its nodes. SC ¼
n 1X SCðiÞ n i¼1
ð103Þ
n ! 1X ! SC w ¼ SC w ðiÞ n i¼1
ð104Þ
Undirected cyclic coefficient CyclC: The undirected cyclic coefficient as published by (Kim et al. 2005) Cyclic topology in complex networks. X 2 1 CyclCðiÞ ¼ jNi j ðjNi j 1Þ ðj;kÞ2N N 2 þ disti ðj; kÞ i
i
j6¼k
ð105Þ With 8 length of the shortest path from j to k > > > < that does not contains i; disti ðj; kÞ ¼ > if such a path exists > > : 1; otherwise ð106Þ
Brain Struct Funct
Directed cyclic coefficient CyclC! : A publication about the directed cyclic coefficient is unknown. The directed cyclic coefficient is implemented here as follows: CyclC! ðiÞ ¼
jNiout j
1 jNiout \ Niin j 1 2 þ disti ðj; kÞ in
jNiin j
X
ð107Þ
ðj;kÞ2Niout Ni j6¼k
! Directed weighted cyclic coefficient CyclC w : A publication about the directed weighted cyclic coefficient is unknown. The directed weighted cyclic coefficient is implemented here as follows: ! CyclC w ðiÞ ¼
1 jNiout j jNiin j jNiout \ Niin j X 1 wij þ wki þ distwi ðj; kÞ out in
ð108Þ
ðj;kÞ2Ni Ni j6¼k
with distwi ðj; kÞ is the weighted version of disti ðj; kÞ with the weighted path length. Cyclic network coefficient CyclC! : The cyclic coefficient of the network is the average cyclic coefficient of its nodes: n 1X CyclC! ¼ CyclC! ðiÞ ð109Þ n i¼1 Hubness and authoritativeness: A hub is a node that points to many authorities and an authority is a node that has numerous input connections from many hubs (Kleinberg 1999; Sporns et al. 2007). The hubness HubðiÞ of a node i is: X HubðiÞ ¼ AuthðjÞ ð110Þ out j2Ni
with the authoritativeness AuthðiÞ X AuthðiÞ ¼ HubðjÞ j2Niin
ð111Þ
An iterative algorithm is used to calculate a fixed point of these equations. Vulnerability V: The vulnerability V is the maximum relative decrease of the global efficiency removing a single node. GE GEðiÞ V ¼ max ð112Þ i2N GE where GEðiÞ is the global efficiency of the graph (Nnfig; fðj; kÞ 2 Ejj 6¼ i 6¼ kg) that originates by removal of node i and all edges adjacent to i. The weighted version is analog using the weighted global efficiencies.
Random models The following random graph models are compared to the real network of the intrinsic amygdala connectivity. By comparing the average path length and the cluster coefficient of the models with the real network it is feasible to determine a model that is most similar to the real network. Erdo¨s Re´nyi graph: ð113Þ
Gðn; pÞ
where n is the number of vertices and p is the probability that an edge ði; jÞ exists, for all i; j. The degree distribution of the Erdo¨s Re´nyi random graph is binomial in terms of n1 k PðdegðvÞ ¼ kÞ ¼ p ð1 pÞn1k ð114Þ k Watts–Strogatz graph: The small-world model of Watts– Strogatz is a random graph generation model that provides graphs with small-world properties. The network (initially it has a non-random lattice structure) is build by linking each node to its hki closest neighbors using a rewiring probability p. Hence, an edge has the probability p that it will be rewired as a random edge. The number of rewired links can be estimated by: pE ¼ pNhki=2
ð115Þ
Barabasi–Albert graph: The Barabasi–Albert graph is used to generate preferential attachments between nodes. The probability pi that the new node is connected to node i is ki pi ¼ P
j kj
ð116Þ
The degree distribution of a Barabasi–Albert network is scale free following the power law distribution of the form: PðkÞ k3
ð117Þ
Eipert graph: The modified Eipert model (EN: Eipert network) is based on the Barabasi–Albert graph. However, the algorithm starts at a fixed number of nodes and edges are added iteratively. Ozik–Hunt–Ott graph: The Ozik–Hunt–Ott model (OHO) (Ozik et al. 2004) is a small-world randomization approach that was modified for directed networks and a fixed number of edges. The OHO-model uses a growing mechanism in which all connections are made locally to topographical nearby regions. Rewiring graph: The rewiring-models connects each target of an edge of a network to another target node. Power law: PðkÞ ¼ a kc
ð118Þ
D is the deviation (error) of an empirical distribution of degrees from the power law function. A small D value
123
Brain Struct Funct
means that the empirical distribution is similar with the power law function.
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ORIGINAL ARTICLE
The connectome of the basal ganglia Oliver Schmitt • Peter Eipert • Richard Kettlitz Felix Leßmann • Andreas Wree
•
Received: 27 February 2014 / Accepted: 30 October 2014 Ó Springer-Verlag Berlin Heidelberg 2014
Abstract The basal ganglia of the laboratory rat consist of a few core regions that are specifically interconnected by efferents and afferents of the central nervous system. In nearly 800 reports of tract-tracing investigations the connectivity of the basal ganglia is documented. The readout of connectivity data and the collation of all the connections of these reports in a database allows to generate a connectome. The collation, curation and analysis of such a huge amount of connectivity data is a great challenge and has not been performed before (Bohland et al. PloS One 4:e7200, 2009) in large connectomics projects based on meta-analysis of tract-tracing studies. Here, the basal ganglia connectome of the rat has been generated and analyzed using the consistent cross-platform and generic framework neuroVIISAS. Several advances of this connectome meta-study have been made: the collation of laterality data, the network-analysis of connectivity strengths and the assignment of regions to a hierarchically organized terminology. The basal ganglia connectome offers differences in contralateral connectivity of motoric regions in contrast to other regions. A modularity analysis of the weighted and directed connectome produced a specific grouping of regions. This result indicates a correlation of structural and functional subsystems. As a new finding, significant reciprocal connections of specific network motifs in this connectome were detected. All three principal basal ganglia pathways (direct, indirect, hyperdirect) could be determined in the connectome. By identifying these pathways it was found that there exist many further equivalent pathways possessing the same length and mean O. Schmitt (&) P. Eipert R. Kettlitz F. Leßmann A. Wree Department of Anatomy, University of Rostock, Rostock, Germany e-mail: [email protected]
connectivity weight as the principal pathways. Based on the connectome data it is unknown why an excitation pattern may prefer principal rather than other equivalent pathways. In addition to these new findings the local graphtheoretical features of regions of the connectome have been determined. By performing graph theoretical analyses it turns out that beside the caudate putamen further regions like the mesencephalic reticular formation, amygdaloid complex and ventral tegmental area are important nodes in the basal ganglia connectome. The connectome data of this meta-study of tract-tracing reports of the basal ganglia are available for further network studies, the integration into neocortical connectomes and further extensive investigations of the basal ganglia dynamics in population simulations. Keywords Connectome Connectomics Basal ganglia Caudate putamen Striatum Substantia nigra Neuroontology Digital atlasing Tract tracing Multiscale Network analysis Graph analysis Abbreviations A All (all inputs and outputs) AD Average degree Ac Accumbens nucleus AC Amygdaloid complex AGl Lateral agranular prefrontal cortex AGm Medial agranular prefrontal cortex Aut Authoritativeness AW Average weight BC Betweenness centrality BG Basal ganglia C Circle CC Cluster-coefficient CE Closeness centrality
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chain CL CM CNS CPu DG Dic Dii Dis DNN Doc Doi Dos DR EC Ent HIPP Hub I in INN L LGP LHb MDL MDM MDS MGP MRF O out PC PCA PF Pir PL PRC Pub RADin RADout Rec Rel Sic Sii Sis SG SNC SNR Soc Soi Sos
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Chain pattern of a motif Centrolateral thalamic nucleus Central medial thalamic nucleus Central nervous system Caudate putamen Degree Direct input from contralateral Direct input from ipsilateral Direct input from ipsi- and contralateral Direct neighbor network Direct output to contralateral Direct output to ipsilateral Direct output to ipsi- and contralateral Dorsal raphe nucleus Eigenvector centrality Entorhinal cortex Hippocampus Hubness In (input to a region; used in tables) Symmetric input connection to a central node of a motif Indirect neighbor network Laterality Lateral globus pallidus Lateral habenular nucleus Mediodorsal thalamic nucleus lateral part Mediodorsal thalamic nucleus medial part Metric multidimensional scaling Medial globus pallidus Mesencephalic reticular formation Out (Output of region; used in tables only) Symmetric output connection from a central node of a motif Paracentral thalamic nucleus Principal component analysis Parafascicular thalamic nucleus Piriform cortex Path length Page rank centrality Number of articles Radiality of the input Radiality of the output Reciprocal Reliability Subtree input from contralateral Subtree input from ipsilateral Subtree input from ipsi- and contralateral Subgraph centrality Substantia nigra compact part Substantia nigra reticular part Subtree output to contralateral Subtree output to ipsilateral Subtree output to ipsi- and contralateral
SP SPN STh VA VL VM VTA
Length of shortest path Spiny neurons of the CPu Subthalamic nucleus Ventro anterior thalamic nucleus Ventrolateral thalamic nucleus Ventromedial thalamic nucleus Ventral tegmental area A10
Introduction The basal ganglia (BG) are composed of the four major nuclei neostriatum, globus pallidus (GP), substantia nigra (SN) and subthalamic nucleus (STh). The canonical basal ganglia system consists of the BG and of those cortical areas that are connected with BG (Gerfen and Bolam 2010). In rodents the neostriatum is a single nucleus, the caudate putamen (CPu) which is divided by the internal capsule into the caudate nucleus and putamen in higher vertebrates. The GP consists of an external segment or lateral globus pallidus (LGP) and the internal segment or medial globus pallidus (MGP). The MGP in rodents is also called the entopeduncular nucleus. The SN consists of the pars compacta (SNc) and the pars reticulata (SNr). These nuclear complexes are related to the dorsal aspects of the BG. The ventral aspects contain the accumbens nucleus (Ac), the ventral pallidum and the medial aspects of the STh and SN. The dorsal aspect of the basal ganglia has motor and associative functions, whereas the ventral aspect is related to limbic functions. Additional motor components that are strongly associated with the dorsal aspect of the BG are the frontal cortex especially the primary motor cortex (lateral agranular prefrontal cortex) and subnuclei of the thalamus (Gerfen 2004). Principal afferents of the BG arise from the ipsi- and contralateral cerebral cortex (layer 5 glutamatergic neurons), intralaminar thalamic nuclei such as the centromedian and parafascicular nucleus, the dorsal raphe nucleus (DR) and the amygdaloid complex (AC). The most extensive input to the BG is glutamatergic from the cerebral cortex and thalamus. These projections terminate in the heads of spines of the GABAergic spiny neurons (SPN) of the CPu. The latter are the principal projection neurons innervating the SN and GP. These SPN possess a dense local axon collateral system that innervates other SPN and interneurons of the CPu. At the level of macroconnectomes and macrocircuits all three major pathways of the BG are represented by the connectome data in this study. The direct pathway is an excitatory pathway which has the same source region (CPu) and BGoutput region (STh) as the inhibitory indirect pathway. The macrocircuitry of the BG is dominated by these two
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principal pathways (direct, indirect) (Alexander and Crutcher 1990; DeLong 1990; Smith et al. 1998) and a hyperdirect pathway (Kita and Kitai 1994) which will be investigated in the introductory part of the ‘‘Results’’. These pathways along with the cerebral cortex and thalamus form three loops (cortical-basal ganglia loops). It is known that the GABAergic neurons in the LGP influence the activity at every level of the BG by their extensive axon collaterals (Tepper et al. 2007). Single cell labeling studies (Kita and Kitai 1994; Bevan et al. 1998; Sato et al. 2000) have indicated that LGP projection neurons have axonal collaterals to the STh and BG output nuclei. Additionally, up to one-third of LGP neurons project to the CPu (Kita and Kitai 1994; Bevan et al. 1998) and to its GABAergic interneurons which innervate the spiny projection neurons (Kita 1993; Koos and Tepper 1999; Bennett and Bolam 1994). Besides the major inputs to and outputs from the BG nearly all areas of the neocortex are interconnected with regions of the BG (Gerfen and Bolam 2010). In terms of functional segregation, frontal cortical areas and intralaminar thalamic nuclei are involved in planning and execution of movement behavior. Outputs of the BG to the intermediate layers of the superior colliculus are involved in the generation of eye and head movements and output to the pedunculopontine nucleus is necessary for orienting movements of the body. A functional and hodological grouping of subregions the BG into four categories has been proposed (Parent 1986; Alexander and Crutcher 1990; DeLong 1990; Smith et al. 1998; Gerfen and Bolam 2010). The input nuclei are the CPu and STh which receive cortical inputs. The output nuclei are the MGP and SNR which project to the thalamus, midbrain and brainstem. The LGP is a relay nucleus connecting the input and output nuclei. The modulator nucleus is the SNC. The major objectives of this study are 1. 2. 3. 4.
the collation of connectivity data, a weighted and directed analysis of the BG connectome, the analysis at two levels of detail and the intrinsic, extrinsic, and differential analysis
which are specified in the following. In addition to this general organization of the BG abundant information of interconnections and densities of connections between topographic, functional, cytoarchitectonic and chemoarchitectonic subdivisions of BG regions (intrinsic BG connections) is available. More specifically, afferent, efferent intrinsic and extrinsic connections of the BG regions are described in 789 reports (Lanciego and Wouterlood 2011) (objective 1). These connectivity data can be collated for a meta-study as shown by others (Bakker et al. 2012; Sporns et al. 2000b; Stephan et al. 2001b; Press et al. 2001) to
analyze and characterize the resulting network in terms of graph theory (Sporns 2002, 2011; Rubinov and Sporns 2010; Young et al. 1994, 1996; Young 1992b). One particular objective of this study is a weighted and directed analysis of the BG connectome with regard to strengths of connections in combination with ipsi- and contralateral connectivity (objective 2). Connectivity data sets of parts of nervous systems have been developed and analyzed by several groups (Felleman and Essen 1991; Young 1992a, 1993; Sporns et al. 2000a, 2002, 2004; Sporns and Ko¨tter 2004; Honey et al. 2007; Modha and Singh 2010; Sugar et al. 2011), however, these data do not provide a complete collection of BG connections. Most of the connections in these data sets are compiled into neocortical connectomes within the framework of meta-studies of tract-tracing publications. The sources of information in this study originate exclusively from peerreviewed tract-tracing publications where anterograde and/ or retrograde tracers were applied (lesion studies were not evaluated). Only those publications were considered that describe connectivity in juvenile or adult rats. Connections between neuronal regions are handled with the help of a consistent neuroontology (Schmitt and Eipert 2012) that can be updated frequently in parallel to the fast progress of the identification of connections in tract-tracing publications. In most tract-tracing studies that describe new details of afferents and efferents, modifications of nomenclatures and overlap of regions occur which must be interpreted and integrated into connectome projects (Schmitt and Eipert 2012). Therefore, it is necessary to propose refinements of subdivisions of regions that are documented in stereotaxic atlases (Paxinos and Watson 2004, 2009; Swanson 2003). Such subdivisions can be presented in acyclic graphs or hierarchies. This approach allows to expand and aggregate regions and to apply such a selection of super- and subregions for defining a connectome and partial connectomes, respectively, without loosing non-standard terminologies of publications. This technique was applied here to define a core connectome of the basal ganglia (BG1) and a more detailed connectome that includes especially limbic connections (BG2) (objective 3) (the composition of the BG1and BG2-networks is described in the ‘‘Materials and methods’’ as well as in the ‘‘Conditional connectomics’’). The BG-connectome was analyzed in three contexts (objective 4): (1) embedded in regions that directly connect to it (Vlachos et al. 2012) (connectome embedding), to measure extrinsic connectivity; (2) stand-alone, to measure intrinsic connectivity (3) differential, to compare the role of local regions in the other two contexts (differential connectomics). A further aim of this investigation was to collate the connectivity data (Burns and Cheng 2006) of all peer-
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reviewed non-viral tract tracing publications (Bohland et al. 2009a) in which connections of regions of the BG were described. Viral tract tracing was not considered in this work because some neurotrophic viruses propagate transsynaptically in a time-dependent manner. The density and reliability of collated connections should be as large as possible. Such a BG data source has been proposed recently (Looi et al. 2013), but was not available, so far. Now, this BG connectome is available within a database (http://neuroviisas.med.uni-rostock.de/connectome/index. php) and can be downloaded for sharing (Van Horn and Gazzaniga 2013; Koslow 1997, 2005; Ko¨tter 2003), (https://neuinfo.org-/datasharing/index.shtm). Based on these tract-tracing data, the intrinsic, extrinsic, bilateral and weighted connectivity of the BG connectome was characterized in terms of global and local parameters, as well as multivariate statistics to determine regions of specific importance of intrinsic and extrinsic BG connectomes of the rat brain. By applying modularity analysis it was intended to investigate a possible correlation of functionally similar regions and their connectivity patterns. A population-based large-scale simulation of a thalamocortical network of the whole human cortex (Izhikevich and Edelman 2008) has provided evidence for the propagation of an excitatory wave from the visual cortex to frontal areas following a spike injection into the occipital areas. Because macroscopic connectivity in this model was primarily based on tractographic data of DTI-measurements, one assumption of this model was a complete reciprocity of all macroscopic interconnections of cortical regions. The hypothesis that reciprocal connections in tract tracing studies are highly specifically distributed has been investigated. The result of this analysis of reciprocal connections is of particular interest for modeling connectivity at macroscopic levels based on DTI-measurements and tractographic analysis.
Materials and methods In a meta-analysis of 4,513 peer reviewed tract-tracing publications describing non-viral tract-tracing experiments in juvenile and adult rats we found 3,265 articles that provide connectivity data that are useful for a mesoscale connectome analysis (Schmitt et al. 2012b; Burns and Cheng 2006; Burns et al. 2008a, b; Bota and Swanson 2007b, 2010). Nearly 800 of these 3,265 publications are considered as the data core because they contain connectivity data of particular regions of the basal ganglia. The inquiry of these publications was performed through PubMed (http://www.ncbi.nlm.nih.gov/pubmed) using search expressions (Schmitt et al. 2012b). Among the 3,265 articles numerous studies were found that applied tract tracing
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techniques for verifying and confirming experiments in adult laboratory rats as controls. Also these studies were taken into consideration, however, only those connections of them were collated which were observed in control rats. The sources, targets, semiquantitative weights (strengths of connections), type of tracer, directions of transport and features of somata (transmitters, enzymes, receptors) were transferred to tables. The raw data were verified by at least two experts in neuroanatomy. The verified data tables were imported into neuroVIISAS. This software is a platformindependent generic framework (Schmitt and Eipert 2012) developed in JAVATM for advanced connectome analysis (Bullmore and Sporns 2009; Stam and Reijneveld 2007; Albert and Barabasi 2002; Sporns et al. 2000a, 2004), visual analytics (Ware 2013; Archambault et al. 2008; Wong et al. 2006a, b) and nervous system specific 3Dmultiscale visualizations using stereotaxic coordinate systems (Hjornevik et al. 2007; Gustafson et al. 2004, 2007; Ju et al. 2006; Bjaalie 2002). The advantage of neuroVIISAS is that different modalities like ontologies, digital atlasing, connectomics and neuron population simulation (Feng et al. 2005; Burns et al. 2006; Gewaltig and Diesmann 2007; Moore et al. 2007; Niggemann et al. 2008; Lee et al. 2008; Gouws et al. 2009) are integrated into one flexible framework (Martone et al. 2004). The neuroVIISAS installation package can be downloaded from (http://neu roviisas.med.uni-rostock.de). All computations and visualizations were realized exclusively with the neuroVIISAS framework (chord diagrams were generated using the circular layout engine CIRCOS (http://circos.ca) through the CIRCOS–neuroVIISAS-interface). In most reports, connections are described in the form of ordinal categories (1, 2, 3, 4), symbols (*, **, ***) or expressions like: sparse, moderate, dense. Occasionally, authors have performed a stereologic analysis of retrogradely marked perikarya or a densitometric quantification of axonal terminals. There exist no standardization or normalization between these ordinal categories and estimations of the density of connections between independent studies or different reports. Obviously, connection weights are not discrete categories (Hilgetag and Grant 2000) and extend over several orders of dimensions (Markov et al. 2011, 2014). The latter determined a lognormal distribution of the fraction of neurons following injections of different tract tracers. Lognormal distributions of corticocortical connectivity was also found by Oh et al. (2014), ErcseyRavasz et al. (2013) and Wang et al. (2012). Therefore, ordinal categories were transformed to a logarithmic scale. This was realized in neuroVIISAS by a transformation table to approximate the distribution of the connection strengths logarithmically (Appendix Fig. 10). We would like to point out that these estimated connections strengths should be considered as a hypothetical estimate.
Brain Struct Funct
In general, connections are described in tract tracing studies in terms of ipsi- and contralateral courses rather than by providing exact stereotaxic coordinates (The sagittal axes in the rat atlas of Paxinos and Watson (2007) have the same values for the left and right side. In the rat atlas of Swanson (2003) one hemisphere is considered, only.) or an explicit indication of a left and/or right side injection into a specific region. In neuroVIISAS it is possible to manage injections and labelled sites either with explicit laterality, or with a symmetric representation where the labelled sites are either ipsi- or contralateral. To judge the reliability R of the data of connections, a reliability estimation of connectivity data is introduced. Because the connection data and not the measurement of connections itself constitute R it should be considered as an observation score. If this observation score is large then the probability that the connection really exists is large. An important assumption for calculating R is, that at least most of the tract tracing reports of a set of regions of interest have been evaluated and connectivity data are available for computation. The observation score of a connection is estimated by adding reliability weights. Reliability weights (Table 1) are defined for Table 1 Reliability weights used for estimating the reliability parameter R Variable
Case
Value
t
a/r
0.25
t
r
0.5
t
a
0.5
t
r?a/r
0.7
t
a?a/r
0.7
t
a?r
1.0
t w
a?r?a/r 3:0 unknown
1.0 0.7
w
2:0 fibers of passage
0.0
w
1:0 not clear
0.8
w
0:5 exists
w
0.0 not present
w
0.5 very light
1.0
w
1.0 light/sparse
1.0
w
1.5 light/moderate
1.0
w
2.0 moderate/dense
1.0
w
2.5 moderate/strong
1.0
w
3.0 strong
1.0
w
4.0 very strong
1.0
0.9 -1.0
Value reliability weight of connection strength, t variable of reliability weight for transport directions of tracers, w variable of reliability weight for strengths of connections, a anterograde tracer transport, r retrograde tracer transport, a/r bidirectional tracer transport, a?r?a/ r means that a connection has been proved by an anterograde, a retrograde and a bidirectional transported tracer
–
–
types of tracer transport directions (anterograde, retrograde) (t weight of the transport direction of a tracer) and the weight or strength of a connection (w weight of the connection strength).
The reliability R is defined as follows: X X R¼ wþ þ tþ þ w t
ð1Þ
Identical connections (connections between identical regions) may be described severalfold in a particular report in different animals or ‘‘cases’’ and/or in different reports. Since all these data are available in our database we are able to calculate sums of w of such case-based and noncase-based experimental observations. However, if a connection has been proved only by a retrograde or anterograde method then a smaller t ¼ 0:5 is assigned instead of a prove by a anterograde and retrograde method in two independent experiments (t ¼ 1). A bilateral transport (a/r) of a tracer is considered also in this weighting scheme (Table 1). The variables have subscripts and þ which indicate an observation of an existing connection (wþ , tþ ) or an explicit description that a connection does not exist (w , t ). Hence, explicitly not existent connections are weighted by negative reliability weights. The number of P observations of a specific connections is added ( w). However, the value of t is determined by identifying different directions of tracer transports within all connections that were added. For example, a specific connection that was observed in 10 different tract tracing experiments by applying anterogradely transported tracers obtains a w ¼ 10 and a t ¼ 0:5 (R ¼ 10:5). If the connection that was observed 10 times was found using 7 anterograde and 3 retrograde tracers then t ¼ 1 (R ¼ 11). And if the connection is observed 9 times (wþ ¼ 9) with 7 retrograde and 2 anterograde tracers (tþ ¼ 1) and in addition this specific connection has not been found in one experiment (w ¼ 1) with an anterograde tracer (t ¼ 0:5) then R ¼ 8:5. If there are many descriptions in publications that a specific connection does not exist, it gets a strong negative observation score. Finally, these scores are presented in a reliability matrix. We must emphasize that the observation score is defined for the data of a connection between a specific source or efferent region and a target or afferent region. So, it is not a score for connections rather than for the underlying data of the connections. Some connections and regions may receive more attention in tract tracing research so that frequently investigated or popular regions may have higher observation scores. This aspect is explicated in ‘‘Discussion’’. Regions of the nervous system of this particular project of the rat connectome are organized hierarchically (Schmitt and Eipert 2012), in graph theory such a hierarchy is called
123
Brain Struct Funct
a tree. An example of a hierarchy is shown in the Appendix Fig. 11. The root region of this hierarchy can be the nervous system of the rat. If a region is divided into smaller parts, these subregions are the childs of this parent region, which could be further subdivided. A region and all its descendents give rise to a so called subtree. Regions without descendents are called leafs. With regard to this concept the terms direct input or direct output of a region sum up all those connections that were described to end at (afferents), respectively, start from (efferents) this specific region. The terms subtree input or subtree output specify all those connections that end in, respectively, start from any region of the subtree of a given region including the region itself. The BG1 network consists of 10 regions (SNC, SNR, VA, VL, LGP, MGP, CPu, STh, AGl, AGm). These core regions of the BG are involved in the processing of motor functions. There exist several additional regions which are strongly interconnected with the BG1 regions, however, they are merely indirectly connected with regard to the principal BG pathways (direct, indirect, hyperdirect). To obtain a more comprehensive network, conditions for the selection of further regions can be applied (conditional connectomics). These selected regions belong to specific thalamic nuclei (lateral habenula among other regions), mescencephalic nuclei like the ventral tegmental area as well as to the limbic system like the ventral striatum (ventral CPu, accumbens nucleus). Further regions like the piriform cortex, hippocampus, entorhinal cortex and mesencephalic reticular formation (MRF) were selected in combination with their thalamic relay nuclei. In the following we will refer to this network as the BG2 network. The BG2 network is more extensive and contains with regard to the BG1 regions 15 further regions (VTA, MRF, CM, CL, PC, PF, MDL, MDM, VM, LHb, AC, Ac, HIPP, Ent, Pir). These two network configurations of the BG can be considered in a wider sense as a motoric BG connectome (BG1) and an associative BG connectome (BG2). The 10 regions of the BG1 connectome are localized on levels 9–14 (most on level 12) and the regions of the BG2 connectome on levels 8–14 of the hierarchy. The complex spatial arrangement of regions (Fig. 1) and their connections (Fig. 2) becomes visible by applying interactive 3Dvisualization of neuroVIISAS. The BG1 network was embedded in a surrounding network of regions that all have at least one input and one output connection to a region of the BG1 network. All regions of the surrounding network are located at a comparable hierarchical level like the BG1 region. The regions or parts of the selected regions of the BG1 or BG2 network do not overlap with or are not included in other regions (Bohland et al. 2009b). A further hypothesis to be tested in this study is, that specific distributions of reciprocal connections could
123
characterize the three principal pathways of the BG. Such specific distributions were analyzed by a recently developed pathway analysis of the BG-connectome data. Moreover, it was tested if additional routes from a source to a target region may exist in the directed BG connectome, a phenomenon which may entail a routing problem in structural connectomics. To determine pathways from source to target regions a sequential search of interconnected regions (pathway analysis) was applied in neuroVIISAS. The search-technique allows to add individual regions that should be passed in pathways from the provided source to the target regions. The result is shown in a table that contains weights, reliability and reciprocal connections within the detected pathways. To share the connectivity data of the BG, the rat connectome database can directly be queried through a web interface (http://neuroviisas.med.uni-rostock.de/connectome/ index.php).
Results Principal pathways The well-known principal pathways of the BG can be identified in the connectome. The adjacency matrix of the core regions of the BG is shown in Fig. 3a (BG1 network). Each of the following pathways can be derived from this matrix. The direct excitatory pathway [AGl, AGm] ! [CPu] ! [SNR, MGP] ! [VA, VL] ! [AGl, AGm] of the BG can be reconstructed using the adjacency matrix of principal regions (DeLong 1990; Alexander and Crutcher 1990; Albin et al. 1989) (Fig. 4a). The indirect inhibitory pathway [AGl, AGm] ! [CPu] ! [LGP] ! [STh] ! [SNR, MGP] ! [VA, VL] ! [AGl, AGm] of the BG can also be built using the same selection of regions (DeLong 1990; Alexander and Crutcher 1990; Albin et al. 1989) (Fig. 4b). The hyperdirect inhibitory pathway [AGl, AGm] ! [STh] ! [SNR, MGP] ! [VA, VL] ! [AGl, AGm] of the BG can be built by means of the same selection of regions (Bosch et al. 2012; Nambu et al. 2002) (Fig. 4c), too. The number of tract-tracing experiments that describe a certain connection is indicated by the number above each edge. Hence, the connection of the SNC to the CPu is described in 44 tract-tracing publications (Fig. 4a–c). The adjacency matrix of the ipsi- and contralateral BG network suggests that there could be additional pathways from AGl over the CPu back to AGl. Therefore, a pathway analysis was applied to detect additional routes or parallel pathways in the BG network (Table 2). If the maximum pathlength is
Brain Struct Funct Fig. 1 The regions of the BG2 network in stereotaxic space. a The stereotaxic space of the regions of the BG2 network. This view shows the region from dorsal, respectively, from top. Cortical regions are shown in red, subcortical parts in light red, thalamic regions are light magenta and SN are presented in dark magenta. The same color scheme was used for the coloring of abbreviations of regions in matrices. b Abbreviations of the regions of one hemisphere. c View from rostral directly onto the accumbens nucleus and ventral striatum. d View from caudal where midbrain structures and the turquoise MRF appear
restricted to 4 (because the minimum length of the principal pathways is 4), 653 ipsilateral pathways exist from AGl to AGl (circular pathway) in the BG2 network. If the CPu is determined as the first region in a pathway from the AGl back to the AGl, then 74 ipsilateral pathways (8 of these pathways have less than 4 edges) are found. The latter were sorted with regard to the reliability value (see ‘‘Reliability of connections’’). 6 pathways have reliability values that are larger than 10 and all of them have the SNR as the third region within the pathway. Most variability was found for thalamic regions (VM, PF, VL, PC, CM, CL) as output region to AGl. The direct pathway of this group has a reliability of 12. The indirect pathway may use MDL and PF as thalamic output regions to AGl or the VTA as an efferent region to AGl (reliability values are lying between 4 to 5). The hyperdirect pathway has a reliability of 7.5 and may use different output regions (MDL, PF, VM, VTA) to the AGl, too. Besides variations of BG-output nuclei of the thalamus, there are principal changes of input nuclei within these pathways, e.g. AGl has projections to AGm, CL, MDL, MGP, MRF, PC, PF, SNR, VL and VM. The sequence of connections of directly connected regions within a pathway were examined with regard to reciprocal connections. Then the patterns of reciprocity of connections within all principal pathways of the BG (only those regions were considered that were selected in the
BG2-network) were analyzed. In the first part, MGP was selected as a connecting region from the BG to thalamic and mesencephalic regions that have connections to AGl. In the second part SNR was selected as a connecting region from the BG to thalamic and mesencephalic regions that have connections to AGl. Interestingly, it was found in the first part of the analysis that PF is the sole region with reciprocal connections to MGP and AGl within the output of the BG2 network in all three principal BG-pathways. There were no reciprocal connections reported from VA, VL, VM and MDL to MGP. However, MGP seems to have a reciprocal connection with VTA, however, VTA is nonreciprocally connected with AGl. In the second part of this reciprocal pattern analysis with regard to SNR it was found that in addition to PF also VM has reciprocal connections to SNR and to AGl in all three principal BG pathways. Again VTA has a reciprocal connection to SNR and the non-reciprocal connection to VTA in this SNR BG-output pathway. Contralateral connectivity The core regions of the basal ganglia show larger connection strengths in ipsi- than in contralateral connections. The AGl and AGm possess more contralateral connections than the core regions of the BG (Fig. 3b). Reciprocal
123
Brain Struct Funct Fig. 2 The weighted connections and color-coded weighted DGAll parameter of the bilateral BG2 network. a The ipsilateral connections of the BG2 network are shown using a 3-axis expansion technique of neuroVIISAS. b A unilateral view with labels of regions. c Bilateral regions in a comparable view like in (a) with labels
connections of the BG core regions occur exclusively ipsilaterally, however, two reciprocal connections of the contralateral AGl with VL and CPu were identified. Ipsiand contralateral connections are visualized in 3D in Fig. 2. Stronger and weaker ipsi- and contralateral connectivity of AGl is clearly visible in the low-resolution BG1-network visualization using the chord diagrams of CIRCOS (Appendix Fig. 12). However, the CIRCOS visualization is not suitable for the high-resolution BG2 network (Appendix Fig. 13). Reliability of connections The observation scores of the BG network are presented in Fig. 3c. The most reliable connection in the BG network is the ipsilateral SNC ! CPu projection (43.6) followed by
123
the ipsilateral SNR ! CPu (24.1) projection. The largest reliability of a contralateral connection is observed for SNC ! CPu (5). Further, large reliabilities exist for ipsilateral efferences of STh to SNR (14) and MGP (10). Global and local connectome analysis of the BG2 network The ipsilateral BG2 network consists of 25 regions and 235 connections and the adjacency matrix has a line density of 39.167 %. The connections of thalamic regions to mesencephalic and diencephalic regions of the adjacency matrix are sparse, a phenomenon that is well presented by the adjacency matrix (Fig. 3d) as well as in the distance matrix that show relatively larger distances from these regions to other regions of the BG network (Appendix Fig. 14). The
Brain Struct Funct
a
b
c
d
Fig. 3 Matrix representations of the BG connectome. a Adjacency matrix of the core regions of the motoric basal ganglia. Numbers and gray values are indicating the number of experiments that describe connections. Source regions are shown in the rows and target regions in the columns. b The bilateral connectivity of the core regions of the motoric basal ganglia. The weighted adjacency matrix displays the mean weights of connections. Weights are ranked as shown in the color table. Altogether, the ipsilateral connectivity is denser than the contralateral. Relative strong connections exist between contralateral cortical regions, only. c The reliability of the weighted connectivity of the core regions of the motoric basal ganglia. The reliability of the weighted bilateral connections is calculated using the number of
experiments that describe connections (if different experiments within a particular publication occur than all experiments are considered) and variability of weights. The maximum is 43.6 for SNC ! CPu and the minimum is 0.1. d The adjacency matrix of the ipsilateral BG2 network with regions that are not primarily involved in motoric processing (BG2). The intrinsic connectivity of thalamic regions (magenta) is very weak. The regions of the midbrain and MRF have abundant connections to most other part of the BG2 network. Especially SNR has strong connections to nearly all thalamic regions in this selection. The amygdalar complex and the CPu have comparable number of output connections. The variability of weights of the bilateral BG2 network is shown in the Appendix Fig. 17
central connectional position of the CPu and the substantial contralateral connections have been visualized using an organic planar graph visualization (Fig. 4d). The parafascicular thalamic region has a conspicuously dense connectivity to other regions. This is confirmed by computing the communicability (Estrada and Hatano 2008) (Appendix Fig. 15). Communicability is a measure for the number of paths from a source to a target region. Here, PF has significantly larger values than other thalamic regions. Furthermore, the LGP has relatively small communicability values. A related matrix is the generalized topological overlap matrix GTOM (Yip and Horvath 2007). It is a
measure of pairwise interconnectedness that is proportional to the number of neighbors that a pair of nodes share in common. Interestingly, VTA and MRF possess a noticeable amount of common neighbors resulting in relative large GTOM values (Appendix Fig. 16). The global analysis of the BG2 network reveals a network architecture close to a small-world network because the small worldness is 1.258. The average cluster coefficient of 0.661 is larger than in the Erdo¨s Re´nyi, Watts– Strogatz or Barabasi–Albert randomizations. The scale-free property (power law distribution of edges) of the BG2 network can be asserted because with 2.2, the deviation D
123
Brain Struct Funct Fig. 4 The three principal pathways of the motoric BG. a Direct pathway. b Indirect pathway. c Hyperdirect pathway. The levels of the regions within the hierarchy (the hierarchical organization of regions is shown in Appendix Fig. 11) are shown at the left side of the graphs. The number of publications of specific parts of a pathway are displayed by the numbers on the connections and the maximal weight by the same colors as shown in Fig. 3b. Thick lines indicate reciprocal connections. d The bilateral BG2 network with weighted edges in a planar graph visualization with an organic edge layout. The CPu is localized around the center of regions of each hemisphere. _L region of the ipsilateral hemisphere, _R region of the contralateral hemisphere
a
b
c
d
from the power function is relatively small. The mean scale-free property as well as mean global values of the rewiring randomization resemble the real BG2 network values more closely, than those of other randomizations (Appendix Fig. 18). The weighted modularity analysis (Hagmann et al. 2008) distinguishes 5 modules (Fig. 5). In module 1 most regions are strongly interconnected with the CPu. Module 2 contains the primary motor cortex and important core regions of the BG (SNR, STh, LGP) as well as specific thalamic regions and MRF. Module 3 and 5 consist of regions that have memory functions and are involved in emotional processing. Module 4 contains MGP and VL (VA has light connections only). Based on the
123
weighted modularity analysis modules 1, 2 and 4 contain regions that are associated with motoric function and modules 3 and 5 those regions involved in memory and emotional processing. Hence, the computed modules partly reflects the functional separation of regions. Some parameters of the local-network analysis of the BG2 network are summarized in Table 6 (a complete list of all local parameters of this analysis is shown in Appendix Table 15). After sorting the regions by DGall , it turns out that AC has most inputs and outputs (42), followed by the CPu (41) and VTA (40). If the analysis is done on the weighted rather than binarized data, the top three regions are CPu, VTA and SNR (Appendix Table 7). The
Brain Struct Funct Table 2 Pathway analysis starting with projections from the AGl to the CPu Path
PL
Rec
AW
Pub
Rel
AGl ! CPu ! SNR ! VM ! AGl
4
1
3
13
13.13
AGl ! CPu ! SNR ! PF ! AGl
4
1
3
12
12.95
AGl ! CPu ! SNR ! VL ! AGl
4
0
3
11.75
12.02
AGl ! CPu ! SNR ! PC ! AGl
4
0
3
11.25
11.73
AGl ! CPu ! SNR ! CM ! AGl
4
0
3
11
11.48
AGl ! CPu ! SNR ! CL ! AGl
4
0
3
10.75
10.98
AGl ! CPu ! SNR ! MDL ! AGl
4
0
2.75
10
9.75
AGl ! CPu ! SNR ! VTA ! AGl
4
0
2.25
9.25
9.75
AGl ! CPu ! SNR ! VA ! AGl
4
0
2.25
9
9.30
AGl ! CPu ! SNC ! CM ! AGl
4
0
3
7.25
7.63
AGl ! CPu ! SNC ! CL ! AGl AGl ! CPu ! MGP ! VL ! AGl
4 4
0 0
3 3
7.5 8.5
7.53 7.50
AGl ! CPu ! SNC ! VTA ! AGl
4
0
2.75
6.75
7.13
AGl ! CPu ! SNC ! AGm ! AGl
4
0
2.38
7
7.03
AGl ! CPu ! SNC ! PC ! AGl
4
0
2.75
7
7.03
AGl ! CPu ! SNC ! PF ! AGl
4
1
2.38
6.75
7.03
AGl ! CPu ! MGP ! PF ! AGl
4
1
2.75
7.75
6.97
AGl ! CPu ! SNC ! VM ! AGl
4
0
2.38
7.75
6.97
AGl ! CPu ! SNC ! MDL ! AGl
4
0
2
6.25
6.32
AGl ! CPu ! Ac ! VTA ! AGl
4
0
2.25
5.5
6.05
AGl ! CPu ! VL ! AGl
3
1
2.67
5.33
5.93
AGl ! CPu ! MGP ! VM ! AGl
4
0
2.5
7
5.85
AGl ! CPu ! STh ! PF ! AGl
4
1
2.25
4.75
5.78
AGl ! CPu ! AGl
2
1
2.5
5
5.55
AGl ! CPu ! MRF ! CL ! AGl
4
0
2
4.75
5.55
AGl ! CPu ! AGm ! VL ! AGl
4
1
2.5
4.75
5.32
AGl ! CPu ! VTA ! Ent ! AGl AGl ! CPu ! VM ! AGl
4 3
0 1
2.38 2.67
5.25 4.67
5.32 5.17
AGl ! CPu ! MRF ! PF ! AGl
4
1
2
4.25
5.07
AGl ! CPu ! MGP ! MDL ! AGl
4
0
2.5
6.25
4.88
AGl ! CPu ! MGP ! VTA ! AGl
4
0
2
5.75
4.88
AGl ! CPu ! AGm ! AGl
3
1
2.33
4.33
4.87
AGl ! CPu ! PC ! AGl
3
1
2.33
4.33
4.87
AGl ! CPu ! PF ! AGl
3
1
2.33
4.33
4.87
AGl ! CPu ! VTA ! AGl
3
0
2.33
5
4.83
AGl ! CPu ! MGP ! VA ! AGl
4
0
2
5.75
4.80
AGl ! CPu ! MRF ! VL ! AGl
4
0
1.25
4.25
4.78
AGl ! CPu ! VTA ! CL ! AGl
4
1
2.25
4.75
4.75
AGl ! CPu ! VTA ! AGm ! AGl
4
1
2.25
5
4.72
AGl ! CPu ! VTA ! VM ! AGl
4
1
2.5
4.75
4.72
AGl ! CPu ! AC ! VL ! AGl
4
0
2.25
4.25
4.70
AGl ! CPu ! STh ! MDL ! AGl AGl ! CPu ! STh ! VTA ! AGl
4 4
0 0
1.75 1.75
4 4.25
4.70 4.61
AGl ! CPu ! MRF ! PC ! AGl
4
0
2
3.75
4.57
AGl ! CPu ! MRF ! VM ! AGl
4
1
1.75
3.75
4.57
AGl ! CPu ! CM ! AGl
3
0
2.33
4
4.53
AGl ! CPu ! AGm ! CL ! AGl
4
1
2.5
4
4.53
AGl ! CPu ! AGm ! VM ! AGl
4
1
2.5
4
4.50
123
Brain Struct Funct Table 2 continued Path
PL
Rec
AW
Pub
Rel
AGl ! CPu ! AGm ! PC ! AGl
4
1
2.25
3.75
4.28
AGl ! CPu ! PF ! AGm ! AGl
4
1
2
3.75
4.28
AGl ! CPu ! CM ! AGm ! AGl AGl ! CPu ! PC ! AGm ! AGl
4 4
1 1
2 2.25
4 4
4.25 4.25
AGl ! CPu ! PF ! VM ! AGl
4
0
2.25
3.75
4.25
AGl ! CPu ! VTA ! CM ! AGl
4
0
2.25
4.75
4.15
AGl ! CPu ! MRF ! CM ! AGl
4
0
2
3.5
4.13
AGl ! CPu ! VTA ! PF ! AGl
4
1
2.5
4.75
4.13
AGl ! CPu ! MDL ! AGm ! AGl
4
0
1.75
4
4.05
AGl ! CPu ! AGm ! CM ! AGl
4
0
2.25
3.5
4.03
AGl ! CPu ! AGm ! PF ! AGl
4
1
2.25
3.5
4.03
AGl ! CPu ! VL ! AGm ! AGl
4
1
2.5
3.5
4.03
AGl ! CPu ! VM ! AGm ! AGl
4
1
2.5
3.75
4.03
AGl ! CPu ! PF ! VTA ! AGl
4
0
2
3.5
4.00
AGl ! CPu ! MDL ! AGl
3
0
1.33
3.67
3.93
AGl ! CPu ! MRF ! AGm ! AGl
4
0
1.5
3.5
3.88
AGl ! CPu ! MRF ! MDL ! AGl
4
0
1.75
3
3.88
AGl ! CPu ! VTA ! MDL ! AGl AGl ! CPu ! AGm ! MDL ! AGl
4 4
1 1
1.75 2.25
4 3.5
3.80 3.78
AGl ! CPu ! AGm ! VTA ! AGl
4
0
2
3.25
3.72
AGl ! CPu ! MRF ! VTA ! AGl
4
0
1.5
3.25
3.72
AGl ! CPu ! AC ! CM ! AGl
4
0
2
3.25
3.71
AGl ! CPu ! AC ! VTA ! AGl
4
0
1.5
3.75
3.61
AGl ! CPu ! VM ! VTA ! AGl
4
0
2.13
3
3.53
AGl ! CPu ! PC ! VTA ! AGl
4
0
1.75
3
3.50
AGl ! CPu ! MDL ! VTA ! AGl
4
0
1.25
3
3.33
PL path length, Rec reciprocal, AW average weight, Pub number of articles, Rel reliability
importance of a region in this network is estimated by the Shapley-value (Keinan et al. 2004; Ko¨tter et al. 2007). The lowest Shapley-value (largest importance) was calculated for CPu (0:309), then the AGl (0:277) and AGm (0:243) also have smaller values than other regions. The betweenness centrality (BC) (Freeman 1977) is a coefficient that divides the number of shortest paths between regions passing a particular region by the number of all shortest paths. It provides a useful measure for the interconnectedness and embedding of a particular region in the network (Vlachos et al. 2012). Again the CPu is most important and has the largest BC of 0.092, followed by the AC (0.074) and the AGm (0.069). The BC has been visualized in 3D in the bilateral BG network by assigning the BC values with a color code to each region (Fig. 6). The same ranking of CPu (1), AC (0.997) and AGm (0.817) holds for the authoritativeness as well (Kleinberg 1999). The local parameters have been visualized in parallel coordinates (Appendix Fig. 19) after sorting by their similarities. The presentation of the high-dimensional parameter space clearly indicates the central role of the CPu in
123
the BG2 network. Also the cortical input areas AGl and AGm are lying close to the CPu-curve in this visualization. A further finding concerns the laterality (LA ) of connections (Appendix Tables 8, 9). The AGm and AGl have a laterality of 0.62 and 0.607 (ipsilateral connections divided by DGall ). A smaller laterality value was found for MRF (0.552). Interestingly, the core regions of the BG all have lateralities larger than 0.7. This phenomenon is also found when considering reciprocal laterality (LR ). Rich-club analysis and knotty-centers A rich club is a set of high-degree nodes (rich nodes) which are connected with each other even more frequently than it would be expected by their high degree. Nodes with numerous links are much more likely to form well interconnected subgraphs (clubs) than low degree nodes. The rich-club phenomenon can be computed by the rich-club coefficient / (Zhou and Mondragn 2004). /ðkÞ measures the fraction of edges actually connecting nodes with degree [ k out of the maximum number of possible connections
Brain Struct Funct
(Colizza et al. 2006). Rich-club analysis has been applied successfully to detect structural linkages of hubs of the cerebral cortex (Shanahan et al. 2013; Harriger et al. 2012; van den Heuvel and Sporns 2011). The diagram of richclub coefficients in dependence on node degrees /ðkÞ of the bilateral BG2 network allows to detect a rich-club
phenomenon of nodes with degrees k 44 (Fig. 7). These nodes have a / 0:8. The feeder nodes are those nodes of the BG2-network that have direct connections to the richclub nodes. This connectivity is shown in Appendix Fig. 20. It visualizes the dense connectivity of rich nodes and a substantial connectivity between rich-club nodes and feeder nodes. CPu, AGl, AGm and SNR are members of the rich club. Interestingly, the non-canonical BG regions AC, VTA and MRF are also rich-club nodes. The knottycentrality (KC) quantifies the connectivity within a topologically central connective core (Shanahan and Wildie 2012). The knotty-center consists of the following regions found in the ipsi- and contralateral hemispheres (Fig. 21): VTA, MRF, CPu, AGl, AGm, AC. The thalamic output regions do not belong to the knotty-set. However, knottyset members MRF and VTA are massively connected with cortical as well as subcortical hub regions. Motif analysis
Fig. 5 The weighted-modularity analysis of the ipsilateral BG2 network. Five modules have more weighted connections within than among the modules. Thick lines indicate large weights
Motif analysis has been applied by cortical and subcortical connectomes in different species to determine such subgraphs that are significantly more abundant or considerably less frequent than in suitable randomizations (Sporns and Ko¨tter 2004; Sporns 2006; Schreiber and Schwo¨bbermeyer 2004, 2005). The motifs of the ipsi- (Fig. 8a) and bilateral (Fig. 8b) BG2 networks show characteristic distribution patterns. The frequency of 13 3-node motifs and 3 complex 5-node motifs was determined. The 5-node motifs consist of 2 reciprocal edges and two non-reciprocal edges which
Fig. 6 The weighted connections and color-coded weighted betweenness (BC) parameter of the bilateral BG2network. The color code of connections weights is the same as in Fig. 3d. The weighted betweenness centrality BC of the regions has been integrated into this 3D-view. The CPu belongs to the class of maximalweighted BC values indicating its connectional importance in the BG2-network AGm and Pir have smaller values. The large amount of contralateral projections is also visible
123
Brain Struct Funct Fig. 7 The rich-club coefficients / of the ipsi and contralateral BG2 network in dependence of the ranks k of the regions. 1,000 rewiring randomizations were applied and mean rich-club coefficients were computed. A secondary yaxis in green shows the coefficient /ðkÞ=rand/ðkÞ
are oriented in 3 ways: chain-like (chain), centripetal (in) to and centrifugal (out) from the center vertex to the vertices with reciprocal edges. The frequency of motifs in the real ipsilateral BG2 network can be compared with the frequencies of motifs in 1,000 rewiring randomizations with the same number of nodes and edges. It turns out that the motifs 3-09 and 3-13 are significantly more abundant in the real BG2 network than in randomizations (Fig. 8a). In the bilateral BG2 network 3-04, 3-06, 3-09, 3-13 motifs are more abundant than in 1,000 rewiring simulations. All these 3-node motifs have at least one reciprocal edge. In contrast to the ipsilateral network, the 5-node motifs are very abundant (Fig. 8b). The contribution of regions to motifs is shown in Table 3 and Appendix Table 10. AC, CPu and VTA are most often elements of the fully reciprocal motif 3-13 (Table 3). The 5-node motifs only appear sporadically. The MGP is most often (326) an element of the centripetal 5-node motif and 333 times an element in the chain 5-node motif. The centrifugal 5-node motif is built by AGl in the majority of cases. In the bilateral BG2 network MRF participates most often in the full reciprocal motif 3-13. Multivariate analysis Modularity analysis provides a classification of regions based on their number of connections in modules and in between modules. The result of a weighted modularity analysis has been described before (Fig. 5). Metric multidimensional scaling (MDS) arranges regions that have similar connectivity patterns closer together than to regions that have more different connections. The result of a MDS in a normalized and quadratic diagram, respectively, is shown in Appendix Fig. 24 for the ipsilateral and in Appendix Fig. 25 for the bilateral BG2 network. Most of the thalamic regions are located within the lower left quadrant whereas LHb and MDM are located in the upper left quadrant. Cortical motor regions, MGP, CPu, SNR and
123
MRF are in the lower right quadrant. SNC and VTA are located close together in the upper right quadrant. Ent, Pir and HIPP which also share functional features are relatively closely arranged in the upper left quadrant. In the bilateral BG2-network cortical motor, SNC, SNR, VTA, CPu and AC are located close together in the lower quadrants whereas MRF appears as an outlier in the lower quadrants. All other regions are located in the upper quadrants. The principal component analysis was applied to the DGAll , CluCAll , CluC2 , AvgDGnb , VCDG , Loc as proposed by (Echtermeyer et al. 2011) (Appendix Table 13, the definitions of these parameters are summarized in the Appendix) to determine those two dimensions that have the largest influence in the bilateral BG2 network (Fig. 9a, b). It turns out that 5 parameters of the first component (first row of Appendix Table 13) have similar absolute values and determine the distribution of regions in the first dimension. The second dimension (second row of Appendix Table 13) is determined by the largest absolute value by CluC2 . This means, that the variance within the y-axis of the PCA-plane is dominated by the variance of CluC2 . The gray values of the PCA-plane diagram (Fig. 9a) indicate the densities of regions within a ring. The lightest gray value in the middle covers most regions. The STh was selected and the connectional relations of STh in the BG2 network are shown in the circular layout in Fig. 9b. In the inner ring 24 directly connected neighbors are located and in the outer ring 25 indirect neighbors are shown. The number of indirect and direct neighbors is well balanced and typical for all regions within the center of the PCA plane. This result is important since the target of deep brain stimulation is the STh. The CPu shows another pattern of direct–indirect-neighbor connectivity. The number of direct neighbors is much larger than the number of indirect neighbors (Fig. 9c). Interestingly, the number of indirect neighbors of VL (smaller first component than within the CPu) is larger than the number of direct neighbors (Fig. 9d).
Brain Struct Funct
a
b Fig. 8 Overview of the distributions of motifs in the ipsi and contralateral BG2 network. a The frequencies of motifs in the ipsilateral BG2 network. 13 directed 3-node motifs and 3 directed 5-node motifs have been searched: syOut (‘‘Out’’ with regard to the central node of the motif), syIn and syCh. The frequency of a motif in the real network is indicated by the blue point. The BG2 network was randomized using a rewiring procedure. Each randomization contains exactly the same number of regions and connections. The rewiring randomization was repeated 1,000 times, after each randomization the frequencies of motifs were determined, indicated by a small black dot. The red bar indicates the variation of frequency within the standard
syOut
syIn
syCh
syOut
syCh
syIn
deviation. The motifs are sorted by their z values. Hence, motifs that have larger frequencies in the real network appear on the right side of the diagram. Most motifs that have reciprocal edges are more abundant in the real network than in the 1,000 rewiring simulations. b The frequency of motifs in the contralateral BG2 network. The complex motifs syOut, syCh and syIn are much more often represented in the contralateral BG2 network than in the ipsilateral case. The differences of frequencies of motifs in the real BG2 network and in rewiring BG2 networks are stronger in the contralateral than in the ipsilateral case
123
Brain Struct Funct Table 3 Frequency of motifs of the bilateral BG2 network Region
3-01
3-02
3-03
3-04
3-05
MRF_L
121
MRF_R
109
27
2
228
26
28
3
220
23
AC_L
32
17
15
184
AC_R
32
16
15
VTA_L
41
28
11
AGm_L
22
36
VTA_R
41
31
AGm_R
23
CPu_R
3-06
3-07
3-08
3-09
3-10
3-11
3-12
3-13
chain
in
out
36
3
115
98
33
20
164
125
7
0
0
42
2
112
103
31
18
157
125
2
0
0
16
56
2
54
134
19
16
154
114
72
130
8
191
16
56
2
53
162
21
16
157
111
68
136
8
166
10
48
0
54
110
25
15
139
95
78
54
25
24
145
16
112
0
25
151
21
26
105
93
188
273
82
13
162
10
51
0
51
104
24
15
131
90
113
60
25
42
28
143
18
120
1
25
140
22
27
102
88
212
254
77
30
29
30
103
33
176
2
44
94
35
41
127
80
135
42
45
CPu_L
31
28
33
103
33
172
2
41
93
37
39
131
80
114
51
58
AGl_L AGl_R
8 8
7 8
5 5
76 73
1 1
135 136
1 1
2 2
198 184
15 15
4 4
80 72
78 76
249 294
104 108
131 173
PF_L
25
24
15
150
9
65
0
31
100
8
7
67
62
145
160
33
PF_R
22
20
15
145
8
64
0
29
102
7
5
63
62
136
184
30
STh_L
20
45
14
144
6
60
3
23
81
18
13
60
60
265
161
61
STh_R
17
42
12
141
6
63
2
21
83
16
11
55
60
243
173
69
SNR_L
77
55
39
142
13
91
5
63
54
33
13
58
56
193
191
63
SNR_R
78
56
39
141
13
90
5
63
53
33
13
58
56
190
204
51
SNC_R
58
31
20
140
29
77
1
55
41
23
25
68
55
168
153
96
SNC_L
58
30
20
141
29
78
1
55
42
23
25
68
55
151
135
107
MGP_R
34
50
23
105
12
67
2
34
62
17
6
52
49
333
326
105
MGP_L
34
48
23
108
12
67
2
34
63
17
6
52
49
329
299
100
Ac_L
32
45
20
140
19
51
2
17
44
28
16
68
33
201
165
75
LHb_R
33
57
14
124
20
67
6
30
48
23
7
60
31
251
155
111
LHb_L
33
57
14
127
20
66
6
30
48
23
7
60
31
242
149
93
Ac_R
35
52
24
137
21
53
2
20
40
28
15
63
30
192
163
87
VM_R VM_L
51 51
30 29
13 13
99 101
26 26
72 73
2 2
31 31
83 83
11 11
6 6
56 56
27 27
149 141
68 68
34 34
CM_R
70
39
17
110
24
66
4
27
88
16
10
49
21
108
54
20
CM_L
67
37
19
104
24
64
4
28
85
21
12
51
21
101
55
19
Ent_L
25
43
29
76
10
79
3
16
67
14
12
51
19
281
158
112
PC_R
52
20
12
77
2
54
0
6
98
7
5
29
14
120
74
51
PC_L
52
20
12
77
2
53
0
6
99
7
5
29
14
103
73
56
HIPP_L
37
58
33
82
24
117
2
23
35
21
22
62
13
258
120
135
HIPP_R
37
53
32
91
27
112
4
21
39
23
22
59
10
277
189
143
Pir_L
56
47
18
88
29
71
3
44
37
22
16
46
10
249
110
100
CL_R
81
43
18
75
8
53
3
19
59
21
9
30
8
96
51
63
CL_L
82
44
18
75
8
52
3
19
61
21
9
30
8
91
48
67
MDL_R
76
37
8
103
18
24
0
29
62
4
2
35
7
84
145
39
MDL_L
76
38
8
103
18
23
0
29
63
4
2
35
7
71
136
39
Pir_R
59
55
21
85
32
71
4
42
37
24
13
44
6
250
91
96
LGP_L LGP_R
46 44
30 31
7 7
89 87
6 6
24 24
2 2
12 12
19 19
8 8
3 3
25 25
6 6
194 183
193 204
29 26
VL_R
105
28
23
113
22
49
0
46
45
7
8
26
6
62
191
30
VL_L
105
29
23
113
22
48
0
46
46
7
8
26
6
59
181
35
MDM_L
77
19
2
108
6
18
0
20
31
1
8
19
5
159
140
48
MDM_R
78
20
2
105
6
12
0
20
32
1
7
15
2
91
116
17
123
Brain Struct Funct Table 3 continued Region
3-01
3-02
3-03
3-04
3-05
3-06
3-07
3-08
3-09
3-10
3-11
3-12
3-13
chain
in
out
Ent_R
4
19
4
28
0
13
0
0
28
1
0
2
1
57
63
34
VA_L
30
22
17
58
1
36
1
2
0
6
1
0
0
0
67
0
VA_R
30
22
17
58
1
35
1
2
0
6
1
0
0
0
70
0
Regions are sorted using motif 3-13 frequencies chain symmetricCHAIN, in symmetricIN, out symmetricOUT (see legend of Fig. 8)
a
b
c
d
Fig. 9 Principal component analysis of the ipsi and contralateral BG network. a PCA plane of the ipsi- and contralateral BG2 network. The regions of the BG2 network are distributed in the PCA-plane (defined by the first (abscissa) and second component (ordinate) of the PCA) as strongly overlapping pairs. The gray values within a ring present the density of regions in the ring. Most regions are located in the innermost ring. This presentation of regions in the PCA plane need to be considered together with the values of the local network parameters (Appendix Table 13) and their contribution to the first (abscissa) and second component (ordinate). b STh was selected from the bilateral BG2 network. The direct (first) neighbors of STh are located on a ring around STh. The connections of these regions are
dark blue. The indirect (second) neighbors of STh are on the external ring. Connections between second neighbors are light blue and connections between second and first neighbors are yellow. The number of direct and indirect neighbors is similar as well as the number of connections inside the sets of neighbors. c CPu was selected from the bilateral BG2 network. The number of first neighbors of the CPu is much larger than the number of second neighbors. Also the density of connections between second neighbors is much more different from the knotty-center presentation of STh (Appendix Fig. 21). d VL was selected from the bilateral BG2 network. The number of indirect neighbors is much larger than the number of direct neighbors of the VL
123
Brain Struct Funct Table 4 Vulnerability of regions in the ipsilateral BG2 network Region Caudate putamen
Vulnerability 2.2792
Amygdaloid complex
2.1273
Ventral tegmental area A10
1.7475
Mesencephalic reticular formation
1.5196
Medial agranular prefrontal cortex
1.4057
Substantia nigra compact part
0.8360
Lateral agranular prefrontal cortex
0.7600
Substantia nigra reticular part
0.7220
Subthalamic nucleus
0.6461
Medial globus pallidus
0.6081
Accumbens nucleus Parafascicular thalamic nucleus
0.3802 0.2663
Lateral habenular nucleus
-0.3034
Piriform cortex
-0.3414
Entorhinal cortex
-0.4173
Ventromedial thalamic nucleus
-0.5313
Hippocampus
-0.6072
Central medial thalamic nucleus
-0.6452
Mediodorsal thalamic nucleus lateral part
-0.9870
Centrolateral thalamic nucleus
-1.1010
Lateral globus pallidus
-1.1010
Ventrolateral thalamic nucleus
-1.2529
Paracentral thalamic nucleus
-1.3289
Mediodorsal thalamic nucleus medial part
-1.3668
Ventro anterior thalamic nucleus
-2.5062
Differential connectomics: vulnerability Vulnerability analysis provides an estimate of the effect of removing a region from a network. If the CPu is removed, then the mean distance between all resting regions increases by 2.279 % and if AC is removed then the mean distance increases by 2.127 %. A sorted list of vulnerabilities of regions is shown in Table 4. The number of connections or the degree of regions is positively correlated with most local network measures especially the centrality parameters (Appendix Table 12). The number of connections of the ipsilateral BG2 network was visualized using the K-core analysis where regions are sorted with regard to their number of connections and those regions which have the smallest number of connections are removed until the regions with the largest number of connections remain in the center of the K-core visualization (Appendix Fig. 22). This has been performed for the ipsilateral BG2 and the bilateral BG2 network (Appendix Fig. 23). It turns out that CPu, AGl, AGm, VTA, MRF and PF are the core regions of the BG2 network. The MDS of the bilateral BG2 network is relatively symmetric. In this case, all thalamic regions of the ipsilateral hemisphere are
123
arranged in the upper left quadrant. Ent, Pir and HIPP are in between these thalamic regions. SNR, SNC, CPu, AGl and AGm are located in the lower left and lower right quadrant, respectively, for the contralateral regions. If MDS is performed on the weighted connectivity matrix, then the regions are moving closer together in direction of the midline of the MDS diagram (not shown). Connectomics of an embedded BG-network An embedding of the BG1 network into a network of regions which have direct input and output connections to the BG1 regions was performed to determine the importance and ranks of BG1 regions in the context of extrinsic direct and indirect neighbors. The results of the analysis are shown in Appendix Table 14. Changes of ranks of regions were found when comparing the BG1 network with the network of all regions that are directly connected to the BG1 network. In terms of BC, the CPu, AGl and STh are covering the first three ranks. In the direct neighbor network (DNN) the AGl, AGm and LGP have ranks 1, 2 and 3. In the indirect neighbor network (INN) CPu, SNC and AGl have ranks 1, 2 and 3. With regard to EC, the SNC, CPu and MGP are on the first three ranks in the BG1 network, in the DNN and INN these are SNC, SNR and MGP. Hence, with regard to EC there is not so much change in the ranks as compared with the BC parameter. Interestingly, the ranking of regions by means of PRC, SC, Hub, Aut, RADout and RADin in the BG1 network, DNN and INN is similar: CPu, AGl, STh, MGP, SNC or SNR are covering the first three ranks. A further interesting finding is that VL is considered as an important output region of the BG, however, the region rankings of the embedded BG1-network indicate variable ranks of 4 or 7, only. Furthermore, the CPu changes its rank in the DNN when considering the RADin parameter from 1 in the BG1-network to 7. Global extrinsic connectivity of the BG2 network The global extrinsic connectivity of the BG2 network has been determined ipsilaterally (Table 5) and bilaterally (Appendix Table 11). Extrinsic connections are those where at least the source or target is not a member of the set of regions of BG2. The CL has most direct ipsi- and contralateral inputs (284) in the BG2 network. The AGl has most direct outputs followed by MRF and AGm. However, AGm has the most ipsilateral outputs (516). When considering all connections of subregions of a region of BG2, then the MRF has most inputs (78,691) followed by AC (5,424) and CPu (1,536). The ranks of regions changes when regarding their subtree outputs: AC is 4,903, for MRF it is 1,893 and for AGl it is 1,375. In the bilateral
Brain Struct Funct Table 5 Global extrinsic connections of regions of the ipsilateral BG2 network Region
Dic
Dii
Dis
Doc
Doi
Dos
Sic
Sii
Sis
Soc
Soi
Sos
Centrolateral_thalamic_nucleus_L
398
446
844
10
68
78
413
470
883
10
103
Accumbens_nucleus_L
277
378
655
5
74
79
373
829
1,202
17
304
321
Central_medial_thalamic_nucleus_L
278
322
600
18
70
88
288
340
628
18
184
202
Paracentral_thalamic_nucleus_L
242
309
551
10
81
91
245
413
658
10
89
99
Mesencephalic_reticular_formation_L
226
311
537
235
334
569
3,418
4,273
7,691
1,383
3,510
4,893
Amygdaloid_complex_L
148
375
523
15
71
86
951
4,473
5,424
933
3,970
4,903
Ventrolateral_thalamic_nucleus_L
154
344
498
12
52
64
175
384
559
13
56
69
82
311
393
31
79
110
479
1,057
1,536
35
172
207
Parafascicular_thalamic_nucleus_L
124
213
337
42
136
178
160
377
537
127
291
418
Ventromedial_thalamic_nucleus_L
91
244
335
37
54
91
128
356
484
44
161
205
Ventral_tegmental_area_A10_L Medial_agranular_prefrontal_cortex_L
78 70
238 131
316 201
138 130
231 416
369 546
127 112
476 212
603 324
188 203
440 584
628 787
Hippocampus_L
37
137
174
27
46
73
263
1,030
1,293
228
631
859
Entorhinal_cortex_L
35
137
172
36
63
99
166
900
1,066
80
546
626
Lateral_habenular_nucleus_L
30
128
158
44
99
143
70
226
296
52
129
181
Subthalamic_nucleus_L
25
124
149
31
83
114
38
181
219
42
127
169
Substantia_nigra_compact_part_L
32
101
133
48
126
174
47
136
183
58
181
239
Lateral_agranular_prefrontal_cortex_L
30
95
125
284
381
665
89
436
525
574
801
1,375
Piriform_cortex_L
17
103
120
27
53
80
86
300
386
73
191
264
Mediodorsal_thalamic_nucleus_medial_part_L
15
91
106
1
14
15
15
91
106
1
14
15
Mediodorsal_thalamic_nucleus_lateral_part_L
22
82
104
2
18
20
22
83
105
2
18
20
Substantia_nigra_reticular_part_L
20
73
93
50
133
183
20
86
106
74
200
274 91
Caudate_putamen_L
113
Medial_globus_pallidus_L
15
53
68
17
59
76
16
58
74
18
73
Lateral_globus_pallidus_L
10
22
32
0
4
4
10
22
32
0
5
5
Ventro_anterior_thalamic_nucleus_L
12
12
24
0
10
10
15
14
29
0
10
10
The regions were sorted by the direct input sum (Dis) Dic direct input from contralateral, Dii direct input from ipsilateral, Dis direct input from ipsi- and contralateral, Doc direct output to contralateral, Doi direct output to ipsilateral, Dos direct output to ipsi- and contralateral, Sic subtree input from contralateral, Sii subtree input from ipsilateral, Sis subtree input from ipsi- and contralateral, Soc subtree output to contralateral, Soi subtree output to ipsilateral, Sos subtree output to ipsi- and contralateral
BG2 network (Appendix Table 11) CL has most inputs (838) followed by Ac (645) and CM (587). The AGl has most outputs (640) followed by MRF (561) and AGm (524). Most subtree (for details see ‘‘Materials and methods’’) inputs are going to MRF (7,589), AC (5,291) and HIPP (1,358). The output from subtrees is dominated by AC (4,821) followed by MRF (4,700) and AGl (1,307). Local extrinsic connectivity of the BG2 network In this section extrinsic regions are determined that have abundant connections to the intrinsic regions of the BG2 network. Furthermore, extrinsic regions were determined that receive many connections from intrinsic regions. To identify these extrinsic regions which are densely interconnected with intrinsic regions, a search through the whole connectome was performed to filter out those regions which are located around hierarchical level 9 and
possess direct ipsilateral inputs or outputs to the intrinsic regions. Those regions that have no input or no output were removed from this selection. After this conditional expansion of the network, the interactive hierarchical table method of neuroVIISAS was used to determine extrinsic regions that have abundant inputs to or get outputs from intrinsic regions. It turns out that the medulla oblongata provides most inputs to the CPu, followed by the lateral thalamic nuclear group, medial geniculate nucleus, ventral thalamus and intralaminar nuclei of the thalamus. Most outputs of the CPu are going to intrinsic nuclei. However, the lateral thalamic group and the substantia innominata nucleus basalis complex receive a few outputs of the CPu. Abundant inputs to the SNC derive from the medulla oblongata, tuberomammillary nucleus and the pedunculopontine tegmental nucleus. Most external outputs from SNC are going to the ventral thalamus. The medial global pallidus does not have many extrinsic outputs nor does it
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Brain Struct Funct
receive many extrinsic inputs. The subthalamic nucleus receives abundant extrinsic inputs from the pedunculopontine tegmental nucleus and the ventral thalamus. The sole outputs of MGP are going to the olfactory tubercle. The accumbens nucleus receives many inputs from mesocortical regions, the medulla oblongata as well as from hypothalamic zones. Most extrinsic outputs are going to hypothalamic zones, the olfactory tubercle and the medulla oblongata. The VL receives abundant connections from the cerebellum and hypothalamic zone. VL projects extrinsically to the medulla oblongata and mesocortical regions.
Discussion The two levels (BG1, BG2) of the mesoscale connectome of the BG were investigated with regard to intrinsic, extrinsic, weighted, ipsi- and contralateral configurations. Connectome embedding and differential connectomics were applied to determine the importance of regions (ranking) in terms of their connectivity in the network. The routing problem and reciprocity patterns The three principal pathways (direct, indirect, hyperdirect) were revealed in the BG connectome. By applying pathway analysis, many further possible pathways from the AGl over the CPu back to AGl were detected in the ipsilateral BG2 network. So far, alternative pathways in the orofacial system (Mascaro et al. 2005) and the pedunculopontine nucleus of the interface between the BG and brainstem nuclei (Martinez-Gonzalez et al. 2013) have been described. Which pathway or set of pathways are used predominantly in terms of biosignal processing and function cannot be worked out by network analytical methods. Maybe this routing problem can be resolved by considering the coding features of spike patterns (intrinsically bursting, regularspiking, fast-spiking) or inhibitory and excitatory functions of local circuits (Tiesinga et al. 2008; Steriade 2004). In an ongoing meta-study, the transsynaptic pathways after virus (especially pseudorabies) injections are being collated and will be compared with the pathway results of monosynaptic tracing studies. Certain reciprocal and non-reciprocal patterns between MGP and SNR to thalamic and further on to the AGl were detected. Continuous reciprocal links of VM and PF between BG output nuclei and AGl were determined. However, VTA is reciprocally connected with SNR and MGP but non-reciprocally with AGl. Rat connectome studies There are not so many publications available which report graph theoretic analyzes of partial connectomes of the rat
123
nervous system. A major connectional analysis of some parts of the rat central nervous system was made (Burns 1997) and a sophisticated web-page (BAMS2 ) for connectivity retrieval of the rat CNS is available (http://bran cusi.usc.edu). Connectome studies focusing on specific parts or functional systems of the rat CNS were performed in terms of the nucleus of the solitary tract (Palombi et al. 2006), the hippocampus (Burns and Young 2000), the thalamocortical connectivity (Costa and Sporns 2006), the brainstem reticular formation (Humphries et al. 2006) and the retrosplenial cortex (Sugar et al. 2011). Moreover, there exist an increasing amount of meta-studies of connectomes of many other organisms from Caenorhabditis elegans (Towlson et al. 2013) up to primates (Ko¨tter 2004; Stephan et al. 2001a; Passingham et al. 2002). The unique feature of the rat project within neuroVIISAS is (1) the preservation of hierarchical terminologies in the form of an advanced neuroontology (Bota and Swanson 2007a), (2) the explicit collation of ipsi- and contralateral connections, (3) the extraction of collateral projections, (4) transneuronal and transsynaptic connectivity, (5) the multimodal integration of digital atlases, (6) and the integration of populationbased simulations using NEST (Gewaltig and Diesmann 2007). Moreover, the rat connectome project encompasses about 90 % of the available tract-tracing literature of the rat nervous system indexed in PubMed (http://www.ncbi. nlm.nih.gov/pubmed). A list of all reports is available on the webpage http://neuroviisas.med.uni-rostock.de/refer ences.html. In the future, the databases Web of Science, Scopus and Google Scholar will be queried for searching those reports that are not indexed in PubMed. Nevertheless, it is unknown how many connections at the scale of nuclei exist (ground truth of connections) and how dynamic changes of connectivity strengths influences variability of the estimates of connectivity strengths. In particular, the question of completeness becomes evident with regard to contralateral connectivity because in most tract-tracing studies contralateral connections are not systematically investigated. This means that especially contralateral connectivity should be considered as underestimated. In the macaque Barbas et al. (2005) provide evidence that ipsiand contralateral connectivity depend on topological proximity and the structural type of linked areas. Furthermore, they found that projections of both hemispheres are highly correlated with the structural architecture of cortical areas and that the contralateral connectivity mirrors ipsilateral connectivity while being weaker. This suggests that contralateral connectivity brings out a specific laterality pattern which should be investigated further. Hence, the successive accumulation of contralateral connections in the rat connectome project should elucidate the many aspects of contralateral connectivity at the cortical and subcortical level of the central nervous system.
Brain Struct Funct
Reliability of connections
Network granularity
It should be emphasized that some connections and regions may receive more attention in tract-tracing research so that frequently investigated or popular regions may have higher observation scores. This means, that the observation score introduced here should be taken with caution. There could be different reasons like disease (Parkinson syndrome, Huntington-disease and so forth), function (whisker cortex) or accessibility (size of source and target regions in tract tracing experiments, e.g., CPu) why connections or regions are investigated more often than others. Furthermore, most experimental tract-tracing studies typically have been hypothesis driven and restricted to the analysis of one or a few brain regions. The number of tract tracing studies that aim to map connections of a larger functional or structural system of the CNS is rather limited. However, the mouse connectome project at the Allen Institute (http://www. mouseconnectome.org) performs a high-throughput mapping of the interconnectivity of the whole mouse brain based on the tract-tracing technique (Oh et al. 2014). Nevertheless it is useful to have the reliability score available, so that one can filter the connectivity matrix to only contain results that have been observed multiple times. It is reasonable to consider specific reliability weights of connections between two regions in a particular report and in independent reports. In such a way, it could be possible to judge inherent observer biases of connections. The analysis of observation scores shows large values for most connections of the BG network data. Especially, the ipsilateral connections have larger reliability values than the contralateral connections. Obviously, the ipsilateral connectivity of the bilateral BG1 network is stronger than the contralateral connectivity. Furthermore, the laterality of cortical regions is larger than those of the subcortical regions. In the chord diagram of the BG1 network the strong connectivity of SNC to CPu attracts attention. However, in the vulnerability analysis the removal of the SNC, comparable with its neuro-degenerative loss of dopaminergic neurons in the SNC in Parkinson disease, shows an unexpected 6th position in the vulnerability ranking, only. This is in agreement with the modularity role of the SNC, indicating that the SNC is not a complex integrator or hub-region (hub = 0.672) (Sporns et al. 2007). This is also confirmed by the relatively large Shapley-value of 0.008 in the bilateral BG2 network (Small or negative Shapley-values indicate an increase of importance of nodes in a network. Because the Shapley-value is a sum of the differences of connected components, negative values occur. The formal definition can be found in the Appendix.).
The BG2 network contains functionally important limbic regions and specific thalamic, mesencephalic and brainstem regions. In particular, the brainstem and mesencephalic regions show a very dense connectivity to most BG2 regions. However, diencephalic regions of the thalamic nuclei have dense connections to cortical areas and specific subcortical regions of the BG. The larger granularity of the BG2 network emphasizes the impressive contralateral connectivity as shown in the planar network diagram (Fig. 4). This means that the laterality should be considered in dependence of the network granularity. The global connectivity of the BG2 network indicates that it has a small world (Sporns 2006; Bassett and Bullmore 2006) and scale-free structure (Stam and Reijneveld 2007). However, it remains to be seen whether the conclusion will hold when the complete rat connectome is considered (LimaMendez and van Helden 2009). Connectome visualization and thalamic regions The BG2 network was also visualized in 3D-stereotaxic coordinate space (Paxinos and Watson 2007) in neuroVIISAS (Fig. 1) in combination with local-network parameters and weighted connections (Fig. 2). Before computing the surfaces of regions, the contours of all regions of the stereotaxic atlas were manually edited and related to the ontology (Schmitt and Eipert 2012). The surfaces were determined by the marching-cube algorithm of VTK (http://www.vtk.org/) in neuroVIISAS. In the BG2network the CPu has a strong ipsilateral connectivity whereas cortical regions can be distinguished by many contralateral connections. The thalamic PF region is a particular region which has many direct (adjacency matrix) and indirect (communicability matrix) connections to most other regions of the BG1- and BG2-network. In contrast, VA appears to be a region with marginal connectivity only. Modularity analysis and MDS The modular grouping that was found by modularity analysis and MDS indicates a relation of the connectional architecture with functional entities. The regions of the ipsilateral and contralateral network were also grouped by applying metric multidimensional scaling (MDS). Those regions that have similar connections are located close together in the MDS-diagram (Appendix Fig. 24) and those which have different connections are located far away from each other. In the ipsilateral BG2 network nearly all thalamic regions are located in one quadrant of the MDS-
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Brain Struct Funct
diagram and the motor cortical regions, CPu, MGP and SNR are located together with MRF in another quadrant. A trend towards such a thalamic—basal ganglia—grouping was also found in the bilateral BG2 network (Appendix Fig. 25). Local parameters of the basal ganglia connectome Most local parameters of the ipsilateral BG2 network show relatively high correlations with the DGAll parameter. Especially the centrality parameters build a group with similar numeric trends in the parallel coordinate visualization (Appendix Fig. 19). Comparable patterns of correlations of local network parameters were also described by Bounova and de Weck (2012). However, not for each region does the correlation of DGAll (Appendix Table 12) and another parameter like the Shapley-value and EC-value show comparable trends. The CPu has rank 2 with regard to DGAll , however, for EC it has rank 3 (0.92) and for the Shapley-value it has rank 1 (0:31) (Table 6). AGl has a Shapley-value of 0:28 and rank 2, whereas, the EC-value is 0.685 and has a rank of 11, only. The DGAll of AGl is 31 which equals rank 8. AGm shows comparable values: its Table 6 Local parameters of the ipsilateral BG2 network
The regions are sorted using the degree all. Abbreviations and parameter definitions are described in the Appendix. A complete list of all local parameters is shown in the Appendix Table 15 A all, DG degree, I in, O out
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Shapley-value is 0:24 (rank 3) and the EC-value is 0.693 (rank 10), however, its DGAll is 34 with rank 5, only. This means that the total number of connections of a particular region does not fully determine its importance. Furthermore, connectional aspects like connections to other important regions (authoritativeness) and local clustering influence its importance on the connectional level. The weighted-ipsilateral BG2 network provides other ranks, yet the CPu still belongs to the group of regions with highest ranks. The VTA and SNR have ranks 2 and 3 with regard to DGAll . The CPu and VTA stay on ranks 1 and 2 when ECvalues are considered. However, SNR has only rank 7 and MRF gets up to rank 3. The Shapley-value cannot be computed for weighted networks. Hence, some numeric similarities can be observed when connectional roles of particular regions are considered in a non-weighted and weighted BG2 network. A complex region at midbrain and brainstem levels like the MRF with many large weights may strongly change the ranks of importance of regions in a particular functional system like the BG. This means important regions of an isolated functional system or subnetwork like the BG2 that is embedded in a significantly more complex network may change their significances of
Region
REC
DGA
DGO
DGI
BC
Shapley
Hub
Aut
AC
20
42
22
20
0.0736
-0.16
0.96
1
CPu
18
41
20
21
0.0923
-0.31
0.89
1
VTA
17
40
23
17
0.051
-0.1
1
0.89
MRF
15
38
23
15
0.0287
-0.01
1
0.87
AGm
15
34
17
17
0.0694
-0.24
0.72
0.82
SNC
11
32
21
11
0.0177
0.03
0.93
0.67
SNR
11
32
19
13
0.0279
-0.04
0.81
0.77
AGl
14
31
15
16
0.0487
-0.28
0.67
0.76
MGP
11
30
17
13
0.0239
-0.02
0.74
0.77
STh
12
30
17
13
0.0175
0.02
0.78
0.78
PF
11
28
17
11
0.0073
0.06
0.78
0.73
Ac
10
28
13
15
0.0214
0.01
0.58
0.82
LHb
8
22
11
11
0.0057
0.07
0.51
0.71
Pir
5
22
8
14
0.0105
0.07
0.36
0.8
VM Ent
7 8
21 21
9 9
12 12
0.0034 0.0086
0.11 0.05
0.45 0.42
0.77 0.65
CM
5
20
10
10
0.0042
0.12
0.48
0.66
HIPP
6
20
11
9
0.0048
0.11
0.52
0.55
MDL
4
17
4
13
0.0025
0.19
0.2
0.8
CL
4
16
8
8
0.0017
0.14
0.38
0.55
LGP
4
16
5
11
0.0017
0.16
0.23
0.69
PC
5
14
6
8
0.0009
0.15
0.31
0.56
MDM
4
14
4
10
0.0013
0.2
0.19
0.59
VL
3
14
3
11
0.0008
0.25
0.15
0.7
VA
0
5
2
3
0.0001
0.42
0.1
0.18
Brain Struct Funct
regions heavily. This was proved by embedding the lesscomplex BG1 network (contains all BG-core regions) into (1) the extrinsic direct neighborhood network and (2) into the indirect neighborhood network. It was found that some regions-ranks of the direct and indirect networks are constant in terms of EC whereby SNC has rank 1 in this comparison. For the parameters PRC, SC, Hub, Aut, RADout and RADin region-ranks change a little, however, the most important regions stay in the upper ranks. The low ranking of VL does not reflect its significance with regard to its BG output role. This finding is in line with the high variability of regions which have direct input to the AGl in the pathway analysis and supports the hypothesis that the projections from BG-core regions are relatively divergent to different thalamic nuclei. In the non-weighted bilateral BG2 network the MRF, AGm, AGl and VL have the largest DGAll values. However, the Shapley-value of the CPu is conspicuously small (0:37) as well as the value for the AGl (0:37), followed by the AC (0:35). Again, another context like the bilateral selection of BG2 regions changes the importance and ranks of particular regions. With this direct confirmation of the change of importance of brain regions in different contexts of networks it is emphasized that connectomeanalysis depends strongly on the selection of intrinsic regions and extrinsic regions of particular functional subsystems of nervous systems. Motif-analysis A further local feature is the frequency by which regions contribute to particular motifs (Fig. 8a, b; Table 3 and Appendix Table 10). This was investigated in the ipsilateral and contralateral BG2 network. The most impressing result of this analysis was the significantly higher frequency of some motifs (3-12, 3-13) with reciprocal edges. A significantly higher number of complex 5-node motifs was found in the BG2 networks. Those regions, that have high degrees are more frequently found to build reciprocal motifs such as AC, CPu, VTA and MRF in the ipsilateral BG2 network and MRF, AC, VTA and AGm in the bilateral BG2 network. The frequencies in Table 3 of the bilateral BG2-network were sorted with regard to the fully reciprocal motif 3-13. However, regions with high abundance in motifs 3-12 and 3-13 do not have high abundance in the complex 5-node motifs. It is striking that regions with many contralateral connections like AGl and AGm contribute significantly more often to the symmetricINmotif. Surprisingly, the frequency of the MGP-contribution to the symmetricIN-motif is maximal which means that MGP is most often involved in building the symmetricINmotif.
Principal component analysis Principal component analysis (PCA) was performed to determine significant parameters out of a set of 6 local parameters that reflect local connectivity (Echtermeyer et al. 2011). In the PCA-plane the thalamic regions have a relatively small component 1 and a larger component 2, while limbic regions tend to be localized in the center of the PCA-plane and BG-core regions appear to have a relatively larger component 1 (Fig. 9). Again, this distribution of regions can be interpreted as a grouping with functional meaning. Within the PCA-analysis it was found that the CPu has a very large amount of direct connections to most of the BG2 regions (Fig. 9c). Knotty centers and rich-club analysis The knotty-center in the bilateral BG2 network contains the cortical regions AGl and AGm, followed by the CPu and AC as well as MRF and VTA (Appendix Fig. 21). By comparing the knotty-center with the rich-club regions (Appendix Fig. 20) of the bilateral BG2 network, the SNR is the sole region which is a rich-club member, however, not in the knotty-center. Some of these regions were also found in the K-core analysis. This analysis has assigned in the ipsilateral BG2 network the CPu, STh, MGP, AC, PF, MRF, VTA, SNR and SNC to the inner core and in the bilateral BG2 network AGm was assigned to the interior core, however, AGl, SNR, SNC and STh to the first shell. Partly, these groupings provide comparable results for CPu, AC, MRF and VTA, however, some differences for SNC, SNR, AGl, AGm, MGP and STh were found as well. Representation and interpretation of ipsi- and contralateral connections So far, no evidence has been reported of an asymmetry of structural connectivity in the rat brain. However, Chida and Toyosawa (1994) reported a functional asymmetry of the rat hippocampus using small animal EEG. Only, a few studies exist that describe asymmetries of olfactoric processing (Riddle and Purves 1995) and the volume of whisker representing fields of the primary somatosensory cortex in rats (Parthasarathy and Bhalla 2013). In the peripheral nervous system of the rat a significant asymmetry of connections exists, especially for the innervation of impar organs of the abdomen. To allow the representation of peripheral and central connections for extensive connectivity analysis of the complete nervous system of an organism it is mandatory to realize side-dependent connectivity mapping. Finally, the well-known structural as well as functional asymmetry of connections of the human
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Brain Struct Funct
CNS support the concept of a generic framework allowing left- and right-side representations of connectivity. Organization of connections in a hierarchy of regions A further important challenge of the approach described before by Schmitt and Eipert (2012) is the hierarchical organization of subdivisions of regions in the rat connectivity project in neuroVIISAS. The concept of hierarchy is based on spatial scales and semantic relations. The concept is extended by defining variants of the hierarchy to map connections in terms of functional subdivisions. This approach allows the consistent representation of connections of very fine details such as terminal fields as well as connections of functional subsystems like, e.g., the sensory and motoric barrel cortices. Such a scale-based organization of connectivity data has several advantages with regard to multiscale connectivity analysis and connectome based population simulations. By controlling the granularity of a connectome through spatial scales it is also possible to filter and compare connectomes at coarse scales where most connectivity data have larger reliability values. This scaling-technique can be used in combination with functional and chemoarchitectonic data (data are not presented here) which were collated along with the connection data. Such an exceptional database of experimental raw data could be the basis for more realistic large scale simulations of the rat physiom [http://neuroviisas.med.unirostock.de/ratphysiome.html (Bai et al. 2006)]. One important perspective is to use these tract tracing based connectome data of the BG and to combine it with functional attributes at the microcircuit level (Diesmann et al. 1999) to define more realistic and population-based models for simulation studies (Gewaltig and Diesmann 2007). As shown before by (Schmitt and Eipert 2012) this step can be performed also in neuroVIISAS using a hybrid neuron model (Thibeault and Srinivasa 2013) in the NEST simulation engine (Gewaltig and Diesmann 2007).
connectivity reflects, at last partly, functionality. We have worked out a problem by pathway analysis termed the routing problem because multiple pathways from a particular source to a target were found in the data. So far, it had not been possible by network analytical methods to determine a pathway between a source and a target node that is the principal route or a restricted set of functional routes in vivo. Now a comprehensive data set is available and can be shared by the neuroscience community to be used for simulation studies in computational neuroscience as well as in connectomics and neuroinformatics. Acknowledgments The authors thank Klaus-Peter Schmitz (Department of Biomedical Engineering, University of Rostock) for the support of the neuroVIISAS project. We thank Frauke Winzer, Susanne Lehmann, Antje Schu¨mann, Jennifer Meinhardt, Ann-Christin Klu¨nker for their faithful work on extending the database and mappings. All work was supported by the Faculty of Mathematics and Natural Sciences and of the Faculty of Medicine of the University of Rostock.
Appendix A The appendix contains in the first part figures, in the second part tables and the third part formal definitions of matrices, graph-theoretical parameters and randomization models. See Figs. 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26.
Summary The connections of the basal ganglia and important regions directly connected with the basal ganglia were collated in a standardized and systematic way from all available articles which have documented results of tract-tracing experiments in normal juvenile and adult rats. The connectome was analyzed with regard to laterality in a ipsilateral and contralateral way as well as with regard to connection weights in a weighted and non-weighted configuration. Functional groups like thalamic, limbic, basal ganglia core regions and a motor associated region can be deduced from the connectional patterns by multivariate methods. Hence,
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Fig. 10 Ordinal categories of connection strengths are plotted on the abscissa and the hypothetical estimates of connection densities shown on the left ordinate which scales the blue straight line. The right ordinate is linearly scaled for the red graph. The largest slope lies between the semiquantitative values 3.5 and 4. This logarithmic transformation has been used for adapting the ordinal categories to a distribution of connection densities that was observed by others as well [Hilgetag and Grant (2000); Markov et al. (2011, 2014); Oh et al. (2014); Ercsey-Ravasz et al. (2013); Wang et al. (2012)]. The function for transforming ordinal categories of connection strengths to the logarithmic distributed connection densities is f ðxÞ ¼ elnð10Þx4lnð10Þ
Brain Struct Funct
Fig. 11 The regions of the BG2 network are organized in a hierarchy. It is presented as a triangle hierarchy (top) and a region hierarchy (left). The sequence of regions of the hierarchy (from left to right) is
the same as that in the adjacency matrix. All regions of the connectomes that were investigated are organized in a hierarchy
Appendix B
disjunctive region that contains neuron perikarya (sources of physiological action potential) and/or axonal terminals (targets of physiological action potentials). Set of indexed nodes: The set of all indices of nodes is
Tables 7, 8, 9, 10, 11, 12, 13, 14, 15.
Appendix C: Formal definitions of parameters, matrices and concepts The definitions of expressions, parameters, matrices and simulation models (random graph models) used in this article are summarized in the following. More detailed descriptions, algorithms and proofs are provided elsewhere (Newman 2010; Rubinov and Sporns 2010; Newman et al. 2006; Ko¨tter 2007; Brandes and Erlebach 2005; Ko¨tter 2003). Basic definitions Node, vertex: The smallest subunit of a network. With regard to connectomes a node is a circumscribed or
N ¼ f1; 2; 3; . . .; ng
ð2Þ
Number of nodes: The number of nodes (regions, vertices) is n ¼ jNj
ð3Þ
Edge: A directed edge ði; jÞ 2 N N is the line that connects vertices i and j with source i and target j. The set of directed edges E is E NN
ð4Þ
Edges: The number of edges (connections, links) is ¼ jEj
ð5Þ
Set of edges:
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Brain Struct Funct
Fig. 12 Chord diagram visualization of the bilateral BG1 network using CIRCOS. The diagram consists of 4 nested rings. The outer ring of arcs indicates the relative frequency of inputs and outputs to a particular region. The second ring is the input arc ring and the third ring is the output ring of arcs. The inner ring indicates the absolute
numbers of connections followed by a thin layer of arcs with an input part and output part for each region. The extensive connectivity of the CPu with SNC and SNR is visualized prominently. Stronger contralateral connections can be easily found by the thicker red connections between the CPu and contralateral AGl
L ¼ fði; jÞ 2 Eji 6¼ jg
Weighted matrix: The weighted matrix W is
ð6Þ
W ¼ ðwij Þni;j¼1
The set of all not self-referencing edges is ‘ ¼ jLj
ð7Þ
Graph: G ¼ ðN; EÞ
ð8Þ
Adjacency matrix: The adjacency matrix (connectivity matrix) A is A¼
ðaij Þni;j¼1
123
where aij ¼
1
if ði; jÞ 2 E
0
else
ð9Þ
ð10Þ
whereas wij is the weight of the edge ði; jÞ that connects i and j. 0 wij 1. Path: A sequence of vertices (v1 ; . . .vk ) is a path from (v1 to vk ) if 8i 2 f1; . . .; k 1g : ðvi ; viþ1 Þ 2 E. The length of a path v1 ; . . .; vk is k 1. Distance matrix: The distance matrix D is D ¼ ðdij Þni;j¼1 where
ð11Þ
Brain Struct Funct
Fig. 13 Chord diagram visualization of the bilateral BG2 network using CIRCOS. The amount of connections is relatively large. The strong connections of the CPu and AC are diverging into many different regions
dij ¼ dði; jÞ length of the shortest path from i to j; ¼ 1;
if such a path exists else
ð12Þ Generalized topology matrices (GTOM): Let Nm ðiÞ be the m step neighborhood of node i: Nm ðiÞ ¼ fj 6¼ ij minfdði; jÞ; dðj; iÞg mg
ð13Þ
Nmout ðiÞ ¼ fj 6¼ ijdði; jÞ mg
ð14Þ
Nmin ðiÞ ¼ fj 6¼ ijdðj; iÞ mg
ð15Þ
then the GTOM-matrix of step m is defined as
n m GTOMðmÞ ¼ ðgm ij Þi;j¼1 with gij 8 jNm ðiÞ \ Nm ðjÞj þ aij aji > > > > < minfjNm ðiÞj; jNm ðjÞjg þ 1 aij aji ¼ > > > > : 1
if i 6¼ j if i ¼ j
ð16Þ out The definitions of GTOMin ðmÞ and GTOMðmÞ with the directed in out m step neighborhoods Nm ðiÞ and Nm ðiÞ are analog. Degree all (degall , DGAll ): Self-references of nodes are not considered for all three degree measures. degðiÞ ¼ degall ðiÞ
123
Brain Struct Funct
Fig. 14 The distance matrix of the ipsilateral BG2 network. The distances between regions correspond to the smallest number of connections that are necessary to connect pairs of regions. The thalamic paths are in most cases longer (lighter gray values) than paths of midbrain regions or MRF
Fig. 15 The communicability matrix of the ipsilateral BG2 network. The number of shortest paths through a pair of nodes is presented by the communicability matrix (Estrada and Hatano 2008). Light gray values are indicating large communicability values. The CPu, AC, STh, PF, MRF and VTA are contributing to shortest paths of the BG2network
degðiÞ ¼
n X j¼1 j6¼i
Degree out:
123
aij þ aji
Fig. 16 The generalized topology overlap matrix (GTOM) of the ipsilateral BG2 network. The pairwise interconnectedness in relation to the number of neighbors that a pair of nodes share in common is presented by the GTOM. Large values indicate many similarities of neighbors and connections of a pair of nodes. E.g., the pair CL and CPu has a relatively large GTOM value like the pair STh and VTA
Fig. 17 The variability of connection strengths of the bilateral BG2 network. Large-standard deviations of connection weights are indicated by light gray values. The variability of connection weights of the SNC has relatively smaller values in comparison to VTA, HIPP and AGl
ð17Þ degout ðiÞ ¼
n X j¼1 j6¼i
aij
ð18Þ
Brain Struct Funct
Fig. 18 The global parameters of the ipsilateral BG2 network. The first row (beginning with ‘‘Nodes’’) presents the general parameters of the BG2 network. The line density is 52.333 %. If each regions would be connected with each other region then the line density is 100 %. Cy number of shortest cycles and CyC is the cycle coefficient, AvgHD is the average hierarchical level of the regions in the BG2 network. Six-different randomizations have been performed. Each
Degree in: degin ðiÞ ¼
n X
Latout ðiÞ ¼ aji
ð19Þ
j¼1 j6¼i
Reciprocal edge count RecðiÞ: RecðiÞ is the number of reciprocal edges adjacent to a node i. X RecðiÞ ¼ aij aji ð20Þ j2N i6¼j
degIPSI out ðiÞ degout ðiÞ
ð23Þ
Laterality of the reciprocal edge count LatRec ðiÞ: The laterality of the reciprocal edge count is the fraction of ipsilateral reciprocal edges. For a node i 2 N IPSI N. X 1 LatRec ðiÞ ¼ aij aji ð24Þ RecðiÞ IPSI j2N i6¼j
Neighborhoods: Out-neighbors of i: IPSI
N be a subset of nodes and Laterality: Let N degIPSI ðiÞ the degree of the node i in the subset N IPSI . Then the lateralities are defined as follows: degIPSI all ðiÞ degall ðiÞ
ð21Þ
degIPSI in ðiÞ Latin ðiÞ ¼ degin ðiÞ
ð22Þ
Latall ðiÞ ¼
randomization was repeated 1,000 times using exactly the same number of regions and connections as in the real BG2 network. The global network parameters of the real BG network are shown in the first column followed by mean values of the randomizations. The rewiring randomization is showing the smallest differences with regard to global parameters when comparing with the real network
Niout ¼ fj 2 Nnfigjaij ¼ 1g
ð25Þ
In-neighbors of i: Niin ¼ fj 2 Nnfigjaji ¼ 1g
ð26Þ
All neighbors of i: Ni ¼ Niout [ Niin
ð27Þ
123
Fig. 19 The local parameters of the ipsilateral BG2 network in parallel coordinates. Parallel coordinate visualization provides an overview of a high-dimensional parameter space. Parameters are sorted by similarity. The CPu, AGl and AGm are highlighted by dashed lines. The CPu has almost largest or smallest values with regard to centrality measures. For abbreviations of parameters see Appendix
Brain Struct Funct
123
Niþ ¼ Ni [ fig
ð28Þ
Network parameters Communicability matrix G: Gpq ¼
1 ðAk Þ X pq
k!
k¼0
¼ ðeA Þpq
ð29Þ
Modularity measure: Let M ¼ fM1 ; . . .; Mm g be a partition of N. Mi is a group, module or cluster of vertices. With 1 X ðajk þ akj Þ; ei ¼ ð30Þ ‘ j;k2M i
j\k
the fraction of edges that fall within group Mi N and 1X X ðajk þ akj Þ; ai ¼ ð31Þ 2‘ j2M k2Nnfjg the fraction of ends of edges that are attached to vertices in group Mi , the Modularity Q¼
m X
ðei a2i Þ;
ð32Þ
i¼1
whereas a2i is the fraction of edges that would connect vertices within group Mi if they were connected at random. A large modularity implies that the fraction of edges that fall within groups is larger than expected in the random case. The partition is generated by a ‘‘Greedy’’ optimization algorithm. Starting with a partition where every single node has its own group, stepwise those two groups are joined that increase Q most. The algorithm ends if there are no more such groups. The weighted case is similar, only the aij are replaced by wij and ‘ is replaced by the sum of the edge weights X ! wij ‘w ¼ ð33Þ i;j2N i6¼j
The method of Newman and Girvan (2004) was used. Global efficiency GE: X 1 1 GE ¼ ð34Þ nðn 1Þ i;j2N dðijÞ i6¼j
! ! GE and GE w analog with d! ði; jÞ and d w ði; jÞ Directed global efficiency: X 1 1 GE! ¼ nðn 1Þ i;j2N d! ðijÞ !
i6¼j
Harmonic mean HM:
ð35Þ
Brain Struct Funct
Fig. 20 The rich-club regions of the ipsi and contralateral BG2 network have a rich-club coefficient / [ 0:8 respectively a rank of 44: The direct neighbors or feeders are shown in the upper arc. Their interconnections are presented by blue edges. The connection of
HM ¼
1 GE
ð36Þ
The directed and weighted versions use the directed and weighted global efficiencies. Local efficiency: The local efficiency indicates how strong neighbors of nodes are interconnected. For each node i the inverse lengths of the shortest paths of the neighbors of i that are passing i are added. The local efficiency is this sum divided by the maximal possible sum of paths between neighbors that are connected with i. The efficiency of the network (global efficiency) is the average local efficiency of all nodes. Directed local efficiency: P 1 j;k2Ni djk ðNi Þ X 1 j6¼k ! LE ¼ ð37Þ n i2N ni ðni 1Þ
rich nodes with feeder nodes is shown by yellow edges and the interconnectivity of rich nodes by black edges. The indices _L and _R are indicating the sides of the hemispheres
Weighted directed local efficiency: ! P 1 w j;k2Ni djk ðNi Þ X ! 1 j6 ¼ k LE w ¼ n i2N ni ðni 1Þ
ð38Þ
ni [ 1
! whereby ni ¼ jNi j and djk ðNi Þ, respectively, djkw ðNi Þ is the length of the shortest path between j and k that contains only neighbors of i. Directed assortativity coefficient r ! : P r
!
¼ 1 2
P
ði;jÞ2L
degout ðiÞ degin ðjÞ 4‘1
2 ði;jÞ2L ðdegout ðiÞ
hP
þ degin ðjÞ2 Þ 4‘1
ði;jÞ2L ðdegout ðiÞ
hP
þ degin ðjÞÞ
ði;jÞ2L ðdegout ðiÞ
i2
þ degin ðjÞÞ
i2
ð39Þ ! Directed and weighted assortativity coefficient r w :
ni [ 1
i2 w w w ðdeg ðiÞ þ deg ðjÞÞ ij ! out in ði;jÞ2L rw ¼ P hP i2 2 2 w w w w 1 1 w ðdeg ðiÞ þ deg ðjÞ Þ w ðdeg ðiÞ þ deg ðjÞÞ ij ij out in out in ði;jÞ2L ði;jÞ2L 2 4‘ P
w w 1 ði;jÞ2L wij ðdegout ðiÞ degin ðjÞÞ 4‘
hP
ð40Þ
123
Brain Struct Funct
Fig. 21 The regions of the ipsi- and contralateral BG2 network that belong to the knotty-center in an arc diagram visualization. The knotty-center regions are on the lower arc. Direct neighbors of knotty centers are shown in the upper arc. Their interconnections are
presented by blue edges. The connections of the knotty-center with their direct neighbors are shown by yellow edges and the interconnectivity of rich-nodes by black edges. The indices _L and _R are indicating the sides of the hemispheres
Fig. 22 The K-core analysis of the ipsilateral BG2 network. The connections of CPu are highlighted. In the outer circles are those regions which have the smallest number of connections and in the center are regions that have the largest number of connections
Fig. 23 The K-core analysis of the ipsi- and contralateral BG2 network. The connections of the CPu are highlighted
123
Brain Struct Funct
The correlation of the degrees of nodes that are connected: 1 r 1. Large positive values imply that nodes are mainly connected to nodes with similar degrees. Large negative values imply that nodes with a large degree are mainly to nodes that have a small degree. If r 0 there is no relation detectable. Average path length = characteristic path length ðdÞ: With P ¼ fði; jÞ 2 N Njdði; jÞ\1g, the set of paths. 8 X > < 1 dði; jÞ; P 6¼ £ ð41Þ d ¼ jPj ði;jÞ2P > : 0; P¼£ In the weighted case the distances dði; jÞ are replaced by the weighted distances. Average directed degree deg: deg ¼
Fig. 24 Metric multidimensional scaling (MDS) applied to the ipsilateral BG2 network. Regions that are adjacent to each other have more similarities with regard to their connectivity than with regions that are further away from each other. The 8 thalamic regions that are located within the lower-left quadrant are relatively similar with regard to connectivity. AGm and AGl in the lower-right quadrant have also a large similarity. Interestingly, SNR is adjacent to MRF, however, the second major output node of the BG is MGP which is not to far from SNR. Limbic regions like Ac, Ent, HIPP and Pir are located in the upper left quadrant
2‘ n
ð42Þ
Heterogeneity VC: Coefficient of variation (VC) of the degreeall parameter. X 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdegall ðiÞ degÞ2 HVC ¼ ð43Þ deg i2N If HVC ¼ 0, all nodes have the same degree. The larger HVC the more diverse are the node degrees. In the weighted case the versions of the degrees are used. The heterogeneity measure of Estrada (2010) was not implemented because it is not defined for directed and weighted graphs. Line density Ld: Ld ¼
‘ n ðn 1Þ
ð44Þ
Without self-referencing edges. Rich-club coefficient /ðkÞ: With Nk ¼ fi 2 NjdegðiÞ [ kg and Ek ¼ fði; jÞ 2 Nk Nk jði; jÞ 2 Eg Gk ¼ ðNk ; Ek Þ is the subgraph of G ¼ ðN; EÞ containing all vertices with a degree greater than k. The rich-club coefficient /ðkÞ of a graph G is defined as the line density of the subgraph Gk : /ðkÞ ¼ LdðGk Þ
ð45Þ
Diameter Diam: Diam ¼ maxfdði; jÞjdði; jÞ\1g
ð46Þ
Katz index: The Katz index (Katz status index, Katz centrality) is a measure for the direct and indirect input of a node (Foster et al. 2001). CKatz ðiÞ ¼ Fig. 25 MDS applied to the bilateral BG2 network. The major output regions SNR and MGP are located in the upper- and lower- left quadrant, however, not too far from each other. Most limbic and all thalamic regions are located in the upper-left quadrant (left hemisphere) or in the upper-right quadrant (right hemisphere)
1 X X
ak ðAk Þji
ð47Þ
k¼1 j¼1
The attenuation factor a has to be smaller than the reciprocal of the absolute value of the largest eigenvalue of A. For a better readability and comparability of the results, in
123
Brain Struct Funct Fig. 26 Counts of cyclic connections or loops in the bilateral BG2 network. The logarithm (Log) of the frequency of contributions of regions to a cycle of size 2–7 is presented by the y-axis. Every second region (all regions of the left hemisphere) are labeled. For a cycle from MRF back to MRF passing 4 regions within the BG2 network there exist 8,255 possible paths or 41,191,973 paths when 7 regions are allowed
neuroVIISAS the Katz centrality is multiplied by the mean in ðiÞ of all nodes with CKatz ðiÞ [ 0. Hence, of the quotient Cdeg Katz ðiÞ
the values lie in the same range as the indegrees. Number of triangles: X ðaij þ aji Þðaik þ aki Þðajk þ akj Þ t! ðiÞ ¼ j;k2Nnfig j\k
ð48Þ
The maximum number of possible triangles that can be deviated from a complete reciprocal triangle is 8. Weighted number of triangles: X 1 1 1 1 1 1 ! t w ðiÞ ¼ ðw3ij þ w3ji Þðw3ik þ w3ki Þðw3jk þ w3kj Þ ð49Þ j;k2Nnfig j\k
Instead of the sum of triangles (t! ðiÞ) the sum of geometric means of edge weights of each triangle is calculated. The 1
following example provides ðwij wjk wik Þ3 as the summand:
Directed transitivity: The general definition of transitivity (T) is the sum of number of triangles around all nodes divided by the maximum possible sum of triangles around all nodes. P t! ðiÞ ! T ¼ P i2N ð50Þ i2N tmax ðiÞ Directed and weighted transitivity: ! P ! t w ðiÞ w i2N T ¼P i2N tmax ðiÞ
ð51Þ
whereby tmax ðiÞ ¼ degðiÞ ðdegðiÞ 1Þ 2 recðiÞ with degðiÞ ¼ number of adjacent edges of i and recðiÞ ¼ number of reciprocal edges of i (the two directions of one reciprocal edge are considered as one reciprocal edge). The degree deg and the reciprocity rec are defined as: X degðiÞ ¼ aij þ aji ð52Þ j2Nnfig
recðiÞ ¼
X
aij aji
j2Nnfig
For the directed and weighted case: 1 wij [ 0 aij ¼ 0 else
123
ð53Þ
ð54Þ
Brain Struct Funct Table 7 Local parameters of the weighted-ipsilateral BG2 network Region
DGA
DGO
DGI
CDC
Katz
SPC
Triag
CyclC
EccO
CPu
26.25
12.25
14.0
0.53333
12.17519
0.16667
503.6196
0.66462
375
VTA
20.875
12.5
8.375
0.4012
8.04604
0.25
436.14384
0.55527
0.41667
SNR
19.125
11.0
8.125
0.42484
8.51141
0.29167
323.26377
0.66477
0.41667
MRF
18.875
12.375
AC
18.0
9.625
6.5
0.34437
6.58671
0.25
399.3736
0.56189
0.5
8.375
0.46528
7.82869
0.25
402.70699
0.49854
0.45
SNC
17.375
11.125
6.25
0.35971
6.90902
0.25
317.85196
0.59338
0.5
STh
16.875
9.0
7.875
0.46667
7.93142
0.25
313.32162
0.64057
0.41667
AGm
16.25
9.75
6.5
0.4
6.16135
375
246.8733
0.51577
0.41667
MGP
15.5
8.75
6.75
0.43548
6.93033
0.33333
270.12611
0.59842
0.41667
AGl
14.5
7.625
6.875
0.47414
6.36241
0.16667
237.70384
0.56222
0.5
PF
13.875
8.5
5.375
0.38739
5.68001
375
263.27964
0.5863
0.5
Ac
12.75
6.25
6.5
0.5098
6.29047
0.25
227.32308
0.55463
0.5
Pir
11.0
4.5
6.5
0.59091
6.51555
0.25
157.8558
0.56038
0.54167
VM
10.375
4.375
6.0
0.57831
6.2192
0.33333
154.06266
0.5957
0.5
HIPP CM
10.25 10.0
5.25 4.75
5.0 5.25
0.4878 525
4.75053 5.51932
0.25 375
133.31931 131.01033
0.57456 0.59741
0.5 0.5
LHb
10.0
4.875
5.125
0.5125
5.42794
0.25
164.73628
0.50748
0.5
VL
9.0
1.5
7.5
0.83333
7.67315
0.2
81.79831
0.73374
625
MDL
8.75
1.5
7.25
0.82857
7.35711
0.41667
101.08281
0.58321
625
LGP
8.75
2.625
6.125
0.7
6.23351
0.41667
106.87098
0.65648
0.54167
CL
8.5
4.0
4.5
0.52941
4.73061
0.41667
96.16541
0.62436
0.5
Ent
8.375
3.25
5.125
0.61194
4.82634
0.25
113.68525
0.43629
0.66667
PC
7.0
2.5
4.5
0.64286
4.89828
0.41667
71.94521
0.60614
0.54167
MDM
6.0
1.5
4.5
0.75
4.34144
0.58333
56.44901
0.52978
625
VA
2.5
1.0
1.5
0.6
1.51929
625
7.97699
0.70278
625
Lev
Loc
Region
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG
CPu
0.41667
0.29079
0.26399
0.25321
0.31398
0
12.54348
366
0.3729
0.96524
VTA
0.5
0.27075
0.30556
0.27075
0.27191
0
12.92935
0.37537
0.2589
0.98554
SNR
0.5
0.25804
0.41506
0.25952
0.33326
0.4375
12.70238
0.43298
0.23739
0.8122
MRF
0.5
0.2747
0.37619
0.2747
0.29024
0
13.0163
0.38224
0.21062
0.98553
AC
0.5
0.28436
0.30362
0.28436
0.23942
0
13.26136
0.37947
0.17976
0.93529
SNC
0.5
0.28185
0.4625
0.28185
0.32768
125
13.2381
0.39005
0.1655
0.86018
STh
0.45
0.33686
0.40545
0.32557
0.37036
0.03333
14.38889
0.34253
0.10538
0.74536
AGm
0.5
0.23989
0.27574
0.25
0.22607
375
12.54605
0.45852
0.17302
0.68563
MGP
0.54167
0.3102
0.42468
0.29496
0.31854
0.23125
13.50658
0.40979
0.1091
0.76064
AGl
0.5
0.3125
0.27552
0.29412
0.26353
0.24405
13.53676
0.43892
0.08385
0.64739
PF
0.5
0.34835
0.45455
0.34835
0.35869
0.04167
14.90441
0.32782
-0.00814
0.68479
Ac Pir
0.5 0.5
0.37099 0.32639
0.34286 0.36882
0.31127 0.33869
0.30886 0.31955
0.3 0.20238
13.70139 14.35294
0.4002 0.36632
0.00466 -0.09458
0.72482 0.68825
VM
0.5
0.41667
0.42898
0.38805
0.37946
0.11528
15.75893
0.29619
-0.18057
0.54
HIPP
625
0.31439
0.41319
0.33333
0.32676
0.20139
14.39167
0.37484
-0.12721
0.56576
CM
0.5
0.31389
0.42778
0.34345
0.35408
0.0625
15.2
0.32213
-0.17637
0.52686
LHb
625
375
0.46023
0.3875
0.3362
0.16493
15.325
0.31079
-0.18284
0.62755
VL
0.41667
625
0.40568
0.40568
0.46476
0.11298
17.03409
0.25543
-0.28858
0.38215
MDL
0.5
0.34375
0.40064
0.40064
0.38289
0.07273
16.52885
0.26345
-0.28663
0.4676
LGP
0.54167
0.64375
0.43523
0.45265
0.46065
0.16761
16.5
0.28107
-0.28201
0.49548
CL
0.54167
0.41295
0.49107
0.45265
0.41451
0.14489
16.75
0.25151
-0.30743
0.48319
123
Brain Struct Funct Table 7 continued Region
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG
Lev
Loc
Ent
0.58333
0.34201
0.32576
0.35897
0.2814
0.18977
15.03846
0.34071
-0.24859
0.47373
PC
0.5
0.40417
0.47098
0.4566
0.41829
0.13929
18.22222
0.20137
-0.43369
0.3041
MDM VA
0.58333 0.66667
375 625
0.38611 0.16667
0.38611 0.4625
0.32442 0.39885
0.23214 0.22807
15.2875 18,325
0.26926 0.23199
-0.41216 -0.75126
0.34497 0.14264
Region
CEO
CEI
BC
EC
SC
PRC
FC
Stress
CPu
4.20438
4.4376
0.23583
1
243.76129
1
0.50716
182
1.41218
VTA
3.94521
3.25424
0.05577
0.91751
154.20322
0.58706
0.39402
45
1.86591
SNR
3.86577
3.65482
0.0939
0.69998
130.14345
0.53169
0.33929
72
0.64769
MRF
3.86577
2.88
0.01208
897
129.7987
0.41633
0.31682
10
0.78248
AC SNC
3.37632 3.91837
3.13043 3.40828
0.04026 0.02159
0.68099 0.80884
107.60925 116.2092
0.64706 0.38099
0.38934 0.2869
30 26
0.65307 0.74325
STh
3.81457
3.42043
0.0237
0.77207
125.52561
0.4873
0.38685
23
0.26545
AGm
3.72574
2.83744
0.07555
0.67544
88.77745
0.70423
0.38213
64
1.49691
MGP
3.34884
3.18232
0.0226
0.59287
86.04644
0.43381
0.31835
22
-0.02124
AGl
3.40828
2.83744
0.01736
0.66608
85.19876
0.64031
0.38948
17
1.02062
PF
3.40828
2.8263
0.00645
0.72779
99.67888
0.32626
0.31893
10
0.45657
Ac
3.21788
2.83744
0.00543
0.56331
77.21747
0.43082
0.27614
7
-0.18659
Pir
2.96907
2.97214
0.01585
0.35111
45.73839
0.45033
0.21232
14
0.74636
VM
2.89447
3.22148
0.00218
0.3923
55.50867
0.36658
0.27266
5
-0.40825
HIPP
2.66667
2.67907
0.01479
0.35601
34.50464
0.35927
0.24167
10
0.37318
CM
3.16484
2.86567
0.00494
0.39959
45.3308
0.30888
0.22619
7
-0.64639
LHb
3
2.71698
0.00453
0.42068
54.47565
0.29981
0.25625
5
-0.65307
VL
2.26772
3.64557
0.00595
0.17041
24.99282
0.46127
0.16023
5
-1.12268
MDL
2.16867
3.23595
0.00177
0.14331
22.56384
0.43209
0.14423
4
-1.49273
LGP CL
2.67907 2.78261
3.08021 2.8263
0 0.00203
0.26767 0.35679
38.75035 35.96419
0.35154 0.27445
0.20786 0.24242
0 6
-0.5946 -0.69016
Ent
2.37037
2.65438
0.00242
0.26855
26.31024
0.36863
0.25
2
PC
2.65438
2.75598
0.00112
0.26166
28.23121
0.28222
0.2934
4
-1.12268
MDM
2.18347
2.75862
0.00192
0.10016
9.94491
0.28595
0.16944
4
-1.30614
VA
2.19847
2.20015
0
0.11114
5.36512
0.1381
0.15
0
-2.21914
Region
ZI
ZA
PCO
PCI
PCA
RadO
RadI
ZO
0
CenO
CenI
CPu
1.63169
1.63651
0.64973
0.64541
0.64907
1.42882
1.44132
3
10
VTA SNR
1.45048 0.46465
1.89849 0.59929
0.6392 0.66219
0.63711 0.61917
0.63839 0.6512
1.41319 1.40799
1.35938 1.39306
-5 -4
-16 -10
MRF
-0.42875
0.49136
0.65177
0.65385
0.65471
1.40799
1.31944
-5
-19
0.72524
0.73019
0.66352
0.66251
0.66397
1.37049
1.34722
-11
-17
AC SNC
-0.31337
0.23626
0.64083
0.5952
0.62771
1.41146
1.37326
-3
-11
STh
0.65916
0.49556
0.63735
0.59965
0.63473
1.40451
1.37431
-8
-15
AGm
0.77174
1.44677
0.65976
0.66198
0.66225
1.39826
1.31424
-9
-17
MGP
0.46465
0.23626
0.66245
0.55967
0.63541
1.36806
1.35243
-11
-16
AGl
1.37199
1.25569
0.59446
0.62281
0.6088
1.37326
1.31424
-11
-18
PF
-0.79964
-0.17863
Ac
0.48349
Pir
-0.96699
VM
0.1715
HIPP
0.48349
123
0.63149
0.64467
0.64216
1.37326
1.31285
-12
-18
0
0.5632
0.63018
609
1.3559
1.31424
-18
-17
0.29208
0.27778
0.64793
0.58264
1.32986
1.33021
-20
-16
-0.27298
0.65959
0.63281
0.6538
1.32118
1.35625
-18
-19
0.43811
0.52608
0.54
0.55205
1.29167
1.2934
-16
-15
Brain Struct Funct Table 7 continued Region
ZI
ZA
PCO
PCI
PCA
RadO
RadI
CenO
CenI
CM
-1.62924
-1.03731
0.65374
0.63492
0.66375
1.35069
1.31771
-16
-18
LHb
-1.69223
-1.02226
0.65746
0.56871
0.62625
1.33333
1.29861
-16
-18
VL MDL
0.77174 -0.48349
-0.65514 -1.31434
0.44444 0.5
0.65778 0.65161
0.64815 0.66286
1.22569 1.20556
1.39236 1.35764
-23 -22
-12 -14 -19
0.65916
0.02881
0
0.46647
0.35714
1.2934
1.34201
-25
CL
LGP
-0.89689
-0.85283
0.53125
0.54938
0.55536
1.30729
1.31285
-21
-22
Ent
0.96699
0.29208
0.27219
0.51636
0.44108
1.24479
1.28993
-21
-16
PC
-1.02899
-1.22839
0.66
0.64815
0.65561
1.28993
1.30382
-22
-21
MDM
-0.96699
-1.31434
0.44444
0.62346
0.60069
1.20868
1.30417
-24
-19
VA
-1.86942
-2.20123
0.5
0.44444
0.48
1.21181
1.21215
-25
-25
The regions are sorted using the weighted degree all. Abbreviations and parameter definitions are described in the Supporting Information A all, AD average degree, C circle, CC cluster-coefficient, CE closeness centrality, DG degree, I in, O out, SP length of shortest path
Table 8 Local parameters of the bilateral BG2 network Region
HD
REC
DGA
DGO
DGI
CDC
LA
LO
LI
LR
Katz
SPC
MRF_R
12
24
67
41
26
0.39
0.55
0.54
0.58
0.63
25.16
1
MRF_L
12
24
68
42
26
0.38
0.56
0.55
0.58
0.63
25.2
1
AGm_R
14
21
55
27
28
0.51
0.58
0.59
0.57
0.67
26.34
1
AGl_R
14
22
48
24
24
0.5
0.6
0.58
0.63
0.59
23.25
1
AGm_L
14
22
56
28
28
0.5
0.61
0.61
0.61
0.68
26.39
1
AGl_L
14
23
50
25
25
0.5
0.62
0.6
0.64
0.61
23.96
1
VL_L
12
4
22
5
17
0.77
0.64
0.6
0.65
0.75
18.12
1
VL_R
12
4
22
5
17
0.77
0.64
0.6
0.65
0.75
18.08
1
HIPP_L
10
8
31
18
13
0.42
0.65
0.61
0.69
0.75
11.15
1
HIPP_R
10
8
31
18
13
0.42
0.65
0.61
0.69
0.75
11.11
1
AC_R
8
26
64
37
27
0.42
0.66
0.59
0.74
0.77
24.82
1
CM_L
11
8
30
13
17
0.57
0.67
0.77
0.59
0.63
19.12
2
LHb_L
10
10
33
16
17
0.52
0.67
0.69
0.65
0.8
16.61
1
CM_R
11
8
30
13
17
0.57
0.67
0.77
0.59
0.63
19.08
2
LHb_R AC_L
10 8
10 25
33 62
16 36
17 26
0.52 0.42
0.67 0.68
0.69 0.61
0.65 0.77
0.8 0.8
16.56 24.76
1 1
CPu_R
11
21
58
23
35
0.6
0.69
0.87
0.57
0.86
31.66
1
PC_L
11
7
20
9
11
0.55
0.7
0.67
0.73
0.71
13.3
2
VM_L
11
9
30
12
18
0.6
0.7
0.75
0.67
0.78
19.37
2
PC_R
11
7
20
9
11
0.55
0.7
0.67
0.73
0.71
13.27
2
VM_R
11
9
30
12
18
0.6
0.7
0.75
0.67
0.78
19.33
2
VTA_R
11
21
54
33
21
0.39
0.7
0.67
0.76
0.76
20.72
1
CPu_L
11
21
58
23
35
0.6
0.71
0.87
0.6
0.86
31.67
1
PF_R
11
15
38
21
17
0.45
0.71
0.76
0.65
0.73
17.29
2
VTA_L
11
22
56
34
22
0.39
0.71
0.68
0.77
0.77
21.41
1
Pir_R
10
6
28
12
16
0.57
0.71
0.58
0.81
0.67
16.27
1
PF_L
11
15
39
22
17
0.44
0.72
0.77
0.65
0.73
17.33
2
SNR_L
12
13
44
26
18
0.41
0.73
0.73
0.72
0.85
17.68
2
CL_L
11
5
22
10
12
0.55
0.73
0.8
0.67
0.8
13.02
2
SNR_R CL_R
12 11
13 5
44 22
26 10
18 12
0.41 0.55
0.73 0.73
0.73 0.8
0.72 0.67
0.85 0.8
17.64 12.98
2 2
123
Brain Struct Funct Table 8 continued Region
HD
REC
Ent_L
11
Pir_L
10
SNC_L SNC_R
12 12
Ac_R
11
10
34
14
20
0.59
0.76
0.86
0.7
0.9
18.73
1
STh_R
9
14
37
19
18
0.49
0.78
0.84
0.72
0.86
18.09
1
STh_L
9
14
38
20
18
0.47
0.79
0.85
0.72
0.86
18.14
1
Ac_L
11
11
35
15
20
0.57
0.8
0.87
0.75
0.91
18.75
1
MDL_L
12
5
21
5
16
0.76
0.81
0.8
0.81
0.8
16.92
2
MDL_R
12
5
21
5
16
0.76
0.81
0.8
0.81
0.8
16.84
2
VA_L
12
0
6
2
4
0.67
0.83
1
0.75
0
4.23
3
MGP_L
12
12
36
22
14
0.39
0.83
0.77
0.93
0.92
VA_R
12
0
6
2
4
0.67
0.83
1
0.75
0
MGP_R
12
12
36
22
14
0.39
0.83
0.77
0.93
0.92
13.95
1
Ent_R
12
2
6
3
3
0.5
0.83
0.67
1
1
2.76
1
MDM_R
12
3
14
3
11
0.79
0.86
1
0.82
1
11.32
2
MDM_L
12
4
16
4
12
0.75
0.88
1
0.83
1
12.1
2
LGP_L LGP_R
12 12
4 4
16 16
5 5
11 11
0.69 0.69
1 1
1 1
1 1
1 1
10.96 10.91
2 2
Region
Triag
CyclC
EccO
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG
MRF_R
1,975
0.07
2
3
0.39
0.56
0.39
0.45
0.1
36.4
0.39
MRF_L
2,021
0.07
2
3
0.39
0.56
0.38
0.45
0.05
36.23
0.39
AGm_R
1,279
0.07
2
2
0.43
0.45
0.39
0.44
0.34
36.29
0.48
940
0.07
2
2
0.43
0.42
0.42
0.42
0.38
36.58
0.44
AGl_R AGm_L AGl_L VL_L
DGA
DGO
DGI
CDC
LA
LO
LI
LR
Katz
SPC
9
28
14
14
0.5
0.75
0.64
0.86
0.89
13.85
1
7
29
12
17
0.59
0.76
0.67
0.82
0.71
17.02
1
13 13
42 42
27 27
15 15
0.36 0.36
0.76 0.76
0.78 0.78
0.73 0.73
0.85 0.85
15.25 15.24
1 1
14.03
1
4.22
3
1,324
0.07
2
2
0.42
0.45
0.39
0.44
0.34
36.41
0.47
988 296
0.07 0.09
2 3
3 2
0.42 0.85
0.41 0.64
0.41 0.64
0.41 0.65
0.39 0.18
36.26 50.28
0.44 0.23
VL_R
296
0.09
3
2
0.85
0.64
0.64
0.65
0.18
50.17
0.23
HIPP_L
510
0.08
2
3
0.58
0.55
0.52
0.56
0.25
42.3
0.41
HIPP_R
479
0.08
2
3
0.52
0.54
0.52
0.52
0.25
42.17
0.41
AC_R
1,714
0.07
2
3
0.4
0.48
0.4
0.43
0.31
36.79
0.41
CM_L
522
0.09
3
2
0.61
0.65
0.57
0.61
0.18
46
0.31
LHb_L
634
0.1
3
3
0.59
0.64
0.55
0.61
0.26
42.52
0.32
CM_R
498
0.09
2
2
0.53
0.65
0.54
0.58
0.19
44.82
0.36
LHb_R
634
0.1
3
3
0.59
0.64
0.55
0.61
0.27
42.39
0.32 0.39
AC_L
1,724
0.08
2
3
0.42
0.53
0.42
0.46
0.26
37.76
CPu_R
1,423
0.07
2
2
0.5
0.42
0.41
0.44
0.28
37.43
0.4
PC_L
266
0.12
3
3
0.71
0.75
0.72
0.73
0.24
53.77
0.17
VM_L
564
0.1
3
2
0.69
0.67
0.62
0.66
0.18
47.33
0.28
PC_R
266
0.12
2
3
0.71
0.75
0.72
0.73
0.23
53.69
0.18
VM_R VTA_R
564 1,434
0.1 0.08
2 2
2 3
0.69 0.46
0.67 0.59
0.62 0.46
0.66 0.51
0.19 0.2
47.14 39.82
0.28 0.38
CPu_L
1,433
0.07
2
2
0.5
0.42
0.41
0.44
0.28
37.49
0.4 0.32
PF_R
0.1
2
3
0.58
0.65
0.56
0.61
0.22
43.91
1,514
0.08
2
3
0.46
0.56
0.46
0.5
0.2
39.59
0.37
Pir_R
418
0.08
3
3
0.49
0.66
0.56
0.56
0.28
42.36
0.36
PF_L
865
0.09
3
3
0.56
0.65
0.54
0.6
0.22
43.38
0.32
VTA_L
123
838
Brain Struct Funct Table 8 continued Region
Triag
CyclC
EccO
SNR_L
916
0.08
3
3
0.39
0.64
0.39
0.49
0.37
36.23
0.44
CL_L
293
0.1
3
3
0.64
0.68
0.63
0.65
0.26
46.71
0.27
SNR_R CL_R
916 293
0.08 0.1
3 3
3 3
0.39 0.64
0.64 0.68
0.39 0.63
0.49 0.65
0.39 0.27
36.1 46.41
0.44 0.27
Ent_L
453
0.09
2
3
0.6
0.63
0.58
0.61
0.27
44.11
0.34
Pir_L
460
0.08
3
3
0.51
0.65
0.58
0.58
0.29
42.64
0.36
SNC_L
948
0.09
3
3
0.47
0.75
0.48
0.56
0.29
39.76
0.41
SNC_R
948
0.09
2
3
0.47
0.75
0.48
0.56
0.3
39.59
0.41
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG
Ac_R
641
0.1
3
3
0.63
0.56
0.51
0.58
0.37
39.21
0.41
STh_R
804
0.1
2
3
0.61
0.63
0.55
0.62
0.25
42.83
0.34
STh_L
837
0.1
3
3
0.6
0.63
0.54
0.61
0.25
42.42
0.35
Ac_L
679
0.1
3
3
0.6
0.57
0.51
0.58
0.37
39.38
0.4
MDL_L
284
0.12
3
3
0.75
0.69
0.69
0.69
0.25
48.31
0.25
MDL_R
284
0.12
3
3
0.75
0.69
0.69
0.69
0.25
48.19
0.26
20
0.15
3
3
1
0.33
0.67
0.67
0.34
49.83
0.15
VA_L MGP_L
728
0.1
3
3
0.49
0.78
0.48
0.59
0.39
37.79
0.42
20
0.15
3
3
1
0.33
0.67
0.67
0.35
49.5
0.16
MGP_R Ent_R
728 18
0.1 0.12
2 3
3 3
0.49 0.83
0.78 0.67
0.48 0.58
0.59 0.69
0.39 0.42
37.63 39
0.42 0.37
MDM_R
138
0.11
3
3
0.83
0.79
0.79
0.78
0.29
49.91
0.28
MDM_L
180
0.11
3
3
0.92
0.77
0.77
0.78
0.29
48.5
0.3
LGP_L
202
0.15
3
3
1
0.82
0.83
0.87
0.38
44.17
0.3
LGP_R
202
0.15
3
3
1
0.82
0.83
0.87
0.36
43.83
0.31
VA_R
The regions are sorted using the degree all. Abbreviations and parameter definitions are described in the Supporting Information A all, AD average degree, C circle, CC cluster-coefficient, DG degree, I input, L laterality, O out, O output, R reciprocal, SP length of shortest path
Table 9 Local parameters of the bilateral BG2 network Region
Lev
Loc
CCO
CCI
BC
EC
SC
PRC
FC
Stress
Shapley
MRF_R
0.32
0.88
0.86
0.67
0.04
0.98
25,442,898.34
0.58
0.58
735
-0.21
MRF_L
0.33
0.91
0.88
0.67
0.04
1
26,026,701.64
0.58
0.56
763
-0.21
AGm_R
0.25
0.62
0.69
0.7
0.04
0.64
17,358,891.41
0.85
0.66
649
-0.22
AGl_R
0.18
0.49
0.66
0.66
0.04
0.55
13,243,875.48
0.89
0.85
516
-0.37
AGm_L
0.26
0.63
0.7
0.7
0.05
0.66
18,011,872.88
0.83
0.68
675
-0.22 -0.4
AGl_L
0.2
0.5
0.67
0.66
0.04
0.57
14,180,107.63
0.92
0.86
552
VL_L
-0.37
0.31
0.51
0.6
0
0.15
2,727,904.72
0.41
0.26
26
0.21
VL_R
-0.37
0.32
0.51
0.6
0
0.15
2,700,497.46
0.41
0.26
26
0.21
HIPP_L
-0.1
0.43
0.61
0.54
0.01
0.51
5,787,286.52
0.34
0.45
181
0.07
HIPP_R
-0.09
0.43
0.61
0.54
0.01
0.49
5,588,129.56
0.34
0.45
217
0.03
AC_R
0.3
0.76
0.8
0.67
0.07
0.89
22,779,869.63
0.78
0.69
942
-0.35
CM_L
-0.18
0.39
0.57
0.6
0.01
0.34
6,998,306.99
0.43
0.46
129
0.09
LHb_L
-0.1
0.45
0.59
0.6
0.01
0.48
8,298,279.27
0.39
0.52
141
0.03
CM_R
-0.15
0.38
0.58
0.6
0.01
0.32
6,632,895.27
0.43
0.46
230
0.09
LHb_R AC_L
-0.1 0.27
0.46 0.75
0.59 0.79
0.6 0.66
0.01 0.04
0.48 0.89
8,215,504.87 22,732,362.73
0.4 0.73
0.52 0.68
138 698
0.03 -0.25
CPu_R
0.25
0.74
0.65
0.78
0.05
0.55
18,102,852.8
1
0.59
726
-0.37
123
Brain Struct Funct Table 9 continued Region
Lev
Loc
CCO
CCI
BC
EC
SC
PRC
FC
Stress
Shapley
PC_L
-0.45
0.23
0.54
0.55
0
0.28
3,976,306.75
0.31
0.59
34
0.16
VM_L
-0.2
0.4
0.56
0.61
0
0.35
7,402,875.66
0.43
0.49
83
0.1
PC_R VM_R
-0.45 -0.2
0.23 0.4
0.55 0.57
0.56 0.61
0 0
0.28 0.35
3,930,742.69 733,2016.86
0.32 0.44
0.59 0.49
35 82
0.16 0.1
VTA_R
0.18
0.68
0.75
0.62
0.03
0.88
18,725,331.26
0.55
0.64
429
-0.18
CPu_L
0.25
0.74
0.65
0.78
0.05
0.56
18,415,083.57
0.99
0.59
714
-0.37
-0.04
0.44
0.64
0.59
0.01
0.59
10,995,770.08
0.38
0.68
195
-0.01
0.2
0.7
0.77
0.63
0.03
0.9
19,880,271.57
0.57
0.65
485
-0.2
Pir_R
-0.16
0.44
0.56
0.58
0.01
0.29
4,746,984.35
0.45
0.4
123
0.09
PF_L
-0.02
0.46
0.64
0.58
0.01
0.61
11,479,685.57
0.38
0.65
205
-0.02 -0.03
PF_R VTA_L
SNR_L CL_L SNR_R
0.14
0.54
0.67
0.59
0.02
0.58
11,064,009.13
0.45
0.49
336
-0.33
0.31
0.54
0.56
0
0.27
3,797,240.62
0.3
0.42
67
0.17
0.14
0.54
0.67
0.6
0.02
0.57
10,923,968.28
0.45
0.49
347
-0.03
CL_R
-0.33
0.31
0.54
0.56
0
0.27
3,721,042.61
0.3
0.42
67
0.17
Ent_L
-0.19
0.37
0.58
0.57
0.01
0.4
5,650,397.26
0.42
0.55
125
0.06
Pir_L
-0.15
0.45
0.56
0.59
0.01
0.29
5,000,332.71
0.46
0.43
116
0.09
SNC_L
0.07
0.58
0.68
0.58
0.01
0.68
10,703,603.42
0.37
0.48
260
0
SNC_R Ac_R
0.07 -0.03
0.59 0.48
0.69 0.57
0.58 0.61
0.01 0.01
0.67 0.4
10,568,914.69 7,650,218.92
0.38 0.48
0.48 0.49
266 150
0 0.02
STh_R
-0.04
0.46
0.62
0.6
0.01
0.56
10,461,940.53
0.46
0.65
252
-0.02
STh_L
-0.02
0.48
0.62
0.6
0.01
0.59
10,996,004.71
0.46
0.63
263
-0.03
Ac_L
-0.02
0.49
0.58
0.61
0.01
0.43
8,161,457.86
0.48
0.52
166
MDL_L
-0.37
0.31
0.52
0.59
0
0.17
3,135,070.85
0.41
0.31
31
0.2
MDL_R
-0.37
0.31
0.52
0.59
0
0.17
3,078,873.46
0.41
0.31
32
0.2
VA_L
-0.78
0.09
0.43
0.48
0
0.05
249,113.28
0.15
0.27
1
0.47
0.02
0.49
0.64
0.56
0.01
0.57
8,355,735.62
0.38
0.54
217
0.02
-0.78
0.09
0.43
0.48
0
0.05
241,958.15
0.15
0.27
1
0.47
MGP_L VA_R MGP_R
0
0.03
0.49
0.64
0.56
0.01
0.57
8,210,767.81
0.38
0.54
229
0.02
Ent_R
-0.71
0.11
0.47
0.44
0
0.1
268,736.61
0.13
0.58
4
0.31
MDM_R
-0.54
0.22
0.48
0.54
0
0.09
1,061,603.37
0.27
0.27
8
0.27
MDM_L
-0.48
0.25
0.49
0.54
0
0.11
1,396,820.87
0.3
0.33
11
0.23
LGP_L
-0.44
0.3
0.46
0.52
0
0.14
1,710,007.94
0.29
0.39
6
0.21
LGP_R
-0.44
0.3
0.46
0.53
0
0.14
1,670,074.84
0.3
0.39
6
0.21
Region
ZO
ZI
ZA
PCO
PCI
PCA
RadO
RadI
CenO
CenI
Hub
Aut
MRF_R
1.3
0.64
1.15
0.65
0.64
0.65
2.84
2.51
-1
-10
0.98
0.87
MRF_L
1.4
0.71
1.25
0.66
0.65
0.66
2.86
2.51
1
-10
1
0.88
AGm_R
0.53
1
0.75
0.62
0.64
0.63
2.55
2.57
-15
-7
0.67
0.81
AGl_R
0.53
1.36
0.88
0.53
0.52
0.53
2.49
2.49
-18
-11
0.61
0.59
AGm_L
0.59
1.09
0.82
0.64
0.65
0.65
2.57
2.57
-14
-7
0.69
0.81
AGl_L
0.79
1.47
1.11
0.56
0.59
0.58
2.51
2.49
-17
-11
0.63
0.61
-1.24 -1.38
-0.42 -0.43
-1.03 -1.12
0.48 0.48
0.64 0.6
0.62 0.58
2.02 2.02
2.35 2.35
-39 -39
-18 -18
0.15 0.15
0.65 0.65
HIPP_L
0.72
0.52
0.72
0.62
0.52
0.59
2.37
2.14
-24
-25
0.49
0.4
HIPP_R
0.38
0.52
0.49
0.64
0.47
0.59
2.37
2.16
-24
-24
0.47
0.4
AC_R
1.42
1.5
1.64
0.64
0.55
0.61
2.76
2.51
-5
-10
0.89
0.78
CM_L
-0.63
-0.42
-0.6
0.64
0.62
0.64
2.24
2.35
-31
-19
0.35
0.65
LHb_L
-0.23
-0.8
-0.46
0.62
0.65
0.64
2.31
2.33
-27
-19
0.41
0.59
VL_L VL_R
123
Brain Struct Funct Table 9 continued Region
ZO
ZI
ZA
PCO
PCI
PCA
RadO
RadI
CenO
CenI
Hub
Aut
CM_R
-0.8
-0.43
-0.72
0.64
0.58
0.62
2.27
2.35
-30
-19
0.34
0.65
LHb_R
-0.23
-0.79
-0.46
0.59
0.62
0.62
2.31
2.33
-27
-19
0.41
0.59
AC_L CPu_R
1.07 1.3
1.01 2.08
1.18 1.68
0.65 0.42
0.58 0.64
0.63 0.58
2.73 2.47
2.49 2.71
-6 -19
-11 0
0.88 0.54
0.78 1
PC_L
-1.24
-1.18
-1.31
0.67
0.6
0.65
2.16
2.18
-35
-27
0.27
0.45
VM_L
-0.63
0.34
-0.32
0.63
0.59
0.6
2.22
2.37
-32
-18
0.35
0.67
PC_R
-1.19
-1.15
-1.26
0.64
0.51
0.59
2.18
2.2
-34
-26
0.27
0.45
VM_R
-0.61
0.28
-0.32
0.57
0.54
0.55
2.24
2.37
-31
-18
0.35
0.67
VTA_R
1.3
1
1.28
0.62
0.54
0.59
2.67
2.39
-9
-16
0.81
0.67
CPu_L
1.4
2.23
1.82
0.47
0.65
0.61
2.47
2.71
-19
0
0.54
1
PF_R
0.53
-0.08
0.34
0.54
0.55
0.55
2.43
2.31
-22
-21
0.54
0.64
VTA_L
0.72
-0.94
0.04
0.62
0.57
0.6
2.69
2.41
-8
-15
0.83
0.7
Pir_R
0.38
-0.94
-0.19
0.49
0.51
0.56
2.2
2.29
-32
-21
0.3
0.54
PF_L
0.59
-0.04
0.39
0.6
0.58
0.6
2.43
2.29
-22
-22
0.56
0.65
SNR_L
0.99
0.34
0.82
0.6
0.59
0.6
2.51
2.31
-18
-21
0.64
0.61
-0.74
0.56
0.54
0.56
2.16
2.2
-35
-26
0.26
0.48
CL_L SNR_R
-0.63
-0.8
0.91
0.28
0.75
0.57
0.55
0.56
2.51
2.33
-18
-20
0.63
0.6
CL_R Ent_L
-0.61 0.72
-0.79 0.52
-0.72 0.72
0.46 0.52
0.44 0.49
0.46 0.52
2.16 2.29
2.2 2.24
-34 -28
-26 -23
0.25 0.37
0.47 0.44
Pir_L
0.38
0.04
0.26
0.49
0.55
0.56
2.2
2.31
-32
-20
0.3
0.57
SNC_L
1.2
-0.42
0.68
0.59
0.6
0.6
2.53
2.27
-16
-22
0.67
0.54
SNC_R
1.11
-0.43
0.61
0.57
0.55
0.56
2.55
2.27
-15
-22
0.67
0.54
-1.02
0.52
-0.42
0.5
0.54
0.53
2.24
2.37
-29
-17
0.32
0.65
0.34
0.28
0.34
0.53
0.55
0.54
2.39
2.35
-23
-18
0.47
0.63
Ac_R STh_R STh_L
0.38
0.34
0.39
0.59
0.59
0.59
2.39
2.35
-23
-18
0.49
0.63
Ac_L
-0.32
0.52
0.04
0.55
0.58
0.57
2.27
2.37
-29
-17
0.34
0.65
MDL_L
-1.45
-0.04
-1.03
0.64
0.59
0.61
2.06
2.31
-40
-21
0.13
0.59
MDL_R
-1.38
-0.08
-0.99
0.56
0.54
0.54
2.08
2.31
-38
-21
0.13
0.58
VA_L
-1.45
-2.31
-1.88
0
0.38
0.28
1.69
1.92
-47
-37
0.06
0.14
0.52
0.44
0.5
2.43
2.2
-21
-25
0.53
0.47
0
0.38
0.28
1.67
1.92
-47
-37
0.06
0.14
MGP_L
0.99
0.34
0.82
-1.57
-2.59
-2.06
0.91
0.28
0.75
0.48
0.36
0.44
2.45
2.22
-20
-24
0.53
0.47
Ent_R MDM_R
-1.37 -1.72
-2.4 -0.94
-2.03 -1.57
0 0.44
0.44 0.58
0.28 0.56
1.86 1.92
1.71 2.14
-44 -42
-39 -27
0.06 0.08
0.1 0.43
MDM_L
-1.37
0.04
-0.88
0.38
0.54
0.51
1.94
2.16
-41
-26
0.1
0.46
LGP_L
-0.84
-0.42
-0.74
0
0.4
0.3
1.84
2.08
-42
-32
0.13
0.37
LGP_R
-0.99
-0.43
-0.86
0
0.3
0.22
1.82
2.1
-42
-31
0.12
0.36
VA_R MGP_R
The regions are sorted using the degree all. Abbreviations and parameter definitions are described in the Supporting Information A all, AD average degree, C circle, CC cluster-coefficient, DG degree, I in, L laterality, O out, SP length of shortest path
123
Brain Struct Funct Table 10 Frequency of motifs of the ipsilateral BG2 network Region
3-01
3-02
AC&1
0
1
27
0
CPu&1
5
2
20
2
VTA&10
0
1
40
2
MRF&23
0
0
44
2
STh&3
9
3
41
AGm&1
5
3
37
SNR&25
9
6
42
PF&8
3
6
45
SNC&26
2
0
43
MGP&12
17
2
42
5
AGl&3 Ac&5
2 14
2 7
26 34
0 7
LHb&4
10
1
47
5
12
2
13
21
8
1
27
26
2
5
VM&17
9
3
24
8
13
1
14
36
3
1
34
18
2
1
Ent&12
7
3
26
0
24
0
5
34
5
8
32
15
1
6
HIPP&12
8
6
28
8
29
0
12
22
6
8
29
11
2
2
CM&24
11
10
26
10
16
3
15
27
12
4
30
10
0
0
PC&22
4
4
30
0
11
0
3
47
4
1
15
9
3
2
LGP&18
9
4
52
6
6
2
12
11
8
3
25
6
3
5
13
7
27
15
15
2
28
11
14
11
32
6
3
2
MDL&28
9
3
36
11
8
0
22
28
3
2
25
5
0
3
MDM&26
8
1
38
1
11
0
13
22
1
7
13
5
1
3
CL&24
12
5
25
5
14
1
7
28
13
5
23
5
0
0
VL&31
7
2
43
4
7
0
22
23
1
4
15
3
0
1
VA&13
7
5
30
0
25
1
1
0
5
1
0
0
0
7
Pir&20
3-03
3-04
3-05
3-06
3-07
3-08
3-09
3-10
3-11
3-12
3-13
chain
5
0
12
30
0
9
53
1
38
12
0
65
83
0
0
3
73
68
0
1
0
28
34
0
8
3
66
65
0
0
2
0
40
17
8
3
54
65
0
0
4 2
8
0
22
0
16
29
8
6
38
48
0
0
6
63
5
11
36
46
3
2
16
7
3
32
16
20
1
33
45
0
3
5 10
9
0
17
27
6
2
34
45
0
0
14
0
34
10
10
9
39
44
0
0
11
1
21
21
10
3
38
42
5
8
30 15
1 1
0 8
57 24
8 19
3 5
41 41
39 32
0 0
0 0
in
out
Regions are sorted using motif 3-13 frequencies chain symmetricCHAIN, in symmetricIN, out symmetricOUT (see legend of Fig. 8)
Table 11 Global extrinsic connections of regions of the bilateral BG2 network Region
Dic
Dii
Dis
Doc
Doi
Dos
Centrolateral_thalamic_nucleus_L
392
446
838
7
68
75
403
470
873
7
103
Centrolateral_thalamic_nucleus_R
392
446
838
7
68
75
403
470
873
7
103
110
Accumbens_nucleus_R
265
380
645
4
74
78
308
853
1,161
11
309
320
Accumbens_nucleus_L
265
378
643
4
74
78
318
829
1,147
11
304
315
Central_medial_thalamic_nucleus_L
265
322
587
12
70
82
270
340
610
12
184
196
Central_medial_thalamic_nucleus_R
265
322
587
12
72
84
270
340
610
12
188
200
Paracentral_thalamic_nucleus_L
235
309
544
6
81
87
238
413
651
6
89
95
Paracentral_thalamic_nucleus_R
235
309
544
6
81
87
238
413
651
6
89
95
Mesencephalic_reticular_formation_L
222
311
533
227
334
561
3,316
4,273
7,589
1,188
3,510
4,698
Mesencephalic_reticular_formation_R
222
311
533
227
334
561
3,316
4,273
7,589
1,188
3,512
4,700
Amygdaloid_complex_R Amygdaloid_complex_L
144 144
378 375
522 519
4 4
74 71
78 75
632 633
4,659 4,473
5,291 5,106
525 528
4,296 3,970
4,821 4,498
Ventrolateral_thalamic_nucleus_L
144
344
488
8
52
60
160
384
544
9
56
65
Ventrolateral_thalamic_nucleus_R
144
344
488
8
52
60
160
384
544
9
56
65
Caudate_putamen_L
54
311
365
21
79
100
279
1,057
1,336
22
172
194
Caudate_putamen_R
54
311
365
21
79
100
273
1,064
1,337
22
172
194
123
Sic
Sii
Sis
Soc
Soi
Sos 110
Brain Struct Funct Table 11 continued Region
Dic
Dii
Dis
Doc
Doi
Dos
Sic
Sii
Sis
Soc
Soi
Sos
Ventromedial_thalamic_nucleus_L
82
244
326
33
54
87
105
356
461
40
161
201
Ventromedial_thalamic_nucleus_R
82
244
326
33
54
87
105
356
461
40
161
201
Parafascicular_thalamic_nucleus_L Parafascicular_thalamic_nucleus_R
111 111
213 213
324 324
35 35
136 137
171 172
140 140
377 377
517 517
112 112
291 292
403 404
Ventral_tegmental_area_A10_L
60
238
298
99
231
330
96
476
572
124
440
564
Ventral_tegmental_area_A10_R
60
238
298
99
235
334
96
476
572
124
446
570
Medial_agranular_prefrontal_cortex_R
50
138
188
106
418
524
85
219
304
159
586
745
Hippocampus_R
20
162
182
8
47
55
138
1,220
1,358
130
649
779
Medial_agranular_prefrontal_cortex_L
50
131
181
106
416
522
85
212
297
159
584
743
Entorhinal_cortex_L
27
137
164
14
63
77
154
900
1,054
21
546
567
Hippocampus_L
25
137
162
8
46
54
154
1,030
1,184
130
631
761
Lateral_habenular_nucleus_L
24
128
152
37
99
136
55
226
281
40
129
169
Lateral_habenular_nucleus_R
24
128
152
37
99
136
55
226
281
40
129
169
Subthalamic_nucleus_L
19
124
143
23
83
106
29
181
210
34
127
161
Subthalamic_nucleus_R
19
124
143
23
83
106
29
181
210
34
129
163
Substantia_nigra_compact_part_L
24
101
125
30
126
156
37
136
173
33
181
214
Substantia_nigra_compact_part_R
24
101
125
30
126
156
37
136
173
33
181
214
Piriform_cortex_R Lateral_agranular_prefrontal_cortex_R
11 22
112 99
123 121
11 257
55 383
66 640
34 71
309 440
343 511
37 504
227 803
264 1,307
Lateral_agranular_prefrontal_cortex_L
22
95
117
257
381
638
71
436
507
504
801
1,305
Piriform_cortex_L
11
103
114
11
53
64
34
300
334
38
191
229
Mediodorsal_thalamic_nucleus_medial_part_L
11
91
102
0
14
14
11
91
102
0
14
14
Mediodorsal_thalamic_nucleus_medial_part_R
11
91
102
0
16
16
11
91
102
0
16
16
Mediodorsal_thalamic_nucleus_lateral_part_L
17
82
99
0
18
18
17
83
100
0
18
18
Mediodorsal_thalamic_nucleus_lateral_part_R
17
82
99
0
18
18
17
83
100
0
18
18
Substantia_nigra_reticular_part_L
14
73
87
34
133
167
14
86
100
45
200
245
Substantia_nigra_reticular_part_R
14
73
87
34
133
167
14
86
100
45
200
245
Medial_globus_pallidus_L
12
53
65
11
59
70
13
58
71
11
73
84
Medial_globus_pallidus_R
12
53
65
11
59
70
13
58
71
11
73
84
Entorhinal_cortex_R
3
40
43
0
5
5
9
68
77
0
22
22
Lateral_globus_pallidus_L
4
22
26
0
4
4
4
22
26
0
5
5
Lateral_globus_pallidus_R
4
22
26
0
4
4
4
22
26
0
5
5
12 12
12 12
24 24
0 0
10 10
10 10
14 14
14 14
28 28
0 0
10 10
10 10
Ventro_anterior_thalamic_nucleus_L Ventro_anterior_thalamic_nucleus_R
The regions were sorted by the direct input sum (Dis) Dic direct input from contralateral, Dii direct input from ipsilateral, Dis direct input from ipsi- and contralateral, Doc direct output to contralateral, Doi direct output to ipsilateral, Dos direct output to ipsi- and contralateral, Sic subtree input from contralateral, Sii subtree input from ipsilateral, Sis subtree input from ipsi- and contralateral, Soc subtree output to contralateral, Soi subtree output to ipsilateral, Sos subtree output to ipsi- and contralateral
123
123 0.971
0.829 -0.558 0.957
-0.062
-0.090
-0.006
0.181
-0.009
0.030
-0.043
-0.152
0.002 0.147
-0.090
0.080
DGA
DGO
DGI
CDC
LA
LO
LI
LR
Katz LC
Triag
CyclC
0.015 0.024
CCI BC
0.114
-0.017
-0.012
-0.139
-0.139
PRC
FC
Stress
ZO
ZI
-0.172
-0.110
CCO
-0.084
-0.127
Loc
EC
-0.059
Lev
SC
0.106
VCDG
-0.138
0.285
-0.018
CCT
-0.093
-0.012
CCA
CC2
-0.086
CCI
ADnb
-0.833
0.094
CCO
0.822
0.822
0.944
0.814
0.834
0.969
0.915
0.801 0.904
0.922
0.907
0.926
0.686
-0.760
-0.827
-0.445
-0.808
-0.146
EccI
-0.672
0.019
-0.289
EccO
-0.741
0.047
-0.433
-0.487
-0.573
-0.624
0.855
0.930
1.000
-0.018
1.000
-0.018
REC
HD
HD
REC
Region
0.863
0.863
0.941
0.697
0.811
0.978
0.936
0.833 0.878
0.953
0.973
0.969
0.719
-0.780
-0.219
-0.863
-0.867
-0.402
-0.859
-0.166
-0.631
-0.810
0.981
0.860 -0.589
0.076
-0.486
-0.511
-0.615
-0.636
0.884
0.955
1.000
0.971
-0.062
DGA
0.896
0.896
0.881
0.693
0.629
0.950
0.988
0.636 0.786
0.989
0.935
0.926
0.709
-0.787
-0.199
-0.824
-0.862
-0.320
-0.867
0.031
-0.616
-0.748
0.957
0.678 -0.547
0.036
-0.375
-0.529
-0.560
-0.803
0.705
1.000
0.955
0.930
-0.090
DGO
0.648
0.648
0.858
0.573
0.945
0.839
0.679
0.987 0.860
0.718
0.851
0.856
0.600
-0.622
-0.209
-0.763
-0.711
-0.456
-0.685
-0.446
-0.535
-0.756
0.834
0.987 -0.545
0.125
-0.570
-0.386
-0.586
-0.253
1.000
0.705
0.884
0.855
-0.006
DGI
Table 12 Correlation coefficients of local parameters of the bilateral BG2 network
-0.807
-0.807
-0.532
-0.657
-0.217
-0.620
-0.821
-0.179 -0.418
-0.767
-0.616
-0.677
-0.625
0.673
0.009
0.622
0.729
0.107
0.776
-0.278
0.476
0.555
-0.627
-0.221 0.442
-0.048
0.148
0.477
0.377
1.000
-0.253
-0.803
-0.636
-0.624
0.181
CDC
-0.423
-0.423
-0.577
-0.450
-0.548
-0.593
-0.545
-0.603 -0.540
-0.616
-0.536
-0.576
-0.305
0.285
0.503
0.735
0.653
0.491
0.649
0.416
0.493
0.779
-0.565
-0.612 0.412
0.281
0.830
0.814
1.000
0.377
-0.586
-0.560
-0.615
-0.573
-0.009
LA
-0.448
-0.448
-0.452
-0.469
-0.388
-0.480
-0.520
-0.356 -0.413
-0.587
-0.460
-0.505
-0.373
0.340
0.339
0.603
0.565
0.213
0.634
0.156
0.433
0.700
-0.464
-0.405 0.589
0.052
0.373
1.000
0.814
0.477
-0.386
-0.529
-0.511
-0.487
0.030
LO
-0.246
-0.246
-0.446
-0.280
-0.493
-0.464
-0.365
-0.630 -0.417
-0.408
-0.412
-0.447
-0.121
0.114
0.506
0.576
0.488
0.523
0.440
0.526
0.330
0.578
-0.436
-0.606 0.106
0.365
1.000
0.373
0.830
0.148
-0.570
-0.375
-0.486
-0.433
-0.043
LI
0.118
0.118
-0.044
0.094
0.038
-0.004
0.071
0.076 -0.054
0.052
0.151
0.182
0.347
-0.195
0.131
0.193
0.078
0.652
-0.103
0.110
0.064
-0.106
0.033
0.117 -0.404
1.000
0.365
0.052
0.281
-0.048
0.125
0.036
0.076
0.047
-0.152
LR
0.598
0.598
0.812
0.559
0.920
0.816
0.655
0.988 0.807
0.703
0.823
0.831
0.529
-0.541
-0.292
-0.721
-0.667
-0.389
-0.675
-0.494
-0.511
-0.747
0.808
1.000 -0.496
0.117
-0.606
-0.405
-0.612
-0.221
0.987
0.678
0.860
0.829
0.002
Katz
Brain Struct Funct
-0.211
-0.043
-0.194
-0.135
-0.019
-0.103
-0.016
-0.096
LC
PCI
PCA
RadO
RadI
CenO
CenI
Hub
Aut
Region
0.589
0.106
-0.404
-0.496
1.000 -0.537
0.680
0.401
0.071
0.595
0.114
0.596
0.564
0.026
LO
LI
LR
Katz
LC Triag
CyclC
EccO
EccI
CCO
CCI
CCA
CCT
CC2
-0.682
0.412
LA
VCDG
0.442
CDC
0.623
0.808
-0.545
DGI
ADnb
0.033
-0.547
DGO
0.957
0.642
-0.747
-0.273
-0.810
-0.822
-0.401
-0.786
-0.107
-0.617
-0.725
-0.537 1.000
-0.436
-0.464
-0.565
-0.627
0.834
0.957
0.981
-0.558
-0.589
REC
-0.090
Triag
0.925
0.820
0.934
0.790
0.912
0.521
0.481
0.517
DGA
0.147
-0.109
HD
0.747
-0.013
ZA
PCO 0.861
REC
HD
Region
Table 12 continued
-0.722
0.647
0.228
0.884
0.853
0.341
0.906
0.284
0.514
1.000
0.680 -0.725
-0.747
-0.106
0.578
0.700
0.779
0.555
-0.756
-0.748
-0.810
-0.741
0.080
CyclC
0.949
0.857
0.962
0.833
0.949
0.601
0.545
0.584
0.897
0.765
DGA
-0.446
0.422
0.216
0.580
0.534
0.382
0.538
0.210
1.000
0.514
0.401 -0.617
-0.511
0.064
0.330
0.433
0.493
0.476
-0.535
-0.616
-0.631
-0.672
0.019
EccO
0.997
0.673
0.995
0.648
0.967
0.518
0.409
0.554
0.863
0.618
DGO
-0.074
-0.010
0.204
0.224
0.160
0.286
0.051
1.000
0.210
0.284
0.071 -0.107
-0.494
0.110
0.526
0.156
0.416
-0.278
-0.446
0.031
-0.166
-0.146
-0.289
EccI
0.695
0.987
0.731
0.969
0.743
0.619
0.657
0.521
0.783
0.855
DGI
-0.779
0.757
0.105
0.877
0.898
0.260
1.000
0.051
0.538
0.906
0.595 -0.786
-0.675
-0.103
0.440
0.634
0.649
0.776
-0.685
-0.867
-0.859
-0.808
0.094
CCO
-0.813
-0.231
-0.795
-0.200
-0.788
-0.290
-0.094
-0.433
-0.666
-0.278
CDC
-0.208
0.345
-0.057
0.658
0.542
1.000
0.260
0.286
0.382
0.341
0.114 -0.401
-0.389
0.652
0.523
0.213
0.491
0.107
-0.456
-0.320
-0.402
-0.445
-0.086
CCI
-0.590
-0.656
-0.594
-0.619
-0.642
-0.747
-0.684
-0.693
-0.449
-0.402
LA
-0.796
0.856
-0.008
0.966
1.000
0.542
0.898
0.160
0.534
0.853
0.596 -0.822
-0.667
0.078
0.488
0.565
0.653
0.729
-0.711
-0.862
-0.867
-0.827
-0.012
CCA
-0.559
-0.438
-0.562
-0.389
-0.610
-0.586
-0.429
-0.605
-0.419
-0.299
LO
-0.721
0.769
0.057
1.000
0.966
0.658
0.877
0.224
0.580
0.884
0.564 -0.810
-0.721
0.193
0.576
0.603
0.735
0.622
-0.763
-0.824
-0.863
-0.833
-0.018
CCT
-0.398
-0.637
-0.400
-0.639
-0.446
-0.666
-0.722
-0.565
-0.314
-0.352
LI
0.293
-0.297
1.000
0.057
-0.008
-0.057
0.105
0.204
0.216
0.228
0.026 -0.273
-0.292
0.131
0.506
0.339
0.503
0.009
-0.209
-0.199
-0.219
-0.138
0.285
CC2
0.021
0.109
0.076
0.108
0.134
0.134
0.101
0.167
0.198
0.309
LR
-0.930
1.000
-0.297
0.769
0.856
0.345
0.757
-0.010
0.422
0.647
0.623 -0.747
-0.541
-0.195
0.114
0.340
0.285
0.673
-0.622
-0.787
-0.780
-0.760
-0.093
ADnb
0.674
0.987
0.708
0.986
0.735
0.673
0.700
0.581
0.735
0.822
Katz
Brain Struct Funct
123
123
0.709
0.600
-0.625
-0.305
-0.373
-0.121
0.347
0.529
-0.682
CDC
LA
LO
LI
LR
Katz
LC
VCDG
Region
DGI
-0.538
Aut
DGO
-0.569
Hub
0.719
-0.590
CenI
DGA
-0.487
CenO
0.686
-0.605
RadI
0.106
-0.426
RadO
REC
-0.397 -0.373
PCI PCA
HD
-0.618
-0.480
FC
PCO
-0.527
PRC
-0.537
-0.516
SC
-0.598
-0.570
EC
ZA
-0.489
BC
ZI
-0.486
CCI
-0.511
-0.556
CCO
-0.598
-0.622
Loc
ZO
-0.645
Lev
Stress
LC
Region
Table 12 continued
-0.645
0.831
0.182
-0.447
-0.505
-0.576
-0.677
0.856
0.926
0.969
0.926
-0.059
Lev
0.947
0.797
0.953
0.767
0.919
0.531
0.502 0.504
0.840
0.682
0.828
0.828
0.940
0.652
0.742
0.996
0.942
0.864
0.779
0.959
0.968
0.913
Triag
-0.622
0.823
0.151
-0.412
-0.460
-0.536
-0.616
0.851
0.935
0.973
0.907
-0.127
Loc
-0.757
-0.794
-0.786
-0.749
-0.839
-0.762
-0.732 -0.654
-0.783
-0.679
-0.753
-0.753
-0.723
-0.558
-0.711
-0.720
-0.722
-0.679
-0.728
-0.771
-0.774
-0.849
CyclC
-0.556
0.703
0.052
-0.408
-0.587
-0.616
-0.767
0.718
0.989
0.953
0.922
-0.110
CCO
-0.630
-0.536
-0.648
-0.494
-0.633
-0.303
-0.359 -0.161
-0.615
-0.544
-0.583
-0.583
-0.658
-0.604
-0.572
-0.634
-0.619
-0.639
-0.516
-0.634
-0.564
-0.588
EccO
-0.486
0.988
0.076
-0.630
-0.356
-0.603
-0.179
0.987
0.636
0.833
0.801
0.015
CCI
0.013
-0.493
0.001
-0.480
-0.034
-0.265
-0.136 -0.392
-0.088
-0.322
0.046
0.046
-0.204
-0.062
-0.470
-0.125
0.067
-0.267
-0.519
0.015
-0.107
-0.153
EccI
-0.489
0.807
-0.054
-0.417
-0.413
-0.540
-0.418
0.860
0.786
0.878
0.904
0.024
BC
-0.872
-0.693
-0.890
-0.673
-0.932
-0.669
-0.731 -0.493
-0.840
-0.632
-0.860
-0.860
-0.753
-0.684
-0.644
-0.783
-0.849
-0.682
-0.641
-0.869
-0.823
-0.912
CCO
-0.570
0.655
0.071
-0.365
-0.520
-0.545
-0.821
0.679
0.988
0.936
0.915
-0.172
EC
-0.313
-0.442
-0.300
-0.368
-0.242
-0.081
0.010 -0.197
-0.307
-0.299
-0.272
-0.272
-0.557
-0.330
-0.563
-0.440
-0.280
-0.616
-0.456
-0.289
-0.300
-0.310
CCI
-0.516
0.816
-0.004
-0.464
-0.480
-0.593
-0.620
0.839
0.950
0.978
0.969
-0.084
SC
-0.855
-0.701
-0.865
-0.639
-0.862
-0.496
-0.506 -0.448
-0.817
-0.613
-0.837
-0.837
-0.830
-0.653
-0.692
-0.820
-0.830
-0.776
-0.659
-0.839
-0.818
-0.882
CCA
-0.527
0.920
0.038
-0.493
-0.388
-0.548
-0.217
0.945
0.629
0.811
0.834
0.114
PRC
-0.822
-0.763
-0.832
-0.703
-0.832
-0.547
-0.530 -0.490
-0.803
-0.644
-0.799
-0.799
-0.858
-0.634
-0.774
-0.818
-0.783
-0.830
-0.726
-0.813
-0.797
-0.857
CCT
-0.480
0.559
0.094
-0.280
-0.469
-0.450
-0.657
0.573
0.693
0.697
0.814
-0.017
FC
-0.228
-0.279
-0.226
-0.308
-0.267
-0.549
-0.483 -0.495
-0.043
-0.041
-0.039
-0.039
-0.102
0.081
-0.005
-0.274
-0.248
0.005
-0.254
-0.297
-0.267
-0.121
CC2
-0.511
0.812
-0.044
-0.446
-0.452
-0.577
-0.532
0.858
0.881
0.941
0.944
-0.012
Stress
-0.759
-0.555
-0.774
-0.488
-0.731
-0.178
-0.203 -0.180
-0.800
-0.617
-0.814
-0.814
-0.765
-0.621
-0.626
-0.731
-0.745
-0.727
-0.525
-0.731
-0.777
-0.824
ADnb
Brain Struct Funct
-0.121
-0.824
-0.446
-0.074
-0.779
-0.208
-0.796
-0.721
0.293
-0.930
1.000 0.816
0.713
0.665
0.509
0.649
0.673
0.623
0.610
0.580
0.675
0.768
0.768
0.680
0.794
0.346
0.244 0.307
0.714
0.500
0.721
0.563
0.692
EccO
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG Lev
Loc
CCO
CCI
BC
EC
SC
PRC
FC
Stress
ZO
ZI
ZA
PCO
PCI
PCA RadO
RadI
CenO
CenI
Hub
Aut
0.913
0.923
0.837
0.944
0.823
0.952
0.517 0.618
0.635
0.933
0.804
0.892
0.892
0.866
0.700
0.786
0.903
0.913
0.804
0.801
0.915
0.949
0.816 1.000
-0.857
-0.882
-0.310
-0.912
-0.153
-0.588
-0.849
0.642
-0.722
Lev
CyclC
VCDG
Triag
Region
Table 12 continued
0.923
0.821
0.941
0.801
0.923
0.514 0.572
0.545
0.885
0.735
0.862
0.862
0.890
0.570
0.733
0.949
0.922
0.809
0.795
0.936
1.000
0.713 0.949
-0.777
-0.267
-0.797
-0.818
-0.300
-0.823
-0.107
-0.564
-0.774
0.968
Loc
0.989
0.699
0.992
0.675
0.977
0.467 0.590
0.624
0.839
0.621
0.860
0.860
0.874
0.671
0.627
0.952
0.980
0.774
0.656
1.000
0.936
0.665 0.915
-0.731
-0.297
-0.813
-0.839
-0.289
-0.869
0.015
-0.634
-0.771
0.959
CCO
0.629
0.989
0.667
0.982
0.689
0.685 0.639
0.532
0.725
0.821
0.584
0.584
0.809
0.524
0.935
0.786
0.610
0.817
1.000
0.656
0.795
0.509 0.801
-0.525
-0.254
-0.726
-0.659
-0.456
-0.641
-0.519
-0.516
-0.728
0.779
CCI
0.770
0.812
0.783
0.757
0.741
0.412 0.395
0.330
0.777
0.736
0.708
0.708
0.978
0.668
0.901
0.880
0.734
1.000
0.817
0.774
0.809
0.649 0.804
-0.727
0.005
-0.830
-0.776
-0.616
-0.682
-0.267
-0.639
-0.679
0.864
BC
0.989
0.654
0.989
0.632
0.968
0.413 0.541
0.589
0.848
0.600
0.884
0.884
0.837
0.705
0.586
0.935
1.000
0.734
0.610
0.980
0.922
0.673 0.913
-0.745
-0.248
-0.783
-0.830
-0.280
-0.849
0.067
-0.619
-0.722
0.942
EC
0.941
0.803
0.946
0.771
0.913
0.511 0.534
0.506
0.823
0.676
0.808
0.808
0.948
0.695
0.762
1.000
0.935
0.880
0.786
0.952
0.949
0.623 0.903
-0.731
-0.274
-0.818
-0.820
-0.440
-0.783
-0.125
-0.634
-0.720
0.996
SC
0.622
0.927
0.654
0.895
0.655
0.525 0.486
0.403
0.747
0.837
0.610
0.610
0.850
0.652
1.000
0.762
0.586
0.901
0.935
0.627
0.733
0.610 0.786
-0.626
-0.005
-0.774
-0.692
-0.563
-0.644
-0.470
-0.572
-0.711
0.742
PRC
0.701
0.548
0.706
0.510
0.714
0.281 0.336
0.407
0.655
0.547
0.645
0.645
0.673
1.000
0.652
0.695
0.705
0.668
0.524
0.671
0.570
0.580 0.700
-0.621
0.081
-0.634
-0.653
-0.330
-0.684
-0.062
-0.604
-0.558
0.652
FC
0.867
0.813
0.877
0.763
0.835
0.449 0.451
0.402
0.826
0.725
0.784
0.784
1.000
0.673
0.850
0.948
0.837
0.978
0.809
0.874
0.890
0.675 0.866
-0.765
-0.102
-0.858
-0.830
-0.557
-0.753
-0.204
-0.658
-0.723
0.940
Stress
Brain Struct Funct
123
123
-0.583
0.046
-0.860
-0.272
-0.837
-0.799
-0.039
-0.814
0.768
0.892
0.862
0.860
0.584 0.708
0.884
0.808
0.610
0.645
0.784
0.000
1.000
EccO
EccI
CCO
CCI
CCA
CCT
CC2
ADnb
VCDG
Lev
Loc
CCO
CCI BC
EC
SC
PRC
FC
Stress
ZO
ZI
0.598 -0.598
0.828
0.118
LR
Katz LC
-0.753
-0.246
LI
CyclC
0.309
-0.448
Triag
-0.352
-0.423
LO
0.671
1.000
0.725
0.547
0.837
0.676
0.600
0.821 0.736
0.621
0.735
0.804
0.680
-0.617
-0.041
-0.644
-0.613
-0.299
-0.632
-0.322
-0.544
-0.679
0.682
0.822 -0.537
-0.299
-0.402
-0.278
0.855
0.618
LA
0.896
DGO
0.765
0.648
0.863
DGA
0.747
-0.013
-0.807
0.822
REC
CDC
-0.139
HD
ZI
DGI
ZO
Region
Table 12 continued
0.953
0.671
0.826
0.655
0.747
0.823
0.848
0.725 0.777
0.839
0.885
0.933
0.794
-0.800
-0.043
-0.803
-0.817
-0.307
-0.840
-0.088
-0.615
-0.783
0.840
0.735 -0.618
0.198
-0.314
-0.419
-0.449
-0.666
0.783
0.863
0.897
0.861
-0.109
ZA
0.445
0.953
0.402
0.407
0.403
0.506
0.589
0.532 0.330
0.624
0.545
0.635
0.346
-0.203
-0.483
-0.530
-0.506
0.010
-0.731
-0.136
-0.359
-0.732
0.502
0.581 -0.397
0.167
-0.565
-0.605
-0.693
-0.433
0.521
0.554
0.584
0.517
-0.211
PCO
0.276
0.445
0.449
0.281
0.525
0.511
0.413
0.685 0.412
0.467
0.514
0.517
0.244
-0.180
-0.495
-0.490
-0.448
-0.197
-0.493
-0.392
-0.161
-0.654
0.504
0.700 -0.373
0.101
-0.722
-0.429
-0.684
-0.094
0.657
0.409
0.545
0.481
-0.043
PCI
0.406
0.276
0.451
0.336
0.486
0.534
0.541
0.639 0.395
0.590
0.572
0.618
0.307
-0.178
-0.549
-0.547
-0.496
-0.081
-0.669
-0.265
-0.303
-0.762
0.531
0.673 -0.426
0.134
-0.666
-0.586
-0.747
-0.290
0.619
0.518
0.601
0.521
-0.194
PCA
0.883
0.406
0.835
0.714
0.655
0.913
0.968
0.689 0.741
0.977
0.923
0.952
0.714
-0.731
-0.267
-0.832
-0.862
-0.242
-0.932
-0.034
-0.633
-0.839
0.919
0.735 -0.605
0.134
-0.446
-0.610
-0.642
-0.788
0.743
0.967
0.949
0.912
-0.135
RadO
0.584
0.883
0.763
0.510
0.895
0.771
0.632
0.982 0.757
0.675
0.801
0.823
0.500
-0.488
-0.308
-0.703
-0.639
-0.368
-0.673
-0.480
-0.494
-0.749
0.767
0.986 -0.487
0.108
-0.639
-0.389
-0.619
-0.200
0.969
0.648
0.833
0.790
-0.019
RadI
0.899
0.584
0.877
0.706
0.654
0.946
0.989
0.667 0.783
0.992
0.941
0.944
0.721
-0.774
-0.226
-0.832
-0.865
-0.300
-0.890
0.001
-0.648
-0.786
0.953
0.708 -0.590
0.076
-0.400
-0.562
-0.594
-0.795
0.731
0.995
0.962
0.934
-0.103
CenO
0.610
0.899
0.813
0.548
0.927
0.803
0.654
0.989 0.812
0.699
0.821
0.837
0.563
-0.555
-0.279
-0.763
-0.701
-0.442
-0.693
-0.493
-0.536
-0.794
0.797
0.987 -0.569
0.109
-0.637
-0.438
-0.656
-0.231
0.987
0.673
0.857
0.820
-0.016
CenI
0.894
0.610
0.867
0.701
0.622
0.941
0.989
0.629 0.770
0.989
0.923
0.923
0.692
-0.759
-0.228
-0.822
-0.855
-0.313
-0.872
0.013
-0.630
-0.757
0.947
0.674 -0.538
0.021
-0.398
-0.559
-0.590
-0.813
0.695
0.997
0.949
0.925
-0.096
Hub
0.560
0.894
0.736
0.441
0.803
0.785
0.646
0.950 0.700
0.690
0.822
0.798
0.459
-0.460
-0.430
-0.643
-0.610
-0.268
-0.642
-0.460
-0.434
-0.727
0.788
0.969 -0.462
0.179
-0.630
-0.368
-0.597
-0.204
0.944
0.653
0.826
0.759
-0.094
Aut
Brain Struct Funct
0.954
0.649
0.669
1.000
Cluster coefficient (triangle based): The triangle based cluster coefficient (Fagiolo 2007) of a node n is the number of triangles around n divided by the maximum possible number. In this version of the cluster coefficient reciprocal edges to a neighbor of a node n can affect the cluster coefficient of node n. In the other version only edges between neighbors of n have an influence to the cluster coefficient of node n. t! ðiÞ tmax ðiÞ ! ! t w ðiÞ w CT ¼ tmax ðiÞ
0.710
0.995
0.984
0.648
0.669
0.685 0.995 1.000 0.686
1.000
0.964 0.648 0.686 1.000
0.710
0.724 0.972 0.985 0.723
0.984
0.736 0.545 0.572 0.715
0.739
0.760 0.421 0.445 0.706
0.699
0.635 0.586 0.612 0.635
0.720
0.688 0.859
0.603
0.765 0.613
0.879 0.730
0.749
0.655 0.825
0.836
Aut CenO RadI
CenI
Hub
Brain Struct Funct
CT! ¼
ð55Þ ð56Þ
Cluster coefficient: Number of edges between the neighbors of a node divided by the maximum possible number. C ! ðiÞ refers to all neighbors of i. X 1 ajk C ! ðiÞ ¼ ð57Þ jNi j ðjNi j 1Þ j;k2N i
! ðiÞ refers to the out-neighbors of i. Cout X 1 ! Cout ðiÞ ¼ out ajk out jNi j ðjNi j 1Þ j;k2N out
0.972
0.739
0.985
0.723
1.000
0.673
0.519
0.724
0.876
0.677
RadO
j6¼k
ð58Þ
i
0.545
! Cin ðiÞ refers to the in-neighbors of i. X 1 ! Cin ðiÞ ¼ in ajk jNi j ðjNiin j 1Þ in j;k2Ni j6¼k
ð59Þ
In the weighted case the aij are replaced by the wij . Average cluster coefficient: n 1X C! ¼ C! ð60Þ n i¼1 i Most local parameters show high correlations with DGA
0.421 Aut
0.894
0.613
0.859
0.586
0.699 0.720 Hub
0.610
0.836
0.749
0.603
0.572 0.445 CenI
0.899
0.655
0.879
0.612
0.715 0.706 CenO
0.584
0.825
0.730
0.635
0.673 0.519
1.000 0.880
0.876
0.724
0.496
0.677 0.883
0.406 RadO
RadI
0.479
0.942
0.880 1.000 PCA
0.276
0.421
0.355
0.704
0.942 0.704 PCI
0.445
0.491
0.505
1.000
0.479 0.355
0.496 0.421
0.505 1.000
0.491 1.000
0.862 0.953
0.671 ZA
PCO
0.862
PCI ZO Region
Table 12 continued
ZI
ZA
PCO
PCA
j6¼k
and n ! 1X ! Cw ¼ Ciw n i¼1
ð61Þ
Small worldness S: C Crand
S¼
d
ð62Þ
drand
Centrality: Pn degmax degðiÞ n degmax 2 ‘ ¼ CD ¼ i¼1 ðn 1Þ ðn 2Þ ðn 1Þ ðn 2Þ
ð63Þ
This centrality (degree centrality) is defined for an undirected network based on undirected degrees. A directed or
123
Brain Struct Funct Table 13 The local parameters of the bilateral BG2 network were applied to the PCA Component
DGAll
CluCAll
CluC2
AvgDGnb
VCDG
Loc
Share (%)
1
0.440
-0.439
0.154
-0.457
0.438
0.434
74.978
2
-0.280
0.101
0.860
-0.166
0.216
-0.312
19.582
3
-0.040
0.818
-0.148
-0.182
0.434
0.292
2.705
4
-0.425
-0.296
-0.417
-0.049
0.626
-0.404
1.875
5
-0.521
-0.056
-0.133
-0.713
-0.408
0.180
0.621
6
-0.523
-0.191
0.151
0.468
0.117
0.659
0.24
The first two components determine the axes of the PCA-plane Share The share of variance of the data represented by the component
weighted version is not available yet. For the calculation the directed network is transferred to an undirected one. Circle length LC: dði; iÞ; dði; iÞ\1 LCðiÞ ¼ ð64Þ 0; dði; iÞ ¼ 1
! 1 X degwall ðjÞ degNB w ðiÞ ¼ jNi j j2Ni
ð70Þ
Variation coefficient of neighbor degree: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P 2 1 ðdeg ðjÞ degNBðiÞÞ all j2N jNi j i
ð71Þ
Eccentricity out: Eccentricity out, the output eccentricity of the vertex i is the maximum distance from i to any vertex.
VCðiÞ ¼
Eccout ðiÞ ¼ maxfdði; jÞjj 2 Ng
The weighted case is analogue. Locality index of node i ðLocðiÞÞ: The locality index of node i is the fraction of edges adjacent to nodes in Niþ whose source and target lie in Niþ . P P k2Niþ ajk j2N þ
ð65Þ
Eccentricity in: Eccentricity in, the input eccentricity of the vertex i is the maximum distance from i to any vertex. Eccin ðiÞ ¼ maxfdðj; iÞjj 2 Ng
ð66Þ
Cluster-coefficient of second neighbors: The cluster-coefficient of second neighbors (Hierarchical directed cluster coefficient of second (indirect) neighbors) C2 ðiÞ is the number of edges between the 2nd neighbors of node i, divided by the maximum possible number of edges. In the weighted case it is the sum of weights of the edges between the 2nd neighbors of node i, divided by the maximum possible sum. With 
