Relations Between the Geometry of Cortical Gyrification and White-Matter Network Architecture James A. Henderson and Peter A. Robinson

Abstract

A geometrically based network model of cortico-cortical white-matter connectivity is used in combination with diffusion spectrum MRI (DSI) data to show that white-matter cortical network architecture is founded on a homogeneous, isotropic geometric connection principle. No other special information about single connections or groups of connections is required to generate networks very similar to experimental ones. This model provides excellent agreement with experimental DSI frequency distributions of network measures—degree, clustering coefficient, path length, and betweenness centrality. In the model, these distributions are a result of geometrically induced spatial variations in the values of these measures with deep nodes having more hublike properties than superficial nodes. This leads to experimentally testable predictions of corresponding variations in real cortexes. The convoluted geometry of the cortex is also found to introduce weak modularity, similar to the lobe structure of the cortex, with the boundaries between modules having hublike properties. These findings mean that some putative discoveries regarding the structure of white-matter cortical networks are simply artifacts and/or consequences of geometry. This model may help provide insight into diseases associated with differences in gyrification as well as evolutionary development of the cortex. Key words: connectivity;

cortex; geometry; gyrification; networks; white matter

Introduction

T

he cortex contains a thin sheet of neuronal gray matter on its surface. This sheet surrounds a very large number of long-range white-matter neuronal fibers, many of which are projections between different regions of the cortical gray-matter surface (Braitenberg and Schu¨z, 1998). Currently, there is much interest in studying these connecting fibers as a network of connections between small portions of the gray-matter surface (Bullmore and Sporns, 2009, 2012; Kaiser, 2011; Sporns, 2011). Many real-world networks have been found to have scalefree architectures where the degree frequency distribution follows a decaying power-law function (Boccaletti et al., 2006). However, for the cortex, orders of magnitude differences in node degrees would imply vast differences in the properties of different regions of the cortex. Instead, the cortex appears much more structurally homogeneous, though still with distinct differences ranging from neuronal content to associations with high-level processing tasks (Toga et al., 2006). Another feature of many networks is modularity or community structure, where the network is divided into

groups of nodes that connect strongly between themselves and less strongly to other groups. A variety of studies on structural and functional cortical networks have identified modular structure (Chen et al., 2008; Meunier et al., 2009, 2010; Zhou et al., 2006); however, a detailed understanding of how detected modules are associated with the underlying properties of the network is still lacking. Recently, modeling of cortical networks using a spherical surface (Henderson and Robinson, 2011, 2013) has shown evidence that much of the above-observed modular structure, and the concept of a strongly modular cortex has been misconstrued. Based upon a standard modularity measure, experimental diffusion spectrum MRI (DSI) data were found to be no more modular than the isotropic, homogeneous sphere model that had equally high modularity score with similar modular division. This model implied that most intramodule connectivity is not derived from an explicitly modular connection rule but is predominately a result of an underlying network of homogeneous, isotropic, short-range local connectivity in which nearby regions are linked most strongly and further regions less so (Henderson and Robinson, 2011, 2013). However, a separate, weak modular structure

School of Physics, University of Sydney, Sydney, New South Wales, Australia. Brain Dynamics Center, Sydney Medical School–Western, University of Sydney, Westmead, New South Wales, Australia.

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can still exist and is likely to be present because of secondary perturbations overlaid on the homogeneous, isotropic base network structure, though the properties of such perturbations are not known. Network modularity is widely thought to be an important factor in cortical function. The cortex performs many highly specialized functions, and somehow the outputs of these specializations are integrated at different locations, processed, and further dispersed in a complex network arrangement (Brett et al., 2002). Presumably, long-range white-matter connections play an important role in this integration and dispersion of information, and given the specialization of function, it is plausible to expect that there would correspondingly be a high degree of specialization of white-matter connections. Here we investigate the relative importance at the network level of uniform network-wide properties versus the details of individual connections and nodes. Specifically, if we rewire a cortical network using its network average connection kernel, we ask whether we get a similar network in terms of its distributions of network measures. In addition to modularity, other important network measures, mean degree < k > , mean clustering coefficient < CC > , and mean path length < L > , are used to characterize and understand complex networks (Honey et al., 2010). Modeling using plane and spherical geometries has established strong links between these properties and a combination of 2D geometry with a homogeneous, isotropic local connection rule (Henderson and Robinson, 2011, 2013). However, the impact on network structure of convoluted cortical geometry is unknown. Gyrification or folding of the cortex begins before birth and extends into childhood. Gyrification is necessary to fit the large 2D cortical surface into the limited cranial volume. The arrangement of a thin 2D neuronal sheet (gray matter) over a volume of axonal connections (white matter) is thought to be caused by an evolutionary preference for minimization of axon wiring length and volume (Ruppin et al., 1993). The mechanism producing gyrification has not been conclusively determined; however, it is considered to be primarily a result of tension in cortical fibers causing buckling of the cortical surface and/or variations in cellular growth rates (Armstrong et al., 1995; Geng et al., 2009; White et al., 2010). Differences in cortical geometry have been associated with sex, ethnicity, and age, as well as a variety of disorders such as attention deficit/hyperactivity, Alzheimer’s, bipolar, major depressive, obsessive compulsive, and schizophrenia (Li et al., 2007; Mirakhur et al., 2009; Tymofiyeva et al., 2012; White and Hilgetag, 2011; White et al., 2010; Wobrock et al., 2010; Zhang et al., 2009; Zilles et al., 2001). Studies that provide understanding of how connectivity relates to the geometry of the cortex may thus play a role in understanding these disorders. In this article we present a network model of cortico-cortical fibers based upon real cortical surface geometry and an isotropic, homogeneous connection kernel, meaning that each node is connected to the network with the same connection rule. The model produces very similar network measure frequency distributions to those observed in an experimental human DSI cortical network (Hagmann et al., 2008). This implies that these distributions are related to the gyrification of the cortical surface and not a result of specialization of single connections. This geometry produces spatial distribu-

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tions in the model network measures leading to experimentally testable predictions for the variation of network measure values across the cortical surface. It is found that nodes deep within the brain have more hublike properties. The modular properties of the model are then investigated. It is found that the model network has similarities in modular division to the lobe structure of the cortex, implying that it is the gyrification of the cortex that leads to its modular division. Further, we find that the set of nodes forming the boundaries between modules has hublike properties, forming connector hubs between modules. This leads to a more detailed understanding of the architecture of cortical networks—a network generated predominantly using a single network-wide local connection rule that has modularity induced by its underlying convoluted geometry. This geometry produces regions that have hublike properties and subsequently link together and form the boundaries between modules. Methods

Here we outline the methods of construction for the network model used in the later analysis, as well as a description of the network measures, k, CC, L, b, and Q, that are applied to the model. Briefly, the construction of the network model consists of parcellating the cortical surface into nodes of similarly sized areas. We calculate the shortest distances between all pairs of nodes through the cortical white-matter volume. Connections are distributed with a connection kernel that is homogeneous (the same at all nodes) and isotropic (the same in all directions) and decays exponentially with the minimum separation of nodes (fiber length) through the white matter. The model is compared with an experimental DSI connectivity data set from Hagmann and associates (2008), consisting of 998 connected nodes, 9 of which have zero connections and are disconnected from the network. We interpret these as errors and use only the 989 connected nodes. Parcellation of the cortical surface into nodes

In this work we use the 20,484-vertex canonical cortical surface mesh available with the SPM8 software package (Ashburner et al., 2012). This is a surface formed by the white- and gray-matter interface. The surface contains a corpus callosum, which will allow connections between hemispheres to be formed; however, no other commissural fibers are present in the model, although other much smaller commissures are known to exist (Raybaud, 2010; Wakana et al., 2004). In applying network theory to the cortex, the cortex must be parcellated (discretized) into regions of interest that correspond to nodes used in the network model. In investigating geometric effects on cortical networks, the choice of geometry in the parcellation is important. We now outline a parcellation procedure that aims to create a parcellation with 989 regions, equal the number in the DSI data that we compare to, and fewer than the number of vertexes in the surface model. The procedure aims to minimize variation in shape and surface area of the regions of interest to simplify and reduce confusion between effects caused by the (arbitrary) choice of parcellation geometry and those caused by the cortical geometry. The iterative procedure is as follows:

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1. Begin with all vertexes in the cortical surface model being individual nodes. 2. Calculate the surface area corresponding to each node. 3. Remove the node with the lowest surface area and assign each of its composite surface vertexes to their corresponding closest (via a path on the cortical surface) remaining node. 4. Compute the area of each node based on its updated composite surface model vertexes from step 3. 5. Go to step 3 until the desired number of nodes remains. Calculation of the minimum fiber length paths

In this model we assume that cortical axon fibers are distributed according to the shortest possible distance sv, between their end points through the cortical white-matter volume. The following procedure is used to calculate the minimal distance between nodes (generated as in the section Parcellation of the Cortical Surface into Nodes) through the white matter. 1. Begin with a surface model of the cortex. 2. Generate a set of gridpoints with uniform spacing filling the interior volume of the surface model. 3. Form these gridpoints into a network by connecting each gridpoint to its neighboring gridpoints. 4. The connectivity model surface nodes from the parcellation are connected to their nearest points in this internal volume grid, just under the surface. 5. The minimum distance, sv, between model network nodes measured via paths through the cortical whitematter volume is calculated by searching for the shortest path through this grid network. Connection kernels

Here we describe how the network nodes in the model are connected together to form a network. The DSI connectivity data for the human cortex published by Hagmann and associates (2008) were shown by Henderson and Robinson (2013) to have an approximately exponentially decreasing probability of connection with Euclidean (straight line) separation of nodes. It should be noted that this fit is accurate for approximately two orders of magnitude in probability as can be seen in Figure 4b. After this, there are more connections present in the data than are given by this fit; however, they are comparatively few in number. Unfortunately, the cortical surface model for the experimental DSI data is not available to us, so the minimal distances through the white matter between nodes are unknown and a probability distribution based on it cannot be calculated. Hence, we assume the same decaying exponential functional form for the probability distribution with sv: P(sv ) = Ce

ssv 0

,

(1)

where C and s0 are parameters of the model. These parameters are fitted to the experimental DSI data by calculating the network measures described in the Network Measures section as functions of C and s0 and then selecting the C and s0 values that produce model networks most closely match-

ing the DSI measure values (see the Model Geometry section). Note that since the DSI data are undirected, we use an undirected model in which a connection, or absence of a connection, between a pair of nodes is decided once; however, a directed model could be constructed by doing this for each direction of connection. Network measures

We compare models and experimental data with the following commonly used network measures that highlight key properties of networks (Bullmore and Sporns, 2009; Honey et al., 2010). In this article, only unweighted, undirected networks are considered. 1. Degree k: For a single-node i, it is the number of connections to other nodes. The mean degree, < k > , is ki averaged over all nodes in the network (Honey et al., 2010). ki = +j Aij ,

(2)

where A is the network connection matrix. 2. Path length L: For a single-node i, Li is the average of the shortest path (fewest connections) to all other nodes in the network. The mean path length < L > is the average of Li over all nodes in the network. This is not a geometrical length; it relates to numbers of synapses or processing steps between nodes (Honey et al., 2010). There is no explicit equation for path length in terms of the connection matrix (CM); it is determined algorithmically. 3. Clustering coefficient CC: For a single-node i, CCi is the ratio of actual number of connections between its neighboring nodes to the maximum number possible: CCi =

1 + Aij Aik Akj , kj (ki 1)=2 j, k

(3)

The mean clustering coefficient < CC > is the average of CCi over all nodes in the network (Honey et al., 2010). 4. Modularity Q: One of the key aims of investigation into cortical network architecture is to quantify the possible presence of modules: groups of nodes that connect strongly to each other and less strongly to other groups. The modularity of a network is often quantified by dividing the network into a set of distinct putative modules and then comparing the number of actual intramodule connections to the number expected if all the connections were redistributed randomly, subject to maintaining the network’s degree distribution. Mathematically this is expressed as " # kiin kjout 1 n Q = +i, j = 1 Aij dei ej , m m

(4)

where m is the total number of connections in the network, dei ej is the Kronecker delta, equal to one when ci = cj and zero otherwise, and ci is the module

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containing node i (Chen et al., 2008; Honey et al., 2010; Leicht and Newman, 2008). A disadvantage of this measure is that the module ci must be specified before Q can be evaluated. 5. Betweenness centrality b: For a single-node i, bi is the fraction of all shortest paths that pass through a given node summed over each possible pair of nodes, excluding the node itself. Mathematically, bi =

1 rjk (i) , + (N 1)(N 2) i6¼j, i6¼k, j6¼k rjk

(5)

where N is the number of nodes in the network, rjk is the total number of shortest paths from node j to node k, rjk(i) is the number of those paths that pass through node i, and the normalization 1/(N 1) (N 2) scales the maximum possible value of bi to 1, when rst(i) = rst for all s and t. The mean betweenness centrality, < b > , is the average of b over all nodes in the network (Honey et al., 2010). Results

The following key results are described in detail below. The model produces very similar network measure frequency distributions to DSI. In the model, these distributions result from variations in the cortical surface geometry and lead to spatial patterns in network properties with deep nodes having more hublike properties than superficial nodes. The variations in cortical surface geometry also introduce weak modularity, similar to cortical lobe structure. The nodes comprising the boundaries between modules are found to be hublike. Model geometry

The parcellation procedure described in the Parcellation of the Cortical Surface into Nodes section produces the parcellation shown in Figure 1a using n = 989 nodes, equal to the number of nodes in the experimental DSI dataset. The sur-

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face model has a total surface area of 1861 cm2; the distribution of node surface areas is shown in Figure 1b and has a mean node area of 188 mm2 and standard deviation of 34 mm2; the variation in node area is small compared with the mean. The relations between the distances sv, through the white matter, and the Euclidean distances, sE, and between sv and the distances on the cortical surface ss are shown in Figure 2. For any two nodes, sE £ sv £ ss. A linear regression with zero constant estimates that sE is on average 60% of sv, and ss is on average 250% of sv. The connectivity parameters, C and s0, were determined by exploring model network measure values in the parameter space, as shown in Figure 3. In parameter space, all three DSI curves intersect in a small region, meaning that the model can very closely replicate all three mean measure values simultaneously. The DSI < L > curve lies just above the DSI < k > curve for C > 20. The DSI < CC > curve intersects the DSI < L > curve at approximately C = 42.0 and s0 = 9.8 mm, and the DSI < k > curve at approximately C = 47.8 and s0 = 9.0 mm. The mean of these two points, at C = 44.9 and s0 = 9.4 mm, was used in the following analysis. The probabilities of connection for different s are shown in Figure 4. All probabilities decay roughly exponentially, as expected from Equation 1. The increased P(sE) at short ranges (sE £ 50 mm) for model versus DSI data in Figure 4b appears mostly because of there being fewer possible connections at short range in the model than in the experiment, as shown in Figure 4a, and the DSI brain being slightly smaller than in the surface model. Figure 4a also shows that the DSI data contain additional longer-range connections above 50 mm that are not included in this model. These connections are probably very long-range inhomogeneously distributed association fascicules (Wakana et al., 2004). Note that the close similarity between model and experiment in Figure 5 occurs only when using the distance sv to distribute connections. With either sE or ss replacing sv in Equation 1, this close similarity does not occur. This means that it is the length of fiber that is important in determining connectivity, not other distance metrics.

FIG. 1. Parcellation of the cortical surface. (a) The cortical surface mesh parcellated into 989 nodes. In this work we use the 20484-vertex canonical cortical surface mesh available with the SPM8 software package (Ashburner et al., 2012). This is a surface formed by the white- and gray-matter interface. (b) The frequency distribution of node surface areas. Nodes are similarly sized, with their variation in area being small with respect to their mean area. Color images available online at www.liebertpub .com/brain

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FIG. 2. Contour (a, c) and scatter (b, d) plots comparing distances through the white matter sv, Euclidean distance sE, and distances on the cortical surface ss between all pairs of nodes. The dashed lines in all frames indicate points at which both distances are equal. Red lines show linear fits with zero intercept; sE = 0.6sv; ss = 2.5sv. Color images available online at www.liebertpub.com/brain Network measure frequency distributions

The frequency distributions of k, CC, L, and b are shown in Figure 5 for the DSI data, this cortical surface model, and a similar spherical surface model that connects nodes distributed on a spherical surface also with probability that decays exponentially with distance, although with different parameters (Henderson and Robinson, 2013). The spherical surface model with a single set of parameters fitted to data was able to simultaneously produce approximately Gaussian-shaped distributions with similar mean values to DSI (though with noticeably lower CC). However, the DSI measure distributions differ significantly from the near-Gaussian sphericalmodel distributions. Hence, the spherical model does not capture the features of the network that produce the broader variation of measures in DSI. In contrast, our cortical surface model is able to simultaneously and remarkably closely reproduce the DSI distributions for all four measures considered here. Note that all four frames in Figure 5 are for the same model parameters C = 44.9 and s0 = 9.4. For each measure the mean DSI and cortical surface model values are very similar; < kDSI > =

36.1 versus < kmodel > = 38.7, < LDSI > = 3.07 versus < Lmodel > = 3.14, < CCDSI > = 0.47 versus < CCmodel > = 0.47, and < bDSI > = 0.0021 versus < bmodel > = 0.0022. Additionally, the DSI and model skew of the k, CC, and b distributions are also very similar. The DSI k distribution has significantly more low-degree nodes (k < 15) than in our cortical surface model. The range of DSI L values is captured by the model, but the skew of the DSI L distribution is toward slightly lower L than in the cortical surface model. Removing long-range connections in the DSI data that are not present in the model improves the correspondence between the model and DSI L distributions. In the model, < 2 connections with sE > 3 mm are expected for C = 44.9 and s0 = 9.4. Removing connections in the DSI data with sE > 93 mm reduces the left skew of the L distribution, bringing it closer to the model L distribution, without significantly effecting the distributions of other measures. One possible contributing factor to the increased number of very-low-degree nodes in the DSI data compared with the model could be connections missed in the experiment. Differences between the cortical geometries used in the model and found in the DSI data are likely to introduce

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FIG. 3. The dependence of cortical surface model key network measures on the connectivity parameters, C and s0 in Equation 1. In each frame, black contours indicate model mean measure values averaged over an ensemble of networks. Thick green contours indicate human DSI network values: < k > = 36.1, < CC > = 0.468, < L > = 3.06, and < b > = 0.0021. The red crosses indicate model parameters C = 44.91 and s0 = 9.38 that best fit these four measures collectively. (a) < k > . (b) < CC > . (c) < L > . (d) < b > . Color images available online at www.liebertpub.com/brain minor quantitative differences between model and data, but do not impact the qualitative results. Note that for each measure, a small improvement can be made in the correspondence between model and DSI frequency distributions by choosing slightly different parameters that are closer to the DSI measure curve in parameter space in Figure 3. Spatial variation in network measures

The spherical model and DSI distributions in Figure 5 differ significantly in shape. However, there is close similarity

between the cortical surface model and DSI distributions. Since the spherical and cortical surface models differ only in their geometry, this implies that these DSI measure frequency distributions are primarily related to the gyrification in cortical geometry. Figures 6–9 show the spatial variations of k, L, CC, and b, respectively, across the cortical surface. Similar variations would be predicted for experimental data, though such data are not available to us (a surface of the cortex was not made available with the DSI data). In Figure 6 we see that high-k nodes tend to be deeper in the brain while low-k

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FIG. 4. Comparison of probabilities of connection for different measurements s of distances between nodes. Red dots are for the model; black dots are for experimental DSI data. (a) The number of possible connections between any pair of nodes separated by Euclidean distance sE. (b) Probability of connection versus Euclidean distance sE. (c) Probability of connection versus minimal distance through the white matter sv. (d) Probability of connection versus minimal distance along the cortical surface ss. In (c) and (d) no DSI points are shown, as the cortical surface model was not published with the data (Hagmann et al., 2008), so sv and ss are unknown for DSI connections. Color images available online at www.liebertpub.com/brain nodes are more superficial, and similarly for b values in Figure 9. Conversely, Figures 7 and 8 show that nodes with low L and CC values tend to be deeper, while high-L and high-CC nodes are more superficial. High values of both L and CC can be observed in the extremities of lobes (at the tip of the temporal, occipital, and frontal lobes), indicating that the global geometry, not just the local gyrification, is important, since global geometry determines which nodes are more central and hence likely to have lower L. These trends can also be observed in Figure 10, which compares pairs of measures for single nodes with approximate node depth indicated by coloring. These scatter plots are not random and show that in the model there are relations (determined by geometry) between individual measures, and between measures and the depth of individual nodes. The similarity in scatter between the model and DSI indicates that these relations are likely to exist in DSI as well. To summarize, these results imply that deep nodes are more hublike than superficial nodes, as summarized in Table 1. Module properties

Some variation occurs between different instances of the model because of the probability-based method of distributing connections. To account for this variation, the modular

properties of the cortical surface model were investigated using an ensemble of model networks. Figure 11a shows a division of the model cortical surface into modules produced from by following modularization process. Each network in the ensemble is divided into modules using the Newman spectral algorithm (Leicht and Newman, 2008). From this set of modular divisions of different instances of the model, the number of times each pair of nodes is partitioned into the same module is counted. Modules are then constructed by iteratively selecting the largest groups of nodes that are all in the same module in more than 50% of the ensemble. A significant number of nodes remain that do not belong to any of these modules in more than 50% of the ensemble. These nodes have significant variation in their module association and often are located on or near the boundaries of modules detected using the Newman spectral algorithm. The final modules (as shown in Fig. 11) are constructed from groups of nodes that are partitioned together into the same module in more than 50% of the ensemble. Eight large modules are detected as well as 143 nodes (colored gray) that are outside the 8 modules in more than 50% of the ensemble. These modules are approximately symmetric between hemispheres. The single-node probability distributions for the k, L, CC, and b of each module are also displayed in Figure 11. The distributions have all been

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FIG. 5. Frequency distributions of DSI data (black dots), DSI data with connections sE > 93 mm removed (green ‘‘ + ’’), and ensembles of the cortical surface model (red) and spherical model (blue). DSI and cortical surface model distributions are very similar, while the spherical model has similar mean values but different distribution shape. Error bars are one standard deviation. (a) k. (b) L. (c) CC. (d) b. p-Values are calculated using a two-sample Kolmogorov–Smirnov test (Sheskin, 2007) between the DSI and cortical surface model data, giving p-values of pk = 0, pL = 0, pCC = 0.12, and pb = 0.18. Color images available online at www.liebertpub.com/brain normalized to the probability for a single node because the numbers of nodes in each module varies. Overall, all the modules have similar distributions, reflecting the homogeneous, isotropic connection rule that governs all their connections. However, some patterns can be observed by looking at the positions of modules in the sagittal plane. Specifically, corresponding modules in each hemisphere show similarities in their distributions. The central light and dark green nodes have distinctly lower L on average than other modules while also having the lowest CC on average. The frontal orange and purple modules have distinctly lower k on average than other modules. The temporal modules (red and brown) have the highest CC on average, while the occipital modules (blue and cyan) tend to have the highest k and L on average. Overall, the frontal and occipital modules have higher L and lower b than average, while the middle modules tend to have lower L, and higher b than average. These trends

in L and b are global network properties and hence relate to the geometric positions of the modules within the cortex; the middle modules share borders with all modules in their hemisphere, unlike the frontal and rear modules. The trends in the local properties of the network, CC and k, are more strongly related to the local geometry of gyri and sulci. Properties of module-boundary nodes

In order to investigate the modular structure of the network, the properties of nodes at the boundaries of all modules are investigated, that is, nodes that are adjacent to one or more nodes in a different module to itself. Figure 12 shows the k, CC, L, and b probability distributions for a single instance of the cortical surface model divided into two sets of nodes: module-boundary, and nonboundary nodes using a Newman modular division (Leicht and Newman, 2008). Figure 13 shows the relationships between

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FIG. 6. Spatial variation of k for the cortical surface model. The colorbar shows the variation in k values. Deeper nodes tend to have higher k than superficial nodes. Spatial coordinates on the axes are in millimeters. (a) Whole cortex. (b) Sagittal slice. (c) Coronal slice. (d) Axial slice. Color images available online at www.liebertpub.com/brain

pairs of measures for boundary and nonboundary nodes. From these figures, it is clear that the boundary nodes tend to have significantly higher b than nonboundary nodes while also having slightly higher k and slightly lower CC and L, meaning that they are more hublike than nonboundary nodes. Boundary nodes achieve their hublike properties by connecting regions that have large ss compared with sv. Figure 14 shows the expected number of connections versus ss of a single node in an ensemble of model networks, for boundary nodes, nonboundary nodes, and all nodes. The boundary nodes have more connections than nonboundary nodes above about 170 mm, indicating that they are more likely to bridge large distances in the network and thus have increased b. These hubs can therefore be classified as connector hubs, forming connections between modules. High b nodes lying in sulci not close to the module boundaries can be classified as provincial hubs connecting primarily (but not exclusively) within a single module. Discussion

The above results show that white-matter connectivity is closely related to the geometry of the cortical surface. How geometry produces the observed spatial variations in network measures and its implications are discussed below.

Gyrification versus measure distributions

The ability of the spherical model to reproduce the DSI data mean measures indicates that the main contributing factors to these DSI network properties are the 2D geometry and isotropic, homogeneous local connectivity. However, Figure 5 shows that this new cortical surface model is a significant improvement over the spherical surface model in replicating the DSI network properties. The gyrification of the cortex is a secondary factor that acts to perturb the local and global geometry of the network, leading to more variation in network measures and thus flatter frequency distributions (see the section Prediction of Variation of Measures Across the Cortical Surface). This implies the following conclusion about longrange cortical networks: other than a network average connection kernel and geometry, little or no knowledge of single connections or small groups of connections is required to closely reproduce these measures. This conclusion has farreaching implications for both structural and functional cortical networks and their analysis. If white-matter connections are distributed as in this model, randomly according to a decaying exponential (or similar) distribution, and these connections can be rewired while still maintaining cortical function, then this raises interesting questions about cortical function. For example, how would the cortex route information through this network

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FIG. 7. Spatial variation of L for the cortical surface model. The colorbar shows the variation in L values. Deeper nodes tend to have lower L than superficial nodes. High values of L also occur at the extremities: the tips of the temporal, frontal, and occipital lobes. Spatial coordinates on the axes are in millimeters. (a) Whole cortex. (b) Sagittal slice. (c) Coronal slice. (d) Axial slice. Color images available online at www.liebertpub.com/brain if not through developing specific links to and from appropriate sources and destinations? Or is information simply broadcast widely with smaller-scale gray-matter networks left to filter relevant information? Very-long-range connections

The very-long-range connections observed in DSI data (Fig. 4b) but not included in this model do not appear to be necessary to generate a network with similar k, L, CC, and b. These long-range connections are not homogeneously distributed, and if Equation 1 is modified to add a long-range exponential distribution to capture these longer-range connections, then the close correspondence between DSI and model measure frequency distributions in Figure 5 is significantly reduced. In particular since the decay in probability with distance of these connections is relatively slow, they act like additional random connections, and hence L values are reduced significantly below those of the DSI data. In contrast, removing some of the long-range connections in DSI data improves the correspondence between model and DSI L distributions (Fig. 5). This indicates that these longrange connections are very inhomogeneously and anisotropically distributed, connecting relatively few specific areas

and hence shortening relatively few paths through the network. Importantly, removing these connections does not result in a large increase in high L values or < L > in Figure 5b, indicating that they are not to reduce high L outliers (e.g., like random connections in a small-world model). This model shows that the network already has short path length dominated by shorter connections that are still able to bridge large distances in the network. These connections arise through a combination of a large-enough s0, and cortical geometry providing a significant number of locations where sv is much smaller than ss. This then raises a question for future work, if not to substantially reduce path lengths or upper bounds, what are these long-range connections for? Might it be, for example, that they are involved in a network core or ‘‘rich club’’ (van den Heuvel and Sporns, 2011) that requires connections additional to the network formed from an isotropic, homogeneous connection kernel? Also, it has not yet been determined which of these fascicules are accounted for in this model as part of the homogeneous, isotropic connection pattern, and which result from other specialized connection rules. Are any of these bundles of fibers artifacts of constrictions in regions of the white matter and a preference for alignment of fibers to allow higher packing density?

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FIG. 8. Spatial variation of CC for the cortical surface model. The colorbar shows the variation in CC values. Deeper nodes tend to have lower CC than superficial nodes. High values of CC also occur at the extremities: the tips of the temporal, frontal, and occipital lobes. Spatial coordinates on the axes are in millimeters. (a) Whole cortex. (b) Sagittal slice. (c) Coronal slice. (d) Axial slice. Color images available online at www.liebertpub.com/brain

Prediction of variation of measures across the cortical surface

The frequency distributions in Figure 5 are a result of variation in the geometry surrounding each node producing spatial variations in network measures. Figure 15 demonstrates schematically how the convolutions in cortical geometry lead to these spatial variations in measures. In this figure, all nodes (indicated by the colored dots) connect to other nodes (indicated by the colored lines) over a characteristic range via the white matter. Deep nodes (the red node located on a sulcus) connect to a large area, potentially including bridging through the white matter to other gray-matter areas that are not neighboring areas on the cortical surface. Deep nodes thus tend to have a high k and b, and low L and CC. This implies that deep nodes are more hublike than superficial nodes (though not like hubs in a scale-free network with orders of magnitude variation in k), and thus that the core of the network is predominantly deep in the cortex. Conversely, superficial nodes (e.g., the blue and green nodes on the gyri) connect to a small neighboring area on the cortical surface. Hence, superficial nodes tend to have low k and b, and high L and CC. These spatial variations in network measures are demonstrated in the model in Figures 6–9. Given the close similar-

ity between DSI and model frequency distributions that in the model we have shown result from spatial variations, we predict similar spatial variations in real cortexes. This geometrical arrangement of connectivity is also consistent with motif studies in directed cortical network data (Sporns and Ko¨tter, 2004). The independence of connection probability from connection direction is consistent with observed preference for reciprocal connections and hence motifs containing at least one reciprocal connection. Additionally, in three-node motifs the preference for motifs containing two reciprocal connections and other V-shaped motifs is consistent with deep sulcal nodes (red nodes) in this model that connect to nodes that are not near neighbors on the cortical surface. Triangular motifs are consistent with strong connectivity among nodes in a gyrus, for example, the green node and its neighbors. Thus, geometry is also important in determining network motif composition. Gyrification versus modularity

If white-matter cortical networks have a homogeneous, isotropic connection rule, as in the present model (i.e., no other special rules for connections), a further implication is that modularity can only be introduced into the network architecture by variations in the geometry of the cortical

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FIG. 9. Spatial variation of b for the cortical surface model. The colorbar shows the variation in b values. Deeper nodes tend to have higher b than superficial nodes. Spatial coordinates on the axes are in millimeters. (a) Whole cortex. (b) Sagittal slice. (c) Coronal slice. (d) Axial slice. Color images available online at www.liebertpub.com/brain surface. In the cortex, modularity of white-matter connections thus appears to be a geometric perturbation rather than a very distinct, unique modular design in which the modular architecture is unambiguous. In this model a distinct modular design would require obvious geometric bottlenecks to clearly demarcate different geometrical regions, but cortical geometry only contains one obvious geometric bottleneck at the corpus callosum. The detected modules shown in Figure 11 are similar to the four-lobe structure of each hemisphere. In Figure 11, the brown and red modules roughly correspond to the temporal lobes, the cyan and blue modules roughly correspond to a combination of the occipital and parietal lobes, while the light and dark green and purple and orange modules correspond to the frontal lobes, divided into two modules. These similarities are expected from our analysis because the lobes are demarcated by geometrical features that determine connectivity in this model. The modularity score Q (Eq. 4) of a division of a network into modules is maximized by attempting to simultaneously optimize two quantities through the choice of modules. First, the total number of intramodule connections, the potentially positive terms when Aij = 1 in the sum in Equation 4, is maximized; this tends to reduce the number of modules, because larger modules will tend to include more of the available connections as intramodule connections. Second, the density of connections within the modules is maximized (minimiz-

ing the potentially negative terms when Aij = 0 in the sum in Eq. 4); this tends to increase the number of modules, because modules that are too large will be missing many intramodular connections and hence include more of these negative terms in the sum. Dense connectivity occurs in gyri and, consequently, good nodes to choose as module boundaries are sulcal nodes, so that the dense connectivity within gyri is included in modules. This preference for dividing modules with sulcal nodes means that detected modules are naturally bounded by hub nodes, which in the context of modularity are connector hubs. This is what is observed in Figure 12d; the b of the boundary nodes tends to be significantly higher than that of nonboundary nodes. Additionally, the boundary nodes also tend to have lower CC and L, and higher k—all properties of hublike nodes. Thus, we have a network architecture in which the network is weakly divided into modules along boundaries containing connector hubs. The measure probability distributions for modules in Figure 11 are generally similar, reflecting the homogeneous, isotropic connectivity rule that governs the network structure. Any large differences are because of variations in the cortical geometry within and between modules. No one module displays significantly different distributions in all measures; however, there are noticeable differences in single measures. The dark-green, light-green, red, and brown modules located centrally in the sagittal view typically have lower L than other modules because of their central location in the cortex.

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FIG. 10. Comparison between single-node network measure values of DSI and cortical surface model. (a) Coloring of model nodes for frames (b–g); model nodes are colored dots; nodes are ordered according to approximate depth and the order is indicated by color; deep nodes are dark blue and superficial nodes are dark red. DSI nodes are small black dots. (b–g) Scatter plots of node measure values for all pairs of k, L, CC, and b. The range of scatter for the cortical surface model is very similar to DSI in all cases. Color images available online at www.liebertpub .com/brain

Table 1. Summary and Comparison of Network Measures for Superficial and Deep Nodes in the Model

Superficial Deep

k

L

CC

b

Low High

High Low

High Low

Low High

Deep nodes have hublike properties.

The pink and orange frontal modules have lower k than other modules. These large-scale patterns indicate that the largescale lobe geometry of the cortex also affects the network properties. Despite the detection of modularity in this model introduced through the geometry of gyrification, the modularity is relatively weak and it is not clear whether this modularity is an important feature of the network, especially in the context of network function. Using a different modularity metric or slightly relaxing the optimal Q value allows significantly

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FIG. 11. Modular division of the model. (a–d) The brown and red modules roughly correspond to the temporal lobes, the cyan and blue modules roughly correspond to a combination of the occipital and parietal lobes, while the light- and darkgreen, and purple and orange modules correspond to the frontal lobes, divided into two modules, and gray nodes are not consistently associated with any single module. (e–h) Measure distributions for each module. Since modules differ in size, for comparison these distributions are divided by the number of nodes in each module. Modules are indicated by colors as in (a–d); black curves are for all nodes in the network. Dots indicate mean values for each module (e) k, (f) CC, (g) L, and (h) b. Color images available online at www.liebertpub.com/brain different modular divisions. Also, given the similarity in connectivity and network measures of the modules found here, this may indicate that this modularity is not functionally important. However, to determine this more convincingly, other techniques will be required to assess whether this weak modular demarcation remains and is strengthened by dynamics, becoming an important feature in functional networks, or if other effects dominate. A qualitative description of cortical network architecture

The close similarities between the model predictions and DSI distributions imply that additional information on individual connections other than distance through the cortical

volume do not significantly affect the properties of cortical networks investigated here: degree, clustering coefficient, path length, and betweenness centrality. The organization of the DSI data and the model is consistent in their network properties, leading to the following description of long-range white-matter cortical networks: 1. The networks are formed on a (nearly) closed 2D surface. 2. Connections are made according to an isotropic, homogeneous, probabilistic connection rule that connects to all nearest neighbors at short range, and then decay exponentially with the distance between nodes through the white-matter volume. We emphasize that additional

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FIG. 12. Comparison of frequency distributions of module-boundary nodes (red), nonboundary nodes (green), and all nodes (black). Each curve is normalized to a single node. (a) k. (b) L. (c) CC. (d) b. Color images available online at www .liebertpub.com/brain connection rules may well exist but appear to be secondary to this rule. 3. Variation in node properties arises primarily because of the detailed convoluted geometry of the cortex, as well as variation because of the probabilistic connectivity rule. This geometric variation leads to deeper nodes bridging large distances on the cortical surface, producing hublike properties and making them more central in the network. However, these nodes are not as strongly hublike as in some scale-free networks where there can be orders of magnitude differences in k, for example. 4. The detailed cortical geometry perturbs the isotropy and homogeneity of the connection rule to introduce a weak modular structure. These modules tend to be strongly connected through their boundary nodes that are more hublike than nonboundary nodes and form a set of connector hubs. Provincial hubs located on sulci that are distant from module boundaries mostly connect within a single module. We would expect that in species with less gyrification, correspondingly connectivity properties would be-

come spatially more homogenous. Thus, the network properties would be more similar to the spherical model with a lower variance Gaussianlike distribution. This description is in contrast to many other studies that have observed specialization of function in the cortex (Brett et al., 2002) and searched for organizing principles of cortical network architecture on the basis of inhomogeneous, anisotropic specialization of connectivity, using information theoretic approaches (Tononi and Sporns, 2003), and principles such as small worldness, presence of hubs, modularity, and motifs (Bullmore and Sporns, 2009, 2012; Kaiser, 2011; Sporns, 2011; Sporns et al., 2004) that have been described in this model. Instead, the results here are in line with investigations into physical properties of the network such as evolutionary minimization of wiring volume and length in the cortex (Raj and Chen, 2011). This may mean that many putative findings to date could be simply artifact and/or consequences of geometry. Organizing principles such as small worlds and perhaps modularity are still present, but derived from an underlying geometric organization.

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FIG. 13. Comparison of network measures for model module-boundary and nonboundary nodes. (a) Coloring of nodes: red nodes are nodes on the boundary of modules; green nodes are nonboundary nodes. Frames (b–g) show scatter plots of node measure values for all pairs of k, L, CC, and b. Color images available online at www.liebertpub.com/brain

Additionally, this may mean that neural field theory using a uniform connection kernel on a cortical surface may be more widely applicable than previously thought (Deco et al., 2008). Why would connectivity be homogeneous and isotropically distributed? It could be argued on evolutionary principles that connectivity is a measure of the usefulness of a cortical area; the more useful the function of a particular area is, the more other areas will connect to it. Efficient use of resources would then lead to the enlargement of useful, highly connected areas, reducing the spatial density of their connections; conversely, the shrinkage of less useful, less connected areas would increase the spatial density of their connections. This process of size modification would thus tend to homogenize the density of connections and therefore act as a homogenizing connectivity principle at long range while allowing specialization to occur at short

range in the gray matter. Evolutionary or development processes may also have optimized the placement of functional areas relative to sulci and gyri so that areas that require greater network-wide connectivity are positioned deeper in sulci, while areas that require strong connectivity with only a few other similar functional areas are placed more superficially in gyri. Given the importance of detailed cortical geometry on white-matter connectivity and network properties shown here, our model may be useful in providing insight into sex-, ethnicity-, and age-related connectivity differences, and differences in a variety of disorders such as attention deficit/hyperactivity, Alzheimer’s, schizophrenia, bipolar, major depressive, and obsessive compulsive disorders that have been associated with differences in cortical gyrification to normal groups (Li et al., 2007; Mirakhur et al., 2009; Tymofiyeva et al., 2012; White and Hilgetag, 2011; White

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FIG. 14. Expected number of connections in an ensemble of model networks versus ss. Red dots are for nodes on the boundary of modules, green dots are for nonboundary nodes, and black dots are for all nodes. Color images available online at www.liebertpub.com/brain

et al., 2010; Wobrock et al., 2010; Zhang et al., 2009; Zilles et al., 2001). As a starting postulate, since deep nodes have hublike properties, it may be that abnormalities in deep regions are associated with disorders of functional integration, while abnormalities in superficial regions are more associated with disorders in specific functions. It is important to note that the model and DSI data used here contain unweighted, symmetric connections. Future modeling and experiments with weighted, directional connection information may reveal further patterns in connections. These results should also be checked with other datasets obtained using different techniques to confirm that these results are not heavily influenced by artifacts from the DSI and tractography. It may thus be that the network properties considered here are not sensitive to the manner of specialization that occurs in these connections, and thus do not determine the specialization of functional interactions between regions in the cortex. It is likely that other measures, particularly those involving details of brain function, will provide more insight into why particular connections exist. Less dominant connection rules are still likely to be present in white-matter connections. Conclusion

FIG. 15. Schematic demonstrating how the convolutions of the cortical surface affect the connectivity of nodes and lead to spatial variations in network measures. All nodes (indicated by the colored dots) connect to other nodes (three examples indicated by the colored lines) within the same characteristic range measured through the cortical white matter. Deep nodes (the red node located on a sulcus) connect to a larger area than superficial nodes (blue and green), and may also form a bridging connection to another nearby area of cortex because of the large-scale curvature of the cortex. Note that superficial nodes may also be able to form bridging connections if their neighboring gyri are shallow (not shown here). Color images available online at www.liebertpub.com/ brain

We have developed a geometric model of long-range white-matter cortico-cortical connectivity that uses a connection kernel that is constrained by experimental network properties and decays exponentially with distance through the cortical white-matter volume. We show that this model has very similar frequency distributions to experimental DSI connectivity data for k, L, CC, and b measures. The connection rule is homogeneous and isotropic, meaning that there are no special rules for connecting particular cortical regions, details other than the geometry of single connections do not contribute significantly to these observed measures and the overall network architecture, and a network average connection kernel and the cortical geometry is enough to

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reconstruct a very similar network to experiment. Additional connection rules may exist, and this model will allow the differentiation of network properties resulting from these homogeneous, isotropic connections, and properties resulting from other types of connections, thus helping to identify the purpose of different types of connections. In the model, these distributions result because the cortical geometry, particularly folding, produces spatial variations in network measures. Deep nodes are shown in the model to have hublike properties and thus form a network core. The model is found to have a weak modular structure, produced by geometrical features of the cortex, and is similar to the fourlobe structure of each hemisphere. The boundary nodes between modules are shown to be hublike, forming connector hubs in the network. Because of the similarity between model and DSI frequency distributions, we predict similar spatial variations in network measures in real cortexes that can be tested experimentally. These findings are consistent with previous findings from investigations of experimental networks such as small-world architecture, apparent modularity, minimization of wiring length, motif content, and the presence of hubs. Acknowledgments

The Australian Research Council and Westmead Millennium Foundation supported this work. Author Disclosure Statement

No competing financial interests exist. References

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Address correspondence to: James A. Henderson School of Physics University of Sydney Building A28 Sydney New South Wales 2006 Australia E-mail: [email protected]