Neuroscience 267 (2014) 91–101
INTRINSIC MODE FUNCTIONS LOCATE IMPLICIT TURBULENT ATTRACTORS IN TIME IN FRONTAL LOBE MEG RECORDINGS X. HUANG, a,c L. HUANG, b,c T.-P. JUNG, d C.-K. CHENG c* AND A. J. MANDELL e,f
decompositions. Ó 2014 IBRO. Published by Elsevier Ltd. All rights reserved.
a
Department of Computer Science and Engineering, College of Information Engineering, Shanghai Maritime University, Shanghai 201306, PR China b Institute of Electronic Science and Engineering, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu, PR China
Key words: implicit turbulent attractor, Empirical Mode Decomposition, topological entropy, metric entropy, nonuniform entropy, power spectral scaling exponent a.
c Department of Computer Science and Engineering, University of California at San Diego, La Jolla, CA 92093, USA d
Swartz Center for Computational Neuroscience, Engineering, Institute of Engineering in Medicine, University of California at San Diego, La Jolla, CA 92093, USA
INTRODUCTION Most current studies of human MEG recordings are focused on neurophysiological source localization, treating the magnetic field recordings as epiphenomenal reflections of underlying neuroelectrical events (Cohen, 1972; Nolte et al., 2004; Cornwell et al., 2007; Cornwell et al., 2008). Among the techniques used to approximate a solution to ‘‘the inverse problem’’ (given a magnetic field, what and where is its electric event sources) are the beamformer techniques (Vrba and Robinson, 2000; Robinson, 2004). Globally distributed, spontaneous magnetic field fluctuations in the resting condition have often been interpreted as an impediment to techniques of localization being ‘‘. . .high (eigenvalue) ranked background activity. . .interfering magnetic fields. . .(meaningless) intrinsic brain noise. . .’’ (Sekihara et al., 1996, 2008). In contrast, in these studies, the magnetic fields are regarded, not as epiphenomenal signs of underlying neurophysiological source events or functionally irrelevant ‘‘brain noise’’, but rather the B fields are seen as functional components of brain dynamics in themselves, B = cH; where B magnetic flux density, c magnetic permeability and H magnetic field strength. Neocortical magnetic fields associated with neurophysiological events are here regarded as potentially nonlinear ‘‘feedback’’ modulatory near field influences on the thresholds of the neocortical pyramidal neuronal networks that generated them (Chiabrera et al., 1985; Black man et al., 1988; Delparte and Persinger, 2007; Frolich and McCormick, 2010; Ledda et al., 2010). With the nonlinear feedback elements of a dynamical system we posit that the magnetic fields of the continuously active ‘‘resting’’ human cortex without external stimuli (Raichle et al., 2001) can self-organize into non-stationary (transient) turbulent (in time) attractors that reveal themselves by their associated time-scale located, invariant entropic measures (Mandell et al., 2011c).
e
Multi Media Imaging Laboratory, Department of Psychiatry, University of California at San Diego School of Medicine, La Jolla, CA 92093, USA f Fetzer Franklin Fund of the John Fetzer Memorial Trust, Kalamazoo, MI, USA
Abstract—In seeking evidence for the presence and characteristic range of coupled time scale(s) of putative implicit turbulent attractors of dorsal frontal lobe magnetic fields, the recorded nonstationary, nonlinear MEG signals were non-orthogonally decomposed using Huang’s Empirical Mode Decomposition, EMD, (Huang and Attoh-Okine, 2005) into 16 Intrinsic Mode Functions, EMD ? IMFi, i = 1. . .16. Measures known to be invariant in non-uniformly hyperbolic (turbulent) dynamical systems, topological entropy, hT, metric entropy, hM, non-uniform entropy, hU and power spectral scaling exponent, a, were imposed on each of the IMFi which evidenced most clearly an invariant temporal scale zone of IMFi, i = 6. . .11, for hT, which we have found to be the most robust of invariant measures of MEG’s magnetic field turbulent attractors (Mandell et al., 2011a,b; Mandell, 2013). The ergodic theory of dynamical systems (Walters, 1982; Pollicott and Yuri, 1998) allows the inference that an implicit attractor with consistently hT > 0 will also evidence at least one positive Lyapounov exponent indicating the presence of a turbulent attractor with exponential separation of nearby initial conditions, exponential convergence of distant points and disordering, mixing, of orbital sequences. It appears that this approach permits the inference of the presence of chaotic, turbulent attractor and its characteristic time scales without the invocation of arbitrary n-dimensional embedding, phase space reconstructions or (inappropriate) orthogonal
*Corresponding author. Tel: +1-858-534-6184; fax: +1-858-5347029. E-mail address: [email protected] (C.-K. Cheng). Abbreviations: EMD, Empirical Mode Decomposition; IMF, Intrinsic Mode Functions; MEG, Magnetoencephalography. http://dx.doi.org/10.1016/j.neuroscience.2014.02.038 0306-4522/Ó 2014 IBRO. Published by Elsevier Ltd. All rights reserved. 91
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It is for these reasons that we regard the temporal structure of the brain’s magnetic field as important in itself, independent of what it might imply about its sources. We further speculate that the brain’s magnetic fields may mediate ‘‘up’’ states and ‘‘down’’ states of neocortical pyramidal cell excitability thresholds (Cossart et al., 2003; Sachdev et al., 2004). In these methodological studies, the MEG recordings of brain magnetic signals were obtained from archival records recorded at the National Institutes of Mental Health Core MEG Facility (Weinberg et al., 1984; Anninos et al., 1987; Fife et al., 2002). The recordings were made on a 275 channel, superconducting quantum interference devices (SQUID) radial gradiometer system from CTF Systems Inc. of Port Coquitlam, British Columbia, Canada. This data were obtained from previous archival MEG studies focused on relative power using Fourier analysis (Rutter et al., 2009). All subjects involved in these studies had given written informed consent according to protocols approved by the NIH CNS Institutional Review Board.
Hierarchical scaling and measure-theoretic, informational entropies in magnetoencephalic recordings of turbulent brain magnetic fields Recent research has suggested that time-dependent, magnetic field signals of 103–104fT magnetic field strength as recorded from the human brain by magnetoencephalography, MEG, manifest the phase space and measure characteristics of turbulent (chaotic) dynamical systems (Kowalik and Elbert, 1994; Mandell et al., 2011a,b, in press; Robinson et al., 2013; Vakorin et al., 2010). Consistent with this interpretation of B field fluctuations is the presence of a number of quantifiable characteristics of turbulence including positive leading Lyapunov exponents, fractional capacity dimensions and positive entropy generation (Mandell, 2013). In addition and of relevance to this study is the characteristic fractional power law scaling, a, of the MEG signal (de Gennes, 1979; Churilla et al., 1996; Selz and Mandell, 1997; Peng et al., 1998) associated with positive topological and metric entropy generation (Adler and Weiss, 1967; Dinaburg, 1971; Eckmann and Ruelle, 1985). The measures derived from the ergodic (‘‘invariant measure’’) theory of dynamical systems (Cornfield et al., 1982; Walters, 1982; Eckmann and Ruelle, 1985; Pollicott and Yuri, 1998) such as topological entropy hT, metric entropy, hM, and non-uniform entropy, hU = |hT hM| (Ott, 1998), These measures are obtained from discrete and continuous time series in dimension zero and one respectively. However, the hierarchal multi-scale characteristics of the turbulent MEG record (Mandell et al., 2011a) call for a higher dimensional embedding (Cellucci et al., 2003; Pecora et al., 2007). It should be noted that the values of a, hT, hM, and hU are sensitive to the bandwidth of the observations with hT decreasing as a increasing (Mandell and Selz, 1997). Method-exemplifying MEG data reported here were recorded consistently at 600 Hz
with a bandwidth bounded by a 150-Hz cut-off, its ‘‘corner frequency’’ (Rutter et al., 2009). Hierarchical decomposition of the human frontal lobe MEG signal Given the multiplicity of relevant time scales in neurobiological signals (Mandell et al., 1982; Basar et al., 1983; Mandell and Selz, 1992), it becomes necessary to avoid the attendant risk of over-sampling, non-discriminable noise as well as the findings that completely different dynamical structures as nonstationary ‘‘attractive-repellers’’ (Farmer et al., 1983; Nichols et al., 2003) can emerge at different time scales of observation (Ashkenazy et al., 2000).For these reasons, multi-scale techniques have been developed in the computation of singular measures such as the measure-theoretic entropies (Peng et al., 1998; Costa et al., 2005) many of which are a derivative of approximate entropy (Pincus, 1991). The nonlinear and non-stationary properties of brain electromagnetic time series have been approached using a variety of quasi-linear time-frequency techniques such as the Stockwell (Liu et al., 2010) and wavelet transformations (Daubechies, 1992; Jia et al., 2006; Lim et al., 2012) as well as the orthogonally restricted singular value decomposition (Haq et al., 1997). The parametrically free empirical mode decomposition, EMD, that generates intrinsic mode functions, IMFi was introduced by Huang (Huang et al., 1998) to study the multi-scale time-frequency behavior of nonstationary wind and water waves. Among the many subsequent applications to nonstationary and nonlinear time series include those involving the human EEG (Pachori, 2008) and the MEG (Aven et al., 2011; Aven et al., 2012). Empirical Mode Decomposition a simple graphical algorithm constructs new series, Intrinsic Mode Functions, composed of the average sequence of the sequential maxima and minima of the original data. Generating a hierarchy of computationally defined modes, the EMD ? IMFi transformation does not require stationarity or the (theoretical) infinite length of Fourier transformation. Whereas our previous applications of the EMD ? IMFi transformation to human brain MEG series involved the elucidation of the non-orthogonal IMFi amplitude, ‘‘energy’’ distributions across temporal scales, i, and their use in computation of capacity dimensions and Hurst exponents (Aven et al., 1212), the methods described here focus on the distributions of symbolic dynamically derived entropy measures, hT, hM, hU and fractional power law spectral scaling exponents, a. The EMD decomposes the multiscale behavior of the MEG signal into time-scale separated modes indexed as IMFi i = 1. . .n, These in turn are each analyzed with respect to their values for hT, hM, hU and a. Whereas the most relevant time scale(s) of the putative non-stationary attractor would seem to be indicated by the peak amplitude distributions across IMFi (Aven et al., 2012), we have recently shown that in brain space, it can be the case that MEG amplitude (or power) fails to temporally and spatially localize functionally involved
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(B) 16 IMFs of subject B Fig. 1. Characteristic graphs of the EMD decomposition result of two subjects from the test group. The 16 subgraphs correspond to 16 IMFs. A 275 channel, superconducting quantum interference device was used in MEG data collection. The record is sampled at 600-Hz with 150-Hz acquisition cut off. The difference sequences, left minus right, (i.e. Semmetric Sensor Difference Series) ssds(i) signals were taken as the input which can reduce the penetrance noise of electromagnetic field correlates of blink, cough, and movement as well as the cardiac and respiratory artifacts that both symmetric sensors generally share. We used the ssds(i) signals at dorsolateral prefrontal cortex area F14. A 0.6-Hz high-pass and 60-Hz notch filters were routinely applied to the input. Reading left to right for each of the four rows from top to bottom we see the results of EMD ? IMFi, i = 1. . .n, n = 16, transformations of the first 60 s.
brain regions under circumstances in which an entropic measure, rank vector entropy, does so (Robinson et al., 2013). In an analogous way, we use hT, hM, hU and a across IMFi to localize the relevant scales in time not space. This approach exploits the superiority of the MEG over MRI in temporal resolution. Multiple invariant measure-theoretic entropies The theoretical frame work for the analyses of the IMFi is the ergodic theory of dynamical systems (Kolmogorov, 1958; Arnold and Avez, 1968; Eckmann and Ruelle, 1985; Pollicott and Yuri, 1998). Under this aegis, real
valued IMFi series can be reconstructed on smooth manifolds for embedding, partition, symbolic dynamic encoding of transition matrices followed by topological and statistical characterizations (Smale, 1967; Sinai, 1972; Bowen and Ruelle, 1975; Adler et al., 1977; Adler and Marcus, 1979). Whereas ergodic in classical statistical mechanics implies equivalence in the system’s time and space averages (Reichl, 1980), ergodic in this context implies a system with invariant measures across differing initial conditions (Cornfield et al., 1982). The application of the EMD ? IMFi transformation can be viewed as systematic topologicalgeometric sampling with the expectation that the IMFi
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(B) Power spectra of IMFs for subject B Fig. 2. Power spectra in log ordinate-log abscissa plot for two subjects’ 16 IMFs. The power spectra of IMFi can be viewed in order from right to left for i = 1–16. Likewise, the frequency at the peak of each IMFi decreases as index increase. The shapes of the power spectra with same IMF index are similar.
time scales relevant to the putative non-stationary attractor would manifest time scales of the attractor’s invariant measure. There is a family of theorems applicable to chaotic dynamical systems (paraphrased) that says in uniformly (also nonuniformly Young, 1998) divergent, hyperbolic
(no norm 1) dynamical systems, entropy or equivalents are their only invariant measure (Ornstein, 1974; Ornstein, 1989). There appears to be no singular way to compute it such that in its place are an array of quasiequivalent entropies, some of which are of recent vintage (Adler and Weiss, 1967; Adler and Marcus,
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Fig. 3. Curves of the power spectral scaling exponent a of IMFi, i = 1–16 for nine subjects in testing group one. Every curve represents one subject, each IMFi, associated value represents the mean of 20 computations of every 12 s part. The power spectral scaling exponent, a (Montroll and Shlesinger, 1984; Klafter et al., 1992; Bendler et al., 2004) is an empirical, graphical technique in which the axis values in the Fourier-transformed frequency (power) spectral graph are plotted in logarithmic units. The least squares slope of the middle third of the post-maximal frequency decay is calculated using a conventional least squares algorithm to yield an exponential scaling exponent a. As the180-Hz notch filters were applied to the records, we take 180 Hz as the corner frequency. This figure shows the temporal scale locations of the values of a evidencing a decreasing plateau from IMFi, i = 3–8.
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imf index Fig. 4. Curves of topologic entropy hT of IMFi, i = 1–16 for nine subjects in group one. Every curve represents one subject. Each IMFi, associated value represents the mean of 20 computations of every 12 s part. The figure demonstrates a linear plateau region from approximately i = 6–11 consistent with a temporal scale zone with invariant measure (Cornfield et al., 1982). Each IMFi associated group of mean values for the measure represents 20 9 = 180 calculations, for i = 6–11, 6 20 9 = 1080 calculations of hT of group one is almost the same.
1979; Pincus, 1991; Costa et al., 2005). We have used up to fourteen different statistical and graphic estimates related to the system’s measure-theoretic entropy (Mandell, 2013). An informal description of hT, topological entropy, computed as the logarithm of the growth rate of the trace (leading Frobenius-Perron eigenvector) of the transition incidence matrix representation of the MEG series, reflects the exponential rate of emergence of
new orbits (Bowen and Ruelle, 1975; Bowen, 1978), such as the ‘‘loop’’ connections of the brain’s magnetic field fluxes (Mandell et al., in press). The metric entropy, hM, as the sum across rows of the exponentiated Markoff transition matrix of the MEG series R(plog p) describes the logarithmic distribution of probability weights on the orbits (Adler et al., 1977). Lim sup of the entropies is hT and hM (Cornfield et al., 1982; Pollicott and Yuri, 1998). Since in real data, uniform hyperbolicity
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(B) The non-uniform, hU Fig. 5. Curves of metric entropy hM (A) and the non-uniform entropy hU (B) of IMFi, i = 1–16 for nine subjects in testing group one. Every curve represents one subject. Each IMFi, associated value represents the mean of 20 computations of every 12 s part. Whereas hT quantifies the rate of emergence of new orbits, hM quantifies the distribution of the relative occupancies of these new orbits. The non-uniform entropy indicates the degree of non-uniformity of the entropy, hU = |hT hM|.
is seldom seen, hT – hM, non-uniformity of the entropy is computed as |hT hM| = hU.
SURVEY OF METHODS Two hundred and forty seconds of each subject’s dorsal frontal lobe (F14) MEG record were partitioned into twenty parts of 12 s each. This interval was chosen from the literature describing characteristic times of spontaneous thoughts and images in the task free, resting condition (Antrobus et al., 1966; Giambra, 1989). Therefore each IMFi associated value represents the mean of 20 computations. In the aggregate curves for
nine subjects in Fig. 3, Fig. 4, Fig. 5A, B, each IMFi associated group of mean values for the measure represents 20 9 = 180 calculations. In the separate study of 10 additional subjects, Fig. 6, they are the means of 20 10 = 200 computations. It is this statistical redundancy that allowed the discovery of the almost variation free scaling zone of the invariant topological entropic measure, hT, the most reliable reflection of the presence of a turbulent implicit attractor. With respect to the standard techniques for computation of the specific measures, the associated references contain sufficiently operative algorithmic descriptions. It is our purpose here to describe simply, generally and
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imf index Fig. 6. Curves of topologic entropy hT of IMFi, i = 1–16 for 10 subjects in group two. Every curve represents one subject. Each IMFi associated value represents the mean of 20 computations of every 12 s part. The values of topologic entropy hT are derived with the same method as Fig. 4. There are 10 subjects in group two so the graph includes 10 curves. The result in the first group study in which n = 9 subjects vary together closely with the result in the additional second group study involving additional n = 10subjects. The trend demonstrates a linear plateau region from approximately i = 6–11 consistent with a temporal scale zone with invariant measure. Each IMFi associated group of mean values for the measure represents 20 10 = 200 calculations, for i = 6–11, 6 20 10 = 1200 calculations of hT of group two is almost same as 6 20 9 = 1080 calculations of group one.
accessibly how the determinations were made. Note that an eight-partition was used in the computations of the values of hT, hM and hU for each IMFi. EMD transformation EMD ? IMFi transformation (Huang et al., 1998; Huang and Attoh-Okine, 2005; Aven et al., 2011; Aven et al., 2012): for each pass of the EMD, locate all local maxima and interpolate a cubic spline joining them into the upper envelope and all local minima which the cubic spline joins into the lower envelope and the local mean of the two envelopes are subtracted from the original MEG series and the result is plotted as the first, i = 1, IMF1. This is repeated an n number of times, i = 1. . .n, in which n is dependent upon an arbitrary variety of issues including the experimental goal of the transformation. Since we seek a priori unknown temporal region of adjoining IMFi’s in which the entropy measures and scaling are relatively dominant, we extend the series beyond the usual criteria to indices without maxima and minima. The power spectral scaling exponent, a The power spectral scaling exponent, a (Montroll and Shlesinger, 1984; Klafter et al., 1992; Bendler et al., 2004) is another empirical, graphical technique in which the axis values in the Fourier-transformed frequency (power) spectral graph are plotted in logarithmic units. The slope of the middle third of the post-maximal frequency decay is calculated using a conventional least squares algorithm to yield an exponential scaling exponent a
Topological entropy, hT Topological entropy, hT (Adler et al., 1965; Bowen, 1978; Adler and Marcus, 1979): Following the eight partition of the MEG series or an IMFi, a square, symmetric, nonnegative transition-incidence matrix is formed which, given the Hilbert matrix conditions, has a leading Frobenius-Perron eigenvalue, the logarithm of which is the topological entropy, hT (Eckmann and Ruelle, 1985). With respect to our studies of close to 100 MEG subjects, we have found hT to be consistently positive (Mandell, 2013), consistent with the presence of a turbulent magnetic field (Mandell et al., 2011a). As noted above, an algorithmic computation of hT can be calculated as the logarithm of the asymptotic growth rate of the trace of the exponentiated transitionincidence matrix. Entropy hT quantifies the rate of emergence of new orbits, one speculates these may physically be magnetic flux connections. Metric, hM, and non-uniform, hU, entropies Metric, hM, and non-uniform, hU, entropies (Adler et al., 1977; Ott et al., 1994; Weiss, 1995; Ott, 1998; Mandell et al., 2011a): Whereas hT quantifies the rate of emergence of new orbits, hM quantifies the distribution of the relative density of these new orbits. The transition matrix (not transition-incidence matrix) is transformed into a Markoff matrix, each row adding up to 1.0. Then exponentiation of the Markoff matrix reaches its asymptotic expression when all rows are identical. We have the sum of any row as nlog n equals the metric entropy, hM. For equi-distribution across orbits, hM = 1.0 and for maximum topological entropy, hT = 1. For systems with uniform hyperbolicity, each point uniformly
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an intersection of stable and unstable manifolds, |hT hM| = 0.0. It is in this way that |hT hM| > 0.0 indicates the degree of non-uniformity of the entropy, hU = |hT hM|.
RESULTS We derive the values of hT, hM, hU and a as functions of IMFi, i = 1...n. Fig. 1 is a characteristic graph of the result of EMD ? IMFi, i = 1. . .n, n = 16, transformations of 60 s of the MEG record from the F14 sensor frontal lobe region. Reading left to right in each of the four rows we see the results of the method described above in the section ‘EMD transformation’. Here the place in the sequence indicates the mode in the order of the decomposition from left to right and from top to bottom. The graphs show a general decrease in the frequency and complexity of the fluctuations with an increase in mode order. It is in this way that the EMD ? IMFi transformation decomposes the raw MEG signal (recorded at 600 Hz with 150-Hz corner frequency, with a 0.6-Hz high-pass and 60-Hz notch filters) into separated time scales (Huang et al., 1998; Aven et al., 2012). Fig. 2 illustrates the power spectra typical log ordinate-log abscissa plot of a two subjects’ 16 IMFs. The curves demonstrate temporal regions of similarity in the post maximal frequency slopes of decaying power, particularly from IMFi,i = 3–8 (right to left), which are nearly uniform scaling regions. Self-similarity across temporal scales is a generic feature of turbulent attractors of nonlinear dynamical systems (Heisenberg, 1948; Mandelbrot, 1977; Frisch, 1995). This temporal region of near uniform scaling, can be evaluated quantitatively using the distribution of power spectral scaling exponent, a, across IMFi’s. This serves as a component of the aggregate measure evidence for the existence of an implicit attractor. Fig. 3 is a graph of the average (n = 9) IMFi showing the temporal scale locations of the values of a evidencing a decreasing plateau from IMFi i = 3–8 generally consistent with the graphs of the log–log power spectra seen in Fig. 2. It is generally the case that systems (like the brain) with many degrees of freedom and strong interactions characteristically demonstrate fractional power law scaling (Novikov et al., 1997). This is particularly true in systems such as the ‘‘resting’’ human MEG (Mandell et al., 2011a) that manifest turbulent intermittency (Hirsch et al., 1982; Berge et al., 1984). Fig. 4 is a graph of the average (n = 9) of hT as a function of IMFi, demonstrating a linear plateau region from approximately i = 6–11 consistent with a temporal scale zone with invariant measure (Cornfield et al., 1982). The time scale region of the topological entropy measure is right shifted from the plot above, Fig. 3, indicating a more persistent presence of the lim sup of the measure-theoretic entropy, hT at longer time scales. Whereas the boundaries of attractors can be smooth, it is generally the case that turbulent attractors, here the inferred implicit attractor, may have non-uniform fractal basin boundaries (McDonald et al., 1985; Grebogi et al.,
Table 1a. Correlation analyses of group one (n = 9).
fpeak
a hT hM hU
fpeak
a
hT
hM
hU
1.00 0.99 0.75 0.76 0.68
0.99 1.00 0.76 0.78 0.67
0.75 0.76 1.00 0.98 0.96
0.76 0.78 0.98 1.00 0.90
0.68 0.67 0.96 0.90 1.00
Table 1b. Correlation analyses of group two (n = 10).
fpeak
a hT hM hU
fpeak
a
hT
hM
hU
1.00 0.99 0.75 0.76 0.68
0.99 1.00 0.77 0.80 0.68
0.75 0.77 1.00 0.98 0.96
0.76 0.80 0.98 1.00 0.90
0.68 0.68 0.96 0.90 1.00
1987). It is for this reason that the scaling range of a and that of hT are clearly over-lapping but not necessarily isomorphic with respect to the span of the IMFi’s. Fig. 5A is a graph, similar to Fig. 4, of the average (n = 9) of hM as a function of IMFi, demonstrating an irregularly decaying plateau-like region that is less well defined than that of hT. This reflects the greater variation in the relative weight distribution-on the emergent new orbits, the rate of which is indicated by hT. Whereas hT in Fig. 4 demonstrates invariance across the indicated IMFi range, hM decays across the increasing time scale suggesting that the inferred implicit attractor is denser at faster frequencies. The decrease in density at longer time scales, compared with hT, is reflected in the progressive increase in hU = |hT hM| with increasing IMFi seen in Fig. 5B. The clearly invariant measure across IMFi is hT, Fig. 4. It shows remarkable stability across IMFi, i = 6–11, which we interpret as evidence for a temporal scales location of an implicit attractor’s ergodic invariant measure. This finding was examined again using a separate set of observations in 10 (n = 10) additional subjects. The results are graphed in Fig. 6 confirming the invariance of hT across IMFi, i = 6–11. If we assume that the flow of the magnetic flux is continuous, at least C1, the consistent positivity of hT across a finite set of scales is associated with other properties of the putative implicit attractor including the presence of at least one positive Lyapounov exponent indicating both sensitivity to initial conditions and a dynamic that disorders sequences, mixing (Barrreria and Pesin, 2007). We use Figs. 2, 4 and 6 and the hT invariant IMFi, i = 6–11, to estimate the real time scale of the implicit attractor spanning the range of 10–20 Hz. We recall that the Ornstein group of theorems have proven that (paraphrased) in expansive, mixing turbulent attractors, there is only one measure up to equivalence, the system’s entropy (Ornstein, 1974; Ornstein, 1989; Pollicott and Yuri, 1998). Without the availability of a specifically best method for its computation, we have approximated it with multiple simultaneous measures (Mandell, 2013). Although these measures are distinct
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with respect to their computations, it is of interest that as predicted they vary together closely in the first study in which n = 9 subjects (Table 1a) and the additional study involving additional subjects in which n = 10 (Table 1b). The high intercorrelations are consistent with the Ornstein Theorems indicating the equivalency of entropic and related measures of chaotic attractors of nonlinear dynamical systems.
SUMMARY (1) Combining MEG frontal lobe records and their EMF ? IMF time scale decomposition with their associated entropic invariant measures, it was possible to identify and locate in a finite hierarchy of time scales the implicit turbulent magnetic field attractors. (2) In spite of the wide diversity of techniques with which to quantitatively describe the entropic complexity of putative implicit chaotic attractors, the results are highly intercorrelated. This is consistent with theorems suggesting that in such complex systems, there is a singular underlying measure, the measure-theoretic entropy. Its invariance across a finite set of time scales is consistent with the presence of turbulent magnetic field attractors.
Acknowledgements—Appreciation is expressed for the support of Shanghai Municipal Education Commission to Advanced Scholars of Oversea Research and Training plan for University Teachers to Xiaoxia Huang; the Jiansu Oversea Research and Training Program for Prominent University Teachers to Liya Huang; the Fetzer Franklin Fund of the John Fetzer Memorial Trust, Kalamazoo, MI, to Arnold J. Mandell.
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(Accepted 24 February 2014) (Available online 6 March 2014)
INTRINSIC MODE FUNCTIONS LOCATE IMPLICIT TURBULENT ATTRACTORS IN TIME IN FRONTAL LOBE MEG RECORDINGS X. HUANG, a,c L. HUANG, b,c T.-P. JUNG, d C.-K. CHENG c* AND A. J. MANDELL e,f
decompositions. Ó 2014 IBRO. Published by Elsevier Ltd. All rights reserved.
a
Department of Computer Science and Engineering, College of Information Engineering, Shanghai Maritime University, Shanghai 201306, PR China b Institute of Electronic Science and Engineering, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu, PR China
Key words: implicit turbulent attractor, Empirical Mode Decomposition, topological entropy, metric entropy, nonuniform entropy, power spectral scaling exponent a.
c Department of Computer Science and Engineering, University of California at San Diego, La Jolla, CA 92093, USA d
Swartz Center for Computational Neuroscience, Engineering, Institute of Engineering in Medicine, University of California at San Diego, La Jolla, CA 92093, USA
INTRODUCTION Most current studies of human MEG recordings are focused on neurophysiological source localization, treating the magnetic field recordings as epiphenomenal reflections of underlying neuroelectrical events (Cohen, 1972; Nolte et al., 2004; Cornwell et al., 2007; Cornwell et al., 2008). Among the techniques used to approximate a solution to ‘‘the inverse problem’’ (given a magnetic field, what and where is its electric event sources) are the beamformer techniques (Vrba and Robinson, 2000; Robinson, 2004). Globally distributed, spontaneous magnetic field fluctuations in the resting condition have often been interpreted as an impediment to techniques of localization being ‘‘. . .high (eigenvalue) ranked background activity. . .interfering magnetic fields. . .(meaningless) intrinsic brain noise. . .’’ (Sekihara et al., 1996, 2008). In contrast, in these studies, the magnetic fields are regarded, not as epiphenomenal signs of underlying neurophysiological source events or functionally irrelevant ‘‘brain noise’’, but rather the B fields are seen as functional components of brain dynamics in themselves, B = cH; where B magnetic flux density, c magnetic permeability and H magnetic field strength. Neocortical magnetic fields associated with neurophysiological events are here regarded as potentially nonlinear ‘‘feedback’’ modulatory near field influences on the thresholds of the neocortical pyramidal neuronal networks that generated them (Chiabrera et al., 1985; Black man et al., 1988; Delparte and Persinger, 2007; Frolich and McCormick, 2010; Ledda et al., 2010). With the nonlinear feedback elements of a dynamical system we posit that the magnetic fields of the continuously active ‘‘resting’’ human cortex without external stimuli (Raichle et al., 2001) can self-organize into non-stationary (transient) turbulent (in time) attractors that reveal themselves by their associated time-scale located, invariant entropic measures (Mandell et al., 2011c).
e
Multi Media Imaging Laboratory, Department of Psychiatry, University of California at San Diego School of Medicine, La Jolla, CA 92093, USA f Fetzer Franklin Fund of the John Fetzer Memorial Trust, Kalamazoo, MI, USA
Abstract—In seeking evidence for the presence and characteristic range of coupled time scale(s) of putative implicit turbulent attractors of dorsal frontal lobe magnetic fields, the recorded nonstationary, nonlinear MEG signals were non-orthogonally decomposed using Huang’s Empirical Mode Decomposition, EMD, (Huang and Attoh-Okine, 2005) into 16 Intrinsic Mode Functions, EMD ? IMFi, i = 1. . .16. Measures known to be invariant in non-uniformly hyperbolic (turbulent) dynamical systems, topological entropy, hT, metric entropy, hM, non-uniform entropy, hU and power spectral scaling exponent, a, were imposed on each of the IMFi which evidenced most clearly an invariant temporal scale zone of IMFi, i = 6. . .11, for hT, which we have found to be the most robust of invariant measures of MEG’s magnetic field turbulent attractors (Mandell et al., 2011a,b; Mandell, 2013). The ergodic theory of dynamical systems (Walters, 1982; Pollicott and Yuri, 1998) allows the inference that an implicit attractor with consistently hT > 0 will also evidence at least one positive Lyapounov exponent indicating the presence of a turbulent attractor with exponential separation of nearby initial conditions, exponential convergence of distant points and disordering, mixing, of orbital sequences. It appears that this approach permits the inference of the presence of chaotic, turbulent attractor and its characteristic time scales without the invocation of arbitrary n-dimensional embedding, phase space reconstructions or (inappropriate) orthogonal
*Corresponding author. Tel: +1-858-534-6184; fax: +1-858-5347029. E-mail address: [email protected] (C.-K. Cheng). Abbreviations: EMD, Empirical Mode Decomposition; IMF, Intrinsic Mode Functions; MEG, Magnetoencephalography. http://dx.doi.org/10.1016/j.neuroscience.2014.02.038 0306-4522/Ó 2014 IBRO. Published by Elsevier Ltd. All rights reserved. 91
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It is for these reasons that we regard the temporal structure of the brain’s magnetic field as important in itself, independent of what it might imply about its sources. We further speculate that the brain’s magnetic fields may mediate ‘‘up’’ states and ‘‘down’’ states of neocortical pyramidal cell excitability thresholds (Cossart et al., 2003; Sachdev et al., 2004). In these methodological studies, the MEG recordings of brain magnetic signals were obtained from archival records recorded at the National Institutes of Mental Health Core MEG Facility (Weinberg et al., 1984; Anninos et al., 1987; Fife et al., 2002). The recordings were made on a 275 channel, superconducting quantum interference devices (SQUID) radial gradiometer system from CTF Systems Inc. of Port Coquitlam, British Columbia, Canada. This data were obtained from previous archival MEG studies focused on relative power using Fourier analysis (Rutter et al., 2009). All subjects involved in these studies had given written informed consent according to protocols approved by the NIH CNS Institutional Review Board.
Hierarchical scaling and measure-theoretic, informational entropies in magnetoencephalic recordings of turbulent brain magnetic fields Recent research has suggested that time-dependent, magnetic field signals of 103–104fT magnetic field strength as recorded from the human brain by magnetoencephalography, MEG, manifest the phase space and measure characteristics of turbulent (chaotic) dynamical systems (Kowalik and Elbert, 1994; Mandell et al., 2011a,b, in press; Robinson et al., 2013; Vakorin et al., 2010). Consistent with this interpretation of B field fluctuations is the presence of a number of quantifiable characteristics of turbulence including positive leading Lyapunov exponents, fractional capacity dimensions and positive entropy generation (Mandell, 2013). In addition and of relevance to this study is the characteristic fractional power law scaling, a, of the MEG signal (de Gennes, 1979; Churilla et al., 1996; Selz and Mandell, 1997; Peng et al., 1998) associated with positive topological and metric entropy generation (Adler and Weiss, 1967; Dinaburg, 1971; Eckmann and Ruelle, 1985). The measures derived from the ergodic (‘‘invariant measure’’) theory of dynamical systems (Cornfield et al., 1982; Walters, 1982; Eckmann and Ruelle, 1985; Pollicott and Yuri, 1998) such as topological entropy hT, metric entropy, hM, and non-uniform entropy, hU = |hT hM| (Ott, 1998), These measures are obtained from discrete and continuous time series in dimension zero and one respectively. However, the hierarchal multi-scale characteristics of the turbulent MEG record (Mandell et al., 2011a) call for a higher dimensional embedding (Cellucci et al., 2003; Pecora et al., 2007). It should be noted that the values of a, hT, hM, and hU are sensitive to the bandwidth of the observations with hT decreasing as a increasing (Mandell and Selz, 1997). Method-exemplifying MEG data reported here were recorded consistently at 600 Hz
with a bandwidth bounded by a 150-Hz cut-off, its ‘‘corner frequency’’ (Rutter et al., 2009). Hierarchical decomposition of the human frontal lobe MEG signal Given the multiplicity of relevant time scales in neurobiological signals (Mandell et al., 1982; Basar et al., 1983; Mandell and Selz, 1992), it becomes necessary to avoid the attendant risk of over-sampling, non-discriminable noise as well as the findings that completely different dynamical structures as nonstationary ‘‘attractive-repellers’’ (Farmer et al., 1983; Nichols et al., 2003) can emerge at different time scales of observation (Ashkenazy et al., 2000).For these reasons, multi-scale techniques have been developed in the computation of singular measures such as the measure-theoretic entropies (Peng et al., 1998; Costa et al., 2005) many of which are a derivative of approximate entropy (Pincus, 1991). The nonlinear and non-stationary properties of brain electromagnetic time series have been approached using a variety of quasi-linear time-frequency techniques such as the Stockwell (Liu et al., 2010) and wavelet transformations (Daubechies, 1992; Jia et al., 2006; Lim et al., 2012) as well as the orthogonally restricted singular value decomposition (Haq et al., 1997). The parametrically free empirical mode decomposition, EMD, that generates intrinsic mode functions, IMFi was introduced by Huang (Huang et al., 1998) to study the multi-scale time-frequency behavior of nonstationary wind and water waves. Among the many subsequent applications to nonstationary and nonlinear time series include those involving the human EEG (Pachori, 2008) and the MEG (Aven et al., 2011; Aven et al., 2012). Empirical Mode Decomposition a simple graphical algorithm constructs new series, Intrinsic Mode Functions, composed of the average sequence of the sequential maxima and minima of the original data. Generating a hierarchy of computationally defined modes, the EMD ? IMFi transformation does not require stationarity or the (theoretical) infinite length of Fourier transformation. Whereas our previous applications of the EMD ? IMFi transformation to human brain MEG series involved the elucidation of the non-orthogonal IMFi amplitude, ‘‘energy’’ distributions across temporal scales, i, and their use in computation of capacity dimensions and Hurst exponents (Aven et al., 1212), the methods described here focus on the distributions of symbolic dynamically derived entropy measures, hT, hM, hU and fractional power law spectral scaling exponents, a. The EMD decomposes the multiscale behavior of the MEG signal into time-scale separated modes indexed as IMFi i = 1. . .n, These in turn are each analyzed with respect to their values for hT, hM, hU and a. Whereas the most relevant time scale(s) of the putative non-stationary attractor would seem to be indicated by the peak amplitude distributions across IMFi (Aven et al., 2012), we have recently shown that in brain space, it can be the case that MEG amplitude (or power) fails to temporally and spatially localize functionally involved
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(B) 16 IMFs of subject B Fig. 1. Characteristic graphs of the EMD decomposition result of two subjects from the test group. The 16 subgraphs correspond to 16 IMFs. A 275 channel, superconducting quantum interference device was used in MEG data collection. The record is sampled at 600-Hz with 150-Hz acquisition cut off. The difference sequences, left minus right, (i.e. Semmetric Sensor Difference Series) ssds(i) signals were taken as the input which can reduce the penetrance noise of electromagnetic field correlates of blink, cough, and movement as well as the cardiac and respiratory artifacts that both symmetric sensors generally share. We used the ssds(i) signals at dorsolateral prefrontal cortex area F14. A 0.6-Hz high-pass and 60-Hz notch filters were routinely applied to the input. Reading left to right for each of the four rows from top to bottom we see the results of EMD ? IMFi, i = 1. . .n, n = 16, transformations of the first 60 s.
brain regions under circumstances in which an entropic measure, rank vector entropy, does so (Robinson et al., 2013). In an analogous way, we use hT, hM, hU and a across IMFi to localize the relevant scales in time not space. This approach exploits the superiority of the MEG over MRI in temporal resolution. Multiple invariant measure-theoretic entropies The theoretical frame work for the analyses of the IMFi is the ergodic theory of dynamical systems (Kolmogorov, 1958; Arnold and Avez, 1968; Eckmann and Ruelle, 1985; Pollicott and Yuri, 1998). Under this aegis, real
valued IMFi series can be reconstructed on smooth manifolds for embedding, partition, symbolic dynamic encoding of transition matrices followed by topological and statistical characterizations (Smale, 1967; Sinai, 1972; Bowen and Ruelle, 1975; Adler et al., 1977; Adler and Marcus, 1979). Whereas ergodic in classical statistical mechanics implies equivalence in the system’s time and space averages (Reichl, 1980), ergodic in this context implies a system with invariant measures across differing initial conditions (Cornfield et al., 1982). The application of the EMD ? IMFi transformation can be viewed as systematic topologicalgeometric sampling with the expectation that the IMFi
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time scales relevant to the putative non-stationary attractor would manifest time scales of the attractor’s invariant measure. There is a family of theorems applicable to chaotic dynamical systems (paraphrased) that says in uniformly (also nonuniformly Young, 1998) divergent, hyperbolic
(no norm 1) dynamical systems, entropy or equivalents are their only invariant measure (Ornstein, 1974; Ornstein, 1989). There appears to be no singular way to compute it such that in its place are an array of quasiequivalent entropies, some of which are of recent vintage (Adler and Weiss, 1967; Adler and Marcus,
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Fig. 3. Curves of the power spectral scaling exponent a of IMFi, i = 1–16 for nine subjects in testing group one. Every curve represents one subject, each IMFi, associated value represents the mean of 20 computations of every 12 s part. The power spectral scaling exponent, a (Montroll and Shlesinger, 1984; Klafter et al., 1992; Bendler et al., 2004) is an empirical, graphical technique in which the axis values in the Fourier-transformed frequency (power) spectral graph are plotted in logarithmic units. The least squares slope of the middle third of the post-maximal frequency decay is calculated using a conventional least squares algorithm to yield an exponential scaling exponent a. As the180-Hz notch filters were applied to the records, we take 180 Hz as the corner frequency. This figure shows the temporal scale locations of the values of a evidencing a decreasing plateau from IMFi, i = 3–8.
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imf index Fig. 4. Curves of topologic entropy hT of IMFi, i = 1–16 for nine subjects in group one. Every curve represents one subject. Each IMFi, associated value represents the mean of 20 computations of every 12 s part. The figure demonstrates a linear plateau region from approximately i = 6–11 consistent with a temporal scale zone with invariant measure (Cornfield et al., 1982). Each IMFi associated group of mean values for the measure represents 20 9 = 180 calculations, for i = 6–11, 6 20 9 = 1080 calculations of hT of group one is almost the same.
1979; Pincus, 1991; Costa et al., 2005). We have used up to fourteen different statistical and graphic estimates related to the system’s measure-theoretic entropy (Mandell, 2013). An informal description of hT, topological entropy, computed as the logarithm of the growth rate of the trace (leading Frobenius-Perron eigenvector) of the transition incidence matrix representation of the MEG series, reflects the exponential rate of emergence of
new orbits (Bowen and Ruelle, 1975; Bowen, 1978), such as the ‘‘loop’’ connections of the brain’s magnetic field fluxes (Mandell et al., in press). The metric entropy, hM, as the sum across rows of the exponentiated Markoff transition matrix of the MEG series R(plog p) describes the logarithmic distribution of probability weights on the orbits (Adler et al., 1977). Lim sup of the entropies is hT and hM (Cornfield et al., 1982; Pollicott and Yuri, 1998). Since in real data, uniform hyperbolicity
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metrix entropy, hm
0.6 0.5 0.4 0.3 0.2 0.1 0
2
4
6
8
10
12
14
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imf index
(A) The metric entropy, hM 0.4
non uniform entropy, hu
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 0
2
4
6
8
10
12
14
16
imf index
(B) The non-uniform, hU Fig. 5. Curves of metric entropy hM (A) and the non-uniform entropy hU (B) of IMFi, i = 1–16 for nine subjects in testing group one. Every curve represents one subject. Each IMFi, associated value represents the mean of 20 computations of every 12 s part. Whereas hT quantifies the rate of emergence of new orbits, hM quantifies the distribution of the relative occupancies of these new orbits. The non-uniform entropy indicates the degree of non-uniformity of the entropy, hU = |hT hM|.
is seldom seen, hT – hM, non-uniformity of the entropy is computed as |hT hM| = hU.
SURVEY OF METHODS Two hundred and forty seconds of each subject’s dorsal frontal lobe (F14) MEG record were partitioned into twenty parts of 12 s each. This interval was chosen from the literature describing characteristic times of spontaneous thoughts and images in the task free, resting condition (Antrobus et al., 1966; Giambra, 1989). Therefore each IMFi associated value represents the mean of 20 computations. In the aggregate curves for
nine subjects in Fig. 3, Fig. 4, Fig. 5A, B, each IMFi associated group of mean values for the measure represents 20 9 = 180 calculations. In the separate study of 10 additional subjects, Fig. 6, they are the means of 20 10 = 200 computations. It is this statistical redundancy that allowed the discovery of the almost variation free scaling zone of the invariant topological entropic measure, hT, the most reliable reflection of the presence of a turbulent implicit attractor. With respect to the standard techniques for computation of the specific measures, the associated references contain sufficiently operative algorithmic descriptions. It is our purpose here to describe simply, generally and
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topologic entropy, ht
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4
6
8
10
12
14
16
imf index Fig. 6. Curves of topologic entropy hT of IMFi, i = 1–16 for 10 subjects in group two. Every curve represents one subject. Each IMFi associated value represents the mean of 20 computations of every 12 s part. The values of topologic entropy hT are derived with the same method as Fig. 4. There are 10 subjects in group two so the graph includes 10 curves. The result in the first group study in which n = 9 subjects vary together closely with the result in the additional second group study involving additional n = 10subjects. The trend demonstrates a linear plateau region from approximately i = 6–11 consistent with a temporal scale zone with invariant measure. Each IMFi associated group of mean values for the measure represents 20 10 = 200 calculations, for i = 6–11, 6 20 10 = 1200 calculations of hT of group two is almost same as 6 20 9 = 1080 calculations of group one.
accessibly how the determinations were made. Note that an eight-partition was used in the computations of the values of hT, hM and hU for each IMFi. EMD transformation EMD ? IMFi transformation (Huang et al., 1998; Huang and Attoh-Okine, 2005; Aven et al., 2011; Aven et al., 2012): for each pass of the EMD, locate all local maxima and interpolate a cubic spline joining them into the upper envelope and all local minima which the cubic spline joins into the lower envelope and the local mean of the two envelopes are subtracted from the original MEG series and the result is plotted as the first, i = 1, IMF1. This is repeated an n number of times, i = 1. . .n, in which n is dependent upon an arbitrary variety of issues including the experimental goal of the transformation. Since we seek a priori unknown temporal region of adjoining IMFi’s in which the entropy measures and scaling are relatively dominant, we extend the series beyond the usual criteria to indices without maxima and minima. The power spectral scaling exponent, a The power spectral scaling exponent, a (Montroll and Shlesinger, 1984; Klafter et al., 1992; Bendler et al., 2004) is another empirical, graphical technique in which the axis values in the Fourier-transformed frequency (power) spectral graph are plotted in logarithmic units. The slope of the middle third of the post-maximal frequency decay is calculated using a conventional least squares algorithm to yield an exponential scaling exponent a
Topological entropy, hT Topological entropy, hT (Adler et al., 1965; Bowen, 1978; Adler and Marcus, 1979): Following the eight partition of the MEG series or an IMFi, a square, symmetric, nonnegative transition-incidence matrix is formed which, given the Hilbert matrix conditions, has a leading Frobenius-Perron eigenvalue, the logarithm of which is the topological entropy, hT (Eckmann and Ruelle, 1985). With respect to our studies of close to 100 MEG subjects, we have found hT to be consistently positive (Mandell, 2013), consistent with the presence of a turbulent magnetic field (Mandell et al., 2011a). As noted above, an algorithmic computation of hT can be calculated as the logarithm of the asymptotic growth rate of the trace of the exponentiated transitionincidence matrix. Entropy hT quantifies the rate of emergence of new orbits, one speculates these may physically be magnetic flux connections. Metric, hM, and non-uniform, hU, entropies Metric, hM, and non-uniform, hU, entropies (Adler et al., 1977; Ott et al., 1994; Weiss, 1995; Ott, 1998; Mandell et al., 2011a): Whereas hT quantifies the rate of emergence of new orbits, hM quantifies the distribution of the relative density of these new orbits. The transition matrix (not transition-incidence matrix) is transformed into a Markoff matrix, each row adding up to 1.0. Then exponentiation of the Markoff matrix reaches its asymptotic expression when all rows are identical. We have the sum of any row as nlog n equals the metric entropy, hM. For equi-distribution across orbits, hM = 1.0 and for maximum topological entropy, hT = 1. For systems with uniform hyperbolicity, each point uniformly
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an intersection of stable and unstable manifolds, |hT hM| = 0.0. It is in this way that |hT hM| > 0.0 indicates the degree of non-uniformity of the entropy, hU = |hT hM|.
RESULTS We derive the values of hT, hM, hU and a as functions of IMFi, i = 1...n. Fig. 1 is a characteristic graph of the result of EMD ? IMFi, i = 1. . .n, n = 16, transformations of 60 s of the MEG record from the F14 sensor frontal lobe region. Reading left to right in each of the four rows we see the results of the method described above in the section ‘EMD transformation’. Here the place in the sequence indicates the mode in the order of the decomposition from left to right and from top to bottom. The graphs show a general decrease in the frequency and complexity of the fluctuations with an increase in mode order. It is in this way that the EMD ? IMFi transformation decomposes the raw MEG signal (recorded at 600 Hz with 150-Hz corner frequency, with a 0.6-Hz high-pass and 60-Hz notch filters) into separated time scales (Huang et al., 1998; Aven et al., 2012). Fig. 2 illustrates the power spectra typical log ordinate-log abscissa plot of a two subjects’ 16 IMFs. The curves demonstrate temporal regions of similarity in the post maximal frequency slopes of decaying power, particularly from IMFi,i = 3–8 (right to left), which are nearly uniform scaling regions. Self-similarity across temporal scales is a generic feature of turbulent attractors of nonlinear dynamical systems (Heisenberg, 1948; Mandelbrot, 1977; Frisch, 1995). This temporal region of near uniform scaling, can be evaluated quantitatively using the distribution of power spectral scaling exponent, a, across IMFi’s. This serves as a component of the aggregate measure evidence for the existence of an implicit attractor. Fig. 3 is a graph of the average (n = 9) IMFi showing the temporal scale locations of the values of a evidencing a decreasing plateau from IMFi i = 3–8 generally consistent with the graphs of the log–log power spectra seen in Fig. 2. It is generally the case that systems (like the brain) with many degrees of freedom and strong interactions characteristically demonstrate fractional power law scaling (Novikov et al., 1997). This is particularly true in systems such as the ‘‘resting’’ human MEG (Mandell et al., 2011a) that manifest turbulent intermittency (Hirsch et al., 1982; Berge et al., 1984). Fig. 4 is a graph of the average (n = 9) of hT as a function of IMFi, demonstrating a linear plateau region from approximately i = 6–11 consistent with a temporal scale zone with invariant measure (Cornfield et al., 1982). The time scale region of the topological entropy measure is right shifted from the plot above, Fig. 3, indicating a more persistent presence of the lim sup of the measure-theoretic entropy, hT at longer time scales. Whereas the boundaries of attractors can be smooth, it is generally the case that turbulent attractors, here the inferred implicit attractor, may have non-uniform fractal basin boundaries (McDonald et al., 1985; Grebogi et al.,
Table 1a. Correlation analyses of group one (n = 9).
fpeak
a hT hM hU
fpeak
a
hT
hM
hU
1.00 0.99 0.75 0.76 0.68
0.99 1.00 0.76 0.78 0.67
0.75 0.76 1.00 0.98 0.96
0.76 0.78 0.98 1.00 0.90
0.68 0.67 0.96 0.90 1.00
Table 1b. Correlation analyses of group two (n = 10).
fpeak
a hT hM hU
fpeak
a
hT
hM
hU
1.00 0.99 0.75 0.76 0.68
0.99 1.00 0.77 0.80 0.68
0.75 0.77 1.00 0.98 0.96
0.76 0.80 0.98 1.00 0.90
0.68 0.68 0.96 0.90 1.00
1987). It is for this reason that the scaling range of a and that of hT are clearly over-lapping but not necessarily isomorphic with respect to the span of the IMFi’s. Fig. 5A is a graph, similar to Fig. 4, of the average (n = 9) of hM as a function of IMFi, demonstrating an irregularly decaying plateau-like region that is less well defined than that of hT. This reflects the greater variation in the relative weight distribution-on the emergent new orbits, the rate of which is indicated by hT. Whereas hT in Fig. 4 demonstrates invariance across the indicated IMFi range, hM decays across the increasing time scale suggesting that the inferred implicit attractor is denser at faster frequencies. The decrease in density at longer time scales, compared with hT, is reflected in the progressive increase in hU = |hT hM| with increasing IMFi seen in Fig. 5B. The clearly invariant measure across IMFi is hT, Fig. 4. It shows remarkable stability across IMFi, i = 6–11, which we interpret as evidence for a temporal scales location of an implicit attractor’s ergodic invariant measure. This finding was examined again using a separate set of observations in 10 (n = 10) additional subjects. The results are graphed in Fig. 6 confirming the invariance of hT across IMFi, i = 6–11. If we assume that the flow of the magnetic flux is continuous, at least C1, the consistent positivity of hT across a finite set of scales is associated with other properties of the putative implicit attractor including the presence of at least one positive Lyapounov exponent indicating both sensitivity to initial conditions and a dynamic that disorders sequences, mixing (Barrreria and Pesin, 2007). We use Figs. 2, 4 and 6 and the hT invariant IMFi, i = 6–11, to estimate the real time scale of the implicit attractor spanning the range of 10–20 Hz. We recall that the Ornstein group of theorems have proven that (paraphrased) in expansive, mixing turbulent attractors, there is only one measure up to equivalence, the system’s entropy (Ornstein, 1974; Ornstein, 1989; Pollicott and Yuri, 1998). Without the availability of a specifically best method for its computation, we have approximated it with multiple simultaneous measures (Mandell, 2013). Although these measures are distinct
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with respect to their computations, it is of interest that as predicted they vary together closely in the first study in which n = 9 subjects (Table 1a) and the additional study involving additional subjects in which n = 10 (Table 1b). The high intercorrelations are consistent with the Ornstein Theorems indicating the equivalency of entropic and related measures of chaotic attractors of nonlinear dynamical systems.
SUMMARY (1) Combining MEG frontal lobe records and their EMF ? IMF time scale decomposition with their associated entropic invariant measures, it was possible to identify and locate in a finite hierarchy of time scales the implicit turbulent magnetic field attractors. (2) In spite of the wide diversity of techniques with which to quantitatively describe the entropic complexity of putative implicit chaotic attractors, the results are highly intercorrelated. This is consistent with theorems suggesting that in such complex systems, there is a singular underlying measure, the measure-theoretic entropy. Its invariance across a finite set of time scales is consistent with the presence of turbulent magnetic field attractors.
Acknowledgements—Appreciation is expressed for the support of Shanghai Municipal Education Commission to Advanced Scholars of Oversea Research and Training plan for University Teachers to Xiaoxia Huang; the Jiansu Oversea Research and Training Program for Prominent University Teachers to Liya Huang; the Fetzer Franklin Fund of the John Fetzer Memorial Trust, Kalamazoo, MI, to Arnold J. Mandell.
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(Accepted 24 February 2014) (Available online 6 March 2014)
