{"title": "Mapping distinct timescales of functional interactions among brain networks", "book": "Advances in Neural Information Processing Systems", "page_first": 4109, "page_last": 4118, "abstract": "Brain processes occur at various timescales, ranging from milliseconds (neurons) to minutes and hours (behavior). Characterizing functional coupling among brain regions at these diverse timescales is key to understanding how the brain produces behavior. Here, we apply instantaneous and lag-based measures of conditional linear dependence, based on Granger-Geweke causality (GC), to infer network connections at distinct timescales from functional magnetic resonance imaging (fMRI) data. Due to the slow sampling rate of fMRI, it is widely held that GC produces spurious and unreliable estimates of functional connectivity when applied to fMRI data. We challenge this claim with simulations and a novel machine learning approach. First, we show, with simulated fMRI data, that instantaneous and lag-based GC identify distinct timescales and complementary patterns of functional connectivity. Next, we analyze fMRI scans from 500 subjects and show that a linear classifier trained on either instantaneous or lag-based GC connectivity reliably distinguishes task versus rest brain states, with ~80-85% cross-validation accuracy. Importantly, instantaneous and lag-based GC exploit markedly different spatial and temporal patterns of connectivity to achieve robust classification. Our approach enables identifying functionally connected networks that operate at distinct timescales in the brain.", "full_text": "Mapping distinct timescales of functional interactions\n\namong brain networks\n\nMali Sundaresan1\n\ns.malisundar@gmail.com\n\nArshed Nabeel2\n\narshed@iisc.ac.in\n\nDevarajan Sridharan1,2\u2217\nsridhar@iisc.ac.in\n\n1Center for Neuroscience, Indian Institute of Science, Bangalore\n\n2Department of Computer Science and Automation, Indian Institute of Science, Bangalore\n\nAbstract\n\nBrain processes occur at various timescales, ranging from milliseconds (neurons)\nto minutes and hours (behavior). Characterizing functional coupling among brain\nregions at these diverse timescales is key to understanding how the brain produces\nbehavior. Here, we apply instantaneous and lag-based measures of conditional\nlinear dependence, based on Granger-Geweke causality (GC), to infer network\nconnections at distinct timescales from functional magnetic resonance imaging\n(fMRI) data. Due to the slow sampling rate of fMRI, it is widely held that GC\nproduces spurious and unreliable estimates of functional connectivity when applied\nto fMRI data. We challenge this claim with simulations and a novel machine\nlearning approach. First, we show, with simulated fMRI data, that instantaneous\nand lag-based GC identify distinct timescales and complementary patterns of func-\ntional connectivity. Next, we analyze fMRI scans from 500 subjects and show\nthat a linear classi\ufb01er trained on either instantaneous or lag-based GC connectivity\nreliably distinguishes task versus rest brain states, with \u223c80-85% cross-validation\naccuracy. Importantly, instantaneous and lag-based GC exploit markedly differ-\nent spatial and temporal patterns of connectivity to achieve robust classi\ufb01cation.\nOur approach enables identifying functionally connected networks that operate at\ndistinct timescales in the brain.\n\n1\n\nIntroduction\n\nProcesses in the brain occur at various timescales. These range from the timescales of milliseconds for\nextremely rapid processes (e.g. neuron spikes), to timescales of tens to hundreds of milliseconds for\nprocesses coordinated across local populations of neurons (e.g. synchronized neural oscillations), to\ntimescales of seconds for processes that are coordinated across diverse brain networks (e.g. language)\nand even up to minutes, hours or days for processes that involve large-scale neuroplastic changes\n(e.g. learning a new skill). Coordinated activity among brain regions that mediate each of these\ncognitive processes would manifest in the form of functional connections among these regions at\nthe corresponding timescales. Characterizing patterns of functional connectivity that occur at these\ndifferent timescales is, hence, essential for understanding how the brain produces behavior.\nMeasures of linear dependence and feedback, based on Granger-Geweke causality (GC) [10][11]),\nhave been used to estimate instantaneous and lagged functional connectivity in recordings of brain\nactivity made with electroencephalography (EEG, [6]), and electrocorticography (ECoG, [3]). How-\never, the application of GC measures to brain recordings made with functional magnetic resonance\nimaging (fMRI) remains controversial [22][20][2]. Because the hemodynamic response is produced\nand sampled at a timescale (seconds) several orders of magnitude slower than the underlying neural\nprocesses (milliseconds), previous studies have argued that GC measures, particularly lag-based GC,\nproduce spurious and unreliable estimates of functional connectivity from fMRI data [22][20].\n\n\u2217Corresponding author\n\n31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.\n\n\fThree primary confounds have been reported with applying lag-based GC to fMRI data. First,\nsystematic hemodynamic lags: a slower hemodynamic response in one region, as compared to\nanother could produce a spurious directed GC connection from the second to the \ufb01rst [22] [4].\nSecond, in simulations, measurement noise added to the signal during fMRI acquisition was shown to\nproduce signi\ufb01cant degradation in GC functional connectivity estimates [20]. Finally, downsampling\nrecordings to the typical fMRI sampling rate (seconds), three orders of magnitude slower than the\ntimescale of neural spiking (milliseconds), was shown to effectively eliminate all traces of functional\nconnectivity inferred by GC [20]. Hence, a previous, widely cited study argued that same-time\ncorrelation based measures of functional connectivity, such as partial correlations, fare much better\nthan GC for estimating functional connectivity from fMRI data [22].\nThe controversy over the application of GC measures to fMRI data remains unresolved to date,\nprimarily because of the lack of access to \u201cground truth\u201d. On the one hand, claims regarding\nthe ef\ufb01cacy of GC estimates based on simulations, are only as valid as the underlying model of\nhemodynamic responses. Because the precise mechanism by which neural responses generate\nhemodynamic responses is an active area of research [7], strong conclusions cannot be drawn based\non simulated fMRI data alone. On the other hand, establishing \u201cground truth\u201d validity for connections\nestimated by GC on fMRI data require concurrent, brain-wide invasive neurophysiological recordings\nduring fMRI scans, a prohibitive enterprise.\nHere, we seek to resolve this controversy by introducing a novel application of machine learning that\nworks around these criticisms. We estimate instantaneous and lag-based GC connectivity, \ufb01rst, with\nsimulated fMRI time series under different model network con\ufb01gurations and, next, from real fMRI\ntime series (from 500 human subjects) recorded under different task conditions. Based on the GC\nconnectivity matrices, we train a linear classi\ufb01er to discriminate model network con\ufb01gurations or\nsubject task conditions, and assess classi\ufb01er accuracy with cross validation. Our results show that\ninstantaneous and lag-based GC connectivity estimated from empirical fMRI data can distinguish\ntask conditions with over 80% cross-validation accuracies. To permit such accurate classi\ufb01cation, GC\nestimates of functional connectivity must be robustly consistent within each model con\ufb01guration (or\ntask condition) and reliably different across con\ufb01gurations (or task conditions). In addition, drawing\ninspiration from simulations, we show that GC estimated on real fMRI data downsampled to 3x-7x\nthe original sampling rate provides novel insights into functional brain networks that operate at\ndistinct timescales.\n\n2 Simulations and Theory\n\n2.1\n\nInstantaneous and lag-based measures of conditional linear dependence\n\nThe linear relationship among two multivariate signals x and y conditioned on a third multivariate\nsignal z can be measured as the sum of linear feedback from x to y (Fx\u2192y), linear feedback\nfrom y to x (Fy\u2192x), and instantaneous linear feedback (Fx\u25e6y) [11][16]. To quantify these linear\nrelationships, we model the future of each time series in terms of their past values with a well-\nestablished multivariate autoregressive (MVAR) model (detailed in Supplementary Material, Section\nS1).\nBrie\ufb02y, Fx\u2192y is a measure of the improvement in the ability to predict the future values of y given\nthe past values of x, over and above what can be predicted from the past values of z and y, itself (and\nvice versa for Fy\u2192x). Fx\u25e6y, on the other hand, measures the instantaneous in\ufb02uence between x and\ny conditioned on z (see Supplementary Material, Section S1). We refer to Fx\u25e6y, as instantaneous GC\n(iGC), and Fx\u2192y Fy\u2192x as lag-based GC or directed GC (dGC), with the direction of the in\ufb02uence\n(x to y or vice versa) being indicated by the arrow. The \u201cfull\u201d measure of linear dependence and\nfeedback Fx,y is given by :\n\nFx,y = Fx\u2192y + Fy\u2192x + Fx\u25e6y\n\n(1)\n\nFx,y measures the complete conditional linear dependence between two time series. If, at a given\ninstant, no aspect of one time series can be explained by a linear model containing all the values (past\nand present) of the other, Fx,y will evaluate to zero [16]. These measures are \ufb01rmly grounded in\ninformation theory and statistical inferential frameworks [9].\n\n2\n\n\fFigure 1: Network simulations. (A) Network con\ufb01guration H. (Left) Connectivity matrix. Red vs.\nblue: Excitatory vs. inhibitory connections. Deeper hues: Higher connection strengths. Non-zero\nvalue at (i, j) corresponds to a connection from node j to node i (column to row). Sub-network A-B-C\noperates at a fast timescale (50 ms) whereas D-E-F operates at a slow timescale (2 s). (Right) Network\nschematic showing the connectivity matrix as a graph. (B) Network con\ufb01guration J. Conventions are\nthe same as in A. (C) The eigenspectra of networks H (left) and J (right). (D) Simulated time series\nin network con\ufb01guration J with fast (top panel) and slow (bottom panel) dynamics, corresponding to\nnodes A-B and E-F, respectively. Within each panel, the top plot is the simulated neural time series,\nand the bottom plot is the simulated fMRI time series.\n\n2.2 Simulating functional interactions at different timescales\n\nTo test the ability of GC measures to reliably recover functional interactions at different timescales,\nwe simulated fMRI time series for model networks with two con\ufb01gurations of directed connectivity.\nSimulated fMRI time series were generated using a two-stage model (2): the \ufb01rst stage involved\na latent variable model that described neural dynamics, and the second stage that convolved these\ndynamics with the hemodynamic response function (HRF) to obtain the simulated fMRI time series.\n\n\u02d9x = Ax + \u03b5\n\ny = H \u2217 x\n\n(2)\n\nwhere A is the neural (\u201cground truth\u201d) connectivity matrix, x is the neural time series, \u02d9x is dx/dt, H\nis the canonical hemodynamic response function (HRF; simulated with spm_hrf in SPM8 software),\n\u2217 is the convolution operation, y is the simulated BOLD time series, and \u03b5 is i.i.d Gaussian noise.\nOther than noise \u03b5, other kinds of external input were not included in these simulations. Similar\nmodels have been employed widely for simulating fMRI time series data previously [22][2][20].\nFirst, we sought to demonstrate the complementary nature of connections estimated by iGC and dGC.\nFor this, we used network con\ufb01guration H, shown in Fig. 1A. Note that this corresponds to two\nnon-interacting sub-networks, each operating at distinctly different timescales (50 ms and 2000 ms\nnode decay times, respectively) as revealed by the eigenspectrum of the connectivity matrix (Fig. 1C).\nFor convenience, we term these two timescales as \u201cfast\u201d and \u201cslow\u201d. Moreover, each sub-network\noperated with a distinct pattern of connectivity, either purely feedforward, or with feedback (E-I).\nDynamics were simulated with a 1 ms integration step (Euler scheme), convolved with the HRF and\nthen downsampled to 0.5 Hz resolution (interval of 2 s) to match the sampling rate (repeat time, TR)\nof typical fMRI recordings.\nSecond, we sought to demonstrate the ability of dGC to recover functional interactions at distinct\ntimescales. For this, we simulated a different network con\ufb01guration J, whose connectivity matrix\n\n3\n\nDEFABCDestinationSourceFEDCBAACEBDFAFEDCBADestinationSource2s50msConnection strength-200ms-16s-1Node time-constant-6s-16s-15s50msBCD600s600sAmplitude (a.u.)Fast interaction (50ms)Slow interaction (5s)Time (s)Network HNetwork J-0.020Network JImfastintermediateslowRe0.020.0020.002-0.020Network HImfastslowRe0.020.0020.002\fFigure 2: Connectivity estimated from simulated data. (A) iGC and dGC values estimated from\nsimulated fMRI time series, network H. (Leftmost) Ground truth connectivity used in simulations.\n(Top) Estimated iGC connectivity matrix (left) and signi\ufb01cant connections (right, p<0.05) estimated\nby a bootstrap procedure using 1000 phase scrambled surrogates[18]. (Bottom) Same as top panel, but\nfor dGC. (B) dGC estimates from simulated fMRI time series, network J, sampled at three different\nsampling intervals: 50 ms (left), 500 ms (middle) and 5 s (right). In each case the estimated dGC\nmatrix and signi\ufb01cant connections are shown, with the same conventions as in panel (A).\n\nis shown in Fig. 1B. This network comprised three non-interacting sub-networks operating at three\ndistinct timescales (50 ms, 0.5 s, and 5 s node decay times; eigenspectrum in Fig. 1C). As before,\nsimulated dynamics were downsampled at various rates \u2013 20 Hz, 2 Hz, 0.2 Hz \u2013 corresponding to\nsampling intervals of 50 ms, 0.5 s, and 5 s, respectively. The middle interval (0.5 s) is closest to the\nrepeat time (TR=0.7 s) of the experimental fMRI data used in our analyses; the \ufb01rst and last intervals\nwere chosen to be one order of magnitude faster and slower, respectively.\nSuf\ufb01ciently long (3000 s) simulated fMRI timeseries were generated for each network con\ufb01guration\n(H and J). Sample time series from a subset of these simulations before and after hemodynamic\nconvolution and downsampling are shown in Fig. 1D.\n\n2.3\n\nInstantaneous and lag-based GC identify complementary connectivity patterns\n\nOur goal was to test if the ground truth neural connectivity matrix (A in equation 2) could be estimated\nby applying iGC and dGC to the fMRI time series y. dGC was estimated from the time series with\nthe MVGC toolbox (GCCA mode) [1][19] and iGC was estimated from the MVAR residuals [16].\nFor simulations with network con\ufb01guration H, iGC and dGC identi\ufb01ed connectivity patterns that\ndiffered in two key respects (Fig. 2A). First, iGC identi\ufb01ed feedforward interactions at both fast and\nslow timescales whereas dGC was able to estimate only the slow interactions, which occurred at a\ntimescale comparable to the sampling rate of the measurement. Second, dGC was able to identify\nthe presence of the E-I feedback connection at the slow timescale, whereas iGC entirely failed to\nestimate this connection. In the Supplementary Material (Section S2), we show theoretically why\niGC can identify mutually excitatory or mutually inhibitory feedback connections, but fails to identify\nthe presence of reciprocal excitatory-inhibitory (E-I) feedback connections, particularly when the\nconnection strengths are balanced.\nFor simulations with network con\ufb01guration J, dGC identi\ufb01ed distinct connections depending on the\nsampling rate. At the highest sampling rate (20 Hz), connections at the fastest timescales (50 ms)\nwere estimated most effectively, whereas at the slowest sampling rates (0.2 Hz), only the slowest\ntimescale connections (5 s) were estimated; intermediate sampling rates (2 Hz) estimated connections\nat intermediate timescales (0.5 s). Thus, dGC estimated robustly those connections whose process\ntimescale was closest to the sampling rate of the data.\nThe \ufb01rst \ufb01nding \u2014 that connections at fast timescales (50 ms) could not be estimated from data\nsampled at much lower rates (0.2 Hz) \u2014 is expected, and in line with previous \ufb01ndings. However, the\nconverse \ufb01nding \u2014 that the slowest timescale connections (5 s) could not be detected at the fastest\nsampling rates (20 Hz) \u2014 was indeed surprising. To better understand these puzzling \ufb01ndings, we\nperformed simulations over a wide range of sampling rates for each of these connection timescales; the\nresults are shown in Supplementary Figure S1. dGC values (both with and without convolution with\nthe hemodynamic response function) systematically increased from baseline, peaked at a sampling\nrate corresponding to the process timescale and decreased rapidly at higher sampling rates, matching\n\n4\n\nDEFABCACEBDFABSampling Interval500ms50msiGCdGCGround\nTruthGround\nTruth5s0.0600.020\frecent analytical \ufb01ndings[2]. Thus, dGC for connections at a particular timescale was highest when\nthe data were sampled at a rate that closely matched that timescale.\nTwo key conclusions emerged from these simulations. First, functional connections estimated by\ndGC can be distinct from and complementary to connections identi\ufb01ed by iGC, both spatially and\ntemporally. Second, connections that operate at distinct timescales can be detected by estimating\ndGC on data sampled at distinct rates that match the timescales of the underlying processes.\n\n3 Experimental Validation\n\nWe demonstrated the success of instantaneous and lag-based GC to accurately estimate functional\nconnectivity with simulated fMRI data. Nevertheless, application of GC measures to real fMRI data\nis fraught with signi\ufb01cant caveats, associated with hemodynamic confounds and measurement noise,\nas described above. We asked whether, despite these confounds, iGC and dGC would be able to\nproduce reliable estimates of connectivity in real fMRI data. Moreover, as with simulated data, would\niGC and dGC reveal complementary patterns of connectivity that varied reliably with different task\nconditions?\n\n3.1 Machine learning, cross-validation and recursive feature elimination\n\nWe analyzed minimally preprocessed brain scans of 500 subjects, drawn from the Human Connectome\nProject (HCP) database [12]. We analyzed data from resting state and seven other task conditions (total\nof 4000 scans; Supplementary Table S1). In the main text we present results for classifying the resting\nstate from the language task; the other classi\ufb01cations are reported in the Supplementary Material.\nThe language task involves subjects listening to short segments of stories and evaluating semantic\ncontent in the stories. This task is expected to robustly engage a network of language processing\nregions in the brain. The resting state scans served as a \u201ctask-free\u201d baseline, for comparison.\nBrain volumes were parcellated with a 14-network atlas [21] (see Supplementary Material Section\nS3; Supplementary Table S2). Network time series were computed by averaging time series across all\nvoxels in a given network using Matlab and SPM8. These multivariate network time series were then\n\ufb01t with an MVAR model (Supplementary Material Section S1). Model order was determined with\nthe Akaike Information Criterion for each subject, was typically 1, and did not change with further\ndownsampling of the data (see next section). The MVAR model \ufb01t was then used to estimate both an\ninstantaneous connectivity matrix using iGC (Fx\u25e6y) and a lag-based connectivity matrix using dGC\n(Fx\u2192y).\nThe connection strengths in these matrices were used as feature vectors in a linear classi\ufb01er based on\nsupport vector machines (SVMs) for high dimensional predictor data. We used Matlab\u2019s \ufb01tclinear\nfunction, optimizing hyperparameters using a 5-fold approach: by estimating hyperparameters with\n\ufb01ve sets of 100 subjects in turn, and measuring classi\ufb01cation accuracies with the remaining 400\nsubjects; the only exception was for the classi\ufb01cation analysis with averaging GC matrices (Fig. 3B)\nfor which classi\ufb01cation was run with default hyperparameters (regularization strength = 1/(cardinality\nof training-set), ridge penalty). The number of features for iGC-based classi\ufb01cation was 91 (upper\ntriangular portion of the symmetric 14\u00d714 iGC matrix) and for dGC-based classi\ufb01cation was 182\n(all entries of the 14\u00d714 dGC matrix, barring self-connections on the main diagonal). Based on\nthese functional connectivity features, we asked if we could reliably predict the task condition (e.g.\nlanguage versus resting). Classi\ufb01cation performance was tested with leave-one-out and k-fold cross-\nvalidation. We also assessed the signi\ufb01cance of the classi\ufb01cation accuracy with permutation testing\n[14] (Supplementary Material, Section S4).\nFinally, we wished to identify a key set of connections that permitted accurately classifying task\nfrom resting states. To accomplish this, we applied a two-stage recursive feature elimination (RFE)\nalgorithm [5], which identi\ufb01ed a minimal set of features that provided maximal cross validation\naccuracy (generalization performance). Details are provided in the Supplementary Material (Section\nS5, Supplementary Figs. S2-S3).\n\n5\n\n\fFigure 3: Classi\ufb01cation based on GC connectivity estimates in real data.\n(A) Leave-one-out\nclassi\ufb01cation accuracies for different GC measures for the 14-network parcellation (left) and the\n90-node parcellation (right). Within each group, the \ufb01rst two bars represent the classi\ufb01cation accuracy\nwith dGC and iGC respectively. The third bar is the classi\ufb01cation accurcay with fGC (see equation 1).\nChance: 50% (two-way classi\ufb01cation). Error-bars: Clopper-Pearson binomial con\ufb01dence intervals.\n(B) Classi\ufb01cation accuracy when the classi\ufb01er is tested on average GC matrices, as a function of\nnumber of subjects being averaged (see text for details).\n\n3.2\n\nInstantaneous and lag-based GC reliably distinguish task from rest\n\nBoth iGC and dGC connectivity were able to distinguish task from resting state signi\ufb01cantly above\nchance (Fig. 3A). Average leave-one-out cross validation accuracy was 80.0% with iGC and 83.4%\nwith dGC (Fig. 3A, left). Both iGC and dGC classi\ufb01cation exhibited high precision and recall at\nidentifying language task (precision= 0.81, recall= 0.78 for iGC and precision= 0.85, recall= 0.81 for\ndGC). k-fold (k=10) cross-validation accuracy was also similar for both the GC measures (79.4% for\niGC and 83.7% for dGC).\ndGC and iGC are complementary measures of linear dependence, by their de\ufb01nition. We asked if\ncombining them would produce better classi\ufb01cation performance. We combined dGC and iGC in two\nways. First, we performed classi\ufb01cation after pooling features (connectivity matrices) across both\ndGC and iGC (\u201ciGC \u222a dGC\u201d). Second, we estimated the full GC measure (Fx,y), which is a direct\nsum of dGC and iGC estimates (see equation 1). Both of these approaches yielded marginally higher\nclassi\ufb01cation accuracies \u2013 88.2% for iGC \u222a dGC and 84.6% for fGC \u2013 than dGC or iGC alone.\nNext, we asked if classi\ufb01cation would be more accurate if we averaged the GC measures across a\nfew subjects, to remove uncorrelated noise (e.g. measurement noise) in connectivity estimates. For\nthis, the data were partitioned into two groups of 250 subjects: a training (T) group and a test (S)\ngroup. The classi\ufb01er was trained on group T and the classi\ufb01er prediction was tested by averaging GC\nmatrices across several folds of S, each fold containing a few (m=2,4,5,10 or 25) subjects. Prediction\naccuracy for both dGC and iGC reached \u223c90% with averaging as few as two subjects\u2019 GC matrices,\nand reached \u223c100%, with averaging 10 subjects\u2019 matrices (Fig. 3B).\nWe also tested if these classi\ufb01cation accuracies were brain atlas or cognitive task speci\ufb01c. First, we\ntested an alternative atlas with 90 functional nodes based on a \ufb01ner regional parcellation of the 14\nfunctional networks [21]. Classi\ufb01cation accuracies for iGC and fGC improved (87.9% and 90.8%,\nrespectively), and for dGC remained comparable (81.4%), to the 14 network case (Fig. 3A, right).\nSecond, we performed the same GC-based classi\ufb01cation analysis for six other tasks drawn from the\nHCP database (Supplementary Table S1) . We discovered that all of the remaining six tasks could be\nclassi\ufb01ed from the resting state with accuracy comparable to the language versus resting classi\ufb01cation\n(Supplementary Fig. S4).\nFinally, we asked how iGC and dGC classi\ufb01cation accuracies would compare to those of other\nfunctional connectivity estimators. For example, partial correlations (PC) have been proposed\nas a robust measure of functional connectivity in previous studies [22]. Classi\ufb01cation accuracies\nfor PC varied between 81-96% across tasks (Supplementary Fig. S5B). PC\u2019s better performance\nis expected: estimators based on same-time covariance are less susceptible to noise than those\nbased on lagged covariance, a result we derive analytically in the Supplementary Material (Section\nS6). Also, when classifying language task versus rest, PC and iGC relied on largely overlapping\nconnections (\u223c60% overlap) whereas PC and dGC relied on largely non-overlapping connections\n(\u223c25% overlap; Supplementary Fig. S5C). These results highlight the complementary nature of PC\nand dGC connectivity. Moreover, we demonstrate, both with simulations and with real-data, that\n\n6\n\nAccuracy60708090100No. of Subjects01020304050Classi\ufb01cation using dGCClassi\ufb01cation using iGCABAccuracy506070809010014-NetworkdGCiGCfGC90-NodeParcellation Scheme\fFigure 4: Maximally discriminative connections identi\ufb01ed with RFE (A) (Top) iGC connections\nthat were maximally discriminative between the language task and resting state, identi\ufb01ed using\nrecursive feature elimination (RFE). Darker gray shades denote more discriminative connections\n(higher beta weights) (Bottom) RFE curves, with classi\ufb01cation accuracy plotted as a function of the\nnumber of remaining features. The dots mark the elbow-points of the RFE curves, corresponding\nto the optimal number of discriminative connections. (B) Same as in (A), except that RFE was\nperformed on dGC connectivity matrices with data sampled at 1x, 3x, 5x, and 7x of the original\nsampling interval (TR=0.72 s). Non-zero value at (i, j) corresponds to a connection from node j to\nnode i (column to row).\n\nclassi\ufb01cation accuracy with GC typically increased with more scan timepoints, consistent with GC\nbeing an information theoretic measure (Supplementary Fig. S6).\nThese superior classi\ufb01cation accuracies show that, despite conventional caveats for estimating GC\nwith fMRI data, both iGC and dGC yield functional connectivity estimates that are reliable across\nsubjects. Moreover, dGC\u2019s lag-based functional connectivity provides a robust feature space for\nclassifying brain states into task or rest. In addition, we found that dGC connectivity can be used to\npredict task versus rest brain states with near-perfect (>95-97%) accuracy, by averaging connectivity\nestimates across as few as 10 subjects, further con\ufb01rming the robustness of these estimates.\n\n3.3 Characterizing brain functional networks at distinct timescales\n\nRecent studies have shown that brain regions, across a range of species, operate at diverse timescales.\nFor example, a recent calcium imaging study demonstrated the occurrence of fast (\u223c100 ms) and\nslow (\u223c1 s) functional interactions in mouse cortex [17]. In non-human primates, cortical brain\nregions operate at a hierarchy of intrinsic timescales, with the sensory cortex operating at faster\ntimescales compared to prefrontal cortex [13]. In the resting human brain, cortical regions organize\ninto a hierarchy of functionally-coupled networks characterized by distinct timescales [24]. It is\nlikely that these characteristic timescales of brain networks are also modulated by task demands. We\nasked if the framework presented in our study could characterize brain networks operating at distinct\ntimescales across different tasks (and rest) from fMRI data.\nWe had already observed, in simulations, that instantaneous and lag-based GC measures identi\ufb01ed\nfunctional connections that operate at different timescales (Fig. 2A). We asked if these measures\ncould identify connections at fast versus slow timescales (compared to TR=0.72s) that were speci\ufb01c\nto task verus rest, from fMRI recordings. To identify these task-speci\ufb01c connections, we performed\nrecursive feature elimination (described in Supplementary Material, Section S5) with the language\ntask and resting state scans, separately with iGC and dGC features (connections). Prior to analysis\nof real data, we validated RFE by applying it to estimate key differences in two simulated networks\n(Supplementary Material Fig. S2 and Fig. S3). RFE accurately identi\ufb01ed connections that differed in\nsimulation \u201cground truth\u201d: speci\ufb01cally, differences in fast timescale connections were identi\ufb01ed by\niGC, and in slow timescale connections by dGC.\nWhen applied to the language task versus resting state fMRI data, RFE identi\ufb01ed a small subset of\n18(/91) connections based on iGC (Fig. 4A), and an overlapping but non-identical set of 17(/182)\nconnections based on dGC (Fig. 4B); these connections were key to distinguishing task (language)\n\n7\n\nSampling rate1x (0.72s)3x (2.16s)5x (3.60s)7x (5.04s) 13161911211511811316191121151181131619112115118113161911211511810.513161# FeaturesAccuracy1D-DMN\nLECN\nRECN\nA-SAL\nP-SAL\nLANG\nAUD\nSENMOT\nBG\nPREC\nV-DMN\nVISPA\nPR-VIS\nHI-VIS 0.40010.53200.61610.72460.430900000AB\ffrom resting brain states. Speci\ufb01cally, the highest iGC beta weights, corresponding to the most\ndiscriminative iGC connections, occurred among various cognitive control networks, including the\nanterior and posterior salience networks, the precuneus and the visuospatial network (Fig. 5A). Some\nof these connections were also detected by dGC. Nevertheless, the highest dGC beta weights occurred\nfor connections to and from the language network, for example from the language network to dorsal\ndefault mode network and from the precuneus to the language network (Fig. 5B). Notably, these\nlatter connections were important for classi\ufb01cation based on dGC, but not based on iGC. Moreover,\niGC identi\ufb01ed a connection between the language network and the basal ganglia whereas dGC, in\naddition, identi\ufb01ed the directionality of the connection, as being from the language network to the\nbasal ganglia. In summary, dGC and iGC identi\ufb01ed several complementary connections, but dGC\nalone identi\ufb01ed many connections with the language network, indicating that slow processes in this\nnetwork signi\ufb01cantly distinguished language from resting states.\nNext, we tested whether estimating dGC after systematically downsampling the fMRI time series\nwould permit identifying maximally discriminative connections at progressively slower timescales.\nTo avoid degradation of GC estimates because of fewer numbers of samples with downsampling\n(by decimation), we concatenated the different downsampled time series to maintain an identical\ntotal number of samples. RFE was applied to GC estimates based on data sampled at different rates:\n1.4 Hz, 0.5 Hz, 0.3 Hz and 0.2 Hz corresponding to 1x, 3x, 5x, and 7x of TR (sampling period of\n0.72 s, 2.16 s, 3.6 s and 5.04 s), respectively. RFE with dGC identi\ufb01ed 17(/182) key connections\nat each of these timescales (Fig. 4B). Interestingly, some connections manifested in dGC estimates\nacross all sampling rates. For instance, the connection from the precuneus to the language network\nwas important for classi\ufb01cation across all sampling rates (Fig. 5C). On the other hand, connections\nbetween the language network and various other networks manifested at speci\ufb01c sampling rates only.\nFor instance an outgoing connection from the language network to the basal ganglia manifested only\nat the 1.4 Hz sampling rate, to the visuospatial network and default mode networks only at 0.5 Hz, to\nthe higher-visual network only at 0.2-0.3 Hz, and an incoming connection from the anterior salience\nonly at 0.2 Hz. None of these connections were identi\ufb01ed by the iGC classi\ufb01er (compare Fig. 5A\nand 5C). Similar timescale generic and timescale speci\ufb01c connections were observed in other tasks\nas well (Supplementary Fig. S7). Despite downsampling, RFE accuracies were signi\ufb01cantly above\nchance, although accuracies decreased at lower sampling rates (Fig. 4 lower panels) [20]. Thus, dGC\nidenti\ufb01ed distinct connectivity pro\ufb01les for data sampled at different timescales, without signi\ufb01cantly\ncompromising classi\ufb01cation performance.\nFinally, we sought to provide independent evidence to con\ufb01rm whether these network connections\noperated at different timescales. For this, we estimated the average cross coherence (Supplementary\nMaterial, Section S7) between the fMRI time series of two connections from the language network\nthat were identi\ufb01ed by RFE exclusively at 0.2-0.3 Hz (language to higher visual) and 0.5 Hz (language\nto visuospatial) sampling rates, respectively (Fig. 5C). Each connection exhibited an extremum in the\ncoherence plot at a frequency which closely matched the respective connection\u2019s timescale (Fig. 5D).\nThese \ufb01ndings, from experimental data, provide empirical validation to our simulation results, which\nindicate that estimating dGC on downsampled data is a tenable approach for identifying functional\nconnections that operate at speci\ufb01c timescales.\n\n4 Conclusions\n\nThese results contain three novel insights. First, we show that two measures of conditional linear\ndependence \u2013 instantaneous and directed Granger-Geweke causality \u2013 provide robust measures of\nfunctional connectivity in the brain, resolving over a decade of controversy in the \ufb01eld [23][22].\nSecond, functional connections identi\ufb01ed by iGC and dGC carry complementary information, both\nin simulated and in real fMRI recordings. In particular, dGC is a powerful approach for identifying\nreciprocal excitatory-inhibitory connections, which are easily missed by iGC and other same-time\ncorrelation based metrics like partial correlations [22]. Third, when processes at multiple timescales\nexist in the data, our results show that downsampling the time series to different extents provides an\neffective method for recovering connections at these distinct timescales.\nOur simulations highlight the importance of capturing emergent timescales in simulations of neural\ndata. For instance, a widely-cited study [22] employed purely feedforward connectivity matrices with\na 50 ms neural timescale in their simulations, and argued that functional connections are not reliably\ninferred with GC on fMRI data. However, such connectivity matrices preclude the occurrence of\n\n8\n\n\fFigure 5: Connectivity at different timescales.\n(A-B) Discriminative connections identi\ufb01ed\nexclusively by iGC (teal), exclusively by dGC (blue), or by both (yellow). Each connection is\nrepresented as a band going from a source node on the left to a destination node on the right. (C)\n(Top) Discriminative connections identi\ufb01ed by dGC, exclusively at different sampling intervals (1x,\n3x, 5x, 7x TR). (D) (Left) Directed connection between language network and visuospatial network\nidenti\ufb01ed by dGC with fMRI data sampled at 0.5 Hz (sampling interval, 3x TR). (Right) Directed\nconnection between language network and higher visual network identi\ufb01ed by dGC with fMRI data\nsampled at 0.3 Hz (sampling interval, 5x TR). (Lower plots) Cross coherence between respective\nnetwork time series. Shaded area: Frequencies from Fs/2 to Fs, where Fs is the sampling rate of the\nfMRI timeseries from which dGC was estimated.\n\nslower, behaviorally relevant timescales of seconds, which readily emerge in the presence of feedback\nconnections, both in simulations [8][15] and in the brain [17][24]. Our simulations explicitly\nincorporated these slow timescales to show that connections at these timescales could be robustly\nestimated with GC on simulated fMRI data. Moreover, we show that such slow interactions also occur\nin human brain networks. Our approach is particularly relevant for studies that seek to investigate\ndynamic functional connectivity with slow sampling techniques, such as fMRI or calcium imaging.\nOur empirical validation of the robustness of GC measures, by applying machine learning to fMRI\ndata from 500 subjects (and 4000 functional scans), is widely relevant for studies that seek to\napply GC to estimate directed functional networks from fMRI data. Although, scanner noise or\nhemodynamic confounds can in\ufb02uence GC estimates in fMRI data [20][4], our results demonstrate\nthat dGC contains enough directed connectivity information for robust prediction, reaching over\n95% validation accuracy with averaging even as few as 10 subjects\u2019 connectivity matrices (Fig. 3B).\nThese results strongly indicate the existence of slow information \ufb02ow networks in the brain that\ncan be meaningfully inferred from fMRI data. Future work will test if these functional networks\nin\ufb02uence behavior at distinct timescales.\n\nAcknowledgments. This research was supported by a Wellcome Trust DBT-India Alliance\nIntermediate Fellowship, a SERB Early Career Research award, a Pratiksha Trust Young Investigator\naward, a DBT-IISc Partnership program grant, and a Tata Trusts grant (all to DS). We would like to\nthank Hritik Jain for help with data analysis.\n\nReferences\n[1] L. Barnett and A. K. Seth. 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