Multivariate Granger Causality Analysisof fMRI Data

Gopikrishna Deshpande, Stephan LaConte, George Andrew James,Scott Peltier, and Xiaoping Hu*

WHC Department of Biomedical Engineering, Georgia Institute of Technology and EmoryUniversity, Atlanta, Georgia

Abstract: This article describes the combination of multivariate Ganger causality analysis, temporaldown-sampling of fMRI time series, and graph theoretic concepts for investigating causal brain net-works and their dynamics. As a demonstration, this approach was applied to analyze epoch-to-epochchanges in a hand-gripping, muscle fatigue experiment. Causal influences between the activatedregions were analyzed by applying the directed transfer function (DTF) analysis of multivariateGranger causality with the integrated epoch response as the input, allowing us to account for theeffects of several relevant regions simultaneously. Integrated responses were used in lieu of originallysampled time points to remove the effect of the spatially varying hemodynamic response as a con-founding factor; using integrated responses did not affect our ability to capture its slowly varyingaffects of fatigue. We separately modeled the early, middle, and late periods in the fatigue. We adoptedgraph theoretic concepts of clustering and eccentricity to facilitate the interpretation of the resultantcomplex networks. Our results reveal the temporal evolution of the network and demonstrate thatmotor fatigue leads to a disconnection in the related neural network. Hum Brain Mapp 30:1361–1373,2009. VVC 2008 Wiley-Liss, Inc.

Key words: multivariate Granger causality; temporal dynamics of brain networks; graph theoretic anal-ysis; neural effects of prolonged motor performance and fatigue

INTRODUCTION

The role of networks in brain function has been increas-ingly recognized over the past decade [Friston et al., 1993;Sporns et al., 2004]. In functional neuroimaging, brain

networks are primarily studied in terms of functionalconnectivity (defined as temporal correlations betweenremote neurophysiologic events) and effective connectivity(defined as the causal influence one neuronal system exertsover another) [Friston, 1995]. Though the two prominentapproaches to characterizing effective connectivity—struc-tural equation modeling [McIntosh et al., 1994) anddynamic causal modeling [Friston et al., 2003]—have theiradvantages and disadvantages, neither of them incorporateinformation on temporal precedence, which may be con-sidered as a necessary condition for causality. Also, thesetechniques require an a priori specification of an anatomi-cal network model and are therefore best suited to makinginferences on a limited number of possible networks.Recently, an exploratory structural equation modelapproach that does not require prior specification of amodel was described [Zhuang et al., 2005]. However withincreasing number of regions of interest, its computational

Contract grant sponsor: NIH; Contract grant numbers: EB002009;Contract grant sponsor: Georgia Research Alliance.

*Correspondence to: Xiaoping Hu, Ph.D., Wallace H. CoulterDepartment of Biomedical Engineering, Georgia Institute of Tech-nology and Emory University, 101 Woodruff Circle, Suite 2001,Atlanta, Georgia 30322, USA. E-mail: xhu@bme.gatech.edu

Received for publication 10 January 2007; Revised 5 February2008; Accepted 7 April 2008

DOI: 10.1002/hbm.20606Published online 6 June 2008 in Wiley InterScience (www.interscience.wiley.com).

VVC 2008 Wiley-Liss, Inc.

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complexity becomes intractable and the numerical proce-dure becomes unstable. These disadvantages can largelybe circumvented by methods which are based on thecross-prediction between two time series such as Grangercausality [Granger, 1969].With fMRI data, recent studies have applied Granger

causality analysis between a target region of interest (ROI)and all other voxels in the brain to derive Granger causal-ity maps [Abler et al., 2006; Goebel et al., 2003; Roebroecket al., 2005]. A major limitation of applying the target ROIbased approach to neuroimaging data is that it is a bivari-ate method and ignores interactions between other ROIs inthe underlying neuronal network leading to an oversimpli-fication of the multivariate neuronal relationships that existduring the majority of cognitive tasks. Simulations by Kuset al. [2004] have shown that a complete set of observa-tions from a process have to be used to obtain causal rela-tionships between them and that pair-wise estimates mayyield incorrect results. To date, multivariate measures ofGranger causality have been largely limited to electrophys-iological data [Blinowska et al., 2004; Ding et al., 2000;Kaminski et al., 2001; Kus et al., 2004] although multivari-ate autoregressive models have been used to infer func-tional connectivity from fMRI data [Harrison et al., 2003].We have previously presented preliminary forms of thestudy described here [Deshpande et al., 2006a,b].A critical consideration for fMRI data is the limitations

imposed by the hemodynamic response. The fMRIresponse is dictated by the sluggish hemodynamicresponse, which is believed to be spatially dependent[Aguirre et al., 1998; Handwerker et al., 2004; Silva et al.,2002]. Given that the hemodynamic response takes 6–10 s,Granger causality analysis applied to the measured rawtime series sampled with a TR on the order of a secondmay be contaminated by regional differences in the hemo-dynamic response. We alleviate the effect of spatially vary-ing hemodynamic delay by focusing on the causal relation-ships at a temporal scale much coarser than the hemody-namic response. Neuronal processes such as fatigue,learning, and habituation evolve slower than the hemody-namic response and are amenable to a coarse temporalscale causal analysis.Another consideration is that multivariate causality rela-

tionships can be difficult to interpret and to compareacross data sets. With several anatomical regions includedin a network, the possible number of interconnectionsbetween them increases quadratically. The complexity ofthe problem is further increased by our desire to character-ize the temporal evolution of these network interactions.Graph theoretic concepts are well suited to represent theinformation present in these networks. Graphical represen-tations for fMRI-derived causal neuronal networks wereintroduced recently in the context of studying unmeasuredlatent variables in effective connectivity analysis [Eichler,2005]. The utility of graphical models in characterizing thetopology of large networks has been demonstrated in thecase of anatomical networks in macaques [McIntosh et al.,

2006], and functional networks obtained from MEG [Stam,2004] and EEG [Fallani et al., 2006; Sakkalis et al., 2006]. Inthe present study, we have used the graphical representa-tion for effective characterization of the network topology.In addition to utilizing concepts such as clustering [Fallaniet al., 2006; Sakkalis et al., 2006; Stam, 2004], we introducethe application of eccentricity analysis to determine theROIs having a major influence on the network.In this work, we have adapted the directed transfer

function (DTF) which was recently introduced as a causalmultivariate measure for EEG [Kus et al., 2004]. The DTFis based on Granger causality, but is rendered in a multi-variate formulation [Blinowska et al., 2004] and hence iseffective in modeling the inherent multivariate nature ofneuronal networks. For our application, we used the prod-uct of the non-normalized DTF and partial coherence toemphasize the direct connections and de-emphasize medi-ated influences. This procedure has been shown to be ro-bust [Kus et al., 2004] although equally good options suchas conditional Granger causality exist [Chen et al., 2006].Using an extended period of fMRI data collected during afatigue experiment [Peltier et al., 2005], we extracted thearea under each epoch to form a summary time serieswhich captures the epoch-to-epoch variation. The rationalewas that this is more likely to reflect the physiological pro-cess of fatigue and also alleviates the effect of the spatiallyvarying hemodynamic delay. This fact was substantiatedusing simulations. Further, we investigated the changes inthe dynamics of the networks as the subjects progressivelyfatigued, demonstrating the utility of this approach.

MATERIALS AND METHODS

MRI Data Acquisition and Preprocessing

Ten healthy right-handed male subjects performed aprolonged motor task while they underwent functionalmagnetic resonance imaging in a 3T Siemens Trio (SiemensAG, Berlin, Germany). Informed consent was obtainedprior to scanning and the procedure was approved by theinternal review board at Emory University. The subjectsperformed repetitive right-hand contractions at 50% maxi-mal voluntary contraction (MVC) level by gripping a bot-tle-like device [Liu et al., 2002]. Online measurement ofhandgrip force was accomplished by a pressure transducerconnected to the device through a nylon tube filled withdistilled water. For each subject, the target level of 50%MVC was calculated based on the maximal grip forcemeasured at the beginning of the experiment. Visual cues(a rectangular pulse whose profile matched the amplitudeand duration of the handgrip contraction) were generatedby a waveform generator and projected onto the screenabove the subject’s eye in the magnet to guide the subjectsin performing the contractions. Each contraction lasted 3.5s, followed by a 6.5 s intertrial interval (ITI). The total fa-tigue task comprised of 120 contractions lasting 20 min.After the completion of the task, the level of muscle

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fatigue was determined by measuring the MVC handgripforce. The choice of 50% MVC level was made so as to fa-tigue the muscles in �10–15 min for the given length ofcontraction and ITI. Echo planar imaging (EPI) data wasobtained with the following scan parameters: Thirty 4-mmslices (no gap) covering from the top of the cerebrum tothe bottom of the cerebellum, 600 volumes, repetition time(TR) of 2 s, echo time (TE) of 30 ms, a flip angle (FA) of908, and an in-plane resolution of 3.44 3 3.44 mm2.The data analysis for activation detection was carried

out using BrainvoyagerTM 2000 (Ver 4.9 � Rainer Goebeland Max Planck Society, Maastricht, The Netherlands.www.brainvoyager.com). Two subjects were excludedfrom the analysis because of excessive head motion. Subse-quent to motion and slice scan time correction, a referencewaveform derived based on the activation paradigm (Fig.1) was correlated with each detrended voxel time series toproduce activation maps (Fig. 2). The correspondence ofthe activation paradigm with a time series from the pri-mary motor area is illustrated in Figure 1. As shown inFigure 2, six ROIs—contralateral (left) primary motor (M1)cortex, primary sensory cortex (S1), premotor area (PM),ipsilateral (right) cerebellum (C), supplementary motorarea (SMA), and parietal area (P)—were identified fromthe activation maps, and ROI specific average time courseswere obtained. Because of the overlap of activations in M1and S1, these areas were delineated based on the locationof central gyrus [Yousry et al., 1997] by assigning the acti-vations in the precentral gyrus as M1 and that in the post-central gyrus as S1. SMA activation was taken to be medialand the parietal activation included both medial and con-tra-lateral activations in the posterior parietal cortex.To investigate fatigue induced causal influences, the

area under the time course of each epoch was calculatedas a summary measure and a corresponding summarytime series was derived from the mean time series for eachROI (Fig. 3). An epoch was defined as the duration con-taining the contraction time and intertrial interval. Theunderlying hemodynamic response in each epoch corre-sponded to one contraction. Three nonoverlapping seg-ments from the summary time series, each containing 40

points, was input into the multivariate Granger causalityanalysis. The use of these windows allowed us to investi-gate the temporal dynamics of the network.

Simulations

The purpose of the simulations was to show that hemo-dynamic confounds can overwhelm long term effects,leading to erroneous results and this confound can beeliminated by analyzing the summary time series. Twotime series, R1 and R2, were simulated by assuming an

Figure 1.

A time series from M1 overlaid on the activation paradigm. Red: 3.5 s contraction. Blue: 6.5 s

intertrial interval. [Color figure can be viewed in the online issue, which is available at www.

interscience.wiley.com.]

Figure 2.

A sample activation map obtained from the fatigue motor task

showing the regions of interest. n SMA,~M1, * S1, � P,3 PM.

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Figure 3.

Left: Original fMRI time series. Right: Summary time series (yellow patch shows the first time

window). [Color figure can be viewed in the online issue, which is available at www.interscience.

wiley.com.]

Figure 4.

The temporal variation of significance value a (a 5 12P) for all possible connections betweenthe ROIs. The direction of influence, as indicated by the black arrow, is from the columns to the

rows. The red bars indicate the connections that passed the significance threshold of a 5 0.95and the green ones that did not. [Color figure can be viewed in the online issue, which is avail-

able at www.interscience.wiley.com.]

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event-related paradigm consisting of 120 trials with anepoch duration of 20 s. Each event was assumed to lead toa HRF defined by two Gamma functions [Friston et al.,1999] with the following parameters: sampling interval 52 s, dispersion of response 5 1 s, dispersion of undershoot5 1 s, delay of response (relative to onset) 5 6 s, delay ofundershoot (relative to onset) 5 10 s, ratio of response toundershoot 5 6, length of kernel 5 20 s. The HRF for R1was assumed to be 1 s behind that of R2 (i.e., R2 leads R1).The peak amplitudes for the trials in R1 and R2 weremodulated by slowly varying sinusoids, denoted by A1and A2, with A1 leading A2 by one epoch. Because anepoch is 20 s long, R1 leads R2 inspite of the 1 s hemody-namic lead R2 has over R1. Gaussian noise (SNR 5 0, 5, 10,and 100, 500 realizations each) was added to the simulatedsignals to test the effect of random noise in the system.Granger causality analysis was applied to raw time series,R1 and R2, and the summary time series obtained by inte-grating each epoch, C1 and C2. Because C1 and C2 are pro-portional to A1 and A2, respectively, C1 leads C2.

Multivariate Granger Causality Analysis

The principle of Granger causality is based on the con-cept of cross prediction. Accordingly, if incorporating thepast values of time series X improves the future predictionof time series Y, then X is said to have a causal influenceon Y [Granger, 1969]. In the case of any two time series Xand Y, the efficacy of cross-prediction could be inferred ei-ther through the residual error after prediction [Roebroecket al., 2005] or through the magnitude of the predictorcoefficients [Blinowska et al., 2004]. Both approaches areequivalent and the analytical relationship between them isgiven by Granger [1969]. In this section, we describe themultivariate model of Granger causality used in thisstudy.The Granger causality analysis was accomplished using

custom software written in MATLAB (The MathWorks Inc,Massachusetts). A multivariate autoregressive (MVAR)model was constructed from the summary time series ofthe ROIs. In the following, an italic capital letter representsa matrix with components corresponding to the ROIs andthe variable in the parenthesis indicates either time or tem-poral frequency. Let X(t) 5 (x1(t),x2(t),. . .xk(t))

T be the datamatrix and xk correspond to the time series obtained fromthe kth ROI. The MVAR model with model parametersA(n) of order p is given by

XðtÞ ¼Xpn¼1

AðnÞXðt� nÞ þ EðtÞ ð1Þ

where E(t) is the vector corresponding to the residualerror. This form of MVAR is analytically proven to have aunique solution [Caines et al., 1975; Granger, 1969]. Akaikeinformation criterion (AIC) was used to determine themodel order [Akaike, 1974]. Equation (1) can be rewrittenas follows

XðtÞ �Xpn¼1

AðnÞXðt� nÞ ¼ EðtÞ ð2ÞEquation (2) was transformed to the frequency domain

resulting in

Xðf Þ dij �Xpn¼1

aijðnÞe�i2pfn" #

¼ Eðf Þ ð3Þ

We designate aijðf Þ ¼ dij �Ppn¼1

aijðnÞe�i2pfn and A as thematrix corresponding to elements aij. Here, dij is the Dirac-delta function which is one when i 5 j and zero elsewhere.Also, i ¼ 1 � � � k; j ¼ 1 � � � k, where k is the total number ofROIs. Note that time domain matrices are represented bybold letters and their frequency domain counterparts aredenoted by capital letters in normal font.

Xðf ÞAðf Þ ¼ Eðf Þ ð4Þ

Xðf Þ ¼ A�1ðf ÞEðf Þ ¼ Hðf ÞEðf Þ where Hðf Þ ¼ A�1ðf Þ ð5Þ

The transfer matrix of the model, H(f), contains all the in-formation about the interactions between the time seriesand hij(f), the element in the ith row and jth column of thetransfer matrix, is referred to as the non-normalized DTF[Kus et al., 2004] corresponding to the influence of ROI jonto ROI i. Least squares estimation [Tyraskis et al., 1985]was used to solve for the prediction coefficients. This proce-dure imposes a theoretical constraint that the number ofdata points in each time series be more than the number ofMVAR parameters to be estimated (which is the square ofthe number of time series for a first order model) [Kuset al., 2004; Tyraskis et al., 1985].To emphasize direct con-nections and de-emphasize mediated influences, H(f) wasmultiplied by the partial coherence between ROIs i and j toobtain direct DTF (dDTF) [Korzeniewska et al., 2003; Kuset al., 2004]. Simulations provided by Kus et al. and Korze-niewska et al. prove that dDTF is a robust measure capableof de-emphasizing mediated influences. To calculate thepartial coherence, we first computed the cross-spectra using

Sðf Þ ¼ Hðf ÞVH�ðf Þ ð6Þ

where V is the variance of the matrix E(f) and the asteriskdenotes transposition and complex conjugate. The partialcoherence between ROIs i and j is then given by

h2ijðf Þ ¼M2ijðf Þ

Miiðf ÞMjjðf Þ ð7Þ

where the minor Mij(f) is defined as the determinant of thematrix obtained by removing the ith row and jth columnfrom the matrix S.

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The partial coherence between a pair of ROIs indicatesthe association between them when the statistical influenceof all other ROIs is discounted. It lies in the range [0, 1]where a value of zero indicates no direct associationbetween the ROIs. The direct DTF (dDTF) was obtained asthe sum of all frequency components of the product of thenon-normalized DTF and partial coherence as given in theequation below.

dDTFij ¼Xf

hijðf Þhijðf Þ ð8Þ

dDTF as defined above emphasizes the direct connectionsbetween ROIs. Working in the frequency domain offers theadvantage of uncovering interactions in specific frequencybands. In our data, we did not observe distinct frequencyspecific patterns except that Granger causality remainedhigh in the lower frequencies and decreased with increas-ing frequency. This is to be expected considering the factthat the summary time series had higher spectral energyin the low frequency band. Hence we summed all the fre-quency components to obtain one dDTF value for everyconnection. Methodologically, spectral methods have beenshown to be more robust to deviations from the stationar-ity assumption [Granger, 1964].It is to be noted that unlike previously reported studies

[Blinowska et al., 2004; Kaminski et al., 2001; Kus et al.,2004], we avoided normalizing DTF so as to allow directcomparison between the absolute values of the strengthsof influence. Normalization of DTF with respect to inflowsinto any ROI as in Kus et al. [Kus et al., 2004] would makesuch a comparison untenable. As described in the previoussubsection, the calculation of dDTF was carried out usingthe summary time series in three nonoverlapping windowsso as to investigate the temporal dynamics of the network.In addition, connectivity was also computed using the rawtime series for comparison.

Statistical Significance Testing

Analytical distributions of multivariate Granger causalityare not established because they are said to have a highlynonlinear relationship with the time series data [Kaminskiet al., 2001]. Therefore, to assess the significance of theGranger causality reflected by dDTF, we employed surro-gate data [Kaminski et al., 2001; Kus et al., 2004; Theileret al., 1992] to obtain an empirical null distribution. Theoriginal time series was transformed into the frequencydomain and their phase was randomized so as to be uni-formly distributed over (2p, p) [Kus et al., 2004]. Subse-quently, the signal was transformed back to the time do-main to generate the surrogate data. This procedureensured that the surrogate data possessed the same spec-trum as the original data but with the causal phase rela-tions destroyed. dDTF was calculated between the surro-gate data time series representing each ROI. Null distribu-tions were derived for all possible connections between

the ROIs, in each time window and for every subject, byrepeating the above procedure 2,500 times. Therefore, cor-responding to six ROIs (we had 30 possible links betweenthe ROIs) and three time windows, a total of 90 null distri-butions were generated per subject. For each connection,the actual dDTF was compared with its correspondingnull distribution to derive a P-value. To obtain group sig-nificance inference, the P-values from individual subjectswere combined using Fisher’s method [Fisher, 1932] toobtain a single P-value. This procedure was repeated foreach connection in the three temporal windows to obtainsignificant connectivity networks. Using a Jarque-Bera testfor goodness-of-fit to a normal distribution, the distribu-tion of path coefficients within a single time window andthe distribution of the difference in path coefficients weredetermined to be normal. To assess the significance of thechange in path coefficients across different time windows,a paired t-test was performed between windows 1 and 2and between 2 and 3. The significance values were con-trolled for multiple comparisons using Bonferroni correc-tion [Miller, 1991].

Graph Analysis

The causal influences between the ROIs in a networkcould in principle be represented as a weighted directedgraph, whose weights are represented by the dDTF valuefor the corresponding link between the ROIs, the direc-tion of the link being the direction of causal influence andthe ROIs themselves representing the vertices (or nodes)of the network. As mentioned in the introduction, thistype of representation has been used to characterize net-work topology of causal functional networks obtainedfrom MEG data [Stam, 2004] and EEG [Fallani et al., 2006;Sakkalis et al., 2006]. In this study, we focus on clusteringand eccentricity. Although clustering has been used pre-viously [Fallani et al., 2006; Sakkalis et al., 2006; Stam,2004], we have adopted the concept of eccentricity fromgraph theory [Edwards, 2000] and have shown its rele-vance in interpreting the resultant networks.

Mathematical representation of a graph

A graph G is mathematically represented in the form ofa sparse matrix called the adjacency matrix [Skiena, 1990].The adjacency matrix of the directed graph is a matrixwith rows and columns labeled by graph vertices (v), withthe dDTF value corresponding to the influence from vj tovi in the position (vi,vj).

Clustering coefficient

One of the most important aspects of the topology of anetwork is the role of the nodes as either drivers of othernodes or being driven by other nodes. This is assessed bythe total strength of causal influence that is emanatingfrom or incident on the node. Correspondingly, cluster-in

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and cluster-out coefficients [Watts et al., 1998] are definedas,

CinðiÞ ¼Xkj¼1

Gðvi; vjÞ for i ¼ 1 � � � k ð9Þ

CoutðjÞ ¼Xki¼1

Gðvi; vjÞ for j ¼ 1 � � � k ð10Þ

where k 5 6 is the number of nodes in the network. Basi-cally, Cin of a node is the sum of all the corresponding col-umns of G and Cout is the sum of all corresponding rows.While calculating clustering coefficients from the fatiguedata, the mean dDTF averaged over the subjects were usedas entries in the matrix G. Also, the analysis was carried outseparately for each of the three temporal windows.

Eccentricity

The eccentricity E(v) of a graph vertex v in a connectedgraph G is the maximum geodesic distance between v andany other vertex u of G. The geodesic distance betweentwo vertices in a weighted graph is the sum of the causalinfluences along the shortest path connecting them. Weused the Floyd-Warshall algorithm for solving the all-pairsshortest path problem [Cormen et al., 2001] to the find theshortest path between any pairs of nodes. Given thatgraph distance is measured in terms of the strength ofcausal influence, the shortest path between two nodes indi-cates the path along which maximum causal influence isexerted. Eccentricity is related to the individual influenceof a vertex on the overall network performance [Skiena,1990]. A vertex v is said to have a major influence on thenetwork performance if it has the maximum E(v) amongall vertices in the graph. Such a vertex, termed the majornode, wields maximum influence on network behavior.The major nodes in each time window were ascertained toinfer the changing roles of brain regions.

RESULTS

Simulations

Table I lists the results obtained from the simulations.Except when SNR 5 0, dDTF derived using integratedtime series reflects the correct connectivity (C1?C2). Incontrast, the connectivity derived using the raw time series(R2?R1) is incorrect and appears to be more bidirectionalas opposed to unidirectional in case of the integrated timeseries for lowers SNRs. At high SNRs, the ratio of dDTFalong the direction of minor influence to that along themajor influence is lower with the integrated time seriesthan with the raw time series.

Behavioral Data and Preprocessing

There was a significant decrease (P < 0.002) in hand gripforce measured after the motor task as compared to before

the task, indicating that significant muscle fatigue hadoccurred. Of the eight subjects selected for analysis, behav-ioral data was not available for two subjects due to techni-cal difficulties. In the rest of the six subjects, the decrease inhand grip force was 29% 6 11% [Peltier et al., 2005]. Figure2 shows a sample activation map obtained by correlatingthe fMRI time series with the reference waveform and theROIs selected for further analysis. A representative sum-mary time series and the original time series that it isderived from are shown in Figure 3. It can be seen that thesummary time series captures the slow epoch-to-epoch var-iation and in this particular case represents an initialincrease and subsequent stagnation of the epoch response.

Multivariate Granger Causality

A model order of one was assigned based on theAkaike information criterion [Akaike, 1974]. Because asingle time point in the summary time series correspondsto the area under the corresponding epoch, the resultingMVAR model represents epoch-to-epoch prediction. Thetemporal variation of the significance values a (a 5 12P)for connections between all pairs of ROIs is shown in Fig-ure 4. The links that passed the significance threshold ofa 5 0.95 are represented by the bars in red while the con-nections that did not pass the threshold are shown asgreen bars. The network representation of the results inFigure 4 is shown in Figure 5, where the significant con-nections are shown as solid lines with their width reflect-ing the statistical significance of the influence. It is to benoted that the absence of a connection does not necessar-ily imply that there is no causal influence between thecorresponding ROIs. A more lenient threshold or addi-tional statistical power might render an insignificant con-nection significant. The thresholded difference networksbetween windows 1 and 2 and between windows 2 and 3are shown in Figure 6, illustrating the shift in connectiv-ity patterns. As a comparison, the networks obtainedfrom raw time series (using a model order one as deter-

TABLE I. Simulation results for raw time series and

coarse time series

Connection SNR 5 0 SNR 5 5 SNR 5 10 SNR 5 100

Raw time seriesR1?R2 0.03 6 0.04 0.08 6 0.09 0.7 6 0.2 3.7 6 0.1R2?R1 0.05 6 0.08 6.2 6 1.0 8.5 6 1.1 13.9 6 0.5Coarse time seriesC1?C2 0.4 6 0.6 20 6 3.4 28.1 6 2.9 38 6 0.8C2?C1 0.4 6 0.4 0.6 6 0.5 2.1 6 0.9 6.8 6 0.3

Here, the connectivity values indicate the dDTF (summed connec-tivity over the whole spectra) as well as its standard deviationover all the realizations. C1 drives C2 and R1 drives R2 by con-struction. The model obtained by coarse time series is more accu-rate than the one obtained by the raw time series confounded byhemodynamics.

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mined by the application of AIC to raw time series) aredepicted in Figure 7.The computational times for different components of the

implementation on a 2.7 GHz Pentium 5 machine were asfollows. Calculating the summary time series for 600 vol-umes took 2.93 s while dDTF calculation for each timewindow took 1.28 s. The total runtime of the code willdepend on the number of times dDTF is calculated on thesurrogate time series.

Graph Analysis

Clustering coefficients

Table II lists Cin and Cout for the three windows. In thefirst window, M1 was predominantly driven while S1 wasa strong driver. The other areas had a dual role in thesense that they both received and transmitted information.

In the second window, S1 and cerebellum were strongdrivers. In the third window, while S1 and cerebellumremained to be the main drivers, the absolute value of thecoefficients decreased for all ROIs, indicating a reductionof network connectivity. This reduction, also evident inFigure 5, indicates that as muscles fatigued, the connec-tions in the motor network decreased.

Eccentricity

The primary sensory area was the major node in the firstwindow, while the cerebellum was the major mode in thesecond and third windows. This is schematically repre-sented in Figure 5 where the major nodes are marked inblack. This result indicates that S1 wielded maximuminfluence on the network in the first window, and thedominance of influence shifted to the cerebellum in thesecond and third windows.

Figure 5.

A network representation of Figure 4. The significant links are represented as solid arrows and

the P-value of the connections are indicated by the width of the arrows. The major node in

each window is also indicated as dark ovals.

Figure 6.

Thresholded difference networks. Left: window 1–window 2. Right: window 2–window 3. Red

indicates positive difference whereas blue indicates negative difference.

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DISCUSSION

The simulation results support the argument thatGranger causality analysis could be confounded by hemo-dynamic variability which could arise due to a multitudeof factors [Aguirre et al., 1998; Handwerker et al., 2004;Silva et al., 2002]. In the simulation, this is demonstratedby the fact that the simulated connections were incorrectlyidentified by Granger analysis of raw time series but cor-rectly identified by analyzing the summary time series.The analysis of summary time course is robust in the pres-ence of Gaussian noise. Fatigue is a slowly evolvingprocess [Liu et al., 2002a,b] and the causal influencesunderlying fatigue can be assumed to lie on a coarser timescale. Based on the simulation, it is prudent to use thesummary time series in the MVAR instead of the raw timeseries to recover the long term causal influences due to fa-tigue. An MVAR of summary time series represents epoch-to-epoch evolution. Each epoch represents a contractionand the area under it represents the net activity of an ROIdue to the contraction. Hence, epoch-to-epoch causal influ-ence is interpreted as predicting the activities for futurecontractions based on those for the current contraction.According to Wei, subsampling weakens the magnitude

of Granger causality, but preserves the direction and tem-poral aggregation may cause spurious Granger causality.Their conclusion was based on the assumption that thetrue causal influence is at a finer time scale than the sam-pling resolution [Wei, 1982]. In contrast, given that fatigueis a slowly evolving process, the long-term causal influen-ces corresponding to the underlying neurophysiologicalprocesses involving fatigue are occurring at a coarser timescale than the sampling resolution. Therefore Wei’s conclu-sion does not apply to the fatigue data.The Granger causality results presented above reflect a

gradual shift in connectivity patterns across brain regions

during the course of prolonged motor task. During thefirst time window, the network is highly interconnected asillustrated by Figure 5. A high value of Cin for M1 andCout for S1 indicates that the neural network is predomi-nantly driven by feedback mechanisms from the primarysensory cortex. This pattern is consistent with the fine tun-ing of motor responses with sensory feedback [Solodkinet al., 2004]. The drive from the primary sensory to pri-mary motor areas is particularly interesting in light of sim-ilar findings in an electrophysiological study on isometriccontraction in monkeys [Brovelli et al., 2004]. Furthermore,we know that all regions drive the primary motor cortexthrough both direct (SMA, premotor cortex) and indirect(parietal, cerebellum) anatomical pathways [Passingham,1988; Strick et al., 1999]. Structural equation modeling bySolodkin et al. [2004] found that the primary sensory cor-tex weakly drove the primary motor cortex, but did notexert causal influence on other brain regions. However,our results suggest that S1 could have a strong causalinfluence on M1. The fact that neither Cin nor Cout domi-nates each other for the cerebellum and parietal areaspoints to the existence of bidirectional connections betweenthese ROIs and the rest of the network and hence the

Figure 7.

Networks obtained from raw time series for the three windows. The significant links (P

possibility of both top-down and bottom mechanisms ofinfluence.As the motor task progresses into the middle temporal

window, regions that guide motor performance—the cere-bellum, SMA, and premotor cortex—become more promi-nent as indicated by their elevated clustering coefficientsas compared to those in the first window. These regionsare collectively responsible for timing motor responses,response preparation, and sequencing responses [Deiberet al., 1991; Gordon et al., 1998; Ivry et al. 1989; Passing-ham, 1988; Tanji, 1996]. Although S1 is the major node inthe first window, the cerebellum becomes the major nodein the middle window. This shift in the role of the nodesin the network suggests that participants have masteredthe motoric components of the task and are now primarilyfocused on orchestrating these responses. The shift fromthe primary sensory cortex to cerebellum also implies thatparticipants are less reliant on tactile feedback to guideperformance.The network changes yet again during the final stages of

the experiment, but this shift is more subtle than before.The network structure is largely consistent between themiddle and last windows. Most striking is the change inmagnitude; the causal strength of all connections as wellas the clustering coefficients decreases. Whereas the mid-dle window most likely reflects learning (manifesting asboth the strengthening and paring of connections), the lastwindow only shows the weakening of connections. Theseresults are consistent with fatigue, which we have previ-ously demonstrated to reduce interhemispheric connectiv-ity [Peltier et al., 2005]. Although the neural network opti-mized during the middle window remains largely intact,the interregional causal strengths diminishes as fatiguetakes its toll.It is to be noted that decrease in connectivity with fa-

tigue could be associated with a general decrease in activa-tion. However, this is unlikely the case here since it wasfound that while cross correlation with a seed in M1decreased, the activation volume increased with fatigue[Liu et al., 2005]. This is consistent with the fact that thesummary time series (i.e., the area of epochs) also tends toincrease with time. It is conjectured that more cortical neu-rons need to be recruited in strengthening the descendingcommand or processing sensory information during fa-tigue, which lead to increased activation volume. The acti-vation pattern of newly recruited cortical neurons that arenot normally involved in nonfatigue muscle activities maynot closely interact, leading to reduced connectivity.The nodes in the network considered here are not

intended as an exhaustive account of regions mediatingmotor behavior. Only the neural regions demonstrating themost significant activation were examined. Thus, subcorti-cal regions such as the basal ganglia and red nucleus werenot addressed despite their influence on motor perform-ance [Harrington et al., 1998; Liu et al., 1999]. Likewise,thalamic activity was not modeled, even though most ofthe corticocortical, corticocerebellar, and cerebellocortical

anatomical pathways are routed through the thalamus[Jones, 1999].Besides supporting the existing hypothesis on the neural

effects of muscle fatigue, our model demonstrates gradualchanges in neural communication patterns in the pro-longed motor task. We propose that these changes reflectslowly varying neurophysiological alterations caused byfatigue. In addition to obviating the effect of the hemody-namic response on the predictive model, our use of sum-mary measure enabled us to match the temporal scale ofanalysis with the temporal scale at which the underlyingphysiology is likely to evolve. In addition, trial-by-trialvariability in BOLD data and performance measuresmakes both susceptible to the influence of outliers andother statistical pitfalls. Previous studies have circum-vented this limitation with summary time series, such asmean BOLD or mean reaction time by block [Toni et al.,2002] or condition [Tracy et al., 2003]. The present frame-work of analysis is expected to be useful in the investiga-tion of other slowly varying neurophysiological processessuch as learning [Floyer-Lea et al., 2005], habituation [Pflei-derer et al., 2002], chronic pain [Borsook et al., 2006], andtherapeutic effects [Schweinhardt et al., 2006].Because the resulting networks have a complicated to-

pology, a manual perusal of every connection and theirinterpretation is untenable. Therefore we have employedgraph theoretic concepts to unearth possible patterns ofcommunication in the network. This approach gives usefulinsights about the changes in the connectivity patterns andthe contribution of individual and specific groups of ROIsto network behavior. Although we have used only cluster-ing and eccentricity to characterize network topology, sev-eral other options exist within the framework of graphtheory such as connected components and path lengthanalyses [Skiena, 1990] which could potentially be used tocharacterize the network.One question worth asking is whether Granger causality

analysis of the raw ROI time series would lead to similarresults. Such an analysis was performed and led to agreater number of paths that are less significant and ex-hibit no clear driving node as compared to their corre-sponding networks derived from the summary time series.In addition, the networks obtained from the raw time se-ries did not differ significantly between the three time win-dows. This result indicates that analysis using summarytime series is likely more appropriate for capturing thelong-term influences during fatigue while that using theraw time series may be sensitive to influences from TR toTR or hemodynamic effects, which do not vary in the pro-cess of fatigue.A brief discussion of our methodology vis-à-vis SEM is

in order. While Granger causality is data driven, SEMrequires an a priori model. The exploratory SEM approach[Zhuang et al., 2005] does not require an a priori modeland hence is appropriate for comparison with Grangercausality. However, with six ROIs, the exploratory SEMbecomes computationally intractable. Furthermore, for n

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ROIs, SEM can only estimate n(n 2 1)/2 path coefficients;or 15 connections for six ROIs. The first window could notbe analyzed with SEM because it has 18 connections.Therefore, instead of making a comparison with explora-tory SEM, a comparison was made with SEM analysisassuming the connectivity models derived from windows2 and 3 using Granger causality analysis. SEM led to apoor model fit (Root mean square error of approximation>0.25, model AIC >135, Adjusted goodness of fit index

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