Functional connectivity in resting-state fMRI: Is linear correlationsufficient?

Jaroslav Hlinkaa, Milan Palusa, Martin Vejmelkaa, Dante Mantinib,c, Maurizio Corbettac,d,e

aInstitute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodarenskou vezi 2, 18207 Prague, CzechRepublic

bLaboratory for Neuro- and Psychophysiology, Katholieke Universiteit Leuven, 3000 Leuven, BelgiumcInstitute for Advanced Biomedical Technologies, G. D’Annunzio University Foundation, G. D’Annunzio University, 66013

Chieti, ItalydDepartment of Radiology, Washington University, St. Louis, Missouri, USAeDepartment of Neurology, Washington University, St. Louis, Missouri, USA

Abstract

Functional connectivity (FC) analysis is a prominent approach to analyzing fMRI data, especially acquiredunder the resting state condition. The commonly used linear correlation FC measure bears an implicitassumption of Gaussianity of the dependence structure. If only the marginals, but not all the bivariatedistributions are Gaussian, linear correlation consistently underestimates the strength of the dependence.To assess the suitability of linear correlation and the general potential of nonlinear FC measures, we presenta framework for testing and estimating the deviation from Gaussianity by means of comparing mutualinformation in the data and its Gaussianized counterpart. We apply this method to 24 sessions of humanresting state fMRI. For each session, matrix of connectivities between 90 anatomical parcel time seriesis computed using mutual information and compared to results from its multivariate Gaussian surrogatethat conserves the correlations but cancels any nonlinearity. While the group-level tests confirmed non-Gaussianity in the FC, the quantitative assessment revealed that the portion of mutual information neglectedby linear correlation is relatively minor – on average only about 5% of the mutual information alreadycaptured by the linear correlation. The marginality of the non-Gaussianity was confirmed in comparisonsusing clustering of the parcels – the disagreement between clustering obtained from mutual informationand linear correlation was attributable to random error. We conclude that for this type of data, practicalrelevance of nonlinear methods trying to improve over linear correlation might be limited by the fact thatthe data are indeed almost Gaussian.

Keywords: fMRI, functional connectivity, Gaussianity, nonlinearity, correlation, mutual information

1. Introduction

Neuroimaging methods play an important rolein the process of extending our understanding ofbrain structure and function. In this regard, func-tional Magnetic Resonance Imaging (fMRI) holdsa central position, particularly due to its non-invasiveness and its relatively high spatial resolu-tion. Despite the large number of neuroimagingstudies conducted to delineate brain activations re-lated to a wide range of cognitive functions, thefundamental issue of how the brain regions com-municate to each other still remains open.

The concept of brain functional connectivity

(Friston, 1994) is central to the understanding ofthe organized behavior of cortical regions beyondthe simple mapping of their activity. Functionalconnectivity analyses of fMRI data as an approachto study temporal coherence in activity of distantbrain areas, either during task performance or inresting state, has been widely applied in the neu-roscience research. Much recent work has focusedon measuring and interpreting the spontaneous sig-nal fluctuations typically seen during rest (Fox andRaichle, 2007). The initial reservations regardingthe neural origin of these fluctuations (and the re-lated connectivity) are now mostly overcome - someof the strongest evidence for relevance of these sig-

Revised author’s personal version of: Hlinka, J., et al., NeuroImage (2010), doi:10.1016/j.neuroimage.2010.08.042

nal fluctuations is based on the observed direct orindirect electrophysiological correlates (Laufs, 2008;Mantini et al., 2007; Miller et al., 2009), althoughthere is still much to be done in the field of meth-ods for separation of neural and nonneural sourcesof resting signal variation. In the context of thispaper, we stress that in resting state studies, func-tional connectivity analysis commonly plays a cen-tral role (Fox and Raichle, 2007; van Dijk et al.,2010).

The most widely spread method of measuringfunctional connectivity between a pair of regions iscomputing a linear correlation of activity time seriesderived from these regions by e.g. simple spatial av-eraging across all the voxels in the regions. Linearcorrelation is also widely used to obtain so-calledcorrelation maps by correlating the seed voxel orseed region signal with signal from all the othervoxels in the brain, or constrained to gray mat-ter area. From all possible bivariate measures ofassociation, linear correlation is clearly a methodof first choice, reflecting the assumption that therelationship between the fMRI time series can besuitably approximated by a multivariate Gaussianwhite noise process. Additionally, linear correlationis a well-known statistical concept, sufficiently sim-ple to allow wide use and easy communication ofresults between researchers of diverse backgrounds.

On the other hand, from the mid-1980s, nonlinearapproaches to analysis of brain signals are gettingincreased interest of researches who consider non-linearity as an intrinsic property of brain dynam-ics, see e.g. (Stam, 2005) for a review. Hemody-namic nonlinearities are known to affect the bloodoxygenation level-dependent (BOLD) fMRI signal(de Zwart et al., 2009). More specifically, non-linearity of dependence between fMRI time seriesduring resting state has been reported (Lahayeet al., 2003). Use of non-linear measures of func-tional connectivity for the analysis of resting statedata has been proposed (Deshpande et al., 2006;Xie et al., 2008; Maxim et al., 2005), particularly in-cluding measures based on analysis of chaotic non-linear dynamical systems to analyze resting statedata, suggesting that the assumption of linearitymight be oversimplifying.

The question arises, to what extent and in whatcontext is it justified and beneficial to use non-linear measures of functional connectivity and moregenerally any indices such as those motivated bythe assumption of non-linear dynamical system asthe underlying mechanism behind the resting state

fMRI signal. When linear correlation is used as ameasure of functional connectivity, there are someimplicit assumptions made. The first is that theinformation in the temporal order of the samplescan be ignored (both within each timeseries andthe mutual interaction). While the extent of jus-tifiability of this assumption deserves explorationof its own, we keep this interim assumption forthe purposes of this paper, not least in order tokeep the comparison of linear correlation to nonlin-ear measures fair. Accepting for now the assump-tion of no role of temporal order of samples, weask if the instantaneous (zero-lag) dependence be-tween the time series, expressed in the probabil-ity distribution p(X, Y), is fully captured by thelinear correlation r(X,Y). We answer that this istrue under the assumption of bivariate Gaussian-ity of the distribution. Indeed, a multivariate nor-mal (Gaussian) distribution is uniquely defined byits correlation - up to linear shifts and rescaling.To be more precise, bivariate normal distributionis fully characterised by its mean µ = (µx, µy) andits 2 × 2 covariance matrix Cov(X,Y ) – if we al-low for linear shifting and scaling, the remaininginvariant parameter characterizing fully the distri-bution is indeed the correlation r(X,Y ). For a bi-variate Gaussian distribution, the correlation alsouniquely defines the mutual information shared be-tween the two variables X, Y which can be com-puted as I(X,Y ) = IGauss(r) ≡ − 1

2 log(1− r2).On the other side, when the Gaussianity assump-

tion does not hold, the distribution cannot be fullydescribed by the mean and covariance. More, pos-sibly infinitely many higher order moments needto be specified to determine the distribution. Asthe correlation is not sufficient to describe the de-pendence structure, the equation for I(X,Y ) abovecannot hold in general. Interestingly, as we detaillater, the notable properties of normal distributionsenable a derivation of a useful lower bound on mu-tual information valid for a broad class of probabil-ity distributions. In particular, for a bivariate dis-tribution p(X,Y ) with standard normal marginalsp(X), p(Y ), it holds that I(X,Y ) ≥ IGauss(r) =− 1

2 log(1 − r2), where the equality holds exactlyfor bivariate Gaussian distributions. This allows usto quantify the deviation from Gaussianity as thedifference between the total mutual information ofthe two variables I(X,Y ) and the mutual informa-tion IGauss(r) = − 1

2 log(1− r2) that correspond tobivariate Gaussian distribution with the observedcorrelation r.

2

While there are many potential nonlinear FCmeasure candidates, mutual information holds aspecific position among these for its generality. Intheory, it is general enough to capture an arbi-trary form of dependence relation between the vari-ables without any apriori model restrictions on itsform. The properties of mutual information allowus not only to test the suitability of linear correla-tion through probing the Gaussianity of the fMRItime series, but also to construct a quantitative es-timate of connectivity information neglected by theuse of linear correlation. This gives the amount ofadditional information available and bounds the po-tential contribution of non-linear alternatives overthe Pearson correlation coefficient.

We implement the outlined ideas by comparingthe total mutual information between the signalswith the mutual information between the signalsin surrogate datasets. These surrogates are gener-ated in a way that preserves the linear correlation,but cancels any nonlinear information by enforc-ing bivariate Gaussian distribution on the surro-gate signal-pair. This approach allows us to bothtest and quantify the deviation from Gaussianity,providing a principled guide in judging the suit-ability of linear correlation as a measure of FC.The focus on bivariate Gaussianity as the crucialcondition of suitability of use of linear correlationas FC index, along with the illustrative quantita-tive estimation of the deviation from Gaussianityby means of the mutual information neglected bylinear correlation, are the two main contributions ofthis study to the discussion of fMRI functional con-nectivity methods. We apply the presented methodto parcel-average time series obtained from rest-ing state fMRI BOLD signal of healthy subjects,testing and quantifying the deviation from bivari-ate Gaussianity. We complement this by assess-ing the relevance of the detected non-Gaussianitythrough tests of agreement of clustering results ob-tained from original and Gaussianized data.

2. Materials and methods

2.1. Data

Twelve right-handed healthy young volunteers (5males and 7 females, age range 20 – 31 years) par-ticipated in the study. Participants were informedabout the experimental procedures and providedwritten informed consent. The study design wasapproved by the local Ethics Committee of Chieti

University. Subjects lay in a supine position andviewed a black screen with a centered red fixationpoint of 0.3 visual degrees, through a mirror tiltedby 45 degrees. Each volunteer underwent two scan-ning runs of 10 minutes in a resting-state condition.Specifically, they were instructed to be relaxed, butto maintain fixation during scanning. The eye posi-tion was monitored at 120 Hz during scanning usingan ISCAN eye tracker system.

Scanning was performed with a 3T MR scan-ner (Achieva; Philips Medical Systems) locatedat the Institute for Advanced Biomedical Tech-nologies in Chieti, Italy. Functional imageswere obtained using T2-weighted echo-planar imag-ing (EPI) with blood oxygenation level-dependent(BOLD) contrast using SENSE imaging. EPIs(TR/TE=2000/35 ms) comprised 32 axial slicesacquired continuously in ascending order coveringthe entire cerebrum (voxel size=3x3x3.5mm3). Foreach scanning run, intitial 5 dummy volumes al-lowing the MRI signal to reach steady state werediscarded. The next 300 functional volumes wereused for the analysis. A three-dimensional high-resolution T1-weighted image (TR/TE=9.6/4.6ms, voxel size=0.98x0.98x1.2mm3) covering the en-tire brain was acquired at the end of the scanningsession and used for anatomical reference.

Initial data preprocessing was performed usingthe SPM5 software package (Wellcome Departmentof Cognitive Neurology, London, UK) running un-der MATLAB (The Mathworks). The preprocess-ing steps involved the following: (1) correctionfor slice-timing differences (2) correction of head-motion across functional images, (3) coregistrationof the anatomical image and the mean functionalimage, and (4) spatial normalization of all imagesto a standard stereotaxic space (Montreal Neuro-logical Institute, MNI) with a voxel size of 3x3x3mm3.

Ninety parcels from the Automated AnatomicalLabeling (AAL) atlas were used to extract meanBOLD time series after masking out non-gray mat-ter voxels. The anatomical positions of the parcelsare described in (Tzourio-Mazoyer et al., 2002). Ev-ery parcel time series was orthogonalized with re-spect to motion parameters and global mean signaland high-pass filtered at 1/120Hz.

2.2. Analysis

As already mentioned in the Introduction, for abivariate distribution p(X,Y ) with standard nor-

3

mal marginals p(X), p(Y ), it holds that

I(X,Y ) ≥ IGauss = −1

2log(1− r2), (1)

where the equality holds exactly for bivariate Gaus-sian distributions. The inequality (1) stems fromthe fact, that normal distribution is the maximumentropy distribution for a given covariance matrix(or for a given correlation, as we assume withoutloss of generality that σ(X) = σ(Y ) = 1). Fromthe relation between mutual information and en-tropy (I(X,Y ) = H(X) +H(Y )−H(X,Y )) it fol-lows that mutual information of Gaussian distribu-tion IGauss(r) is then minimal from all distribu-tions of given correlation r, under the assumptionof fixed marginal entropies, which is true when themarginals have standard normal distribution. Notethat the assumption of normality of the marginalsis far less restrictive than it might seem. First, ap-proximate data normality is commonly assumed inareas not restricted to fMRI FC analysis. More im-portantly, even if we find particular data deviatingstrongly from normality, any sample distributioncan be monotonously transformed to match normaldistribution.

To assure precise non-Gaussianity estimates, wehave indeed carried out this ‘normalization’ step.First, to each of the values xi of the original vari-able x its sample percentile pi is assigned. Subse-quently, the values xi of the original variable are re-placed by values ni corresponding to the respectivepercentiles pi in the standard normal distributionN(0, 1).

For two discrete random variables X1, X2 withsets of values Ξ1 and Ξ2, the mutual information isdefined as

I(X1, X2) =∑

x1∈Ξ1

∑x2∈Ξ2

p(x1, x2) logp(x1, x2)

p(x1)p(x2),

where the probability distribution function is de-fined by p(xi) = Pr{Xi = xi}, xi ∈ Ξi and thejoint probability distribution function is p(x1, x2)is defined analogously. When the discrete vari-ables X1, X2 are obtained from continuous vari-ables on a continuous probability space, then themutual information depends on a partition ξ cho-sen to discretize the space. Here a simple box-counting algorithm based on marginal equiquanti-zation method (Palus et al., 1993) was used, i.e.,a partition was generated adaptively in one dimen-sion (for each variable) so that the marginal bins

become equiprobable. This means that there is ap-proximately the same number of data points in eachmarginal bin. In this paper we used a simple prag-matic choice of Q = 8 bins for each marginal vari-able Palus and Vejmelka (2007).

For each session, we have computed the mutualinformation (MI) for each pair of parcels, yielding asymmetric 90-by-90 matrix of MI values. Notably,the raw estimates of mutual information suffer fromsome inevitable bias due to finite sample size anddiscretization of the variables. While this bias doesnot affect the statistical testing framework, it is nec-essary to correct for it to allow the use of standardscales for reporting the resulting mutual informa-tion and for more accurate computation of the ne-glected information.

This correction is carried out with the help ofsample mutual information estimates for finite-size(N = 300) random samples from bivariate Gaus-sian distributions with known correlations (andtherefore known mutual informations IGauss =− 1

2 log(1− r2)). These are computed for 50000 bi-variate random samples (each with size N = 300)for each correlation in the range from 0 to 1 (in 200steps of 0.005). The average of the 50000 mutualinformation estimates gives us an approximate ex-pected sample mutual information estimate corre-sponding to each tabulated correlation/mutual in-formation. Thus we obtain a monotonous functiontransforming the known true correlation (or the re-spective mutual information) to its expected nu-merical MI estimate for sample size N = 300. Oncegenerated, the inverse of this monotonous functioncan be used to transform each estimated mutual in-formation to obtain a more accurate bias-correctedestimate of the true mutual information. (The re-lationship between the true population mutual in-formation and the expected finite-sample estimateis shown in Supplementary Figure 1.)

To compare the (total) mutual information tothe portion of information conveyed in the linearcorrelation, for each dataset, 99 random realiza-tions of multivariate time series preserving the lin-ear structure but canceling the nonlinear structurewere constructed, and MI was computed for thesesurrogates. If the original time series dependencestructure was Gaussian (and therefore fully cap-tured by the linear correlation), the MI in the sur-rogates should not differ from the original MI, upto some random error. The alternative case shouldmanifest itself as a decrease in the MI in the surro-gates with respect to the original data.

4

The surrogates were constructed as multivariateFourier transform (FT) surrogates (Prichard andTheiler, 1994; Palus, 1997): realizations of multi-variate linear stochastic process which mimic indi-vidual spectra of the original time series as well astheir cross-spectrum. The multivariate FT surro-gates are obtained by computing the Fourier trans-form of the series, keeping unchanged the mag-nitudes of the Fourier coefficients (the amplitudespectrum), but adding the same random number tothe phases of coefficients of the same frequency bin;the inverse FT into the time domain is then per-formed. The multivariate FT surrogates preservethe part of dependence which can be explained bya multivariate linear stochastic process.

The idea of comparing the MI of data to MI of’linear’ surrogates rather than directly to linear cor-relation of data has two aspects. First, it allowsa direct quantitative comparison of the nonlinearand linear connectivity, while correlation and mu-tual information estimators have generally differ-ent properties. Second, generation of the surro-gates allows direct statistical testing of the differ-ence. However, this procedure generates 99 esti-mates of the linear MI for each parcel pair; one foreach surrogate. While these are useful for hypoth-esis testing, for general presentation of the differ-ence we use the mean value of these 99 values. Inthe following we refer to this as ‘Gaussian’ MI, andit actually closely estimates the MI of a bivariateGaussian distribution IGauss(r) = − 1

2 log(1 − r2),where r stands for the correlation of the two vari-ables (see the close match in Figure 3, red and pur-ple line). The ‘neglected’ MI is estimated by thedifference between data MI and the Gaussian MI:Ineglected(X,Y ) = I(X,Y )− IGauss(r).

2.3. Statistical tests

For each session and each parcel pair, non-Gaussianity was tested at p = 0.05 by comparingthe data MI against the MI distribution of the mul-tivariate FT surrogates. Notably, the parcel-levelresults suffer from heavy multiple comparison prob-lem. As we are primarily interested in the bivariatenon-Gaussianity in general rather than in its spe-cific allocation to particular parcel-pairs, we furtheruse the results of the individual tests in a higher-level analysis.

On a session-level, the number of significantparcel-pairs in a given session was tested againstthe null hypothesis that the number of individ-ual significant entries has a binomial distribution

B(n = 4005, p = 0.05), where n = 4005 = 90(90−1)2

is the number of all parcel pairs and p = 0.05 is thesingle-entry false positive rate under the conditionof pure Gaussianity of the bivariate distributions.

As it may be argued that the assumption of pairindependence is too lenient, but the exact level ofdependence is difficult to establish, we also carriedout robust group level tests. The group level testsinvolved one-sample t-tests. The first was a test ofthe percentages of significant pairs in the 24 ses-sions against the null hypothesis of mean of 5% ofsignificant parcel-pair results; the second was a testof the neglected information session-wise averages(across all parcel-pairs) against the null hypothesisof zero mean.

To explicitly control for any potential bias inthe numerical generation of the surrogate distribu-tions, the group-level tests were complemented bypaired t-tests of the same two quantities againsttheir values obtained from a control set of lin-ear, ‘shadow’ datasets. For each session, a shadowdataset was created as a multivariate FT surro-gate of the marginally normalized original dataset.Thus the shadow dataset preserved only the lin-ear (correlation) structure of the original dataset ofthe respective session. Subsequently, each shadowdataset has undergone the same procedure as orig-inal data, including the initial normalization, gen-eration of multivariate surrogates, computation ofMI and statistical testing of pair-wise MI againstsurrogates. In this way, we have mimicked the fullprocedure of processing the original data using a‘purely linear FC’ shadow dataset, accounting forany potential slight bias in the detection rate in-troduced by numerical properties of the algorithm.Apart from the percentages, we have also testedthe mean neglected information from data versusshadow datasets by mean of a paired t-test. All thegroup-level tests used a p=0.05 significance thresh-old, but we also report the attained significancelevel.

2.4. Relevance for clustering

To explore the relevance of the studied non-Gaussianity for further data analysis, we have as-sessed the difference in clustering of anatomicalparcels based on two functional connectivity meth-ods: mutual information and linear correlation. Toensure a fair comparison, two preparatory stepswere carried out here. Firstly, to prevent artifi-cial mismatch due to merely different scaling ofthe two functional connectivity measures, we have

5

monotoneously rescaled the correlation values byr 7→ − 1

2 log(1 − r2), effectively yielding an esti-mate of the ‘linear’ mutual information IGauss. Al-though transformed to share the scale of mutualinformation, this connectivity measure reflects onlythe linear correlation between the parcel time se-ries, unlike the true mutual information computedby the equiquantization method. Secondly, dueto random error in estimation of functional con-nectivity measures, we could expect some randomlevel of disagreement between connectivity matricesand therefore between clustering based on differentfunctional connectivity measures. To control forthis effect, we have tested the specific effect of non-linearity included in the original data by comparingthe agreement between MI and correlation matrices(represented by IGauss) computed from data withthe agreement of MI and correlation computed froma multivariate linear surrogate dataset. This waywe effectively test whether the clustering disagree-ment includes a contribution due to nonlinearity, oris a mere result of the imperfect estimation of theconnectivity measures from finite-size samples.

2.4.1. Identification of clusters

Functional networks in the parcel time serieswere identified using a spectral clustering techniquecalled Multiclass Spectral Clustering (Yu and Shi,2003). In this technique clusters (correspondingto the sought functional networks) are found byoptimizing the Multiway Normalized Cut objec-tive, which is based on the Normalized Cut objec-tive (Shi and Malik, 2000). Formally the appliedmethod operates on weighted graphs where the ver-tices of the graph correspond to parcels and theedges of the graph are weighted by the functionalconnectivity between the respective parcels. Intu-itively a good clustering is represented by a parti-tion of all vertices to disjoint sets (clusters) whichsatisfies that the sum of weights of edges betweenvertices in different sets is small and the sum ofweights of edges between vertices in the same set islarge. Both of these properties are optimized at thesame time by the applied algorithm - we refer thereader to (Yu and Shi, 2003) for details. Thus theoptimal partition computed by the algorithm hasclusters which are more internally connected andless connected with each other. This correspondsto a grouping of the parcels so that the time seriesinside one cluster are similar and at the same timeseries of parcels in different clusters are not similar.

2.4.2. Agreement of clusterings

After obtaining the two clusterings of the parcelsfrom the two connectivity methods, we have toquantify their agreement. A natural procedure isto view the cluster assignment of a randomly cho-sen parcel as a discrete random variable, with in-teger values ranging from one to the total numberof clusters. Then we can express the agreement be-tween clusterings obtained for the two connectiv-ity methods as the mutual information between thetwo discrete random variables corresponding to thetwo generated clusterings. As the mutual informa-tion between two variables is bounded by entropyof each of them, we further normalize the mutualinformation, dividing it by the minimum of the en-tropies of the two variables. Thus, if the clusteringsare identical, then their normalized mutual infor-mation will be 1, while normalized mutual infor-mation of 0 corresponds to completely independentcluster assignments.

To statistically test the nonlinearity effect, wecompared the agreements obtained for the originaldata and linearized ‘shadow’ data by means of apaired t-test. Visual inspection of the clusteringof the group-average connectivity matrices revealedthat the choice of N=10 clusters provides satisfac-tory results — see Figure 1 for the example cluster-ing results. Therefore, N=10 clusters were used forprocessing each session. For robustness, the analy-sis was repeated for all other cluster counts in therange from N=2 to N=20 with FDR<0.05 correc-tion for multiple comparisons.

3. Results

3.1. Descriptive assessment

In descriptive terms, the data MI has proved verysimilar to the Gaussian MI (see Figure 2). In partic-ular, averaging across all parcel pairs, the data MIranged between 0.04 and 0.10 bits for different ses-sions, while the neglected MI was more than an or-der of magnitude smaller (0.0005-0.0068 bits). Nev-ertheless, the neglected MI was consistently posi-tive, which was not the case for shadow datasets(ranging from -0.0007 to 0.0016 bits).

Independently of the strength of coupling, thedata MI was moreover typically within the range ofsurrogate MI, as illustrated in Figure 3. Here, eachblue dot corresponds to MI of one parcel pair; thesurrogate distribution is represented by red (lightblue, green) lines for mean (1st percentile, 99th

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Figure 1: Clustering of the parcels for the choice of N=10clusters based on average linear correlation matrix. Thesame colour is used for all parcel within a given cluster; anMNI template anatomical map is underlaid. The obtainedclusters generally correspond to known functional networks.

percentile) of the surrogate distribution. Althoughthe session with the most non-Gaussianity is de-picted here, the distribution of computed MI fordata and the corresponding shadow dataset (Fig-ure 4) are almost indiscernible. Also, apart fromthe random error due to MI estimation from shorttime series, which is shared by data and shadowdata, both scatters follow well the theoretical pre-diction of dependence of MI on linear correlation(IGauss = − 1

2 log(1−r2), valid exactly under Gaus-sianity, purple line), which is closely approximatedby the surrogate mean.

3.2. Statistical tests

The percentage of parcel pairs with significantnon-Gaussianity was slightly elevated in all sessionsabove the 5% expected under the null hypothesis(ranging from 5.3 to 10.0% of significant pairs indifferent sessions). If all the parcel pairs were con-sidered independent this would constitute signifi-cant percentage for all but 5 of the sessions con-sidered (comparing to binomial distribution B(n =

4005, p = 0.05), where n = 4005 = 90(90−1)2 is the

number of all parcel pairs).While the assumption of independence of the

pairs might be too lenient, yielding false positiveresults, we also carried out group level tests that

1 2 3 4 5 6 7 8 9 10 11 120

0.05

0.1

0.15

subject numberav

erag

e m

utua

l inf

orm

atio

n (b

its)

Gaussian MINeglected MI

Figure 2: Comparison of the average Gaussian and neglectedinformation. Each stackbar represents values for one session,averaged across all parcel pairs.

−1 −0.5 0 0.5 10

0.5

1

1.5

2

correlation

mut

ual i

nfor

mat

ion

(bits

)

Figure 3: Mutual information as function of correlation in anexample dataset. The session with the most non-Gaussianityis depicted. Each blue dot corresponds to MI of one par-cel pair; red (light blue, green) lines correspond to mean(1st percentile, 99th percentile) of the surrogate distribution.The purple line shows the theoretical mutual information ofan exactly Gaussian distribution with the given correlationIGauss(r) - it is not well visible as it closely matches themean of the surrogate distribution.

7

−1 −0.5 0 0.5 10

0.5

1

1.5

2

correlation

mut

ual i

nfor

mat

ion

(bits

)

Figure 4: Mutual information as function of correlation in anexample surrogate. Each blue dot corresponds to MI of oneparcel pair; red (light blue, green) lines correspond to mean(1st percentile, 99th percentile) of the surrogate distribution.The session with the most non-Gaussianity is depicted.

confirmed the statistical deviation from Gaussian-ity. In particular, the counts of pairs with signifi-cant nonlinearity were significantly higher than the5% expected by chance, as tested by one-sided t-test(t = 6.95,df = 23, p < 10−6). Also, the neglectedinformation averaged over parcel pairs was signif-icantly biased above the mean zero value corre-sponding to full Gaussianity (t = 8.52,df = 23, p <10−7).

As described in subsection 2.3, we have alsocarried out another group-level tests that explic-itly control for any potential biases in the sur-rogate method. The comparison of results fromthe original data with those from control (linear)‘shadow’ datasets confirmed the detection of non-Gaussianity in the data. The counts of pairs withsignificant nonlinearity were significantly higher indata than similar counts obtained from shadowdatasets, when compared on group level by meansof a paired t-test (t = 6.26,df = 23, p < 10−5).Also, the neglected information in data averagedover parcel pairs was positive for all sessions andon average had value 0.0029 bits while the ne-glected information in the shadow datasets fluctu-ated around zero with mean of 0.0006 bits. Thisdifference was also clearly statistically significant(t = 6.51,df = 23, p < 10−5).

3.3. Relevance for clustering

We tested whether the neglected informationhad a significant effect on the overall structure of

the connectivity, as captured in clustering of theparcels. N=10 clusters were computed for each ses-sion using the MI and rescaled linear correlationFC matrices and the agreement of the two clus-terings was computed. This was carried out bothfor normalized data and their linearized ‘control’versions. The agreements were generally quite high(0.80±0.07 on the scale from 0 to 1) and did not dif-fer significantly from those obtained from the ‘con-trol’ linearized data (t = 0.39,df = 23, p = 0.70).This suggest that the mismatch between the clus-terings based on the two measures was (similarlyas in the case of the ‘control’ dataset) attributablegenerally to random error in estimation of the func-tional connectivity indices from finite size samples.Repeating the analysis for all cluster counts in therange N = 2 to N = 20 confirmed the observa-tion of no significant difference (p < 0.05, FDRcorrected).

4. Discussion

The presented study reveals that the bivariatedependence structure of the fMRI BOLD regionaltime series is captured very well by linear correla-tion. Indeed, average mutual information was onlyseveral percent higher than the mutual informationin surrogate data that contained only the linear partof the dependence. This gives explicit and quan-tifiable argument for the intuitive choice of linearcorrelation as a measure of functional connectivityfor fMRI time series.

Nevertheless, we have shown that there is a sta-tistically significant contribution of non-Gaussiandependencies in the data, although the effect isso subtle that testing across many pairs or evenacross many sessions was needed to acquire suffi-cient power for such tests. The detection of non-Gaussian coupling is not surprising in the light ofthe fact that the dynamics of brain activity as wellas the hemodynamic response of the vasculaturecontain many nonlinearities.

We have also carried out tests of the relevanceof the neglected information for applications: themarginality of the difference between full mutualinformation and correlation as FC measures wasconfirmed in comparisons using clustering of theparcels – the disagreement between clustering ob-tained from mutual information in data and itscorrelation-based approximation did not exceed thedisagreement due to random error in estimation ofthe FC.

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For testing Gaussianity of the bivariate depen-dences, we have used a framework based on a com-parison of mutual information evaluated on dataand on surrogate timeseries. Below we discuss themotivation of the method choice from the varietyof methods available in the wider context of mul-tivariate normality testing. Importantly, the prob-lem of testing multivariate normality is more com-plex than that of testing univariate normality andfor principal reasons there is not a unique agree-ment in the statistical community on a single bestmethod for testing multivariate normality. For arecent review of the topic we refer the reader toMecklin and Mundfrom (2004). Many alternativetests for multivariate normality could therefore beconsidered, which may to some extent differ in thepower against particular alternatives. However, theframework used in this paper has several importantfeatures that make it well suited for the currentstudy. The first reason is practical; mutual infor-mation is used both for testing and direct quan-tification of the deviation from Gaussianity. Cru-cially, as mutual information is a widely used andtheoretically grounded assumption–free measure ofdependence, this quantification also has a generaldirect and intuitive interpretation. Secondly, manyavailable multivariate normality tests assume inde-pendence of the samples. This is clearly not valid inthe case of fMRI time series. The use of multivari-ate linear surrogates in our approach controls anybias due to temporal autocorrelation and sidestepsthe need to estimate and correct for the reduceddegrees of freedom of the test. Also, the test basedon mutual information has the advantage of beingconsistent in the sense of being sensitive to virtu-ally any departure from Gaussianity of the depen-dence. This advantageous property is not sharedby many commonly used tests for multivariate nor-mality such as those based on testing the devia-tion of skewness or kurtosis of the distribution fromthe values valid for normal distribution. Finally,the used framework allows to test the dependencestructure itself, independently of the normality ofthe univariate marginal distributions. This is alsonot in general true for other multivariate normalitytests, although adaptations may be possible.

Another topic deserving a discussion is theadopted multi-level testing strategy and its poten-tial alternatives. As we mentioned, the binomialtest used to assess the Gaussianity null hypothesison session-level might be too lenient due to depen-dence among the 4005 lower-level tests. We have

avoided speculative estimation of the test interde-pendence level within each session and rather ro-bustly tested the Gaussianity on the group level.Importantly, this was sufficient to statistically re-ject the Gaussianity null hypothesis. If further evi-dence on the level of individual parcel-pairs or ses-sions was required, the use of criteria for hypothesistesting such as the family-wise error rate and thefalse discovery rate to obtain more detailed and po-tentially more powerful tests could be considered.Nevertheless, their implementation may prove pro-hibitively cumbersome in the current context. Inparticular, these methods generally involve com-paring the smallest (or each) p-value against thecorrected significance level p = 0.05/4005 ∼ 10−6.Such a low p-value is clearly below the resolution ofthe sample histograms capturing the null hypothe-sis with 99 surrogates; the minimal required num-ber of surrogates would be in the order of millionsfor each test, which is technically impractical to in-feasible. Fitting the null distribution from smallernumber of surrogates by e.g. imposing an assump-tion of approximate normality of the MI distribu-tion may seem promising, but may be too crudeand even slight departure from the true null distri-bution might crucially affect the true significancelevels, particularly for the critical values of the or-der of 10−6.

It is important to keep in mind that the observeddeviations from Gaussianity might not reflect onlya stationary non-Gaussianity in neuronal connec-tivity. In the presented framework, deviation fromthe null hypothesis could be caused also by nonsta-tionarity of the signal. Vice versa, non-Gaussianitymight lead to false detection of nonstationarity ifthe test assumed a linear Gaussian generating pro-cess as in (Chang and Glover, 2010). Technically, aprecise isolation of these two effects is extremelychallenging. A practical consideration allows toreasonably reconcile the two alternative interpreta-tions: heavy nonstationarity could indeed increasethe estimated non-Gaussianity, but it would also onits own invalidate the use of simple linear correla-tion, leading to the same conclusion about its suit-ability. On another note, we also should not forgetthat we are working on the level of fMRI BOLD sig-nal rather that with neuronal activations - both theGaussian and non-Gaussian contributions to FC arelikely to be affected to some extent by non-neuralsources of signal variation. Nevertheless, in the endthe estimation of whether this would play in favorof or against the use of linear correlation (and how

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much) seems to be entirely speculative at this point.It can be argued that the amount of non-linear

mutual information detected is likely to depend onthe preprocessing of the data and the acquisitionparameters. In this study, we have used standardacquisition and preprocessing, maybe with the ex-ception of not using any explicit temporal low-passfiltering. The rationale for omission of this step wasto avoid introduction of temporal averaging thatwould further diminish an already small nonlinear-ity in the data. The choice of working with sev-eral tens of gray matter parcels per hemisphere wasmotivated by approach of (Lahaye et al., 2003) aswell as computational feasibility and allowing re-producibility and comparability across subjects anddatasets. On the other side, incoherence of voxel-signals within the parcels might lead to losing someof the nonlinearity on the level of parcel-averaging,and therefore working on the level of temporalsignal of spatial Independent Component Analysis(ICA) components or single voxel signal might intheory be more sensitive to nonlinearity. Exploringthe dependence of the non-Gaussian contribution toFC on the time series extraction method is a sub-ject of future work. Also, the analysis presented inthis paper did not consider any time-lags. This isconsistent with comparing to linear correlation asa measure of FC. The time lags would have to beincluded were we interested in assessment of causalor ‘effective connectivity’ measures (linear versusnonlinear) – this defines a natural follow-up of therecent study.

Apart from some minor technical aspects suchas normalisation of the timeseries, our study differsfrom the previous probes into the potential of non-linear fMRI FC such as (Xie et al., 2008; Maximet al., 2005; Lahaye et al., 2003; Deshpande et al.,2006) mainly in that we explicitly focus on the bi-variate Gaussianity rather than the linearity as-sumption as the condition of suitability of use oflinear correlation as functional connectivity index.The deviation from this condition allowed us toquantify the potential available for arbitrary ‘non-linear’ connectivity measures. This general inter-pretation is allowed by the use of a very general de-pendence measure – mutual information. In theory,this is able to capture virtually any form of statis-tical dependence. Of course, some practical limi-tations stem from the inevitably finite sample size,forcing us to summarize the results across parcelpairs and subjects. Last but not least, we providean illustrative quantitative estimation of the devia-

tion from Gaussianity by means of the mutual infor-mation neglected by linear correlation, that shouldgive a theoretical upper bound on any improvementto be made by an arbitrary nonlinear connectivitymeasure. We note that an earlier attempt towardsquantification of the FC ‘nonlinearity’ was madein (Lahaye et al., 2003), who reported also the ex-plained variances by the higher order (nonlinear upto order 5) terms in the predictor set within a lin-ear regression framework – although these varianceswere not corrected for the effect of extra numberof regressors in the model, and also the consideredmodel could not capture a more general form of de-pendence. A similar approach as in Lahaye et al.(2003) was previously also applied to assessment ofthe non(linear) character of connectivity in EEGdata by Pijn et al. (1990). The h2 coefficient ap-plied in that study expresses the proportion of vari-ance in one variable explained by another variableusing a polynomial fit; the aim was to provide morerobust detection of interactions than simple linearcorrelation in case when the linearity assumption isviolated.

Despite the differences in the aims and method-ology, our observations agree qualitatively with theprevious studies such as (Lahaye et al., 2003; Xieet al., 2008) in that we also reject the specific hy-pothesis of stationary multivariate linear Gaussianprocess as the structure of resting state fMRI sig-nal.

For clarity, we stress again that the above men-tioned Gaussianity condition, rather than mere lin-earity of the process, is the key assumption for suit-ability of use of linear correlation. To fully acknowl-edge this consider that as stated e.g. in (Bickeland Buhlmann, 1996), a linear stationary process(Xt)t∈Z is usually defined by

Xt =

∞∑j=0

φjεt−j ; (t ∈ Z), (2)

where εt is i.i.d with E[εt] = 0, E|εt|2 < ∞ and∑∞j=0 φ

2j <∞.

While the definitions of linear process may differin details, in most cases they are general enough toinclude processes with non-Gaussian distribution,which are then not fully described by their corre-lation structure. This may lead to some confusion,pointed out in (Nagarajan, 2006), who showed thatsome widely used surrogate-based linearity testssuch as those used in (Xie et al., 2008) are actuallysensitive to non-Gaussianity. Careful examination

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of the assumptions of these tests reveals that thenull hypothesis there is that the data come from aprocess that is equivalent to a linearly filtered whiteGaussian noise. Of course, if the general definitionof linearity is complemented by the condition thatthe generating noise is Gaussian, such a linear pro-cess is indeed also Gaussian, with probability distri-bution fully characterised by the mean and covari-ance structure. Discussion of the potential causesof rejection of the null hypothesis of ‘linearity’ inthe FT-surrogate framework by processes that arestill linear in the general sense of having the form(2) is also provided in (Palus, 2007).

For completeness, we note that linearity is alsooften discussed as an alternative to nonlinear, po-tentially chaotic deterministic dynamical systems.In this context caution is warranted with the in-terpretation of many ‘chaotic’ characteristics suchas fractional correlation dimension or Lyapunov ex-ponents when the underlying system might be ofstochastic (non)linear nature rather than determin-istic (non)linear dynamical system, and particularlywhen short time series such as those acquired fromfMRI are being analyzed.

To summarize, we have assessed the suitability oflinear correlation as functional connectivity mea-sure for fMRI time series by testing and quanti-fying the deviation from bivariate Gaussianity us-ing mutual information. The results were as fol-lows: firstly, the quantitative assessment revealedthat the portion of mutual information neglectedby using linear correlation instead of considering anarbitrary non-linear form of instantaneous depen-dence is minor. Secondly, formal group-level testrevealed that the percentage of parcel-pairs withsignificant non-Gaussian dependence contributionis indeed above random. Overall we conclude thatlinear correlation of normalized data well capturesthe full functional connectivity: although existenceof non-Gaussian contribution to functional connec-tivity can be established, practical relevance of non-linear methods trying to improve over linear corre-lation might be limited by the fact that the dataare indeed almost Gaussian. This was documentedby a quantitative estimate of mutual informationneglected due to the use of linear correlation fortypical resting state fMRI data, as well as by com-plimentary comparison of results of clustering anal-ysis.

Acknowledgements

This work has been supported by the EC FP7project BrainSync (EC: HEALTH-F2-2008-200728,CR: MSM/7E08027); and in part by the Insti-tutional Research Plan AV0Z10300504. DM waspartly supported by the Research Foundation Flan-ders (grant FWO A 4/5 SDS 15387).

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