A simple view of the brain through a frequency-specific functional connectivity measure

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A simple view of the brain through a frequency-specific functionalconnectivity measure

R. Salvador,a,b,⁎ A. Martínez,a E. Pomarol-Clotet,b J. Gomar,b F. Vila,a S. Sarró,b

A. Capdevila,a and E. Bullmorec

aFundació Sant Joan de Déu, SJD SSM, Dr. Antoni Pujadas, 42 Sant Boi de Llobregat 08830, Barcelona, SpainbBenito Menni, CASM, Dr. Antoni Pujadas, 38 Sant Boi de Llobregat 08830, Barcelona, SpaincBrain Mapping Unit, University of Cambridge, Department of Psychiatry, Addenbrooke’s Hospital, Cambridge CB2 2QQ, UK

Received 3 April 2007; revised 18 June 2007; accepted 6 August 2007Available online 25 August 2007

Here we develop a measure of functional connectivity describing thedegree of covariability between a brain region and the rest of the brain.This measure is based on previous formulas for the mutual information(MI) between clusters of regions in the frequency domain. Under thecurrent scenario, the MI can be given as a simple monotonous functionof the multiple coherence and it leads to an easy visual representationof connectivity patterns. Computationally efficient formulas, adequatefor short time series, are presented and applied to functional magneticresonance imaging (fMRI) data measured in subjects (N=34)performing a working memory task or being at rest. While restingstate coherence in high (0.17–0.25 Hz) and middle (0.08–0.17 Hz)frequency intervals is bilaterally salient in several limbic and temporalareas including the insula, the amygdala, and the primary auditorycortex, low frequencies (b0.08 Hz) have greatest connectivity in frontalstructures. Results from the comparison between resting and N-backconditions show enhanced low frequency coherence in many of theareas previously reported in standard fMRI activation studies ofworking memory, but task related reductions in high frequencyconnectivity are also found in regions of the default mode network.Finally, potentially confounding effects of head movement and regionalvolume on MI are identified and addressed.© 2007 Elsevier Inc. All rights reserved.

Keywords: Brain connectivity; Mutual information; Frequency domain;Resting state; Default mode network; N-back; Coherence

Introduction

One of the major technical difficulties faced by fMRI brainconnectivity analyses is the high number of interactions that may

⁎ Corresponding author. Benito Menni, CASM, Dr. Antoni Pujadas, 38Sant Boi de Llobregat 08830, Barcelona, Spain. Fax: +34 936240268.

E-mail address: [email protected] (R. Salvador).Available online on ScienceDirect (www.sciencedirect.com).

1053-8119/$ - see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.neuroimage.2007.08.018

potentially be considered and their later graphical representation.Different approaches have been taken to avoid, or to minimize,these challenges. Among them, selecting a single seed region andassessing its degree of covariability with the rest of loci of thebrain, in a bivariate fashion, is one of the simplest and most usedtechniques (see some examples in Biswal et al., 1995; Goebel etal., 1998; Cordes et al., 2001); however, its scope is limited to theselected area.

Alternatively, a subset of candidate regions may be chosen onthe basis of prior knowledge and their connectivity may beanalysed, frequently through complex statistical models. Amongthem, the structural equation models (Bullmore et al., 2000;Schlosser et al., 2006; Kim et al., 2007) and the dynamic causalmodels (Friston et al., 2003; Penny et al., 2004) allow a highdegree of specificity in the nature of the relations assessed, but theyheavily depend on the correctness and completeness of the model.

Principal component analysis, multidimensional scaling andindependent component analysis have been successfully applied, aswell, to fMRI datasets (Bullmore et al., 1996; Friston et al., 1999;Beckmann and Smith, 2004; Salvador et al., 2005a), leading to theisolation of distinct brain networks with high levels of covaria-bility. The major challenge faced by these techniques, though, isthe integration of the individual information at the group level(Welchew et al., 2005). While some solutions have been proposedfor the independent component analysis (see Calhoun et al., 2004and Esposito et al., 2005) further improvement may still berequired, specially to deal efficiently with group comparisons.

In this paper, we propose an alternative way of quantifyingfunctional connectivity. The method is derived from some of thefrequency-based coherence measures previously described bySalvador et al. (2005b, 2007). Such measures can, in turn, beseen as adaptations to a time dynamic framework of some of thegeneral concepts presented by Tononi et al. (1998). These authorsworked out interesting quantities for quantifying different aspectsof clustering and brain integration. However, their models did notinclude, in an explicitly way, the temporal dimension.

280 R. Salvador et al. / NeuroImage 39 (2008) 279–289

Specifically, we give the formulas in the frequency domain forthe mutual information between a region and all other regions ofthe brain, and we show an efficient way to avoid computationalproblems arising from high dimensional datasets. We show that theproposed technique leads to a simple representation of inter-regional connectivity in the brain at specific frequency intervals byanalysis of resting state fMRI data acquired from 34 healthyvolunteers. Additionally we demonstrate significant task-relatedincreases and decreases in MI during performance of a workingmemory task by the same group of subjects; and we addresspotentially confounding effects of regional volume and headmovement on MI.

Methods

The mutual information between a region and the rest of the brain

A set of p regional time series extracted from an fMRI datasetcan be considered, in a rather simple way, as an n-length, finiterealisation of a multivariate stochastic process

X ðtÞ ¼ fX1ðtÞ;X2ðtÞ; N ;XpðtÞg; taZ ð1ÞUnder assumptions of stationarity and joint normality, Salvador

et al. (2007) derived general expressions describing the degree ofconnectivity between two clusters (subsets) of brain regions. Thesewere developed in the frequency domain and were given in termsof mutual information (MI).

Our current interest is in describing the functional connectivitybetween one brain region and the rest of regions of the brain. Moreformally, we want to restrict ourselves to the specific case where asubset a of X contains the mean time series of a single brain region(Xa), and the other subset b contains the time series of theremaining p−1 regions in the brain (Xb={X1, …, Xa− 1, Xa+1, …,Xp}). As it is shown in Appendix A, when the general modelproposed by Salvador et al. (2007) is constrained to this scenario,the mutual information between region a and cluster b, at theFourier frequency ωk, becomes

MIa;b xkð Þ ¼ � 1

2log 1�mCohXa;Xb xkð Þ� �

; ð2Þ

where log() is the natural logarithm. Eq. (2) is a simple monotonicfunction of the multiple coherence between Xa and Xb (Brillinger,1981)

mCohXa ;Xb xkð Þ ¼ V ðxkÞtðfXbðxkÞÞ�1PV ðxkÞfXaðxkÞ ; ð3Þ

with V(ωk) being a vector containing column a of the spectraldensity matrix of Eq. (1) and

PV ðxkÞ being its complex conjugate.

fXb(ωk) is the spectral density matrix of the subset Xb and fXa

(ωk) isthe spectral density value of Xa (a scalar).

In order to summarise the MI over the spectra, average valuesinvariant to the length n of the observed time series, may be givenby

MIa;b ¼ � 12n

Xnk¼1

log 1�mCohXa ;Xb xkð Þ� �; ð4Þ

where k is indexing all Fourier frequencies. Interestingly, thisformula is very similar to the formula of the mutual information

between two univariate stochastic processes a and c (Granger andHatanaka, 1964)

MIa;c ¼ � 12p

Z p

�plog 1� CohXa;Xc kð Þ� �

dk: ð5Þ

Due to the close relation between multiple coherences andmutual information, in the rest of the text we will use,interchangeably, both terms to refer to the functional connectivitylevels observed. All analyses presented, however, are based onestimated MI values.

Sample estimates of mutual information

Given a finite realisation of Eq. (1) (in our framework, a set of ptime series extracted from an fMRI experiment) simple estimatesfor MI may be derived. Specifically, values from the filtered cross-periodograms of the observed time series can be used as estimatesof the spectral densities in Eq. (3), and, later, averages of MI overthe Fourier frequencies can be obtained through Eq. (4). Indeed,we can average over a limited range of frequencies in order toobtain the MI for a certain frequency band.

Very often, the number p of time series will be high and theirlength n moderate, potentially producing technical problems in thecomputation of the MI. Collinearity between some of the timeseries in Xb will alter the inversion in Eq. (3), and will lead tounreliable estimates of the multiple coherence. In these situations, areduction of the dimensionality of fXb

(ωk) through a diagonalisationmay be adequate, i.e.,

fXbðxkÞ ¼ L*ðxkÞDðxkÞLðxkÞ; ð6Þwhere L(ωk) and Δ(ωk) are the matrices containing the eigenvectorsand real positive eigenvalues of fXb

(ωk) (which is Hermitianpositive definite), and L∗(ωk) is the transposed conjugate of L(ωk).Dimensionality reduction will be achieved by taking the s biggesteigenvalues accounting for a large proportion of the variability (say95%) and discarding the rest. For values of r obeying

Xr�1

i¼1

Di; iðxkÞ=Xpi¼1

Di; iðxkÞV0:95; ð7Þ

where eigenvalues are ordered in decreasing magnitude, s will bemax(r). From the s selected eigenvalues and their respectiveeigenvectors, a computationally adequate formula for the modifiedmultiple coherence will be given by

mCohXa;Xb xkð Þ ¼ W ðxkÞtðDsðxkÞÞ�1PW ðxkÞfXaðxkÞ ; ð8Þ

where W(ωk)=Ls∗ (ωk)V(ωk) (see Appendix B for the derivation ofthese equations). Indeed, as it is shown in the same appendix, thisprocedure is equivalent to calculate the multiple coherencebetween Xa and the s principal component series (Brillinger,1981). Again, estimates of Eq. (8) will be plugged-in in Eq. (4) toobtain the MI over the frequencies of interest.

Functional MRI data acquisition and pre-processing

Thirty-seven healthy volunteers were recruited by advertisementwithin the local community, of which 34 were finally included in thestudy (20 males, 14 females, mean age 40.3 years [S.D. 11.9 years]).

Table 1List of brain regions extracted from the template of Tzourio-Mazoyer et al.(2002) and their abbreviations as used in this study

Region Code

Precentral gyrus PreCGSuperior frontal gyrus, dorsolateral SFGdorSuperior frontal gyrus, orbital part ORBsupMiddle frontal gyrus MFGMiddle frontal gyrus orbital part ORBmidInferior frontal gyrus, opercular part IFGopercInferior frontal gyrus, triangular part IFGtriangInferior frontal gyrus, orbital part ORBinfRolandic operculum ROLSupplementary motor area SMAOlfactory cortex OLFSuperior frontal gyrus, medial SFGmedSuperior frontal gyrus, medial orbital ORBsupmedGyrus rectus RECInsula INSAnterior cingulate and paracingulate gyri ACGMedian cingulate and paracingulate gyri DCGPosterior cingulate gyrus PCGHippocampus HIPParahippocampal gyrus PHGAmygdala AMYGCalcarine fissure and surrounding cortex CALCuneus CUNLingual gyrus LINGSuperior occipital gyrus SOGMiddle occipital gyrus MOGInferior occipital gyrus IOGFusiform gyrus FFGPostcentral gyrus PoCGSuperior parietal gyrus SPGInferior parietal, but supramarginal and angular gyri IPLSupramarginal gyrus SMGAngular gyrus ANGPrecuneus PCUNParacentral lobule PCLCaudate nucleus CAUlenticular nucleus, putamen PUTLenticular Nucleus, pallidum PALThalamus THAHeschl gyrus HESSuperior temporal gyrus STGTemporal pole: superior temporal gyrus TPOsupMiddle temporal gyrus MTGTemporal pole: middle temporal gyrus TPOmidInferior temporal gyrus ITG

The same 45 regions were extracted from right and left hemispheres toprovide 90 regional time series in total for each subject.

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All participants gave written informed consent prior to involvementin accordance with the Declaration of Helsinki, and the researchprotocol was approved by the local ethics committee.

Each individual underwent a single scanning session in whichtwo fMRI datasets were acquired under different experimentalconditions. The first dataset was acquired while the individuals wereresting quietly in the scanner with instructions to keep their eyesopen in order to avoid high inter-individual variability in self-generated drowsiness. The second dataset was acquired whilesubjects performed a sequential-letter version of the N-backmemorytask presented in a blocked periodic paradigm. Specifically, four 1-back and four 2-back blocks of 48 s were presented in an alternateway, and interleaved with baseline periods of 16 s. Each stimuluswas presented for 1 s, followed by a non-stimulus period of anothersecond.

In both fMRI experiments, 266 volumes of gradient echo echo-planar imaging (EPI) data depicting the blood-oxygenation-level-dependent (BOLD) contrast were acquired using a GE Signasystem operating at 1.5 T in the Hospital Sant Joan de Déu(Barcelona). Each volume comprised 16 axial planes acquired withthe following parameters: TR=2000 ms, TE=40 ms, flipangle=70°, section thickness=7 mm, section skip=0.7 mm, in-plane resolution=3×3 mm. The first 10 volumes were discarded toavoid T1 saturation effects, leaving 256 volumes available forfurther analysis.

Several modules of the FSL software (reviewed in Smith et al.,2004) were used for the pre-processing of the MR images. The BETmodule was used to extract the signal from the skull, volumes wereco-registered with MCFLIRT, and were finally normalised to a MNIstandard brain image with FLIRT. To minimise movement artefacts,all individuals with an estimated maximum absolute movementgreater than 3.0 mm and/or an average absolute movement greaterthan 0.3 mmwere discarded from the study. This led to the exclusionof 3 of the originally recruited individuals.

Extraction of regional time series

The images of each subject were parcellated with a standardtemplate that divides the whole brain in 90 regions (Tzourio-Mazoyer et al., 2002), which are listed in Table 1. For each of theseregions, the mean time series for all voxels was calculated for bothtasks separately. This led to an individual matrix of 90 columns(regions) by 256 rows (time points) for each one of the subjects andconditions (rest or working memory). As a first step to reduce theinfluence of head movement, each time series was regressed on sixother time series: the three sets of translations (in the x, y, and zdirections) estimated in the image realignment phase, and the threefirst order differences of these translations.

Initial results with the resting data showed a clear relationbetween the size of the region of interest and its MI (Fig. 1). Toavoid this size related effect, the original template was modified.An algorithm that eroded the limits of the regions was applied untilall of them had the volume of the smallest region, i.e., leftamygdala=1.76 cm3.

The implementation of the algorithm was based on two majorsteps. First, reading over the 3D template, all voxels of one regionwith one or more neighbours (in the 3×3×3 window) notbelonging to the same region were tagged as bordering voxels (butnot changed yet). Once all the 3D image was read (or enoughvoxels had been labelled), these bordering voxels were assigned tothe background. This two-fold process was applied, iteratively, as

many times as required to achieve the desired size for the region.The eroded template was used for the analysis of connectivitypatterns in the resting state data. The original template, though, waskept for the comparison between the resting state and the N-backtask, as this analysis was based on relative differences in MI ateach region.

Group analysis of movement related and task-related patterns inMI

An analysis of potential residual effects of movement on MIvalues was carried out at the group level for the resting state data.

Fig. 1. Scatterplots showing relationships between MI and regional volume. Left panel: MI (y-axis) versus regional volume (x-axis) based on the originaltemplate image. Right panel: MI estimated for the eroded image (y-axis) versus regional volume in the original image (x-axis). As expected, the strongassociation with volume is clearly reduced when MI is calculated using the eroded template.

282 R. Salvador et al. / NeuroImage 39 (2008) 279–289

Average absolute values, of the first order differences intranslations in the three x, y, and z-directions, from all individualswere used as predictors in a linear model

E½logðMMIÞ� ¼ b0 þ b1logðmov:xÞ þ b2logðmov: yÞþ b3logðmov: zÞ ð9Þ

Logarithmic transformations were chosen as they lead to anormal distribution of the MI model residuals, they achievedlinearity between variables, and they corrected the skeweddistribution of movement values.

The comparison of MI patterns between the resting conditionand the N-back task was performed with an extension of Eq. (9)including the factor condition (task) and the effect of individual(ind) as a block

E½logðMMIÞ� ¼ b0 þ b1task þ b2indþ b3logðmov: xÞþ b4logðmov: yÞ þ b5logðmov: zÞ: ð10Þ

The significance of the effect of interest (task) was assessed withstandard F tests. A Bonferroni threshold with α=0.05/90=0.00056and a less restrictive familywise threshold with α=0.005 (oneexpected false positive every 200 tests under the null hypothesis)were used to account for the multiple comparisons. The R 2.2.1package (R Development Core Team, 2005) was used for allstatistical calculations including the frequency analyses at theindividual level.

Results

Resting state functional connectivity

Fig. 2 shows brain maps of mutual information (MI), averagedover all individuals, at three different frequency bands (highfrequencies [0.17–0.25 Hz], middle frequencies [0.08–0.17 Hz]and low frequencies [b0.08 Hz]). This figure is complemented byTable 2, where further information on the estimates and theirvariability for the set of regions with highest levels of MI is given.

As shown by the wide quantile ranges listed in Table 2,individual levels of MI are quite variable. However, averagedvalues of MI show distinctive patterns of regional connectivity

when analysed at the different frequency bands (Fig. 2), indicatinghighest levels of functional connectivity in the low frequencyinterval (b0.08 Hz).

At high and middle frequencies, the greatest coherence was inthe primary auditory cortex (Heschl’s gyrus), the insula and theamygdala (all bilaterally). High levels of MI in different parts ofthe frontal lobes were segregated, as well, by frequency. Thus,while coherence in the orbito-frontal regions was more salient athigh frequencies, dorsal prefrontal connectivity was enhanced atlow frequencies. The occipital cortex had substantial levels of MI,at low and high frequencies, especially in primary visual andrelated regions (calcarine and lingual cortices).

A summary of the results from the analysis of residual effects ofmovement on the MI values is given in Fig. 3. In this figure thesignificance of movement on the MI is plotted against the averageMI for each of the 90 regions. In all significant models, coefficientsfor movement were positive (MI always increased with move-ment). Although a residual effect of movement was present inmany brain regions, especially at the high frequencies, there wasno strong linear association between movement and MI in anyfrequency interval; see Fig. 3. While the lack of strong associationsdiscounts the role of movement as a unique or major factor insetting the averaged MI, areas with significant movement effectsinclude some of those listed in Table 2 (see last column of thistable). At high frequencies, many areas comprising most of theorbitofrontal and occipital regions are affected by movement, andin middle frequencies a reduced but similar set of regions is found.Consequently, the high levels of MI observed in these regionsshould be regarded with caution.

Task-related differences in MI

The most salient task-related differences in MI were found inthe low frequency interval, which included the frequencies relatedto blocked presentation of the working memory task; see Table 3and Fig. 4. All areas with significant task-related differences in MIat this frequency range had enhanced functional coherence with theother regions of the brain during the working memory task.Strongest increases in MI were found bilaterally in adjacentstructures of the parietal lobes including the inferior parietal

Fig. 2. Brain maps of MI levels in the resting state for the 90 regions of interest, averaged over all individuals in the sample, at three different frequency bands.Talairach Z coordinates are given for the slices shown. The left side of each image represents the left side of the brain. Mutual information values at lowfrequencies are, in general, higher than those at middle and high frequencies. Although calculations were based on time series extracted by an eroded version ofthe template, the original, non-eroded version, has been used here for presentation purposes.

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lobules, the precunei and some medial parts. Neighbouring regionssuch as both cunei, the superior occipital gyri, and the posteriorcingulate cortices were significant as well, but not all of themreached the restrictive Bonferroni threshold. Other relevant regionssignificant at the more lenient familywise threshold were the leftthalamus, the right supramarignal gyrus, the right middle frontalgyrus, the right supplementary motor area, and the left calcarinecortex.

While less abundant, areas with differential connectivity at themiddle and high frequencies showed both increases and reductionsin MI (see Table 3 and Fig. 4). Reduced MI in the working memorycondition was significant in the anterior cingulate (at the

Bonferroni threshold) and in the left superior temporal pole, theright insula and the left paracentral lobule (at the more lenientfamilywise threshold). At these frequencies, increased coherenceduring task performance was restricted to the middle and superiordorsolateral cortices and the supplementary motor area in the leftfrontal lobe.

Discussion

Here we apply an alternative way of measuring functionalconnectivity in the frequency domain. Under the proposedapproach, the mutual information between a region and the rest

Table 2Sorted list of the 20 brain regions with highest MI levels in the resting condition, at each of the three frequency bands analysed

Region Mean 95% confidence limits 5% quantile 95% quantile Movement

High frequencies (0.17–0.25 Hz) HES.R 0.703 0.601 – 0.806 0.286 1.278 NoHES.L 0.696 0.574 – 0.818 0.238 1.382 NoINS.L 0.676 0.576 – 0.775 0.260 1.175 NoAMYG.R 0.664 0.516 – 0.813 0.257 1.223 NoAMYG.L 0.638 0.529 – 0.748 0.243 1.137 NoINS.R 0.625 0.526 – 0.725 0.286 1.133 NoSPG.L 0.600 0.444 – 0.756 0.228 1.553 YesORBsup.L 0.585 0.466 – 0.704 0.207 1.381 YesORBsup.R 0.582 0.442 – 0.721 0.190 1.253 YesITG.L 0.570 0.430 – 0.711 0.204 1.345 YesORBmid.L 0.562 0.467 – 0.657 0.220 1.104 NoOLF.L 0.560 0.470 – 0.649 0.207 1.061 NoORBmid.R 0.546 0.447 – 0.645 0.220 1.103 YesORBsupmed.L 0.538 0.443 – 0.633 0.227 1.040 YesORBsupmed.R 0.536 0.435 – 0.638 0.230 1.093 YesOLF.R 0.528 0.429 – 0.628 0.232 1.153 NoSOG.L 0.524 0.384 – 0.664 0.208 1.164 YesMFG.L 0.517 0.409 – 0.624 0.209 1.162 YesCAL.L 0.512 0.378 – 0.646 0.247 1.006 YesLING.R 0.511 0.407 – 0.614 0.217 1.240 Yes

Middle frequencies (0.08–0.17 Hz) HES.R 0.739 0.561 – 0.917 0.213 1.926 NoHES.L 0.681 0.538 – 0.825 0.218 1.536 NoINS.R 0.630 0.500 – 0.759 0.202 1.283 NoINS.L 0.627 0.520 – 0.734 0.193 1.155 NoAMYG.R 0.566 0.445 – 0.687 0.192 1.339 NoAMYG.L 0.563 0.436 – 0.691 0.171 1.293 NoLING.R 0.540 0.465 – 0.615 0.253 0.978 NoOLF.L 0.526 0.423 – 0.629 0.220 1.202 NoPHG.L 0.519 0.418 – 0.620 0.231 1.152 NoLING.L 0.518 0.423 – 0.613 0.228 1.144 YesCAL.L 0.516 0.412 – 0.620 0.230 0.908 YesSPG.L 0.513 0.393 – 0.633 0.159 1.193 YesDCG.L 0.508 0.409 – 0.608 0.163 0.915 NoPHG.R 0.504 0.385 – 0.622 0.163 1.184 NoTPOsup.R 0.499 0.414 – 0.585 0.220 0.845 YesOLF.R 0.489 0.388 – 0.591 0.176 1.122 NoCAL.R 0.487 0.380 – 0.595 0.156 0.966 YesSOG.L 0.481 0.388 – 0.575 0.194 0.974 YesORBsup.L 0.473 0.383 – 0.564 0.185 0.970 NoROL.L 0.473 0.381 – 0.564 0.179 0.881 No

Low frequencies (0.002–0.08 Hz) SFGdor.R 0.996 0.846 – 1.146 0.389 1.758 NoSFGdor.L 0.958 0.816 – 1.101 0.400 1.752 NoMFG.L 0.954 0.806 – 1.102 0.328 1.669 YesPreCG.L 0.926 0.767 – 1.085 0.401 1.766 NoSFGmed.R 0.915 0.783 – 1.046 0.340 1.453 NoCAL.L 0.905 0.789 – 1.021 0.348 1.450 NoLING.R 0.895 0.794 – 0.997 0.466 1.396 NoCAU.R 0.884 0.787 – 0.980 0.343 1.249 NoORBsupmed.R 0.876 0.741 – 1.010 0.360 1.611 NoDCG.L 0.868 0.736 – 0.999 0.290 1.625 NoPoCG.L 0.851 0.716 – 0.987 0.346 1.556 NoTPOsup.R 0.845 0.729 – 0.961 0.370 1.571 NoMFG.R 0.835 0.713 – 0.957 0.271 1.304 NoLING.L 0.834 0.727 – 0.942 0.370 1.362 NoORBsup.R 0.830 0.675 – 0.985 0.353 1.847 NoORBinf.R 0.830 0.682 – 0.978 0.309 1.447 YesORBmid.L 0.828 0.695 – 0.961 0.367 1.554 NoINS.L 0.826 0.698 – 0.955 0.256 1.462 NoSMA.L 0.816 0.678 – 0.954 0.267 1.408 NoORBmid.R 0.813 0.697 – 0.929 0.281 1.462 No

Different columns give the mean MI, its confidence interval based on the standard normal theory, the 5% and 95% quantiles of the sample of individual MIvalues, and the presence of residual movement effects at α=0.05.

284 R. Salvador et al. / NeuroImage 39 (2008) 279–289

Fig. 3. Plots showing the relation between the statistical significance ofmovement effects on the MI and the averaged MI for each of the 90 regions,at the three frequency bands analysed. The residual effect of movement issignificant for a substantial number of regions at high frequencies, but is lessmarked at middle and low frequencies. Vertical lines mark the α=0.05threshold, dashed horizontal lines indicate the upper thresholds used for theselection of areas in Table 2. Data come from the resting state condition.

285R. Salvador et al. / NeuroImage 39 (2008) 279–289

of the brain can be given as a simple monotonous function of themultiple coherence. Looking at the results, it becomes apparentthat, apart from its frequency specificity, the value of the methodcomes from its ability to portray different aspects of functionalconnectivity through simple brain maps.

Rather than being an alternative to the other availabletechniques listed in the introduction, we see the proposed methodas a complementary tool. Thus, while independent componentanalyses or, seed based, correlation maps may help isolatingdistinct brain networks with high levels of covariability (and usinga much finer spatial resolution), our technique allows having adiffuse, but global view of connectivity patterns summarised insingle brain maps. Indeed, as it is exemplified through the resting–Nback comparison, it may even be used to complement the resultsderived from standard fMRI activation studies (but this will requireacquiring an extra resting state sequence with the same acquisitionparameters).

Averaged maps from the resting state sequences showdifferent connectivity patterns for the three frequency bands.Although the origin of these patterns is not known, some possibleexplanations may be given. The high values of MI observedbilaterally, at middle and high frequencies, in several temporaland limbic structures can be associated to their known functions.

The insula is believed to process convergent information fromseveral sensory modalities monitoring the state of the body, and ithas been related to the experience of pain and other basicemotions including disgust, anger and fear (Mesulam, 2000;Adolphs, 2002; Baliki et al., 2006). Some authors have evensuggested that its role in mapping visceral states and emotionscould be the basis for conscious feelings (Damasio, 1999). In thatsense, the high levels of connectivity found here could support anintegrative role for this brain structure. Similarly, the amygdalahas been linked to emotions and to their role in learningmodulation (Phelps and LeDoux, 2005; Sigurdsson et al., 2007),and its enhanced coherence is probably related functionally tothat in the insula.

Prominent MI values in the Heschl’s gyrus may have aparsimonious explanation. Strident noise delivered at highfrequencies by the scanner is likely to have an effect on theprimary auditory cortices, and related structures. Likewise, thesignificant levels of connectivity in the primary visual cortex andadjacent regions may be simply explained by direct visualstimulation. Indeed, both vision and audition are among the mostrelevant and highly organised external sensory modalities inhuman brain (Mesulam, 2000), and our results would suggest anotable role for them in brain organisation. However, the presenceof significant movement effects in occipital areas should betaken into consideration here. While some of the structuresmentioned above are still relevant at low frequencies, the highestlevels of low frequency coherence were found in frontalareas. Again, there is no direct explanation for such connectivitylevels but they could be related to integrative attentional andcognitive processes (see, for instance, recent reviews by Rush-worth et al., 2005; Thompson-Schill et al., 2005; D’Esposito andChen, 2006).

As exemplified by the analysis of task related changes, theproposed method may be applied effectively to assess connectivitydifferences. At low frequencies (which included the frequencies ofblocked periodic presentation of the N-back task) enhancedconnectivity occurred in similar areas to those significant instandard fMRI activation studies of the N-back working memoryparadigm (see meta-analysis by Owen et al., 2005) and in otherstudies exploring effects of working memory task difficulty onactivation or connectivity of regions (Honey et al., 2000, 2002).These included the inferior parietal lobule and related posteriorparietal areas, the supplementary motor area, some prefrontalregions, and the thalamus, thus showing a strong relation betweenpatterns of functional connectivity and task effects.

At middle and high frequencies, connectivity enhancement canbe related as well to areas known to be activated by the task, butwe also find task related reductions. Interestingly, these areparticularly prominent in the anterior cingulate. This region is partof what is known as “default mode network”; a cluster of regionsthat reduce their co-activity when the brain becomes engaged inspecific tasks (Raichle et al., 2001; Greicius et al., 2003). Amongother functions, it has been suggested that the anterior cingulatecould be involved in monitoring one’s mental state; an activity thatwould cease as the individual becomes active (Gusnard et al.,2001).

While potentially relevant, the substantive interpretationsprovided in this discussion should be taken with some caution.Several confounding factors, of physiological or methodologicalorigin, might have influenced the results, and should be furtherexplored. As made evident by Fig. 1, using regions of interest is

Table 3Regions showing significant differences in MI between the resting state andthe N-back task for the three frequency bands analysed

Region Coefficient p-value

High frequencies (0.17–0.25 Hz) ACG.R −0.346 7.36e−06 ⁎⁎ACG.L −0.333 0.000147 ⁎⁎

MFG.L 0.070 0.002965 ⁎

TPOsup.L −0.256 0.003010 ⁎

INS.R −0.355 0.003546 ⁎

SFGdor.L 0.110 0.004439 ⁎

PCL.L −0.095 0.004999 ⁎

Middle frequencies (0.08–0.17 Hz) INS.R −0.383 0.001863 ⁎

SMA.L 0.217 0.001920 ⁎

Low frequencies (0.002–0.08 Hz) IPL.L 0.436 1.30e−08 ⁎⁎PCUN.L 0.398 2.09e−06 ⁎⁎IPL.R 0.448 1.06e−05 ⁎⁎PCUN.R 0.437 1.39e−05 ⁎⁎SPG.L 0.298 0.00023 ⁎⁎

SPG.R 0.338 0.000313 ⁎⁎

SOG.R 0.285 0.000335 ⁎⁎

CUN.L 0.283 0.000458 ⁎⁎

PCG.L 0.373 0.000460 ⁎⁎

THA.L 0.256 0.000644 ⁎

SMG.R 0.317 0.000857 ⁎

PCG.R 0.261 0.001193 ⁎

CUN.R 0.255 0.001312 ⁎

SOG.L 0.250 0.002202 ⁎

CAL.L 0.222 0.003781 ⁎

MFG.R 0.214 0.004295 ⁎

SMA.R 0.185 0.004306 ⁎

The estimated coefficients from the model and their statistical significanceare given as well. Negative coefficients indicate reduced connectivity for thetask.

* Significant at the familywise level (α=0.005).** Significant at the Bonferroni threshold (α=0.00056).

Fig. 4. Brain maps showing the regions with significant differences inMI between the resting state and the N-back task. Differences in highand middle frequencies are portrayed in the same images. Decreasedconnectivity in the task is shown in blue (Bonferroni level, α=0.00056)and violet (familywise level, α=0.005). Increased connectivity with thetask is depicted in white (Bonferroni level) and light brown (familywiselevel). The left side of each image represents the left side of the brain.Talairach Z coordinates are given for the slices shown. Table 3 lists theregions portrayed here.

286 R. Salvador et al. / NeuroImage 39 (2008) 279–289

not free of unexpected consequences (although the methodsproposed here can be potentially applied, as well, to voxel basedanalyses). We have also shown that residual movement effects mayinfluence mutual information values, and while they do not seem tobe a leading factor in shaping connectivity levels, they weresignificant in areas with enhanced connectivity at high frequencies(e.g., orbitofrontal and occipital areas). In the comparison betweenresting and N-back conditions, though, we have included averagedmovement parameters as covariates in order to account for theseeffects, leading to some reasonable results.

Finally, the method proposed may benefit from futureimprovements. As it is now, the probabilistic model does notconsider non-linearities, and it is restricted to joint Gaussiandensity functions. Further extensions of the, frequency based,mutual information measures given here to more general settingsmay bring more insight to new brain connectivity studies. Inaddition, other technical aspects, such as the usage of multitapertechniques in the non-parametric estimation of the spectral densityfunctions (a single Bartlett window was applied here to reduceleakage), or a more complex modelling of movement relatedeffects may be useful, as well.

Acknowledgments

This study has been supported by four grants from the SpanishMinistry of Health. Two of them (CP04/00322 and PI051874)

given to Raymond Salvador and Ángel Martínez, another(PI052693) provided to Edith Pomarol-Clotet, and FI05/00322given to Jesús Gomar. It has been partially funded, as well, by aMarie Curie European Reintegration Grant MERG-CT-2004-511069. Additional funding has been received, as well, from theSpanish Ministry of Health, Instituto de Salud Carlos III, Red deEnfermedades Mentales (REM-TAP Network). Ángel Martínez hascontributed significantly to this work and, in consequence, it willbe included as part of his doctoral thesis. Software developmentwas supported by a Human Brain Project grant from the National

287R. Salvador et al. / NeuroImage 39 (2008) 279–289

Institute of Biomedical Imaging and Bioengineering and theNational Institute of Mental Health.

Appendix A. The mutual information can be given as a simplemonotonic function of the multiple coherence

If we apply the discrete Fourier transform (DFT) to a finiterealisation of Eq. (1) (main text), we have a new set of values

Y ðxkÞ ¼ fY1ðxkÞ; Y2ðxkÞ; N ; YpðxkÞg ðA:1Þ

at each ωk (k: 1, …n) Fourier frequency. The mutual informationbetween any pair of subsets a, b of Y at frequency ωk is (Salvador etal., 2007, Eq. (10))

MIa;b xkð Þ ¼ � 12log jfXa;Xb xkð Þj=jfXa xkð ÞOfXb xkð Þj� � ðA:2Þ

where fXa(ωk), fXb

(ωk) and fXaXb(ωk) are the Hermitian positive

definite, (cross-)spectral density matrices of the original processesXa(t), Xb(t) and {Xa(t), Xb(t)} Symbol ∣∣ denotes the determinant.

To find a simplified version for the specific case where acontains a single time series and b contains the remaining p−1time series (Xb={X1,…, Xa−1, Xa+1,…, Xp}) we will work with theratio of determinants of Eq. (A.2).

jfXaXbðxkÞj=jfXaðxkÞOfXbðxkÞj ðA:3Þ

For simplicity, we may sort regions taking a in the first place. Inthat case fXb

(ωk) is the p−1 order matrix obtained by removing row1 and column 1 of fXaXb

(ωk), and fXa(ωk) is the real, positive scalar

in position (1,1) of fXaXb(ωk).

Now, for convenience, we set some definitions. We define D1j

as the determinant of matrix fXaXb(ωk) without row 1 and column j,

and Di,j as the determinant of fXaXb(ωk) without rows 1 and i and

columns 1 and j. V is the column vector containing the values 2,…,p of the first row of fXaXb

(ωk), and V̄ is the column vectorcontaining the values 2, …, p of the first column of fXaXb

(ωk)(which are the conjugate values of V). With all these definitions wecan, at this point, expand the numerator of Eq. (A.3). First, weapply twice the Laplace expansion over a determinant (seedefinition at the end of appendix)

jfXaXb ðxkÞj ¼ fXa ðxkÞjfXb ðxkÞj �Xpj¼2

ð�1Þj�2V ½j� 1�D1;j

¼ fXa ðxkÞjfXb ðxkÞj�Xpj¼2

V ½ j� 1�Xpi¼2

ð�1Þiþj�4Di; j V̄ ½i� 1�

¼ fXa ðxkÞjfXb ðxkÞj � VT

Xpi¼2

ð�1Þi�2Di;2 V̄ ½i� 1�

v

Xpi¼2

ð�1Þiþp�4Di;p V̄ ½i� 1�

0BBBBBBBBB@

1CCCCCCCCCA

¼ fXa ðxkÞjfXb ðxkÞj�VT

D22 �D32 N ð�1Þp�2Dp2

�D23 D33 N ð�1Þp�1Dp3

v v O vð�1Þp�2D2p ð�1Þp�1D3p O ð�1Þ2p�4Dpp

0BB@

1CCA V̄

¼ fXa ðxkÞjfXb ðxkÞj � V tadjðfXb ðxkÞÞt V̄ ðA:4ÞðA.4Þ

with adj being the adjugate matrix (see definition below). Next ifwe take the definition of inverse through the adjugate (see below),we obtain the following equality for Eq. (A.3)

jfXaXb ðxkÞjjfXaðxkÞOfXb ðxkÞj ¼ 1� V tðfXbðxkÞÞ�1 V̄

fXaðxkÞ ¼ 1�mCohXa;Xb xkð Þ;

ðA:5Þ

where mCohXaXb(ωk) is the multiple coherence (Brillinger, 1981).

Therefore, Eq. (A.2) finally becomes

MIa;b xkð Þ ¼ � 12log 1�mCohXa;Xb xkð Þ� �

; ðA:6Þ

which is the mutual information between a (the mean time series ofa single index region), and b (the time series of the rest of the p−1regions in the brain) at frequency ωk.

The Laplace expansion of a determinant: If we define the i,jminor Mij of a n×n matrix B as the (n−1)×(n−1) matrix thatresults from deleting the ith row and the jth column of B, and the i,jcofactor of B as Cij=(−1)

i+j|Mij|, then, the determinant of B can befactorised, by either fixing any of the columns of B (e.g., j),through

jBj ¼Xni¼1

bijCij; ðA:7Þ

or by fixing any row (e.g., i), through

jBj ¼Xnj¼1

bijCij; ðA:8Þ

where bij is the scalar at position i,j of B.Inversion through the adjugate: The adjugate of a n×n matrix B

(denoted here as adj(B)) is a matrix of the same order with elementsin each i,j position given by

adjðBÞij ¼ Cji; ðA:9ÞCij being the cofactor defined above. Then, if B is invertible, it canbe shown that its inverse is related to adj(B) by a simple formula

B�1 ¼ jBj�1adjðBÞ ðA:10Þ

Appendix B. Further aspects on the diagonalisation of fXb

If we consider the multivariate stochastic process Xb for t∈Z,under the assumption of summable covariances, then its spectraldensity will be a continuous function in λ∈ (0, 2π). If L(λ) is theset of eigenvectors from the diagonalisation of fXb

(λ), we can applythese eigenvectors to generate a filter in the time domain(Brillinger, 1981)

l uð Þ ¼ 1

2p

Z 2p

0LT kð Þexp ikuf gdk: ðB:1Þ

This filter can be used to create the principal component series ofXb

UðtÞ ¼Xlu¼�l

lðt � uÞXbðuÞ; ðB:2Þ

288 R. Salvador et al. / NeuroImage 39 (2008) 279–289

which is a set of time series with diagonal spectral densitymatrices, and with maximised variances given by the eigenvalues(Brillinger, 1981).

Now, for a specific Fourier frequency ωk, if we denote Δs(ωk)and Ls(ωk) as the matrices of eigenvalues and related eigenvectorsselected through Eq. (7) (main text), replacing fXb

(ωk) by Δs(ωk) inthe MI calculations will be equivalent to calculating the mutualinformation between Xa(t) (the brain region of interest) and the sprincipal components. In particular, the multiple coherencebetween Xa(t) and Us(t) (the stochastic process containing the sprincipal components) will be given by

mCohXa ;Us xkð Þ ¼ W ðxkÞtðDsðxkÞÞ�1PPW ðxkÞfXaðxkÞ ; ðB:3Þ

where W(ωk) is an s-length column vector containing the cross-spectral densities between Xa(t) and each component of Us(t).

A computationally simple formula can be derived for W(ωk)through few steps. Starting with the cross-covariance functionbetween Xa and Us, we can use Eq. (B.2) to expand it

covfXaðt þ vÞ;UsðtÞgðvÞ ¼

covfXaðt þ vÞ;Xlu¼�l

lðt � uÞXbðuÞgðvÞ; vaZ ðB:4Þ

covfXaðt þ vÞ;UsðtÞgðvÞ ¼Xlu¼�l

lðt � uÞ½covfXaðt þ vÞ;XbðuÞgðvÞ�: ðB:5Þ

Now, by taking Fourier transforms on both sides, we can rewriteEq. (B.5) as

f ðXa;UsÞðxkÞ ¼ Ls*f ðXa;XbÞðxkÞ; ðB:6Þwhich, using the notation of Eq. (3) (main text), is simply

W ðxkÞ ¼ Ls*V ðxkÞ: ðB:7Þ

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