Measuring Phase Synchrony in Brain Signals

Jean-Philippe Lachaux, Eugenio Rodriguez, Jacques Martinerie,and Francisco J. Varela*

Laboratoire de Neurosciences Cognitives et Imagerie Cerebrale, CNRS UPR 640Hopital de La Salpetriere, Paris, France

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Abstract: This article presents, for the first time, a practical method for the direct quantification offrequency-specific synchronization (i.e., transient phase-locking) between two neuroelectric signals. Themotivation for its development is to be able to examine the role of neural synchronies as a putativemechanism for long-range neural integration during cognitive tasks. The method, called phase-lockingstatistics (PLS), measures the significance of the phase covariance between two signals with a reasonabletime-resolution (,100 ms). Unlike the more traditional method of spectral coherence, PLS separates thephase and amplitude components and can be directly interpreted in the framework of neural integration.To validate synchrony values against background fluctuations, PLS uses surrogate data and thus makes noa priori assumptions on the nature of the experimental data. We also apply PLS to investigate intracorticalrecordings from an epileptic patient performing a visual discrimination task. We find large-scalesynchronies in the gamma band (45 Hz), e.g., between hippocampus and frontal gyrus, and local synchronies,within a limbic region, a few cm apart. We argue that whereas long-scale effects do reflect cognitive processing,short-scale synchronies are likely to be due to volume conduction. We discuss ways to separate such conductioneffects from true signal synchrony. Hum Brain Mapping 8:194–208, 1999. r 1999Wiley-Liss,Inc.

Key words: neural synchrony; phase-locking; coherence; EEG; EcoG; epilepsy; gamma-band; deblurring

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INTRODUCTION

Cognitive acts require the integration of numerousfunctional areas widely distributed over the brain andin constant interaction with each other [Friston et al.,1997; Tonini and Edelman, 1998]. It has become a topicof much interest to explore the possibility that suchlarge-scale integration could be mediated by neuronalgroups that oscillate in the gamma range (30–80 Hz,also referred to as 40 Hz) that enter into precisephase-locking over a limited period of time. (Hereafter,

we refer to these phenomena simply as ‘‘synchrony’’ or‘‘phase synchrony’’.) Whence the importance for areliable and robust method for directly measuringsuch phase synchrony in this frequency band forexperimentally recorded biological signals, which arenot spikes, but local field potentials (LFP) of variousdegrees of spatial resolution. The purpose of this report isto introduce, for the first time, an effective method toestimate the instantaneous phase relationship betweentwo neuroelectric or biomagnetic signals and to apply it tointracortically recorded signals in humans. We also intro-duce the required statistical means to test the validity ofthe measured synchrony against the background fluctua-tions in a given experimental situation.

The role of synchronization of neuronal discharges,although not a new idea, has been greatly highlightedby results from microelectrodes in animals [see, e.g.,

Contract grant sponsor: DRET; Contract grant sponsor: HumanScience Frontier.*Correspondence to: Dr. Francisco J. Varela, LENA-CNRS UPR 640,Hopital de La Salpetriere, 47 Bd de l’Hopital, 75651 Paris cedex 13,France. E-mail: fv@ccr.jussieu.frReceived for publication 13 March 1998; accepted 17 May 1999

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Singer and Gray, 1995]. In fact, two scales of phasesynchrony can be distinguished: most animal studiesbased on microelectrode recordings have dealt withshort-range synchronies [e.g., Gray et al., 1989; Neuen-schwander et al., 1996], or between adjacent areas corre-sponding to a single sensory modality [e.g., Konig etal., 1995]. These local synchronies have been inter-preted most commonly as subserving ‘‘perceptual bind-ing.’’ More recently, evidence has also been found forlong-range synchronizations between widely separatedbrain regions [Roelfsema et al., 1997]. This is in agreementwith the more general notion that phase synchronyshould subserve not just binding of sensory attributes,but the overall integration of all dimensions of a cognitiveact, including associative memory, emotional tone, andmotor planning [Damasio, 1990; Varela, 1995].

These multiunit studies in animals have been comple-mented by studies at coarser levels of resolution inhumans and animals. In fact, gamma band responsescan be recorded during visual discrimination protocolson the human scalp [Tallon-Baudry et al., 1997] and insubdural electrocorticograms [Le van Quyen et al.,1997; Lachaux et al., 1999a]. One can distinguish anearly (100 ms) response phase-locked to the stimulusor evoked response, and a later (200 ms) induced re-sponse nonphase-locked to the stimulus. Both theseresponses necessarily imply a degree of local phase-locking, since otherwise no signal would reach thesurface of cortex or scalp with enough amplitude.They would annihilate due to the summation ofphases distributed very broadly. Thus there is someevidence to suggest that synchronization mechanismsas those found in single-unit studies in animals alsooccur at large scales recorded with macroelectrodes;they are long-range synchronies.

However, the quantification of phase synchronybetween macroelectrodes (EEG/MEG or intracorticalrecordings) requires methods that are entirely differentthat the cross-correlograms between spike dischargesthat suffice for microelectrode studies. This is the mainpurpose of the present study. But in this context, it isimportant to distinguish very sharply between syn-chrony (as defined above) and the classical measuresof spectral covariance or coherence that have beenextensively used elsewhere [Bullock and McClune,1989; Bressler et al., 1993; Menon et al., 1996]. Unfortu-nately, coherence is a measure that does not separatethe effects of amplitude and phase in the interrelationsbetween to signals. This point is discussed in detaillater, but the novelty of our results is to a large extentsimply to arrive at a measure where the phase compo-nent is obtained separately from the amplitude compo-nent for a given frequency. Only then can one proceed

to test the synchrony hypothesis for brain integration.In a separate work, we have successfully applied thepresent method for the detection of gamma synchro-nies over the scalp during visual perception [Rod-riguez et al., 1999]. Here, we present the methods inextenso on the basis of simulations and intracorticalrecordings as an ideal test case.

This report is divided into three main sections. Thefirst introduces a new technique for phase-lockingdetection: the phase-locking statistics (PLS). The sec-ond section applies PLS to LFP data from humanintracortical recordings. In the third section, we exam-ine the problem of volume conduction and the choiceof a reference electrode, the two main difficultiesassociated with the interpretation of synchrony results.

QUANTIFICATION OF PHASE-LOCKING

Phase-locking statistics

Here, we introduce our method of detecting syn-chrony in a precise frequency range between tworecording sites. This method uses responses to arepeated stimulus and looks for latencies at which thephase difference between the signals varies little acrosstrials (phase-locking). Given two series of signals x andy and a frequency of interest f, the procedure computesfor each latency a measure of phase-locking betweenthe components of x and y at frequency f (we call thismeasure phase-locking value, or PLV). This needs theextraction of the instantaneous phase of every signal atthe target frequency. The procedure follows three steps(Fig. 1).

Step 1. We band-pass filter (finite impulse responseof length 5 300 ms) each signal between (f 6 2 Hz).

Step 2. We compute its convolution with a complexGabor wavelet centered at frequency f:

G(t, f ) 5 exp 12 t2

2st22 exp 5 j2pft6.

Following Grossman [1989], we chose st 5 7/f. Forinstance, st 5 140 ms for f 5 50 Hz. The phase of thisconvolution f(t, n) is extracted for all time-bins t, trialn [1, . . . , N], and for each of the pair of electrodes.

The phase locking value (PLV) is then defined attime t as the average value:

PLVt 51

N 0on51

N

exp ( ju(t, n))0where u(t, n) is the phase difference f1(t, n) 2 f2(t, n).PLV measures the intertrial variability of this phase

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difference at t: If the phase difference varies little acrossthe trials, PLV is close to 1; it is close to zero otherwise(Fig. 2). This procedure can be repeated for severalfrequencies in order to study a broader frequencyrange.

Step 3. The third step, or ‘‘test,’’ is to build astatistical test based on surrogate data [Theiler et al.,1992, Muller-Gerking et al., 1996] to differentiate signifi-cant PLV against background fluctuations. Formerly,we test our samples of phase differences for random-ness against a unimodal distribution with an unspeci-fied mean direction. The Raleigh test can be used[Fisher, 1993] to test the H0 hypothesis that our samples

are drawn from a uniform distribution. However, inour case, the sampling distribution of a statistic is notknown and we cannot assume uniformity. This can beseen in a simple example. Let us assume two indepen-dent neural populations that start to oscillate at 40 Hzin response to stimulation after a random delay of50–55 ms. The phase of these oscillations at 55 ms willvary from trial to trial from 0–90° (5 ms is about aquarter of a cycle at 40 Hz). Therefore, the phasedifference between these two neural groups will reachvalues only between 290° and 90°, which is notuniform. Yet, a Raleigh test would conclude that thephase differences are not uniformly distributed and

Figure 1.Evaluation of the instantaneous phase of a signal for a frequency f.As a first step, the raw signal s (t) is band-pass filtered to generatef(t) (step 1); the convolution of f(t) with a Morlet wavelet centeredat frequency F provides the envelope a(t) and the instantaneousphase f(t) (step 2). f(t) can also be inferred (bottom left) from acomparison between the latencies of f’s maxima (black points) and

reference time markers (black ticks separated by a constantinterval: 1/f). Both methods give the same results. To test themethods, we generated a signal x(t) 5 sin(2pFt 1 u(t)), withabrupt phase variations, and computed f(t), an estimator of u(t).On the bottom right, inset shows cos(u(t)) (rectangular function)together with cos(f(t)) (smooth approximation).

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would detect a phase-locking between the groups.When the sampling distribution of a statistic is un-known, one must rely on recent techniques of random-ization, or bootstrap [Fisher, 1993]. Our statistical testis based on randomization and is adapted to ourparticular set of data.

The main advantage of this approach is that it doesnot require any a priori hypothesis on the signals. Wetest the H0 hypothesis that the two series of phasevalues f1(n) and f2(n) are independent. For thispurpose, we generate 200 new series of variables,which have the same characteristics as the originalsignal coming from electrode 2, except that we builtthem to be independent of the signals coming fromelectrode 1. These series are created by shuffling thetrials within the measures of electrode 2 to make new

series y8(n) 5 y(shuffle (n)), where (y(i) is the signalrecorded at electrode 2 during trial i (Fig. 2).

For each surrogate series y8, we measure the maxi-mum between x and y8 in time. These 200 values areused to estimate the significance of PLV between theoriginal signals x and y. The proportion of surrogatevalues higher than the original PLV (between x and y)for a time t is called phase-locking statistics (PLS). Itmeasures the probability of having false positives for agiven level of significance. In this study, we used acriterion of 5% (PLS , 5%) to characterize significantsynchrony, but this is, of course, a function of therequired rigor of significance in the context of thesignals being studied. Our method is related to anapproach proposed by Friston et al. [1997] to quantifyMEG data. In fact, they propose to estimate the

Figure 2.Estimation of phase-locking value. Left: Our synchrony index isdirectly related to the intertrial variability of the phase differencesbetween two electrodes (see description of the method fordetails). By averaging these phase differences across the trials, we

obtain a complex value u (for each latency t), which amplitude (abs(u)) is the phase-locking value. Right: Surrogate data are con-structed by shuffling the trials of one of the electrodes (see text fordetails).

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correlation coefficient between two signals at a precisefrequency, which implies a degree of phase-locking.Also, their statistical analysis was markedly differentfrom ours.

Some comments are in order. In step 1, one maywonder if the filtering step is not redundant with thewavelet convolution and if it does not introduceartifacts. It seems that f(t, n) could be obtained directlyby a convolution of the wavelet with the unfilteredsignal (call this method A), instead of using a filteringstep prior to the convolution (method B). In fact, bothmethods were tested and led to slightly differentresults. We could make the following verifications ofmethod B: (1) when transient 40 Hz oscillations easilycould be seen in raw (unfiltered) data, we checked thatthe latencies of their peaks were identical with those ofthe filtered signals, and (2) when the filtered signal hadclear oscillations, its exact phase could be read straight-forwardly from the latencies of the maxima of theseoscillations (this direct lag estimation is slow, buthighly reliable). By comparing this phase with ourevaluation, we could check the exactness of method B.Since method A occasionally gave different results, werelied on method B.

In step 3, the statistical method we use should detectany significant phase synchrony between two elec-trodes, except in one important case: when the phasevalues f1(n) and f2(n) remains constant across thetrials. In that situation, shuffling the trials within themeasures of electrode 2 does not change the phase-locking value, whereas signals are actually synchronous.These false negatives easily can be detected since they areassociated with high PLV. Also, they easily can be detectedusing simpler techniques that estimate the intertrial vari-ability of the phase of each electrode as recently proposed[Tallon-Baudry et al., 1996; Lachaux et al., 1999a].

Why not use coherence?

Most studies that have attempted directly to studythe putative importance of synchrony so far haveemployed a measure traditionally called frequencycoherence (or more specifically, magnitude squaredcoherence, MSC) [Clifford Carter, 1987]. Thus theimprovements provided by PLS are best seen in con-trast to MSC. The most salient differences are twofold.

Coherence can be applied only to stationary signals.Coherence is a measure of the linear co-variancebetween two spectra. In particular, for each frequencyf, the MSC is defined for two zero-mean stochasticprocesses by the squared modulus of their cross-spectral density at frequency f, normalized by their

respective auto-spectra. These spectra can be estimatedfrom finite sets of data by: (1) subdividing the wholedata sets into segments, (2) computing approximationsof the spectra of each segment using a DFT (DiscreteFourier Transform), and (3) averaging these subspectraacross all the segments. The quality of the estimationdepends on the number and size of the segments[Clifford-Carter, 1987]. Segments are usually succes-sive time intervals, defined by a window sliding intime over the whole recording. In that case, a singlemeasure of coherence typically needs several secondsof data, which limits the temporal resolution of themethod. However, for protocols with repeated stimula-tions, coherence can be computed with a much betterresolution because the window that defines the seg-ments can be slided across trials instead of being slidedin time (event-related coherence). Yet, both methodsrequire that each segment of data correspond to thesame process with the same spectral properties. Sincethis assumption of stationarity (in time or across trials)can rarely be validated, we prefer a measure ofphase-locking that does not require stationarity.

Coherence does not specifically quantify phase-relationships. Coherence also increases with ampli-tude covariance, and the relative importance of ampli-tude and phase covariance in the coherence value isnot clear. Since phase-locking is sufficient to concludethat two brain regions interact, it is important todevelop alternative measures for the sole detection ofphase covariance. In fact, there is no clear interpretation forthe changes in coherence between two neural signals,beyond an obvious indication of interdependency.

In addition, the classic statistical analysis of coher-ence is not based on a comparison with trial-shiftedsurrogate data, but rather on a comparison withindependent white noise signals [Clifford-Carter, 1987].Therefore, coherence statistics test the H0 hypothesisthat pairs of neural signals behave like independentwhite-noise signals. Since neural signals are not white-noise signals, this H0 hypothesis might be too strongand might be rejected too easily.

Simulations

We first tested PLS with two series of signalsphase-locked during two well-defined intervals inorder to validate its temporal resolution. These signalswere derived from two independent series of 50signals obtained from intracortical recordings of anepileptic patient (see next section) (s1(t, n 5 1 . . . 50)and s2(t, n 5 1 . . . 50), totaling 50 responses to twodifferent stimuli measured in two different brain loca-

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tions. We applied a finite impulse response band-passfilter (41–45 Hz) to s1 and s2 to generate two new seriess1

v and s2v and computed a third series c1

v 5 Bell · s1v 1

(1 2 Bell ·)· s2v. (Bell was a function always equal to

zero, except during two periods of 75 ms and 200 mswhere it was equal to 1 (Fig. 3.)

Signal s2 was then modified into c2 5 s2 2 s2v 1 c1

v, asignal synchronous with s1 (41–45 Hz) during twoshort intervals. We computed PLS between s1 and c2

(target frequency 5 43 Hz) to test whether this methodcould detect and separate these two episodes. Asshown in Figure 3a, the temporal resolution of PLS wasprecise enough to do so. It can, therefore, be used todetect short episodes of synchrony (in this case, threeoscillation cycles at 40 Hz, 75 msec) with a relativelysmall number of trials (50).

We devised a second simulation intended to illus-trate the inability MSC to separate phase and ampli-tude covariance. We used two series of signals phase-locked during two well-defined intervals, but withindependent amplitudes. We expected that a lack ofamplitude covariance would induce MSC ignore syn-chrony, but not PLS.

This simulation was very similar to the previousone: using two independent series (50 trials) of data s81and s82, we repeated the above procedure to generatetwo series s81v and s82v and a third series c81v 5 Bell 8 ·s81v 1 (1 2 Bell8 ·) · s82v. (In this case, Bell 8 had twononzero periods lasting 200 ms each.) s2 was this timemodified into c2 5 s2 2 s2

v 1 random · c1v, where random

was a value randomly chosen between 0 and 1 anddifferent for each trial. This number was introduced tomake the amplitudes of the signals independent, sothat synchronous periods specified by Bell 8 corre-sponded solely to phase-coupling and not to ampli-tude covariance. The results of this simulation areshown in Figure 3b; as expected, MSC was reduced to thelevel of noise during the first episode of synchrony. Incontrast, PLS detects both episodes of synchrony correctly.

Interestingly, in our simulations MSC did not givefalse positives when there was amplitude covariance,but not phase locking. Thus our tentative conclusion sofar is that MSC is fooled specially by synchronies thatare not accompanied by high amplitude covariances.

MEASURING SYNCHRONIES IN HUMANINTRACORTICAL RECORDINGS

Subjects and recordings

Here, we examine the results of applying PLS tosubdural recordings of an epileptic patient performinga visual discrimination task. Details of the experimen-tal conditions can be found elsewhere [Lachaux et al.,1999a]. This right medial temporal epilepsy patient(subject PI) required intracranial EEG monitoring toconfirm the exact site of the epileptogenic zone beforesurgical treatment. Four electrodes plots, each witheight recording sites, were inserted along occipito-

Figure 3.(a). Estimation of PLV’s temporal resolution. Two periods of highsynchrony are simulated. Synchrony periods are characterized bynonzero values of the bell function (see text for details). Values areplotted as a function of time; values above the horizontal line aresignificant (PLS , 0.05). Synchrony can be detected in segments asshort as 75 ms. (b). Estimation of PLV’s detection resolution. As inFigure 3, two periods of high synchrony are simulated (nonzerovalues of the bell function (see text for details). PLV and MSCvalues are plotted as a function of time; PLV above the horizontalline are significant (PLS , 0.05). During periods of low amplitudeco-variance, MSC gives a false negative.

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limbic and fronto-cingulate trajectories. The positionsof the electrodes obtained from IRM are given in TableI. The task was a classic visual oddball discrimination:a panel of 80 red-light diodes (targets) was lightedrandomly interleaved with 320 green diodes (nontar-gets). The stimuli were turned on for 40 ms, with aninterstimulus interval varying randomly between 800ms and 1,200 ms. The patient had to press a button inresponse to target stimuli only. Electrical data wererecorded relative to an average reference.

In a previous study of gamma emission from compa-rable human data using the same protocol, we hadfound that a specific response around 45 Hz is trig-gered specifically by the stimuli [Lachaux et al., 1999a].Accordingly, we focused our study of PLS only aroundthis frequency.

Results

The matrices of PLS values for all pairs of electrodesand both target and nontarget stimuli are shown inFigure 4 (see legend for conventions). As for thestimulation effect, significant synchronies are slightlymore common between left hippocampus and frontal-cingulate regions. On the whole, however, the syn-chrony patterns are roughly comparable for bothconditions, although slightly increased for the nontar-get condition. It is important to make it clear that ourpurpose here was not to study in detail the neuro-significance of phase-locking related to the visual task

performed by the subject. This would entail a carefulanalysis of the time courses of synchrony and of thedifferences between target vs. nontarget, followingRodriguez and colleagues [1999]. For subject PI, such astudy is made difficult by the clinical constraints forelectrode placement, preventing a choice of positionsthat would be more consonant given the chosencognitive task. Thus we find only a very limitedchange in synchrony over time and between condi-tions. Accordingly, our aim here is more modest: toapply our method to real signals (subdural LFP poten-tials) in order to verify the existence of stimulus-triggered gamma synchronies between electrodes fromsignals of this intermediate level of resolution (ascompared to scalp recordings) and to distinguishbetween short-range and long-range synchrony.

Local synchronies. If we first focus on the short rangefor synchronies found within the same region ofelectrodes (limbic or frontal, left or right), it is easilyseen that the intensity of phase-locking (quantified byPLV) decreases steadily with interelectrode separation.For instance, in the right limbic recordings of this patient(which correspond to a necrotic hippocampus where theepileptic focus was diagnosed), synchrony extends up tointerelectrode distances of up 8 cm (Fig. 5).

However, these observations are an artifact due tothe fact that neighboring electrodes simply will recordthe same field potential due to volume conduction andare thus trivially synchronous. To distinguish betweenvolume conduction and true synchrony is actually themain difficulty that limits the understanding of syn-chrony at short range. The right hippocampus containsdamaged tissues with a higher conductivity thannormal, thus enhancing volume conduction locally. Incontrast, in the left limbic region, synchrony decreasedmuch faster with electrode separation, and it did notextend beyond 2 cm. This result coincides very wellwith the findings by Menon et al. [1996] on normalhuman subdural recordings, showing that shortrange synchrony decreased sharply after 2 cm of in-terelectrode distance. They explained these results bystrong local lateral connections that would inducesynchronization of neighboring neurons within the 2cm radius. In our view, simple passive volume conduc-tion also may create such spurious synchrony (see nextsection).

Long-range synchronies. In contrast with synchronywithin regions (short-range synchrony), synchrony be-tween locations (long-range synchrony) varied in time

TABLE I. Electrodes positions for subject PI*

1. R anterior amygdalia2. R hippocampus—head3. R hippocampus—anterior4. R hippocampus—body5. R white matter6. R white matter7. R temporo-occipital sulcus8. R temporo-occipital gyrus9. L anterior amygdalia

10. L hippocampus—head11. L hippocampus—anterior12. L hippocampus—body13. L white matter/hippo-

campus (posterior)14. L white matter15. L white matter16. L temporo-occipital sulcus17. R medial gyrus18. R anterior cingulate

gyrus

19. R anterior cingulategyrus

20. R frontal superior gyrus21. R frontal superior gyrus22. R frontal superior gyrus23. R front superior gyrus/

arachnoide space24. R perimeningeal space25. L medio orbital gyrus/

olfactory sulcus26. L anterior cingulate

gyrus/cingular sulcus27. L anterior cingulate

gyrus28. L frontal superior gyrus29. L frontal superior gyrus30. L frontal superior gyrus31. L front superior gyrus/

arachnoide space32. L perimeningeal space

* R 5 right hemisphere; L 5 left hemisphere.

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and occurred only between specific pairs of electrodeswith no relation to interelectrode separation. For in-stance, the right-limbic vs. left-frontal quadrant of thematrix in Figure 4 shows the localized emergence anddisappearance of synchronies over time. The timing ofthis pair is quite different from its converse, with theleft-limbic region.

We conclude that these observations are convincingevidence that significant long-range synchronies areestablished during this cognitive task involving deeplimbic and frontal regions. These synchronies cannotbe explained by volume conduction; it seems morelikely that they represent a partial correlate of thefunctional integration mechanism during visual dis-crimination. A detailed account and cognitive analysisof long-range synchronies in intracortical recordings of

several epileptic patients will be presented elsewhere.The present results, along with previous reports[Bressler et al., 1993; Desmedt and Tomberg, 1994;Friston et al., 1997; Roelfsema et al., 1997], strongly supportthe view that gamma synchrony acts as distributed unify-ing mechanism [Rodriguez et al., 1999].

SEPARATING SYNCHRONY FROM VOLUMECONDUCTION

As noted above, synchrony between two macrorecordings is not always due to neural interactions.Whereas animal studies of synchrony are based on thecoincidence of spikes recorded from microelectrodes(single or multiunit recordings), human studies usesurface or implanted electrodes that integrate neural

Figure 4.PLV values for subject PI. Values are presented for six consecutive time windows (stimulation occurs at zero ms). Boxes above diagonal(up-left) correspond to the target stimulation; those under diagonal (down-right) to nontargets. Each box represents the synchrony valuefor an electrode pair. If synchrony does not reach significance, the box is filled by a circle with a dot in the center. When it reachessignificance threshold (PLS , 0.05), the square is filled to an extent proportional to the PLV value.

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activity over large volumes. When the volumes re-corded by two electrodes overlap, the shared neuronalpopulation creates spurious synchrony between thesignals (see Appendix). Here, we discuss ways toreduce such synchrony due to volume conduction (or‘‘conduction synchrony’’) better to identify synchronydue to actual neural couplings (‘‘true synchrony’’). Twofactors are important to examine: (1) volume conduction,and (2) an inappropriate choice of the reference electrode.

Influence of volume conduction on synchrony

Separation between conduction synchrony and truesynchrony. The separation between these two types ofsynchrony is difficult because they can occur at thesame latencies and in the same frequency range (Fig.4). This contradicts a common assumption that conduc-tion synchrony should be broad-band and roughlyconstant in time, whereas true synchrony should bemore specific. Another common assumption is that thephase difference between electrodes should be zero incase of conduction synchrony. This is usually false:even if two electrodes record the same group ofsources, the signals of these electrodes are differentlinear combinations of the sources amplitudes, be-cause each source is recorded by the two electrodes intwo different ways that depend on the relative positionof the source and the electrodes. Therefore, the phase

of the signals recorded by the two electrodes should bedifferent, except if all the sources have the same phase.

One can still suggest a rule of thumb based onintersite comparison. If volume conduction createshigh synchrony between two electrodes, then highsynchrony also should be observed between theirneighbors. In other words, in general, when there issynchrony due to diffusion, there should be diffusion ofsynchrony. This diffusion of synchrony is precisely theeffect shown in Figure 4 within regions, but not betweenthem. Yet, this rule of thumb falls short of providing areliable test to identify conduction synchronies.

Reducing volume conduction. Since there is no univer-sal rule to distinguish true synchrony from conductionsynchrony, one must rely on recordings with highspatial resolution in which the overlap between thebrain volumes recorded by different probes is mini-mum. In this sense, subdural recordings constitute anideal study case. To study normal subjects, the use ofEEG or MEG is the most common option. MEG shouldbe preferred: volume effects are less severe because thehead tissues induce no diffusion of the magnetic field,in contrast with electrical potential [Hamalainen et al.,1993]. Also, the amplitude of the magnetic field de-creases faster with distance than the electrical poten-tial, so that the volume recorded by an EEG electrode islarger than that recorded by a MEG sensor. In addition,the spatial resolution of MEG can be increased beforeestimating synchronies [Friston et al., 1997].

Still, most studies turn to EEG data since MEGremains a rare and expensive technique and becauseEEG can record sources configuration not recorded byMEG. Two main techniques have been used to im-prove the spatial precision of EEG: (1) inverse deblur-ring [Le and Gevins, 1993], and (2) the derivation of thescalp current density profiles (SCD) [Pernier et al., 1988].We tested on a simulation the efficiency of both methodsfor the study of synchrony from surface EEG recordings.

This simulation mimics the experimental setting of aclassic EEG sensory task in which 50 repetitions of thesame stimulus generate independent responses in twocortical locations. We selected two independent seriesof 50 recordings from subject PI and used them asamplitudes of two sources placed in a spherical modelof the head [for details see Lachaux et al., 1997]. Thisthree-layered model comprised: (1) brain (conductiv-ity 5 1, arbitrary units; radius 5 8.5 cm), (2) skull(conductivity 5 0.0128; radius 5 9.2 cm), (3) skin (con-ductivity 5 1; radius, 10 cm) (infinite reference: poten-tial is zero far from the sphere). Electrical sources wereexpressed as two dipoles with radial orientations withrespect to the scalp, located in two superficial symmet-

Figure 5.PLV as a function of distance for the right temporal electrodes ofsubject PI. A regular 1 cm spacing separates eight electrodes. Thesynchrony decreases almost linearly from 1 to 0 when theinterelectrodes distance increases.

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ric positions in each hemisphere (see Fig. 6). Thesurface activity induced by these dipoles was com-puted (forward calculation) using the software BESA[Scherg, 1990] in 55 scalp positions for all 50 trials.

The EEG thus obtained was treated by a variation ofthe deblurring technique originally proposed by Leand Gevins [1993] to calculate the correspondingECoG (backward calculation). Using BESA, we alsocomputed the potential generated by the two dipoleson the cortex (i.e., the true EcoG, forward problem) tocheck the accuracy of the ECoG reconstructed bydeblurring. The dipole model and the correspondingEEG and ECoG are shown in Figure 6.

We then computed the phase-locking values for thescalp and cortical positions lying on a left-right axis(T7-Cz-T8). For comparison, we computed the scalp

current density on this spherical model [Pernier et al.,1988] and the PLV between the SCDs at thesesame scalp positions (Fig. 7, EEG and ECoG; Fig. 8,SCD). Since the sources of this model were indepen-dent, all significant PLS were due to conductionsynchrony.

As shown in Figure 7, deblurring sharpened theborders of synchronous regions and reduced spurioussynchronies (Fig. 7). SCD produced comparable ef-fects, but slightly less detailed. Therefore, these meth-ods seem useful first steps before the estimation ofsynchrony from EEG recordings. In practical cases,SCD may be preferred because its computation ismuch simpler than deblurring. Precise deblurringrequires a description of the geometry and the conduc-tivity of each individual’s skull and skin.

Figure 6.Simulation of an EEG and its corresponding ECoG. Two radial dipoles were placed in symmetric positions in the (T7-Cz-T8) plane,containing both ears and the head’s vertex (c). EEG generated by these dipoles was computed, and ECoG reconstructed using adeblurring technique. (a) EEG cartography for two dipoles with same amplitude (b) corresponding ECoG. (d) EEG and ECoGcontributions of each dipole as a function of laterality along a T7-Cz-T8 axis.

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In a recent study, Nunez and colleagues [1997] testedthe effect of volume conduction on the correlationbetween electrodes and found converging conclusions.Independent distributed sources were simulated in avolume conductor model of the head, the correspond-ing EEG was calculated in 64 scalp positions, and thesquared correlation coefficient was computed as afunction of interelectrode separation. Granted, correla-tion coefficient is not a close estimate of synchrony;Nunez et al. [1997] also suggested using SCD or EcoGestimations to reduce conduction interactions.

Influence of the reference-electrode on synchrony

Apart from volume conduction, the other majorproblem in synchrony estimation is the choice of thereference electrode. This problem uniquely concernsEEG; another reason to prefer the MEG technique. For

electric recordings, a review of the literature on fre-quency coherence provides no general agreement onthe optimal reference. In the present study, we took anaverage reference to study epileptic data and aninfinite reference in the electrocorticogram simulation.

Figure 9 shows results obtained with other references.When subtracting a reference term, it is not clear whethersynchrony should decrease (because a common part of thesignals is removed), or increase (because a common termis artificially added to all the signals). When studying EEG,the computation of scalp current density is often a way tosolve the reference problem, because spatial derivationmakes this term disappear. But our simulations (Fig. 9)show that this processing can create artificial synchronies.However, we used here spline interpolations to estimateSCDs [Pernier et al., 1988]; other approaches are possibleand they may yield a better estimation [Lagerlund et al.,1995]. Nunez and colleagues [1997] proposed a detailed

Figure 7.PLV values for simulated EEG and ECoG. Using the same presenta-tion as for Figure 5, PLV is presented for n electrodes lying on theT7-Cz-T8 lines. Electrodes are referred by their position along thisaxis from left (left ear) to right (right ear). For some electrodepairs, synchrony is significant between a and b (up-left part of the

matrix), but not between b and a (bottom-right part of the matrix).In other words, maps are not rigorously symmetric. This is becausethe surrogate PLV distributions are not strictly the same in the twocases, and thus the 5% significance thresholds may be slightlydifferent.

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study of the influence of reference on interelectrode corre-lation coefficients. They argue that it is impossible topredict the effect of a reference on scalp coherence withoutboth an accurate volume conductor model and a prioriknowledge of all sources locations.

In brief, the question of the distinction betweensynchrony due to volume conduction and synchronyas functional neural integration is still inconclusive.Here, we note some steps in that direction, but nonethat provide a complete solution.

DISCUSSION

Narrow vs. broad frequency bands

One of the main limitations of the method presentedhere is that it depends crucially on the choice of a

specific frequency for analysis in order to separate theamplitude and phase components of the signals. Thisseparation is meaningless if one needs to work with abroad band. Further, choosing a given frequency needsfilters with excellent resolutions in both time andfrequency, which do not change the phase. (Ourchoice, a classic finite impulse response filter, may notbe optimum.)

Yet, a careful time frequency analysis of intracorticaland scalp data shows that their spectral content is verybroad (between 0–80 Hz0, and the relative amplitudesvaries considerably over time during an experimentalsituation [Tallon-Baudry et al., 1997; Lachaux et al.,1999a; Rodriguez et al., 1999]. It is of the greatestinterest to investigate if there are interactions betweenfrequencies located in different bands and what, if any,is their functional significance. PLS permits only a

Figure 8.PLV values for simulated scalp current density. SCD is computed from EEG presented on Figure 6.Up-left: Contributions of each dipole to SCD are presented as a function of laterality along aT7-Cz-T8 axis. Right: PLV values are presented for electrodes lying on this axis.

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limited answer to these questions, since the sameanalysis can be repeated over several bands. However,a complete answer would require a method based on abroad-band calculation of synchrony. Such an ap-proach takes us far away from classical methods, but isan active area of study in mathematical physics.

Average vs. single trial

A second important limitation of the method intro-duced here is that it focuses on the detection ofphase-locking between pairs recordings across trials:i.e., the likelihood that phase-difference between theoscillations of two neural populations remains thesame from trial to trial for a given delay in time. Thisexcludes those synchronies that are not establishedwith a fixed delay from trial to trial. In order to extract

such information, one needs to introduce averaging orsmoothing procedures over time and to work on thebasis of single trials, not averages. An analogy for thestudy of gamma band emission can be helpful here.PLS can be compared to the procedures that detect theevoked gamma emission, but fail to detect the inducedresponses. In fact, induced responses are not stimuluslocked and thus require a trial by trials analysis toconstruct a probability distribution [Tallon-Baudry etal., 1997; Lachaux et al., 1999a]. PLS cannot detect thissecond type of single-trial phase locking if the phase-difference varies in latency between trials.

The detection of this second type of phase-locking insingle trials is possible using a recent technique,smoothed phase locking statistics (SPLS) that we haveintroduced recently [Lachaux et al., 1999b]. The appli-cation of this improved approach will tell, in time, if

Figure 9.Effect of reference electrode on PLV values. We computed PLV for the simulated EEG datapresented in Figure 7, using an average electrode (left), instead of the infinite reference used forFigure 7. We also computed synchrony for subject PI for both stimulation conditions using noreference instead of average reference; spurious synchronies appear for almost all pairs.

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functional neural synchronies are more common thanwe have shown in this report.

NOTE ADDED IN PROOF

After this paper was sent for publication, a recentpublication introduced an alternative method to com-pute phase synchorny between bivariate data based onthe Hilbert transform (Tass et al., Physical Rev Letters81:3291–94, 1998). The two approaches are comparablein several respects. A study of their comparativeadvantages is being carried out.

ACKNOWLEDGMENTS

This work began a few years ago thanks to theessential contribution of Johannes Muller-Gerkin onMSC calculations. Our thanks also to Laurent Hugue-ville for his help on EcoG methods.

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APPENDIX

Source of spurious synchronies

Here, we consider how volume conduction caninduce spurious correlations between electrode poten-tials. For this purpose, we consider a set of micro-electrodes located within a brain area to record local-field potentials produced by an assembly of neuronswith uncorrelated activities.

In response to a stimulation s, each neuron has anactivation p(x, t, s), which is a function of its position x,time t, and s. Suppose to fix ideas that p(x, t, s) is awhite-noise such that the mean value of p(x, t, s)2 overtime is unity. Since the neurons have uncorrelatedactivities, the mean value of p(x, t, s) · p( x8, t, s) acrosstime and trials (we note it 7p(x) · p(x8)8) is then d(x, x1),(i.e., 1 if x 5 x8, 0 if not).

The local-field potential recorded by an electrodelocated in y, can be expressed as a linear combinationof the sources amplitudes:

e(y, t, s) 5 e h(y, x) · p(x, t, s)dx.

We consider further that all the electrodes are roughlylocated at the same distance around the sources, sothat 7e( y)8 is roughly the same for all the electrodes.Thus we get a fair approximation of the correlationbetween the potentials in two positions y and y8 by

computing 7e( y) · e( y8)8. This is done as follows. Forany time t and sample s,

e(y, t, s) · e(y8, t, s)

5 1e h(y, x) · p(x, t, s) · dx2 · 1e h(y8, x8) · p(x8, t, s) · dx82or

e(y, t, s) · e(y8, t, s)

5 e e h(y, x) h(y8, x8) · p(x, t, s) · p(x8, t, s) dxdx8

so that when averaging,

7e(y) · e(y8)8 5 e e h(y, x) h(y8, x8) · 7p(x) · p(x8)8 dxdx8

or,

7e(y) · e(y8)8 5 e e h(y, x) h(y8, x8) · d(x, x8) dxdx8

finally,

7e(y) · e(y8)8 5 e h(y, x) h(y8, x) dx.

Thus due to volume-conduction, 7e( y) · e( y8)8 is non-zero even if there is no correlation at all between thetwo sources of activities.

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