Complex networks: new trends for the analysis of brain connectivity
Mario Chavez, Miguel Valencia, Vito Latora, Jacques Martinerie
CComplex networks: new trends for the analysis of brain connectivity
MARIO CHAVEZ
LENA-CNRS UPR-640. Hˆopital de la Salpˆetri`ere.47 Bd. de l’Hˆopital, 75651 Paris CEDEX 13, France
MIGUEL VALENCIA
LENA-CNRS UPR-640. Hˆopital de la Salpˆetri`ere.47 Bd. de l’Hˆopital, 75651 Paris CEDEX 13, FranceDepartment of Neurological Sciences, Center of Applied Medical Research,University of Navarra, Avda Pio XII 31. 31008, Pamplona, Navarra
VITO LATORA
Dipartimento di Fisica e Astronomia, Universit`a di Catania and INFN,Via S. Sofia, 64, 95123 Catania, ItalyLaboratorio sui Sistemi Complessi, Scuola Superiore di Catania,Via San Nullo 5/i, 95123 Catania, Italy
JACQUES MARTINERIE
LENA-CNRS UPR-640. Hˆopital de la Salpˆetri`ere.47 Bd. de l’Hˆopital, 75651 Paris CEDEX 13, France
Last revised version: October 26, 2018
Abstract
Today, the human brain can be studied asa whole. Electroencephalography, magnetoen-cephalography, or functional magnetic resonanceimaging (fMRI) techniques provide functionalconnectivity patterns between different brain ar-eas, and during different pathological and cog-nitive neuro-dynamical states. In this Tutorialwe review novel complex networks approaches tounveil how brain networks can efficiently man-age local processing and global integration forthe transfer of information, while being at the same time capable of adapting to satisfy chang-ing neural demands.
In recent years, complex networks have providedan increasingly challenging framework for thestudy of collective behaviors in complex systems,based on the interplay between the wiring ar-chitecture and the dynamical properties of thecoupled units [1, 2]. Many real networks werefound to exhibit small-world features. Small-world (SW) networks are characterized by hav-1 a r X i v : . [ phy s i c s . d a t a - a n ] F e b ng a small average distance between any twonodes, as random graphs, and a high clusteringcoefficient, as regular lattices [3, 4, 5, 6]. Thus, aSW architecture is an attractive model for brainconnectivity because it leads distributed neuralassemblies to be integrated into a coherent pro-cess with an optimized wiring cost [7, 8, 9].Another property observed in many networksis the existence of a modular organization in thewiring structure. Examples range from RNAstructures, to biological organisms and socialgroups. A module is currently defined as a subsetof units within a network such that connectionsbetween them are denser than connections withthe rest of the network. It is generally acknowl-edged that modularity increases robustness, flex-ibility and stability of biological systems [10, 11].The widespread character of modular architec-ture in real-world networks suggests that a net-work’s function is strongly ruled by the organi-zation of their structural subgroups.Recent studies have attempted to character-ize the functional connectivity (patterns of sta-tistical dependencies) observed between brainactivities recorded by electroencephalography(EEG), magnetoencephalography (MEG), orfunctional magnetic resonance imaging (fMRI)techniques [12, 13, 14, 15, 16]. Surprisingly,functional connectivity patterns obtained fromMEG and EEG signals during different patholog-ical and cognitive neuro-dynamical states, werefound to display SW attributes [15, 16]; whereasfunctional patterns of fMRI often display a struc-ture formed by highly connected hubs, yieldingan exponentially truncated power law in the de-gree distribution [12, 13, 14]. For a completereview of these issues, reader can refer to theRefs. [17, 18].In functional networks, two different nodes(representing two electrodes, voxels or source regions) are supposed to be linked if some de-fined statistical relation exceeds a threshold.Regardless of the modality of recording activ-ity (EEG, MEG or fMRI), topological featuresof functional brain networks are currently de-fined over long periods of time, neglecting pos-sible instantaneous time-varying properties ofthe topologies. Nevertheless, evidence suggeststhat the emergence of a unified neural process ismediated by the continuous formation and de-struction of functional links over multiple timescales [20, 19, 21].Empirical studies have lead to the hypothesisthat transient synchronization between distantand specific neural populations underlies the in-tegration of neural activities as unified and co-herent brain functions [19]. Specialized brainregions would be largely distributed and linkedto form a dynamical web-like structure of thebrain [20]. Thus, brain regions would be par-titioned into a collection of modules, represent-ing functional units, separable from -but relatedto- other modules. Characterizing the dynami-cal modular structure of the brain may be crucialto understand its organization during differentpathological or cognitive states. An importantquestion is whether the modular structure hasa functional role on brain processes such as theongoing awareness of sensory stimuli or percep-tion.To find the brain areas involved in a givencognitive task, clustering is a classical approachthat takes into account the properties of theneurophysiological time series. Previous stud-ies over the mammalian and human brain net-works have successfully used different methodsto identify clusters of brain activities. Some clas-sical approaches, such as those based on princi-pal components analysis (PCA) and independentcomponents analysis (ICA), make very strong2tatistical assumptions (orthogonality and sta-tistical independence of the retrieved compo-nents, respectively) with no physiological justifi-cation [22, 23].In this Tutorial, we review an approach thatallows to characterize the dynamic evolution offunctional brain networks [24, 25]. We illustratethis approach on connectivity patterns extractedfrom MEG data recorded during a visual stimu-lus paradigm. Results reveal that the brain con-nectivity patterns vary with time and frequency,while maintaining a small-world structure. Fur-ther, we are able to reveal a non-random mod-ular organization of brain networks with a func-tional significance of the retrieved modules. Thismodular configuration might play a key role inthe integration of large scale brain activity, fa-cilitating the coordination of specialized brainsystems during a cognitive brain process. To illustrate our approach, we consider the brainresponses recorded during the visual presenta-tion of non-familiar pictures. Although our ap-proach is applicable to any of the functionalmethods available (EEG, fMRI, MEG), here weuse the magnetoencephalography. This modal-ity of acquisition has the major feature that col-lective neural behaviors, as synchronization oflarge and sparsely distributed cortical assem-blies, are reflected as interactions between MEGsignals [26]. We study the functional connec-tivity patterns associated with dynamic brainprocesses elicited by the repetitive application(trials) of a external visual stimulus [27]. Forthis experiment, a collection of 48 simple struc-tural images and scrambled images were ran-domly shown to epileptic patients for a peri- ode of 150 ms with an inter-stimulus intervalof 2 s. Patients were required to respond bypressing a button each time an image was per-ceived. The event-related brain responses wererecorded (from two patients) with a whole-headMEG system (151 sensors; VSM MedTech, Co-quitlam, BC, Canada) digitized at 1 .
25 kHz witha bandpass of 0 −
200 Hz.The basic steps of our approach are schemat-ically illustrated in Fig. 1. Each of the sig-nals is decomposed into time-frequency compo-nents, as shown in panel a). The relations be-tween two signals j and k are firstly definedin time-frequency space, as shown in panel b).A statistical criterion is then used to define afunctional connectivity matrix for each time-frequency point, panel c). The details of thestatistical criterion we adopted are reported inSection 2.1. In panel d), topological metrics areextracted from the connectivity patterns to ob-tain a time-frequency characterization of brainnetworks. The metrics investigated are analyzedin Section 2.2. Finally, in panel e), at a givenfrequency, or time instant of interest, the modu-lar structure is characterized as discussed in Sec-tions 2.3 and 2.4. To evaluate the features ofbrain connectivity, the obtained functional net-works are compared with equivalent regular andrandom networks. A unified definition of brain connectivity is diffi-cult from the fact that the recorded dynamics re-flect the activities of neural networks at differentspatial and temporal resolutions. Three types ofconnectivity are currently considered: anatomi-cal (description of the physical connections be-tween two brain sites), functional (defined by a3igure 1: General scheme for the extraction of the time-varying brain networks: (a) signals aredecomposed into time-frequency components to compute (b) pair-wise relations; (c) functionalconnectivity matrices are extracted at each point of the time-frequency space, defining (d) thefunctional brain networks used to extract the topological attributes (color codes the nodes degree)and the (e) modular structure (brain sites belonging to each module are arbitrarily colored). Seedetails in the text. 4emporal correlation between distant neurophys-iological events) and effective (causal influencethat a neural system may exert over another).Here, we consider the functional links in brainsignals defined by means of the phase-lockingvalue (PLV) computed between all pairs of sen-sors [28]. To compute the PLV values, we useda complex Morlet’s wavelet function defined as w ( t, f ) = A exp( − t / σ t ) × exp( i πf t ). Nor-malization factor A was set to A = ( σ t √ π ) − / . σ t = m/ πf , m is a constant that defines thecompromise between time and frequency reso-lution, and f is the center frequency of thewavelet. Hence, in time domain, its real andimaginary parts are a cosine and a sine, re-spectively, of which the amplitude envelope isa Gaussian with a standard deviation of σ t . Infrequency domain, the Morlet wavelet is also aGaussian with a standard deviation σ f given m = f /σ f . Here, m was chosen to be 7. Bymeans of this complex wavelet transform an in-stantaneous phase φ triali ( t, f ) is obtained for eachfrequency component of signals i = 1 , . . . , M ateach repetition of the stimulus (trial). The PLVbetween any pair of signals ( i, k ) is inversely re-lated to the variability of phase differences acrosstrials:PLV i,k ( t, f ) =1 N trials (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N trials (cid:88) trial =1 exp j ( φ triali ( t,f ) − φ trialk ( t,f )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where N trials is the total number of trials. Ifthe phase difference varies little across trials, itsdistribution is concentrated around a preferredvalue and PLV ∼
1. In contrast, under the nullhypothesis of a uniformity of phase distribution,PLV values are close to zero.Finally, to assess whether two different sen-sors are functionally connected, we calculated the significance probability of the PLV valuesby a Rayleigh test of uniformity of phase. Ac-cording to this test, the significance of a PLVvalue determined from N trials can be calculatedas p = exp( − N trials PLV ) [29]. To correct formultiple testing, the False Discovery Rate (FDR)method was applied to each matrix of PLV val-ues [30]. With this approach, the threshold ofsignificance P LV th was set such that the ex-pected fraction of false positives is restricted to q ≤ . A ij = 1 if P LV ij > P LV th ; and zero other-wise). Although topological features can alsobe straightforwardly generalized to weighted net-works [32], we obtained qualitative similar re-sults (not reported here) for weighted networkswith a functional connectivity strength betweennodes given by w ij = P LV ij . More refined sta-tistical tools can also be used to estimate time-varying and directed brain networks [31]. A set of metrics can be used to characterize thetopological properties of the functional networkswe have constructed [1, 2]. Here, we use threekey parameters: mean degree (cid:104) K (cid:105) , clustering in-dex C and global efficiency E . Briefly, the degree k i of node i denotes the number of functionallinks incident with the node and the mean de-gree is obtained by averaging k i across all nodesof the network. The clustering index quantifiesthe local density of connections in a node’s neigh-borhood. The clustering coefficient c i of a node i is calculated as the number of links betweenthe node’s neighbors divided by all their pos-5ible connections and C is defined as the aver-age of c i taken over all nodes of the network [3].The global efficiency E provides a measure ofthe network’s capability for information transferbetween nodes and is defined as the inverse ofthe harmonic mean of the shortest path length L ij between each pair of nodes [6]. The node-efficiency E i of the i th node is likewise definedas the inverse of the harmonic mean of the min-imum path length between node i and all othernodes in the network.To asses the small-world behavior of functionalnetworks, we perform a benchmark comparisonof the functional connectivity patterns [3]. Forthis, the clustering and efficiency coefficients offunctional networks are compared with those ob-tained from equivalent random and regular con-figurations. Regular networks were obtained byrewiring the links of each node to its nearest (inthe sensors space) neighbors, yielding a nearest-neighbor connectivity with the same degree dis-tribution as the original network. To create anensemble of equivalent random networks we usethe algorithm described in Ref. [3]. According tothis procedure, each edge of the original networkis randomly rewired avoiding self and duplicateconnections. The obtained randomized networkspreserve thus the same mean degree as the origi-nal network whereas the rest of the wiring struc-ture is random. Many real networks have a modular structure,i.e. their associated graphs are in general glob-ally sparse but locally dense. In these net-works, modules are defined as groups of verticeslinked such that connections between them aredenser than connections with the rest of the net-work. It is currently accepted that a partition P = {C , . . . , C M } represents a good division inmodules if the portion of edges inside each mod-ule C i (intra–modular edges) is high comparedto the portion of edges between them (inter–modular edges). The modularity Q ( P ), for agiven partition P of a network is formally de-fined as [33]: Q ( P ) = M (cid:88) s =1 (cid:34) l s L − (cid:18) k s L (cid:19) (cid:35) , (1)where M is the number of modules, L is thetotal number of connections in the network, l s is the number of connections between vertices inmodule s , and k s is the sum of the degrees of thevertices in module s .To partition the functional networks in mod-ules, we used a random walk-based algo-rithm [34], because of its ability to managevery large networks, and its good performancesin benchmark tests [34, 37]. Similar theoreti-cal frameworks have been recently proposed forspectral coarse-graining [35, 36]. The algorithmis based on the intuition that a random walker ona graph tends to remain into densely connectedsubsets corresponding to modules. Let P ij = A ij k i to be the transition probability from node i tonode j , where A ij denotes the adjacency matrixand k i is the degree of the i th node. This definesthe transition matrix ( P t ) ij for a random walkprocess of length t (denoted here P tij for simplic-ity). One can notice that, if two vertices i and j are in the same community, the probability P tij is high, and P tik (cid:39) P tjk ∀ k .The metric used to quantify the structuralsimilarity between vertices is given by ρ ij = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) l =1 ( P til − P tjl ) k l (2)6his distance has several advantages: it quanti-fies the structural similarity between vertices andit can be used in an efficient clustering algorithmto maximize the network modularity Q . Fur-ther, using matrix identities, the distance ρ canbe written as ρ ij = (cid:80) nα =2 λ tα ( v α ( i ) − v α ( j )) ;where ( λ α ) (cid:54) α (cid:54) n and ( v α ) (cid:54) α (cid:54) n are the n eigen-values and right eigenvectors of the matrix P , re-spectively [34]. This relates the random walk al-gorithm to current methods using spectral prop-erties of the graphs [38, 36]. The random-walkbased approach, however, needs not to explicitlycompute the eigenvectors of the matrix; a com-putation that rapidly becomes intractable whenthe size of the graphs exceeds some thousands ofvertices.To find the modular structure, the algorithmstarts with a partition in which each node inthe network is the sole member of a module.Modules are then merged by an agglomera-tive approach based on a hierarchical clusteringmethod [39]. The algorithm stops when all thenodes are grouped into a single component. Ateach step the algorithm evaluates the quality ofpartition Q . The partition that maximizes Q isconsidered as the partition that better capturesthe modular structure of the network. In the cal-culation of Q , the algorithm excludes small iso-lated groups of connected vertices without anylinks to the main network. However, these iso-lated modules are considered here as part of thenetwork for the calculation of the topological pa-rameters. To evaluate the agreement between modules as-signments at a given time instant or frequencyone can used the adjusted Rand index Ra [41],which is a traditional criterion for comparison of different results provided by classifiers and clus-tering algorithms, including partitions with dif-ferent numbers of classes or clusters. For twopartitions P and P (cid:48) , the original Rand index isdefined as [40] R = a + da + b + c + d ; where a is num-ber of pairs of data objects belonging to thesame class in P and to the same class in P (cid:48) , b is number of pairs of data objects belonging tothe same class in P and to different classes in P (cid:48) , c is the number of pairs of data objects be-longing to different classes in P and to the sameclass in P (cid:48) , and d is number of pairs of data ob-jects belonging to different classes in P and todifferent classes in P (cid:48) . Thus the index R hasa straightforward interpretation as a percentageof agreement between the two partitions and ityields values between 0 (if the two partitions arerandomly drawn) and 1 (for identical partitionstructures).The Rand index, however, has a bias if a par-tition is composed by many clusters, and it cantake a non-null value for two completely ran-dom partitions. The index R can be straight-forwardly corrected for the expected value un-der the null hypothesis according to the follow-ing general scheme: Ra = R − E { R } max { R }− E { R } . Us-ing the generalized hypergeometric distributionas the null hypothesis, the adjusted Rand indexthat corrects for the expected number of nodespairs placed in the same module under two ran-dom partitions is given by [41] Ra = a − ( a + c )( a + b ) a + b + c + d a + b + ca + b + c + d − ( a + c )( a + b ) a + b + c + d (3)which has an expected value of zero under thenull hypothesis, and it takes a maximum value ofone for a perfect agreement of the two partitions.Thus, the adjusted Rand index is a statistics onthe level of agreement or correlation between two7artitions. Fig. 2 shows the topological attributes of func-tional networks elicited by the -unexpected- im-ages. Pictures show the values of the mean de-gree, clustering index and efficiency of networksbetween, calculated at each point of the time-frequency space, 600 ms before and 1 s after theonset of the stimulus.The first crucial observation is that functionalconnectivity patterns are not time-invariant, butinstead they exhibit a rich time-frequency struc-ture during the neural processing. All thetopological features (specially (cid:104) K (cid:105) and C ) ex-hibit high values in a frequency band close to10 Hz, which is a spectral component mostlyinvolved in the processing of visual informa-tion [27]. Whereas the functional networks in thefrequency range of 10 −
30 Hz display large pat-terns of synchronization/desynchronization be-fore the stimuli, a highly connected pattern isinduced by the stimulus at about 250 ms andbetween 15 and 25 Hz, suggesting a connectiv-ity induced by the unexpected sensory stimuli.This is followed by weak connected structuresat frequency bands close to 7 and 15 Hz arisingduring the post-stimulus activities and markingthe transition between the moment of percep-tion and the motor response of the subject. Thetopological features of these connectivity pat-terns were detected as statistically different fromthe pre-stimulus epoch by a Z -test corrected bya FDR at q ≤ .
05. Brain activities above 30 Hzare characterized by a poor global connectivity.Local parameters, k i , c i and E i , for each sensor of the network are shown at three different timeinstants for a frequency of 20 Hz . During theprocessing of the stimulus, a time-space variabil-ity of connectivity is observed. Before the onsetof the stimulus, the networks are characterizedby a very sparse connectivity. Then, a clear clus-tered structure triggered by the stimulus appearsat t = 250 ms, defining two main regions (frontaland occipital) with a high density of connections.After the stimulus, the functional wiring displaysagain a sparse structure. The comparison of the brain networks againstrandom and regular configurations is shownFig. 3. Typically, small-world networks exhibit a E sw greater than regular lattices, but less thanrandom wirings E lat < E sw < E rnd ; while forthe mean cluster index, C rnd < C sw < C lat is ex-pected [3]. Results reveal that, despite the vari-ability observed, functional networks display atopology different from regular and random net-works. Namely, C (cid:104) C rnd (cid:105) > C (cid:104) C lat (cid:105) <
1, whichindicates a SW structure ( (cid:104) ... (cid:105) stays for an aver-age over the ensembles of equivalent networks).Further, (cid:104) E lat (cid:105) E < (cid:104) E rnd (cid:105) E >
1, supportingthe hypothesis of a SW connectivity.It is important to emphasize that, in con-trast with previous studies which have focusedon time-invariant networks [12, 13, 14, 15, 16],our approach reveals a dynamical small-worldconnectivity at multiple time scales. This is aremarkable result, insofar as it suggests that theprocessing of a stimulus involves an optimized(in a SW sense) functional integration of distantbrain regions by a dynamic reconfiguration oflinks.8igure 2: Time-frequency maps of topological features extracted from brain networks associatedto a visual stimulus presentation (arriving at t = 0). (a) mean degree (cid:104) K (cid:105) , (b) clustering index C and (c) efficiency E . The reported values refer to the average over subjects. Dotted lines outlinethe regions revealing a significant change from the pre-stimulus region. Lower row: topographicdistribution of the local parameters for the 20 Hz activities (indicated by the thick dashed line) atthree different time instants.Figure 3: Comparison of functional networks with random and regular configurations: time-frequency maps of (a) C/ (cid:104) C lat (cid:105) , (b) C/ (cid:104) C rnd (cid:105) , (c) (cid:104) E lat (cid:105) /E and (d) (cid:104) E rnd (cid:105) /E . Results of equivalentrandom and regular networks refer to the average of 20 realizations. A potential modularity of brain-webs is sug-gested by the fact that brain networks display a9lustering index larger than that obtained fromrandom configurations [42]. Indeed, the presenceof modules is actually confirmed by the high val-ues of Q obtained for brain networks extractedfrom brain activities at different time instantsand frequencies. Fig. 4 shows the spatial dis-tribution of the modules for different networkswith the following Q values: (a) Q = 0 .
55, (b) Q = 0 .
33, (c) Q = 0 .
53, (d) Q = 0 .
53, (f) Q = 0 .
49 and (g) Q = 0 . Ra = 0 .
39. Then, the largeconnectivity triggered by the stimulus is also ac-companied by an increase in the number of mod-ules, yielding a more complex modular struc-ture. The observed changes are directly relatedwith the specific nature of the task:the detectionand low-level processing of the stimulus involvesthe visual system, but further processing as theidentification and perception of the picture re-quires the mediation of regions as those locatedin frontal regions. Surprisingly antero-posteriorrelations elicit a large and unique module fittingfronto-occipital regions. Although a one-to-oneassignment of anatomo-functional roles to eachdetected module is difficult to define, results re-veal some other interesting modules, as the oneslocated over the motor cortex at f = 18 . Ra = 0 . .
38 obtained for other frequencies.These are remarkable results as they supportthe hypothesis that brain dynamics relies on dif-ferent modular organizations to integrate distantspecialized, but functionally related, brain re-gions. Our findings suggest modularity as anorganization basis leading distributed groups ofspecialized neural assemblies to be integratedinto a coherent process during different cognitiveor pathological states. A modular description ofbrain networks might provide, more in general,meaningful insights into the functional organiza-tion of brain activities during others neural func-tions, such as attention and consciousness.
In this Tutorial we have addressed a fundamen-tal problem in brain networks research: whetherand how brain behavior relies on the coordinationof a dynamic mosaic of functional brain mod-ules during cognitive states . We have proposed amethod to study the time-frequency dependen-cies of functional brain networks, thus offeringan instantaneous description of the brain archi-tecture. Applied to a visual stimulus paradigm,the method reveals that the functional brain con-nectivity evolves in a small-world structure dur-ing the different episodes of the neural process-ing. Furthermore, by using a random walk-basedanalysis, we have identified a non-random modu-10igure 4: Topographical distribution of the modules extracted from brain networks at differenttime instants and frequencies: (a) time instant t = -0.25 s, frequency f =20 Hz; (b) t = 0.25 s afterthe presentation of the stimulus at f =20 Hz; (c) t =0.75 s, f =20 Hz; (e) t =-0.25 s, f =10 Hz; (f)t =0.25 s, f =10 Hz and (g) t =0.75 s, f =10 Hz. Brain sites belonging to each functional brainmodule were arbitrarily colored (there is no color correspondence between the modules of differentnetworks). For the sake of clarity, isolated nodes were colored in black. (d) Time-frequency mapsof mean degree is plotted to help network’s localization in the time frequency space11ar structure in the functional brain connectivity.The present analysis was performed on MEGdata in sensor space, which contains some in-herent spurious correlation between magneticfields on the surface of the brain. Althoughthis caveat does not affect the characterizationof the global network topology, accurate infer-ences about anatomical locations needs a sourcereconstruction of the activity in the cortex. Inthis study, we have reduced the influence of spu-rious correlations by simply excluding the near-est sensors from the computation of PLV values.Our approach may provide meaningful in-sights into how brain networks can efficientlymanage a local processing and a global integra-tion for the transfer of information, while beingat the same time capable of adapting to satisfychanging neural demands. Although the neuro-physiological mechanisms involved in the func-tional integration of distant brain regions arestill largely unknown, a dynamic SW organiza-tion is a plausible solution to the apparently op-posing needs of local specificity of activity ver-sus the constraints imposed by the coordinationof distributed brain areas. The modular struc-ture constitutes therefore an attractive model forthe brain organization as it supports the coexis-tence of a functional segregation of distant spe-cialized areas and their integration during brainstates [4, 5]. We suggest that this network de-scription might provide new insights into the un-derstanding of human brain connectivity duringpathological or cognitive states.Applied to other multivariate data, our ap-proach could provide new insights into the struc-ture of the time-varying connectivity at a certaintime [24]. A modular description of brain net-works might provide, more in general, meaning-ful insights into the functional organization ofbrain activities recorded with others neuroimag- ing techniques (EEG, MEG or fMRI) during di-verse cognitive or pathological states [25]. Inthis study, the functional links have been definedin MEG signals by means of the phase-lockingvalue. We notice, however, that other time-frequency methods (e.g. wavelet cross-spectra)can also be used to detect and characterize atime-varying connectivity of spatially extended,nonstationary systems (e.g. financial or epidemi-ological networks).
The authors would like to thank S. Dupont, A.Ducorps and G. Yvert for clinical and techni-cal support during data acquisition. This workwas supported by the EU-GABA contract no.043309 (NEST) and CIMA-UTE projects.
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