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International Journal of Bifurcation and Chaos, Vol. 19, No. 2 (2009) 665–676c! World Scientific Publishing Company

CLUSTER STRUCTURE OF FUNCTIONALNETWORKS ESTIMATED FROMHIGH-RESOLUTION EEG DATA

ROBERTA SINATRA!

Complex Systems Laboratory,Scuola Superiore di Catania, Catania, Italy

[email protected]@ssc.unict.it

FABRIZIO DE VICO FALLANI†Interdepartmental Research Centre for Models and

Information Analysis in Biomedical Systems,University “La Sapienza”, Rome, Italy

LAURA ASTOLFI†,‡, FABIO BABILONI†,‡,FEBO CINCOTTI† and DONATELLA MATTIA††IRCCS “Fondazione Santa Lucia”, Rome, Italy

‡Department of Human Physiology and Pharmacology,University “La Sapienza”, Rome, Italy

VITO LATORADepartment of Physics and Astronomy,

University of Catania,and INFN Sezione Catania, Italy

Received March 14, 2008; Revised April 9, 2008

We study the topological properties of functional connectivity patterns among cortical areas inthe frequency domain. The cortical networks were estimated from high-resolution EEG record-ings in a group of spinal cord injured patients and in a group of healthy subjects, during thepreparation of a limb movement. We first evaluate global and local e!ciency, as indicators ofthe structural connectivity respectively at a global and local scale. Then, we use the MarkovClustering method to analyze the division of the network into community structures. The resultsindicate large di"erences between the injured patients and the healthy subjects. In particular,the networks of spinal cord injured patient exhibited a higher density of e!cient clusters. In theAlpha (7–12Hz) frequency band, the two observed largest communities were mainly composed ofthe cingulate motor areas with the supplementary motor areas, and of the premotor areas withthe right primary motor area of the foot. This functional separation strengthens the hypothesisof a compensative mechanism due to the partial alteration in the primary motor areas becauseof the e"ects of the spinal cord injury.

Keywords: Community structure; network e!ciency; cortical activity.

!Author for correspondenceThe first two authors have contributed equally to the present paper.

665

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666 R. Sinatra et al.

1. Introduction

Over the last years, there has been an increasinglylarge interest in finding significant features fromhuman brain networks. In particular, the concept offunctional connectivity plays a central role to under-stand the organized behavior of anatomical regionsin the brain during activity. This organization isthought to be based on the interaction between dif-ferent specialized areas of cortical sites. Indeed, sev-eral methods have been proposed and discussed inthe literature, with the aim of estimating the func-tional relationships among the physiological sig-nals [David et al., 2004; Lee et al., 2003] obtainedfrom di"erent neuro-imaging devices such as thefunctional Magnetic Resonance Imaging (fMRI)scanner, electroencephalography (EEG) and mag-netoencephalography (MEG) apparatus [Horwitz,2003]. Recently, a multivariate spectral techniquecalled Directed Transfer Function (DTF) has beenproposed [Kaminski et al., 2001] to determine direc-tional influences between any given pair of channelsin a multivariate data set. This estimator is able tocharacterize at the same time direction and spec-tral properties of the brain signals, requiring onlyone multivariate autoregressive (MVAR) model tobe estimated from all the EEG channel recordings.The DTF index has been demonstrated [Kaminskiet al., 2001] to rely on the key concept of Grangercausality between time series — an observed timeseries x(n) leads to another series y(n) if the knowl-edge of past x(n)s significantly improves the pre-diction of y(n) — [Granger, 1969]. However, theextraction of salient characteristics from brain con-nectivity patterns is an openly challenging topic,since often the estimated cerebral networks have arelative large size and complex structure. Conse-quently, there is a wide interest in the developmentof mathematical tools that could describe in a con-cise way the structure of the estimated cerebral net-works [Tononi et al., 1994; Stam, 2004; Salvador,2005; Sporns, 2002].

Functional connectivity networks estimatedfrom EEG or magnetoencephalographic (MEG)recordings can be analyzed with tools that havebeen already proposed for the treatments of com-plex networks as graphs [Strogatz, 2001; Wang &Chen, 2003; Sporns et al., 2004; Stam et al., 2006a].Such an approach can be useful, since the use ofmathematical measures summarizing graph prop-erties allows for the generation and the testingof particular hypothesis on the physiologic nature

of the functional networks estimated from high-resolution EEG recordings. However, first resultshave been obtained for a set of anatomical brainnetworks [Strogatz, 2001; Sporns et al., 2002]. Inthese studies, the authors have employed two char-acteristic measures, the average shortest path L andthe clustering index C, to extract respectively theglobal and local properties of the network struc-ture [Watts & Strogatz, 1998]. They have found thatanatomical brain networks exhibit many local con-nections (i.e. a high C) and a shortest separationdistance between two randomly chosen nodes (i.e.a low L). Hence, anatomical brain networks havebeen designated as small-world in analogy with theconcept of the small-world phenomenon observedmore than 30 years ago in social systems [Milgram,1967].

Many types of functional brain networks havebeen analyzed in a similar way. Several studiesbased on di"erent imaging techniques like fMRI[Salvador et al., 2005; Eguiluz et al., 2005; Achard &Bullmore, 2007], MEG [Stam et al., 2006a, 2006b;Bassett et al., 2006; Bartolomei et al., 2006] andEEG [Micheloyannis et al., 2006; Stam et al., 2007]have shown that the estimated functional networkscan indeed exhibit the small-world property. In thefunctional brain connectivity context, these prop-erties have been demonstrated to reflect an opti-mal architecture for the information processing andpropagation among the involved cerebral structures[Lago-Fernandez et al., 2000; Sporns et al., 2000]. Inparticular, a high clustering index C is an indica-tion of the presence in the network of a large num-ber of triangles. However, this index alone does notreturn detailed information on the presence of largerconnected clusters of nodes. This fact makes up areal obstacle in the analysis of the network proper-ties especially in the field of the Neuroscience wherethe correlated behavior of di"erent cortical regionsplays a fundamental role in the correct understand-ing of cerebral systems. Methods to detect the com-munity structures in a graph, i.e. tightly connectedgroup of nodes, are now available in the market[Harary & Palmer, 1973]. Communities (or clustersor modules) are groups of vertices that probablyshare common properties and/or play similar roleswithin the graph [Boccaletti et al., 2006]. Hence,communities may correspond to groups of pages ofthe World Wide Web dealing with related topics[Flake et al., 2002], to functional modules such ascycles and pathways in metabolic networks [Guimer& Amaral, 2005; Palla et al., 2005], to groups of

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Cluster Structure of Functional Networks Estimated from High-Resolution EEG Data 667

a!ne individuals in social networks [Girvan & New-man, 2002; Lusseau & Newman, 2004], to compart-ments in food webs [Pimm, 1979; Krause et al.,2003] and so on. Finding the communities withina cerebral network allows identifying the hierarchyof functional connections within a complex architec-ture. This opportunity would represent an interest-ing way to improve the basic understanding of thebrain functioning. Indeed, some cortical regions aresupposed to share a large number of functional rela-tionships during the performance of several motorand cognitive tasks. This characteristic leads to theformation of highly connected clusters within thebrain network. These functional groups consist of acertain number of di"erent cerebral areas that arecooperating more intensively in order to complete atask successfully.

In the present paper, we focus on a study ofthe structural properties of functional networks esti-mated from high-resolution EEG signals in a groupof spinal cord injured patients during the prepa-ration of a limb movement. In particular, we firstinvestigate some indicators of the connectivity ata global and local scale. Then we analyze the net-works by studying their structure in communities,and we compare the results with those obtainedfrom a group of healthy subjects.

2. Methods

2.1. High-resolution EEGrecordings in SCI patients andhealthy subjects

All the experimental subjects participating in thestudy were recruited by advertisement. Informedconsent was obtained for each subject after theexplanation of the study, which was approved bythe local institutional ethics committee. The spinalcord injured (SCI) group consisted of five patients(age 22–25 years; two females and three males).Spinal cord injuries were of traumatic aetiology andlocated at the cervical level (C6 in three cases, C5and C7 in two cases, respectively); patients hadnot su"ered from a head or brain lesion associatedwith the trauma leading to the injury. The con-trol (CTRL) group consisted of five healthy volun-teers (age 26–32 years; five males). They had nopersonal history of neurological or psychiatric dis-order; they were not taking medication, and werenot abusing alcohol or illicit drugs. For EEG dataacquisition, subjects were comfortably seated on areclining chair, in an electrically shielded, dimly lit

room. They were asked to perform a brisk pro-trusion of their lips (lip pursing) while they wereperforming (healthy subjects) or attempting (SCIpatients) a right foot movement. The choice of thisjoint movement was suggested by the possibility totrigger the SCIs attempt of foot movement. In fact,patients were not able to move their limbs; how-ever they could move their lips. By attempting afoot movement associated with a lips protrusion,they provided an evident trigger after the volitionalmovement activity. This trigger was recorded tosynchronize the period of analysis for both the con-sidered populations. The task was repeated every6–7 seconds, in a self-paced manner, and the 100single trials recorded will be used for the estimateof functional connectivity by means of the DirectedTransfer Function (DTF, see following paragraph).A 96-channel system (BrainAmp, BrainproductsGmbH, Germany) was used to record EEG andEMG electrical potentials by means of an electrodecap and surface electrodes, respectively. The elec-trode cap was built accordingly to an extensionof the 10–20 international system to 64 channels.Structural MRIs of the subjects head were takenwith a Siemens 1.5T Vision Magnetom MR system(Germany).

2.2. Cortical activity and functionalconnectivity estimation

Cortical activity from high-resolution EEG record-ings was estimated by using realistic head mod-els and cortical surface models with an averageof 5.000 dipoles, uniformly disposed. Estimation ofthe current density strength, for each one of the5.000 dipoles, was obtained by solving the LinearInverse problem, according to techniques describedin previous papers [Babiloni et al., 2005; Astolfiet al., 2006]. By using the passage through theTailairach coordinates system, twelve Regions OfInterest (ROIs) were then obtained by segmenta-tion of the Brodmann areas on the accurate corticalmodel utilized for each subject. The ROIs consid-ered for the left ( L) and right ( R) hemispheres are:the primary motor areas for foot (MF L and MF R)and lip movement (ML L and ML R); the propersupplementary motor area (SM L and SM R); thestandard premotor area (6 L and 6 R); the cingu-lated motor area (CM L and CM R) and the asso-ciative Brodmann area 7 (7 L and 7 R). For eachEEG time point, the magnitude of the five thousanddipoles composing the cortical model was estimated

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by solving the associated Linear Inverse problem[Grave de Peralta & Gonzalez Andino, 1999]. Then,the average activity of dipoles within each ROI wascomputed. In order to study the preparation toan intended foot movement, a time segment of 1.5seconds before the lips pursing was analyzed; lipsmovement was detected by means of an EMG. Theresulting cortical waveforms, one for each prede-fined ROI, were then simultaneously processed forthe estimation of functional connectivity by usingthe Directed Transfer Function. The DTF is a fullmultivariate spectral measure used to determine thedirected influences between any given pair of sig-nals in a multivariate data set [Kaminski et al.,2001]. In order to be able to compare the resultsobtained for data entries with di"erent power spec-tra, the normalized DTF was adopted. It expressesthe ratio of influence of element j to element i withrespect to the influence of all the other elements oni. Details on the DTF equations in the treatment ofEEG signals have been largely described in previouspapers [Astolfi et al., 2005; Babiloni et al., 2005]. Inthe present study, we selected four frequency bandsof interest (Theta 4–7 Hz, Alpha 8–12 Hz, Beta 13–29 Hz and Gamma 30–40 Hz) and we gathered therespective cortical networks by averaging the val-ues within the respective range. In order to consideronly the functional links that are not due to chance,we adopted a Monte Carlo procedure. In particu-lar, we contrasted each DTF value with a surrogatedistribution of one thousand DTF values obtainedby shu#ing the signal samples in the original EEGdata set. Then, we considered a threshold value bycomputing the 99th percentile of the distributionand we filtered the original DTF values by remov-ing the edges with intensity below the statisticalthreshold.

2.3. Evaluation of global and locale!ciency

A graph is an abstract representation of a net-work. A graph G consists of a set of vertices (ornodes) V and a set of edges (or connections) Lindicating the presence of some sort of interactionbetween the vertices. A graph can be described interms of the so-called adjacency matrix A, a squarematrix such that, when a weighted and directededge exists from the node i to j, the correspond-ing entry of the adjacency matrix is Aij != 0; other-wise, Aij = 0. Two measures are frequently usedto characterize the local and global structure of

unweighted graphs: the average shortest path L andthe clustering index C [Watts & Strogatz, 1998;Newman, 2003; Grigorov, 2005]. The former mea-sures the average distance between two nodes, thelatter indicates the tendency of the network to formhighly connected clusters of nodes. Recently, a moregeneral setup has been proposed to study weighted(also unconnected) networks related to e!ciency[Latora & Marchiori, 2001, 2003]. The e!ciency eij

in the communication between two nodes i and jis defined as the inverse of the shortest distancebetween the vertices. Note that in weighted graphsthe shortest path is not necessarily the path withthe smallest number of edges. In the case when thetwo nodes are not connected, the distance is infiniteand eij = 0. The average of all the pair-wise e!cien-cies eij is the global-e!ciency Eg of the graph G:

Eg(G) =1

N(N " 1)

!

i"=j#V

1dij

(1)

where N is the number of vertices composing thegraph. The local properties of the graph can becharacterized by evaluating for every vertex i thee!ciency of Gi, which is the subgraph induced bythe neighbors of the node i [Latora & Marchiori,2001]. Thus, we defined the local-e!ciency El ofgraph G as the average:

El(G) =1N

!

i

Eg(Gi). (2)

Since node i does not belong to subgraph Gi,the local e!ciency measures how the communica-tion among the first neighbors of i is a"ected bythe removal of i [Latora & Marchiori, 2005, 2007].Hence, the local e!ciency is an indicator of the levelof fault-tolerance of the system. Separate ANOVAswere conduced for each of the two variables Eg andEl. Statistical significance was fixed at 0.05, andmain factors of the ANOVAs were the “between”factor GROUP (with two levels: SCI and CTRL)and the “within” factor BAND (with four levels:Theta, Alpha, Beta and Gamma). Greenhouse &Geisser correction has been used for the protectionagainst the violation of the sphericity assumptionin the repeated measure ANOVA. Besides post-hocanalysis with the Duncan’s test and significancelevel at 0.05 has been performed.

2.4. Detection of community structures

In order to detect the community structures, wehave implemented the Markov Clustering (MCl)

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algorithm [Vandongen, 2000; Enright et al., 2002].It is one algorithm of a few available that workseven with directed graphs and it is based on theproperties of the dynamical evolution of randomwalkers moving on the graph. This approach is alsouseful since it manages to achieve reliable resultswhen the graph contains self-loops, i.e. edges con-necting a node to itself. Since a community is agroup of densely connected nodes, a random walkerthat started in a node of a given community willleave this cluster only after having visited a largenumber of the community’s nodes. Hence, the basicidea implemented in the algorithm is to favor therandom motion within nodes of the same commu-nity. This is obtained by alternating the applica-tion of two operators on the transition matrix of therandom walk: the expansion operator and the infla-tion one. The expansion operator applied on a givenmatrix returns its square power, while the inflationoperator corresponds to the Hadamard power of thesame matrix, followed by a scaling step. In practice,the algorithm works as follows:

1. Take the adjacency matrix A and add a self-loopto each node, i.e. set Aii = 1 for i = 1, 2, . . . ,N ;

2. Obtain from A the transition probability matrixW that describes the random motion: Wij =Aij/

"k Akj. Every element Wij expresses the

probability to go from j to i in one step. W isa stochastic matrix i.e. a matrix of non-negativeelements and where the sum of elements of eachcolumn is normalized to one:

"Ni=1 Wij = 1;

3. Take the square of W (expansion step);4. Take the rth power (r > 1) of every element of

W 2 and normalize each column to one to obtaina new stochastic matrix W $: W $ = [(W 2)ij ]r/("

k [(W 2)kj]r) (inflation step);5. Go back to step 3.

Step 3 corresponds to computing random walksof “higher-lengths”, that is to say random walkswith many steps. Step 4 will serve to enhance theelements of a column having higher values. Thismeans, in practice, that the most probable tran-sition from node j will become even more proba-ble compared to the other possible transitions fromnode j. The algorithm converges to a matrix invari-ant under the action of expansion and inflation. Thegraph associated to such matrix consists of di"er-ent star-like components; each of them constitutes acommunity (or cluster) and its central node can beinterpreted as the basin of attraction of the com-munity. For a given r > 0, MCl always converges

to the same matrix; for this reason it is classifiedas a parametric and deterministic algorithm. Theparameter r tunes the granularity of the clustering,meaning that a small r corresponds to a few bigclusters, while a big r returns smaller clusters. Inthe limit of r # 1 only one cluster is detected. Inthe present study, an analysis at di"erent levels ofgranularity has been performed in order to find thevalue of r which better fits with the experimentaldata. Then we have represented how the average ofthe number of clusters changes as a function of r(Fig. 3). Finally the value r = 1.5 has been chosento study in detail how the nodes are organized inclusters.

3. Results

Figure 1 shows the realistic head model obtainedfor a representative subject. The twelve ROIs usedin the present study are illustrated in color on thecortex model that is gray colored. At the bottom ofFig. 1, we report the adjacency matrices represent-ing the cortical networks estimated, in the Alphafrequency band, from the two analyzed populationsduring the movement preparation. Note that suchnetworks are directed. Consequently, the obtainedadjacency matrix is not symmetric. The level of graywithin each matrix in figure encodes the number ofsubjects that hold the functional connection identi-fied by row i and column j.

As a measure of global and local perfor-mances of the network structure we have evalu-ated the global-e!ciency Eg and the local-e!ciencyEl indices obtained for each frequency band andfor each subject. The average values of Eg and El

derived from the healthy group (CTRL) and fromthe group of patients (SCI) are illustrated for eachband in the scatter plot in Fig. 2. We have per-formed an Analysis of Variance (ANOVA) of theobtained results. The Eg variable showed no sig-nificant di"erences for the main factors GROUPand BAND. In particular, the “between” factorGROUP was found having an F value of 0.83,p = 0.392 while the “within” factor BAND showedan F value of 0.002 and p = 0.99. The ANOVA per-formed on the El variable revealed a strong influ-ence of the between factor GROUP (F = 32.67,p = 0.00045); while the BAND factor and the inter-action between GROUP X BAND was found notsignificant (F = 0.21 and F = 0.91, respectively,p values equal to 0.891 and 0.457). Post-hoc testsrevealed a significant di"erence between the two

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Fig. 1. (Top) Reconstruction of the head model from magnetic resonance images. The twelve regions of interest (ROIs) areillustrated in color on the gray cortex and labeled according to previously defined acronyms. (Bottom left) Adjaceny matrixfor the control group (CTRL) in the Alpha (7–12 Hz) band. The level of gray encodes the number of subjects that hold thefunctional connection identified by the row i and column j. (Bottom right) Adjacency matrix for the patients group (SCI) inthe Alpha band. Same conventions as above.

examined experimental groups (SCI, CTRL) in theAlpha and Beta band (p = 0.01, 0.03, respectively).In particular, the average local-e!ciency of the SCInetworks was significantly higher than the CTRLnetworks for all these bands.

The identification of functional clusters withinthe cortical networks estimated in the control sub-jects and in the spinal cord injured patients duringthe movement preparation was addressed throughthe MCl algorithm (see Methods — Detection ofCommunity Structures (2.4)). In Fig. 3 the averagenumber of clusters detected in the CTRL and SCInetworks is reported as a function of the granular-ity parameter r. As it can be observed, for everyvalue of r, and for both the Alpha and Beta bands,

the average number of clusters is greater for theSCI group than for the CTRL one. One of the mainproblems with the MCl algorithm is the choice ofthe value of the granularity parameter to be used.Usually good values of r are in the range ]1, 3[ [Van-dongen, 2000; Enright et al., 2002]. For the caseunder study here, the presence of a plateau in Fig. 3indicates that there is a region of values such thatthe number of clusters does not strongly depend onr. We have decided to adopt the granularity r = 1.5that is a value in the plateau. For this value of r,the average number of clusters in the Alpha bandis equal to 3.2 for the cortical networks of the con-trol subjects, while it is equal to 5 for the spinalcord injured patients. In the Beta band, the average

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Fig. 2. Scatter plot of the average e!ciency indexesobtained from the estimated cortical networks. Global-e!ciency is on the x-axis; local-e!ciency is on the y-axis.The Greek symbol encodes the frequency band (! Theta, "Alpha, # Beta and $ Gamma) and it represents the averageof the values computed from the control (CTRL, blue-coloredfont) and spinal cord injured group (SCI, red-colored font).

number of cluster is 3.4 for the CTRL networks and4.6 for the SCI networks, as can be observed at thebottom in Fig. 3.

Figure 4 illustrates the partitioning of the cor-tical networks estimated in the Alpha band for arepresentative subject of the control (CTRL) groupand for a representative patient of the spinal cordinjured (SCI) group.

Functional networks are represented as three-dimensional graphs on the realistic head modelof the experimental subjects. The color of eachnode, located in correspondence to each corticalarea (ROI), encodes the cluster to which the nodebelongs. The functional clusters detected within thecortical networks in all the subjects and patientsparticipating in the present study are listed inthe following tables. Table 1 presents the resultsobtained in the Alpha band while Table 2 presentsthe results obtained in the Beta band.

In the Alpha band, the CTRL networks donot present a particularly complicated division intoclusters, since for each subject the large part of thecortical areas belong to a unique large community(Cluster 1). In general, the SCI networks organizedin a larger number of clusters are more clustered,and two main communities can be observed (Clus-ters 1 and 2). The first community is mostly com-posed of the cingulate motor areas (CM L andCM R), the supplementary motor areas (SM L andSM R) and the left primary motor area (MF L).The second community is predominantly composedof the left premotor areas (6 L) and the right pri-mary motor area of the foot (MF R). The remainingROIs tend to form isolated groups.

In the Beta band both the cortical networks ofthe control and spinal cord injured group tend toget organized in two main modules (Clusters 1 and2). In particular, while for both the populations thefirst cluster is principally composed of the cingulate

1 1.5 2 2.5 3 3.5 4 4.5 53

4

5

6

r

# cl

uste

rs

CTRLSCI

1 1.5 2 2.5 3 3.5 4 4.5 53

4

5

6

r

# cl

uste

rs

CTRLSCI

Alpha band

Beta band

Fig. 3. Representation of the mean number of clusters for CTRL (blue circles) and SCI (red squares) groups as a function ofr, granularity parameter of the MCl algorithm, in both the Alpha (top) and Beta bands (bottom).

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Fig. 4. Graphical representation of the identified clusters of ROIs within the functional networks estimated from a represen-tative control (CTRL) subject and spinal cord injured (SCI) patient during the movement preparation in the Alpha band. Thefunctional network is illustrated as a three-dimensional graph on the realistic cortex model. Spheres located at the barycenterof each ROI represent nodes. Black directed arrows represent edges. The graph partitioning is illustrated through the nodescoloring. Nodes with same colors belong to the same cluster.

Table 1. Cortical network partitioning in the Alpha frequency band.

CTRL

Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5 Cluster 6

ALFA CM L,CM R,6 L,6 R, SM R 0 0 0 0SM L,ML L,ML R,MF L,

MF R,7 L,7 RARGI CM L,CM R,6 L,6 R, SM L SM R ML L 0 0

ML R,MF L,MF R,7 L,7 R

CIFE CM L,CM R,6 L,6 R, ML L 7 L 0 0 0SM L,SM R,ML R,MF L,

MF R,7 RMADA CM L,CM R,6 L,SM R, 6 R,SM L,ML R, 0 0 0 0

ML L,7 L,7 R MF L,MF RMAMA CM L,CM R,6 L,6 R, ML R 7 L 7 R 0 0

SM L,SM R,ML L,MF L,MF R

SCI

Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5 Cluster 6

BARO CM L,CM R,6 R 6 L,SM R, SM L,ML L, 7 R 0 0ML R,MF L,MF R 7 L

IRIO CM L,CM R,SM L, 6 L 6 R ML L MF R 7 RSM R,ML R,MF L,7 L

MASI CM L,CM R,6 L,SM L, 6 R,MF R,7 R 7 L 0 0 0SM R,ML L,ML R,MF L

POAL CM L,CM R,6 R,SM R 6 L, SM L, ML L ML R MF L 7 RMF R,7 L

TRDA CM L,CM R,SM L,SM R, 6 L 6 R ML R 7 L 7 RML L,MF L,MF R

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Table 2. Cortical network partitioning in the Beta frequency band.

CTRL

Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5

ALFA CM L,CM R,6 L,6 R, 7 L 7 R 0 0SM L,SM R,ML L,ML R,

MF L,MF RARGI CM L,CM R,6 R,ML L, 6 L,SM L,MF R SM R 7 R 0

ML R,MF L,7 LCIFE CM L,CM R,6 L,SM L, 6 R,ML R 7 L 0 0

SM R,ML L,MF L,MF R,7 R

MADA CM L,CM R,SM L,SM R, 6 L,6 R,MF L ML L 7 R 0ML R,MF R,7 L

MAMA CM L,6 R,ML L,MF R CM R,6 L,SM L, 7 L 7 R 0SM R,ML R,MF L

SCI

Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5

BARO CM L,CM R,6 L,6 R, SM R,ML L 7 L 7 R 0SM L,ML R,MF L,MF R

IRIO CM L,CM R,6 L,SM L, 6 R ML R 7 L 7 RSM R,ML L,MF L,MF R

MASI CM L,CM R,6 L,SM L, 6 R ML L MF L 7 RSM R,ML R,MF R,7 L

POAL CM L,CM R,SM L, 6 L,ML R,MF R 7 L,7 R 6 R ML LSM R,MF L

TRDA CM L,CM R,SM L,SM R, 6 R,ML R 6 L ML L 0MF L,MF R,7 L,7 R

motor areas CM L and CM R, the SM L and SM Rand the MF L and MF R, the second cluster doesnot present a common set of ROIs across the exper-imental subjects neither in the CTRL or the SCIgroup.

4. Discussion

The evaluation of the estimated cortical networkswas addressed by means of a set of measures typi-cal of complex network analysis [Boccaletti et al.,2006; Micheloyannis et al., 2006; Stam & Reijn-eveld, 2007; Hilgetag et al., 2000]. We have firstcomputed global (Eg) and local e!ciency (El), twomeasures that allow characterizing the organiza-tion of the functional flows in both the inspectedpopulations [De Vico Fallani et al., 2007b]. Theresults indicate that spinal cord injuries signifi-cantly (p < 0.05) a"ect only the local proper-ties of the functional architecture of the corticalnetwork in the movement preparation. The globalproperty of long-range integration between theROIs within the network did not di"er significantly

(p > 0.05) from the healthy behavior. The higheraverage value of local e!ciency in the SCI groupsuggests a larger level of the internal organizationand a higher tendency to form modules. In partic-ular, this di"erence can be observed in the two fre-quency bands Alpha (7–12 Hz) and Beta (13–29 Hz)that are already known for their involvement in elec-trophysiological phenomena related to the prepa-ration and to the execution of limbs movements[Pfurtscheller & Lopes da Silva, 1999]. Althoughthe e!ciency indexes describe the network topol-ogy concisely, they are not able to give informationabout the number of modules and their compositionwithin the network. For this reason, the detectionof community structures was addressed by means ofthe Markov Clustering (MCl) algorithm [Vandon-gen, 2000; Enright et al., 2002]. The same methodhas already been used successfully to detect clus-ters in sequence similarity networks [Enright et al.,2002] and in configuration space networks derivedfrom free-energy landscapes [Gfeller et al., 2007].The obtained results reveal a di"erent average num-ber of clusters for the functional networks of the

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spinal cord injured patients and the control sub-jects in both the main spectral contents. In parti-cular, in the Alpha band the SCI network presentsan average number of modules equal to five, whilethe CTRL network appears to be divided into threegroups. This outcome is in accordance with the sig-nificant (p < 0.05) higher level of local-e!ciencyfound in the functional networks of the SCI patientswith respect to the control subjects. A high El valuereflects a high clustering index C and therefore ahigh density of network communities. The corti-cal areas of the control subjects do not present aclear partitioning in di"erent modules. They ratherappear to belong to a unique community, mean-ing that they are all involved, in the same way, inthe exchange of information during the movementpreparation. The analysis of the functional commu-nities within the networks obtained for the spinalcord patients revealed a higher tendency to formseparate clusters. The premotor areas (Brodmann6 L and 6 R), the associative regions (Brodmann7 L and 7 R) and the right primary motor area ofthe foot (MF R) break away from the large modulethat was found in the networks of the CTRL group.In particular, the area MF R and the region 6 Lbelong to the same cluster in at least three experi-mental patients. This result reveals the necessity ofthe SCI networks to hold a more e!cient commu-nication between these frontal premotor and pri-mary motor structures, which are already knownto be active during the successful execution of asimple movement [Ohara et al., 2001]. In the Betaband, the average number of identified clusters inthe SCI networks and in the CTRL networks isless di"erent. Moreover, the ROIs that appeared tobelong to di"erent clusters in the Alpha band are inthis case functionally tied in the same community.In summary, while in the Alpha band the controlgroup mostly presented a unique large cluster; thespinal cord injured patients mainly exhibited twoclusters. These two largest communities are mainlycomposed of the cingulate motor areas with the sup-plementary motor areas and of the premotor areaswith the right primary motor area of the foot. Thisfunctional separation is thought to be responsiblefor the highest level of internal organization in theestimated networks and strengthens the hypothe-sis of a compensative mechanism due to the partialalteration in the primary motor areas because ofthe e"ects of the spinal cord injury [De Vico Fallaniet al., 2007a].

Acknowledgments

The present study was performed with the supportof the COST EU project NEUROMATH (BMB0601), of the Minister for Foreign A"airs, Divisionfor the Scientific and Technologic Development, inthe framework of a bilateral project between Italyand China (Tsinghua University). This paper onlyreflects the author’ views and funding agencies arenot liable for any use that may be made of the infor-mation contained herein.

Notes

In the current work, all the estimated functionalnetworks are treated as unweighted and directedgraphs. They all have the same number of con-nections representing the 25% (for the communitystructure analysis) and the 30% (for the e!ciencyindexes analysis) most powerful links within thenetwork. These particular values belonged to aninterval of thresholds (from 0.1 to 0.5), for whichresults remained significantly stable [De Vico Fal-lani et al., 2007a].

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