Computers in Biology and Medicine 41 (2011) 1178–1186

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Computers in Biology and Medicine

0010-48

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/cbm

EEG-based functional networks in schizophrenia

Mahdi Jalili a,n, Maria G. Knyazeva b,c

a Department of Computer Engineering, Sharif University of Technology, Tehran, Iranb Department of Clinical Neuroscience, Centre Hospitalier Universitaire Vaudois (CHUV), and University of Lausanne, Lausanne, Switzerlandc Department of Radiology, Centre Hospitalier Universitaire Vaudois and University of Lausanne, Switzerland

a r t i c l e i n f o

Keywords:

EEG

Schizophrenia

Functional connectivity

Graph theory

Unpartial cross-correlation

Partial cross-correlation

25/$ - see front matter & 2011 Elsevier Ltd. A

016/j.compbiomed.2011.05.004

esponding author. Tel.: þ98 0 21 6616 4633;

ail address: MJalili@sharif.edu (M. Jalili).

a b s t r a c t

Schizophrenia is often considered as a dysconnection syndrome in which, abnormal interactions between

large-scale functional brain networks result in cognitive and perceptual deficits. In this article we apply

the graph theoretic measures to brain functional networks based on the resting EEGs of fourteen

schizophrenic patients in comparison with those of fourteen matched control subjects. The networks were

extracted from common-average-referenced EEG time-series through partial and unpartial cross-correla-

tion methods. Unpartial correlation detects functional connectivity based on direct and/or indirect links,

while partial correlation allows one to ignore indirect links. We quantified the network properties with the

graph metrics, including mall-worldness, vulnerability, modularity, assortativity, and synchronizability.

The schizophrenic patients showed method-specific and frequency-specific changes especially pro-

nounced for modularity, assortativity, and synchronizability measures. However, the differences between

schizophrenia patients and normal controls in terms of graph theory metrics were stronger for the

unpartial correlation method.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Techniques from graph theory are increasingly being applied tomodel the functional and/or structural networks of the brain [1,2].The brain networks can be studied at different levels ranging frommicro-scale containing a number of interconnected neurons tomacro-scale containing distributed brain regions. To construct thelarge-scale networks, signals recorded from the brain via methodssuch as electroencephalography (EEG), magnetocephalography(MEG), functional magnetic resonance imaging (fMRI), or diffusiontensor imaging (DTI), are used [3–7]. Often, binary (directed orundirected) adjacency matrices are analyzed [1,2], where binarylinks represent the presence or absence of a connection. The firststep in analyzing brain networks is to extract its structure fromthe time-series. Possible methods are cross-correlation, coherence,and synchronization likelihood [3–6]. The next step is to representit in a number of biologically meaningful measures. To this end,measures such as characteristic path length, efficiency of connec-tions, clustering coefficient, modularity, node degree and central-ity, assortativity, and synchronizability are applied [7,8].

Large-scale brain networks, comprising anatomically or func-tionally distinct regions and inter-regional pathways, exhibitspecific non-random patterns with the small-world and/or scale-

ll rights reserved.

fax: þ98 0 21 6601 9246.

free properties [9,10]. Graph theoretical analysis on anatomicaland functional networks of the brain have revealed its economicalsmall-world structure characterized by high clustering (transitiv-ity) and a short characteristic path length [11]. The brain func-tional networks are cost-efficient in the sense that they provideefficient parallel processing for low connection cost [12]. Braindisorders influence the anatomical and functional brain networks.

Brain wirings may show abnormal patterns in schizophrenia(SZ). SZ symptoms affect the patients by manifesting as auditoryhallucinations, paranoid or bizarre delusions and/or disorganizedspeech and thinking in the context of significant social and/oroccupational dysfunction. About 1% of the population worldwidesuffers from different forms of SZ [13]. Additionally, another 3% ofthe population has SZ-type personality disorders. SZ is the fourthleading cause of disability in the developed counties for people atthe age of 15–44.

Schizophrenic patients show the abnormal patterns of brainconnectivity. MRI-based studies on a large group of SZ patientsrevealed the reduced hierarchy of multimodal networks andincreased connection distance [14]. The disruption of effectivesmall-world architecture in many cortical regions, includingprefrontal, parietal, and temporal lobes was shown for functionalnetworks based on EEG [4] and fMRI [15]. Another fMRI studyshowed reduced clustering and small-worldness in SZ, as well asreduced probability of high-degree hubs, and increased robust-ness to networks’ component failures [16]. Nonlinear correlationanalysis of EEG time-series also confirmed lower clustering and

M. Jalili, M.G. Knyazeva / Computers in Biology and Medicine 41 (2011) 1178–1186 1179

shorter path lengths in SZ compared to healthy subjects [17]. Thestructural properties of EEG-based networks activated by workingmemory tasks are also abnormal in SZ patients [18].

In this paper we consider fourteen SZ patients and fourteenmatched control subjects. Brain functional networks are extractedfrom subjects’ EEG time-series through unpartial and partialcross-correlation analysis, the latter being suggested as comple-mentary tools for extracting brain networks [19]. Then, the graphmetrics, such as node-strength, small-worldness, vulnerability,modularity, assortativity, and synchronizability, are calculatedand compared for SZ and control groups.

2. Methodology

2.1. EEG recording

Fourteen schizophrenic patients (mean age 33.5710.1; 11 menincluding) were recruited from the in/outpatient schizophreniaunits of the Psychiatry Department, Lausanne University Hospital.All diagnoses were made according to DSM-IV criteria on the basisof the Diagnostic Interview for Genetic Studies (DIGS) [20], or by aconsensus of two experienced psychiatrists after a systematicreview of medical records. Fourteen healthy control subjects (meanage 33.979.9) without known neurological or psychiatric illness ortrauma and without substance abuse or dependence matched thepatients for age, gender, and handedness. They were recruited fromthe local community based on the DIGS interview [20] or theSymptom Checklist [21] (8 and 6 subjects, respectively). Allparticipants in this study were fully informed about the studyand gave written consent. All the procedures conformed to theDeclaration of Helsinki (1964) by the World Medical Associationconcerning human experimentation and were approved by the localethics committee of the Lausanne University.

The 3–4 min of resting-state eyes-closed EEG data werecollected in a semi-dark room with a low level of environmentalnoise while each subject was sitting in a comfortable chair. Theresting-state EEGs were recorded with the 128-channel GeodesicSensor Net (EGI, USA) with all the electrode impedances keptunder 30 kO. The recordings were made with vertex referenceusing a low-pass filter set to 100 Hz. The signals were digitized ata rate of 1000 samples/s with a 12-bit analog-to-digital converter.They were further filtered (FIR, band-pass of 1–70 Hz, notch at50 Hz), re-referenced against the common average reference(CAR), and segmented into non-overlapping epochs using theNS3 software (EGI, USA).

Artifacts in all channels were edited off-line: first, automati-cally, based on an absolute voltage threshold (100 mV) and on atransition threshold (50 mV), and then by thorough visual inspec-tion, which allowed us to identify and reject epochs or channelswith moderate muscle artifacts not reaching threshold values. Wealso excluded from further analysis the sensors that recordedartifactual EEG in at least one subject. Finally, 101 sensors wereused for further computation. Data were inspected in 1 s epochsand the number of artifact-free epochs entered into the analysiswas 185751 for the patients, and 195745 for the controlsubjects.

We have previously reported some aspects of this dataset,including the whole-head dysconnection maps [22], hypofrontal-ity [23] of alpha rhythm, and asymmetry of functional connectiv-ity [24].

2.2. Constructing brain functional networks

The connectivity matrices of brain networks were extractedfrom EEG time-series. In order to reveal the functional

connectivity taking into account both direct and indirect links,one should consider partial and unpartial correlations [19]. Wecalculated the Pearson correlation coefficient for all possible pairsbetween 101 artifact-free sensors. More precisely, the Pearsoncorrelation coefficient between sensor i and j is obtained as

rij ¼covði,jÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

varðiÞvarðjÞp ð1Þ

where cov (i,j) is the covariance between nodes i and j, and var(i)is the variance of node i. We further calculated the partialcorrelation matrices. The correlation between sensor i and j

partialized to the group of sensors k is obtained as

rijk ¼rij�rjkrikffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�r2

ikÞð1�r2jkÞ

q ð2Þ

where rij is the unpartialized correlation between i and j, rik is thecorrelation between i and k, and rjk is the one between j and k.Note that k can be a group consisting of one or several sensorlocations.

The first-order partial correlation coefficient has been proposedfor constructing brain networks [19]. In other words, the partialcorrelation between two sensors would be the minimum of 100correlation values: one unpartialized correlation and 99 correlationvalues partialized to each of the 99 remaining sensors. In this way,for each epoch in each subject, two 101�101 weighted correlationmatrices were obtained: partialized and unpartialized. As the partialand unpartial correlations were computed for all epochs, they werethen averaged over all artifact-free epochs for each subject. Theseaveraged correlations were used to construct the correspondingfunctional networks pooled in the groups of SZ patients and controlsubjects.

A precise way of computing partial correlation is to consider allpossible choices and then to select the minimum among the obtainedcorrelation values. In other words, to compute the correlationbetween two sensor locations, one should first compute the unpar-tialized correlation (zero-order correlation), all first-order correlations(correlations partialized to single sensors), all second-order correla-tions (those partialized to any combination of two sensors), all third-order correlations (those partialized to any combination of threesensors), and so on. Then, the minimum among these values is thetrue correlation between the two sensors. For many networks(especially if one analyzes a number of subjects each of who isrepresented by many EEG epochs), the computation of such ameasure is an expensive task. We limited the computations to first-order partial correlations.

The next step was to binarize the weighted matrices. Itresulted in binary adjacency matrices, with elements equal to 1,if the absolute correlation value exceeds a threshold TH, or 0, if itdoes not. The choice of the threshold TH has significant influenceon the constructed connectivity structures such that fixing TH lowvalues generates densely connected networks, whereas networksbased on large TH values are sparse. To analyze the networkproperties as a function of TH, the correlation matrices werethresholded at different TH values.

2.3. Network measures

The binary adjacency matrices were summarized in a fewneurobiologically meaningful network metrics [2]. Let us showthe binary adjacency matrix as A¼(aij), where aij indicates thecorresponding element in the (i,j) entry of A. A simple measurefor the network is node degree that is defined by the sum of thelinks connecting a node. More precisely, the degree of node i is

M. Jalili, M.G. Knyazeva / Computers in Biology and Medicine 41 (2011) 1178–11861180

defined as

ki ¼X

j

aij ð3Þ

Alternative nodal information for the original weighted net-work is node-strength. Let us consider a weighted networkW¼(wij), where wij is the weight of the link between node i and j.The node-strength is defined as the sum of the weights of thelinks tipping to it

si ¼X

j

wij ð4Þ

2.3.1. Small-worldness index measuring functional segregation and

integration

Brain’s ability in functional segregation and integration isimportant for binding and information processing. Functionalsegregation is the ability in information processing in a specia-lized manner. A possible measure for quantifying the functionalsegregation is the clustering coefficient defined as [7,25]

C ¼1

N

Xk

Pi,jaijaikajk

kkðkk�1Þð5Þ

where N is the number of nodes of the network under study.Functional integration is an ability of the brain to combine

information processed in distributed brain regions. The frequentgraph theory measure used for this purpose is the global efficiency

defined as [7]

E¼1

NðN�1Þ

Xi,j

1

lijð6Þ

where lij is the length of the shortest path between the nodes i

and j.It has been shown that many real-world networks, including

brain networks, have a structure that is neither pure random norregular but somewhere in between [25,26]. They are indeedsmall-worlds. A measure, called small-worldness, has been pro-posed to capture the network ability of segregation and integra-tion [27], which estimates its clustering coefficient and efficiencycompared to those of a number of properly random networks:

Small-worldness¼C

Crandom

E

Erandomð7Þ

where Crandom and Erandom are the average clustering coefficientand efficiency in the corresponding Erdos–Renyi graph [28]having the same number of nodes and edges as compared tothe original graph. For each case, we created 10 randomizednetworks with the same degree distribution and computed theirmetrics by averaging over these 10 realizations.

2.3.2. Modularity index

Another metric for measuring the ability of a network forsegregation is its modularity, which is observed in many real-world networks. To capture the degree of modularity in thenetwork with predetermined M modules, the following indexhas been proposed [29]:

Q ¼XiAM

eii�XjAM

eij

0@

1A

2264

375 ð8Þ

where the network is fully partitioned into M non-overlappingmodules (clusters), and eij represents the proportion of all linksconnecting nodes in module i with those in module j. Themodularity index is computed by estimating the optimal modularstructure for a given network [2,29].

2.3.3. Measures of centrality:

In order to take into account the significance (centrality) ofnetwork elements, i.e. nodes and edges, their centrality can beconsidered [30]. Let us denote the edge between the nodes i and j

by eij. Edge-betweenness centrality (load or traffic) rij of thenetwork is defined as

rij ¼Xpau

GpuðeijÞ

Gpuð9Þ

where Gpu is the number of shortest paths between nodes p and u

in the graph; and Gpu(eij) is the number of these shortest pathsmaking use of the edge eij.

In a similar way, one can define node-betweenness centrality.Node-betweenness centrality Oi is a centrality measure of node i

in a graph, which shows the number of shortest paths making useof node i (except those between the ith node with the othernodes) [30]. More precisely

Oi ¼X

ja iak

GjkðiÞ

Gjkð10Þ

where Gjk is the number of shortest paths between nodes j and k

and Gjk(i) is the number of these shortest paths making use of thenode i.

2.3.4. Measures of resiliency

Networks may undergo random and/or intentional failures intheir components, and their resiliency against such a failure is ofhigh importance for their proper functioning. If the performanceof the network is associated to its efficiency, the vulnerability of anode would be the amount of drop in the performance when thenode is removed along all tipping edges from the network. Moreprecisely

Vi ¼E�Ei

Eð11Þ

where E is the efficiency of the network and Ei is the efficiency ofthe network after the removal of the nodes i. A measure for thenetwork vulnerability is the maximum vulnerability for all itsnodes

V ¼maxiVi; i¼ 1,2,. . .,N ð12Þ

Another measure for resiliency of networks is based on thedegree–degree correlation, i.e. assortativity of the network. Manyreal-world networks show assortative or disassortative behavior.In assortative networks, nodes with high degree tend to connectto other nodes with high degree, whereas in disassortativenetworks, nodes with high degree tend to be linked to those withlow degree [31]. In order to calculate the degree correlation, onemay use the Pearson correlation of the degrees at both ends of theedges of the network [32]

r¼ð1=MÞ

Pj4 ikikjaij�½ð1=MÞ

Pj4 ið1=2ÞðkiþkjÞaij�

2

ð1=MÞP

j4 ið1=2Þðk2i þk2

j Þaij�½ð1=MÞP

j4 ið1=2ÞðkiþkjÞaij�2ð13Þ

where M is the total number of the edges of the network and ki isthe degree of node i. If r40, the network is assortative, whereasro0 indicates a disassortative network. For r¼0 there is nocorrelation between the node-degrees. The assortative networksare likely to consist of mutually coupled high-degree nodes and tobe resilient against random failures. In contrast, the disassortativenetworks are likely to have vulnerable high-degree nodes.

2.3.5. Measure of synchronizability

Various brain disorders have been linked to abnormalities inbrain synchronization [33]. The eigenratio of the Laplacian matrixof the connection graph has been proposed as a measure for its

M. Jalili, M.G. Knyazeva / Computers in Biology and Medicine 41 (2011) 1178–1186 1181

synchronizability [34]. The Laplacian of a graph A¼(aij) isobtained as L¼D–A, where D¼(dii) is a diagonal matrixwith node-degrees in the diagonal entries. Let us considerthe eigenvalues of the Laplacian matrix as li, ordered as0¼l1rl2ryrlN, where N is the number of nodes in thenetwork. The eigenratio lN/l2 is a measure for the degree ofsynchronization properties of the network and the smaller theeigenratio of a network the better its synchronizability [34].

2.4. Statistical assessments

The assessment for statistical significance of any possibledifferences between the graph metrics of brain networks in SZpatients and normal controls was performed through Wilcoxon’sranksum test. The tests were carried out separately for any valuesof the threshold. The P-values are not corrected for multiplecomparisons.

All the computations were performed in MatLab. We usedsome available tools, i.e. BGL toolbox [35] and connectivity tool-box [36].

3. Application

The computations resulted in two 101�101 weighted con-nection matrices for each subject; one based on unpartial cross-correlations and another one based on partial correlations. As theunpartial correlation values were compared between SZ andcontrol groups, the percentage of sensor pairs with significantlydifferent correlation values (Po0.05, Wilcoxon’s ranksum test)varied from 8.73% in beta band to 4.04% in delta band (Fig. 1). Thepercentage of sensor pairs, where the partial correlations weresignificantly different between SZ patients and controls (Po0.05,Wilcoxon’s ranksum test), varied from 7.72% in beta band to5.03% in delta band (Fig. 1).

In order to obtain a better spatial pattern on the changes wecomputed the node-strength that is the sum of the weights ofedges linking a node to others (Fig. 2). The strength-maps werenot the same for partial and unpartial correlations. The differencemaps (SZ patients vs. controls) showed the patchy changes ofstrength in SZ patients when partial correlations were used to

Fig. 1. Differences in the cross-correlation matrices in SZ patients compared to contro

correlation matrices (101�101 matrices) at Po0.05 (Wilcoxon’s ranksum test) in SZ

whereas nonsignificant entries are left white. The analysis was carried out in different

(13–30 Hz), and gamma (30–70 Hz).

construct the networks. These changes were represented eitherby singletons or by small clusters of sensors. By contrast, therewas a clear pattern of changes for unpartial correlations in alpha,beta, and gamma bands. We found the clusters of sensors over theright frontal and left parietal regions (alpha band) and over theright frontal continued to the central region (gamma band),showing decreased values of node-strength in the SZ patientscompared to the controls. Node-strengths based on unpartialcorrelations in beta band were characterized with increasedvalues in the clusters of sensors located in frontal, left occipital,left temporal, and right temporal (Fig. 2).

In order to analyze the properties of the brain networks, webinarized the weighted correlation matrices by thresholdingthem. The binary adjacency matrices in SZ patients and controlswere then compared for their graph metrics. Functional networksof SZ brains constructed through partial correlations showed nosignificant changes in the small-worldness index as compared tothose of normal controls (Fig. 3). However, there were significantdifferences for the networks based on unpartial correlations. Inparticular, we found that for the high-value TH threshold, thesmall-worldness was different in alpha and beta bands.

The modularity metric is used for measuring the segregationproperties of a network. The modularity index of unpartialnetworks was significantly higher in SZ than in controls(Po0.05, Wilcoxon’s ranksum test) for a large range of THs inthe beta band (Fig. 4). Except for a single TH value in the gammaband with significant difference for unpartial case, we found nochanges in the modularity (Fig. 4). Therefore, the beta band canserve as a marker for abnormal modular structure in SZ functionalnetworks.

Resiliency of brain networks belongs to those properties thatmay be altered by SZ. We considered two markers that are indirectlyrelated to the resilient behavior of a network: vulnerability andassortativity. The vulnerability index showed little changes for thenetworks based on partial correlations. For the case of unpartialcorrelations, there were a few changes especially for the two clustersof TH values in middle ranges in gamma band (Fig. 5), where thevulnerability of functional networks in the SZ patients was lowerthan in controls (Po0.05, Wilcoxon’s ranksum test).

The pattern of changes in the assortativity, i.e., degree–degreecorrelation, in delta band was almost the same for partial and

ls. The graphs show the significant differences for the partial and unpartial cross-

patients vs. normal controls. The significantly different correlations are in black,

frequency bands, including delta (1–3 Hz), theta (3–7 Hz), alpha (7–13 Hz), beta

Fig. 2. Whole-head difference maps of node-strengths in SZ patients vs. normal controls Group-averaged difference node-strength-maps (the strength of a node is defined

in Eq. (4)) for delta, theta, alpha, beta, and gamma bands. The difference maps show the significant (Po0.05, Wilcoxon’s ranksum test) between-group changes. Sensors

with strength values significantly higher in SZ patients than in controls are in red, whereas those with lower values are in blue. There are no significant differences in the

gray regions.

Fig. 3. Functional segregation and integration of brain functional networks in SZ patients with small-worldness index. The graphs show the mean values of the small-

worldness index as a function of the threshold in SZ patients and normal controls for different frequency bands, including the delta, theta, alpha, beta, and gamma. The

brain functional networks were based on partial and unpartial cross-correlation matrices. The blue dots represent the threshold values, where the value of the small-

worldness index is significantly different between SZ and normal groups (Po0.05, Wilcoxon’s ranksum test). (For interpretation of the references to color in this figure

legend, the reader is referred to the web version of this article.)

M. Jalili, M.G. Knyazeva / Computers in Biology and Medicine 41 (2011) 1178–11861182

unpartial case; SZ networks showed decreased assortativity for acluster of high values of the threshold (Fig. 6). It is worthmentioning that the unpartial correlation values between a pairof sensors are often much larger than the corresponding partialcorrelation value. The assortativity of brain networks wasincreased for a cluster of low threshold values in alpha band forunpartial and in beta band for partial case. We also calculated thecentrality measures (node and edge-betweenness centrality mea-sures); however, SZ patients showed no significant differences ascompared to control in these centrality measures.

Finally, we tested the brain networks for their synchronizationproperties. Fig. 7 shows the synchronizability index, i.e. theeigenratio of the Laplacian matrix of the connection graph forthe SZ patients and controls. The networks based on partialcorrelations have altered synchronizability only in the beta bandand for some high values of the threshold. However, for thenetworks based on unpartial correlations, decreased synchroniz-ability, i.e. increased eigenratio, is characteristic in the theta,alpha, beta, and gamma bands. We found a cluster of intermedi-ate threshold values (with a larger range in the alpha and gamma

Fig. 4. Measure of modularity of brain functional networks in SZ patients compared to normal controls. The graphs show the mean values of the modularity index as a

function of threshold in the SZ patients and normal controls. Other designations are as in Fig. 3.

Fig. 5. Measure of vulnerability of brain functional networks in SZ patients compared to normal controls. The graphs show the mean values of the vulnerability index as a

function of threshold in SZ patients and normal controls. Other designations are as Fig. 3.

Fig. 6. Measure of assortativity of brain functional networks in SZ patients vs. normal controls. The graphs show the mean values of the assortativity, i.e., degree–degree

correlation index as a function of threshold in SZ patients and normal controls. Other designations are as Fig. 3.

M. Jalili, M.G. Knyazeva / Computers in Biology and Medicine 41 (2011) 1178–1186 1183

Fig. 7. Measure of synchronizability of brain functional networks in SZ patients compared to normal controls. The graphs show the mean values of the synchronizability

index, i.e., the eigenratio (the largest eigenvalue of the Laplacian matrix of the connection graph divided by its second smallest eigenvalue), as a function of threshold in SZ

patients and normal controls. Other designations are as Fig. 3.

M. Jalili, M.G. Knyazeva / Computers in Biology and Medicine 41 (2011) 1178–11861184

compared to other bands), where the synchronization propertiesof the SZ networks were worse than those of controls.

4. Discussion

Recent advancements in statistical methods for analyzingcomplex networks have influenced studies in brain networkorganization. It has been shown that large-scale brain networksconstructed through recordings such as EEG, MEG, FMRI, and DTIshow attributes such as small-world property, modularity, andscale-free degree distribution [1,36–40]. Graph theory analysis ofbrain signals may give useful information on the mechanisms thatvarious brain disorders influence the brain structural and func-tional organization. SZ is one of those disorders that has beenshown to alter a number of metrics of brain functional networks[4,14–18].

The abnormality in the cotico-cortical connectivity has beenwidely reported in SZ [37–40]. A necessary link between abnor-mal circuitry and basic SZ symptoms is functional connectivity.However, the changes in the anatomical and functional connec-tivity in SZ are not in the same direction [41]. While theanatomical connectivity assessed by DTI showed nearly uniformdecrease in SZ, the functional connectivity captured through fMRIshowed both increased (for some connections such as cingulateand thalamus) and decreased (for some connections such asmiddle temporal gyrus) regimes [41]. The application of state-spaced based synchronization measure (S-estimator) on the dataused for the present study has revealed the coexistence of hypo-and hyper-synchronization clusters in SZ [22]. The abnormalfunctional connectivity and synchronization in SZ has also beenrevealed by other studies (reviewed in Ref. [42]).

Following current views by ‘‘functional connectivity’’ weunderstand cooperation between distributed neural assembliesin the brain. Common ways of assessing the cooperation amongcortical networks are measuring their synchronization, correla-tion, or coherency. Graph theory analysis is another methodproviding a global picture of the functional connectivity andcooperation among distributed brain regions [1,36–40]. In thepresent study we applied the graph theory techniques on theEEGs of SZ patients and compared their statistics with those ofnormal control subjects. The observed wide-spread morphologi-cal abnormalities in SZ such as enlarged ventricles (reviewed in

Ref. [43]), decreased cortical volume or thickness coupled withincreased cell packing density [44–46], and reduced clustering ofneurons [40], suggest the dysconnectivity model of SZ [47]. Thedysconnection hypothesis suggests anomalous structural integ-rity and/or functional connectivity in SZ [47].

The association between anatomical and functional connectiv-ity in the brain signifies a challenging issue in neuroscienceresearch. Segregation and integration in the brain have beenproposed as two potential principles linking these differentmodes of brain connectivity [48]. The interplay of segregationand integration in brain networks may cause information bindingresulting in the generation of information that is simultaneouslyhighly spread and highly mixed. These two principles, i.e. func-tional segregation and integration of brain networks, play impor-tant roles in information processing and proper functionality ofthe brain. The modularity index is one of those measuresfrequently used for characterizing the segregation properties ofcomplex networks and the brain networks have been shown tohave a modular structure [49,50]. We found the beta band as amarker of abnormal modularity of brain networks in SZ patients.The small-worldness is another measure revealing the functionalsegregation and integration in the networks [51–53]. In our data,this measure was significantly different in SZ patients as com-pared to controls for a cluster of high valued thresholds in alphaand beta bands. Previous fMRI studies in SZ patients and healthycontrols also showed disturbed topological properties in the brainfunctional networks in patients, such as a lower strength anddegree of connectivity, a lower efficiency, a lower clusteringcoefficient, and, hence, disrupted small-worldness [15]. Therefore,abnormal small-worldness is associated with partial disorganiza-tion of brain networks in SZ-affected brain.

Networks may undergo random and/or intentional failures intheir components. Many biological networks have shown resilientbehavior against random failures and a number of measures suchas assortativity and vulnerability are important in characterizingsuch a resilient behavior. The pattern of changes for thesemeasures in SZ patients reveals the alternation in the resiliencyof brain functional networks in SZ.

Synchronization is believed to play an important role ininformation processing in the brain at both macroscopic andcellular levels [26,54]. Our results showing a pattern of significantdifferences in the synchronization properties of brain networks inSZ is of high importance, since a number of previous reports have

M. Jalili, M.G. Knyazeva / Computers in Biology and Medicine 41 (2011) 1178–1186 1185

shown that the synchronization among the cortical regions isaltered in SZ [22,33,55–59]. In particular, our previous analysis ondifferent aspects of the dataset reported here showed a synchro-nization landscape in SZ characterized by synchronizationchanges in centro-parietal sensors over the left hemisphere, alarge cluster over the right hemisphere, and a midline cluster oflocations over the centro-parieto-occipital region [22]. The ana-lysis reported here looks at the synchronizability from differentperspective and considers the synchronization properties of thebrain networks rather than looking for a synchronous pattern inthe original EEG signals.

We found that, in general, the differences between SZ patientsand control subjects in terms of graph metrics are more pro-nounced in the cases where unpartial correlations are used ascompared to partial correlations. The unpartial correlation gives anestimate of the (direct or indirect) relationship between twovariables. Because of low computational cost, it can be readilyused to analyze large-scale complex networks such as brainnetworks extracted from fMRI or EEG. By contrast, partial correlation(especially of higher-orders) is often computationally expensive.However, it reduces the prediction of indirect functional connectiv-ity. The methods, therefore, provide different information on thesystem under study and may complement each other. Unpartialcorrelation analysis detects any kind of linear functional connectiv-ity between two sites, which can be direct or indirect through othersites. Whereas, partial correlation analysis tries to minimize detect-ing the functional connectivity resulted by indirect links.

5. Summary

A possible approach to study functional connectivity is tomodel it via the graph theory techniques and to analyze the graphproperties of the networks. Given that different brain statesrespond to different measures, complementary information onthe usefulness of functional brain networks can be obtained bycombining the networks based on partial and unpartial correla-tions. Here we studied multichannel resting-state EEGs of four-teen SZ patients and fourteen matched healthy control subjects interms of their graph properties. First, the pair-wise partial andunpartial cross-correlations were computed for all noise- andartifact-free EEG channels. The average correlation matrices werethen used to extract the connection graph of the brain in SZpatients and normal subjects. Unpartial networks of SZ patientsshowed a distinct pattern in terms of node-strengths. The clustersof sensors with decreased strength were characteristic for thealpha and gamma bands, while those with increased strengthwere found in the beta band.

Then, we thresholded the correlation matrices at differentvalues and computed graph metrics for the constructed networks.We considered three classes of measures among various metricsavailable in graph theory: those related to the segregation/integration, resiliency, and synchronizability of networks. Thesmall-worldness and modularity were among those measuresrepresenting the segregation/integration properties of the brainnetworks. For high threshold values, the small-worldness of SZbrains was significantly higher in alpha band, whereas it waslower in beta band. Abnormal small-worldness has been reportedin SZ patients through fMRI [14,15] and low-resolution EEG [4,15]studies. We found the beta band as a marker for abnormalities inthe modularity of brain networks in SZ. The SZ-specific reductionof modularity in beta band was in agreement with the resultsobtained through fMRI [60].

Brain networks may undergo failures in their components. Wefound that the networks of SZ brains in gamma band are lessvulnerable as compared to normal controls. Increased robustness

of SZ brain networks extracted through fMRI has also beenpreviously reported [60]. We also tested assortativity of brainnetworks. Functional MRI-based brain networks have shownincreased assortativity in SZ patients [14]. Our data showed bothincreased and decreased assortativity, which was band- andmethod-specific. The changes in the centrality measures havealso been reported for SZ brains [17]; however, we failed to showany significant difference in node and edge centrality measures inour data.

There are many works reporting abnormalities in neuraloscillation and synchronization in SZ (reviewed in Ref. [42]).However, we could not find any prior work studying thesynchronization properties of brain networks in SZ. Our datashowed SZ-specific wide-spread decreased synchronizability intheta, alpha, beta, and gamma bands. Temporal synchronization isimportant in information binding and cortical computing in thebrain [54]. Decreased level of synchronizability in SZ may berelated to the dysfunction of cortical networks in this illness.

This study has several limitations, implying that our resultsneed to be considered as preliminary requiring further validation.The studied group was relatively small and needs to be repeatedin larger diverse groups. The study also needs to be performedthrough other imaging techniques such as fMRI, DTI, and/or MEG.In addition to graph theoretical tools, which enable the analysis ofthe network’s topological features, other analysis approaches canalso be done in parallel. For example, approaches focusing on thethree-dimensional structure of brain networks including morpho-metric methods (such as those for measuring wiring length orvolume [61]) can be employed.

Conflict of interest statement

None declared.

Acknowledgments

The authors would like to thank Dr. Kim Q. Do for providinginformation on the SZ patients.

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