EEG-based functional networks in schizophrenia

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

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EEG-based functional networks in schizophrenia Mahdi Jalili a,n, Maria G. Knyazeva b,c a b c

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

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-correlation 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 pronounced 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 to model the functional and/or structural networks of the brain [1,2]. The brain networks can be studied at different levels ranging from micro-scale containing a number of interconnected neurons to macro-scale containing distributed brain regions. To construct the large-scale networks, signals recorded from the brain via methods such as electroencephalography (EEG), magnetocephalography (MEG), functional magnetic resonance imaging (fMRI), or diffusion tensor imaging (DTI), are used [3–7]. Often, binary (directed or undirected) adjacency matrices are analyzed [1,2], where binary links represent the presence or absence of a connection. The first step in analyzing brain networks is to extract its structure from the time-series. Possible methods are cross-correlation, coherence, and synchronization likelihood [3–6]. The next step is to represent it in a number of biologically meaningful measures. To this end, measures such as characteristic path length, efficiency of connections, clustering coefficient, modularity, node degree and centrality, assortativity, and synchronizability are applied [7,8]. Large-scale brain networks, comprising anatomically or functionally distinct regions and inter-regional pathways, exhibit specific non-random patterns with the small-world and/or scale-

n

Corresponding author. Tel.: þ98 0 21 6616 4633; fax: þ 98 0 21 6601 9246. E-mail address: [email protected] (M. Jalili).

0010-4825/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compbiomed.2011.05.004

free properties [9,10]. Graph theoretical analysis on anatomical and functional networks of the brain have revealed its economical small-world structure characterized by high clustering (transitivity) and a short characteristic path length [11]. The brain functional networks are cost-efficient in the sense that they provide efficient parallel processing for low connection cost [12]. Brain disorders influence the anatomical and functional brain networks. Brain wirings may show abnormal patterns in schizophrenia (SZ). SZ symptoms affect the patients by manifesting as auditory hallucinations, paranoid or bizarre delusions and/or disorganized speech and thinking in the context of significant social and/or occupational dysfunction. About 1% of the population worldwide suffers from different forms of SZ [13]. Additionally, another 3% of the population has SZ-type personality disorders. SZ is the fourth leading cause of disability in the developed counties for people at the age of 15–44. Schizophrenic patients show the abnormal patterns of brain connectivity. MRI-based studies on a large group of SZ patients revealed the reduced hierarchy of multimodal networks and increased connection distance [14]. The disruption of effective small-world architecture in many cortical regions, including prefrontal, parietal, and temporal lobes was shown for functional networks based on EEG [4] and fMRI [15]. Another fMRI study showed reduced clustering and small-worldness in SZ, as well as reduced probability of high-degree hubs, and increased robustness to networks’ component failures [16]. Nonlinear correlation analysis of EEG time-series also confirmed lower clustering and

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shorter path lengths in SZ compared to healthy subjects [17]. The structural properties of EEG-based networks activated by working memory tasks are also abnormal in SZ patients [18]. In this paper we consider fourteen SZ patients and fourteen matched control subjects. Brain functional networks are extracted from subjects’ EEG time-series through unpartial and partial cross-correlation analysis, the latter being suggested as complementary tools for extracting brain networks [19]. Then, the graph metrics, such as node-strength, small-worldness, vulnerability, modularity, assortativity, and synchronizability, are calculated and compared for SZ and control groups.

2. Methodology 2.1. EEG recording Fourteen schizophrenic patients (mean age 33.5710.1; 11 men including) were recruited from the in/outpatient schizophrenia units of the Psychiatry Department, Lausanne University Hospital. All diagnoses were made according to DSM-IV criteria on the basis of the Diagnostic Interview for Genetic Studies (DIGS) [20], or by a consensus of two experienced psychiatrists after a systematic review of medical records. Fourteen healthy control subjects (mean age 33.979.9) without known neurological or psychiatric illness or trauma and without substance abuse or dependence matched the patients for age, gender, and handedness. They were recruited from the local community based on the DIGS interview [20] or the Symptom Checklist [21] (8 and 6 subjects, respectively). All participants in this study were fully informed about the study and gave written consent. All the procedures conformed to the Declaration of Helsinki (1964) by the World Medical Association concerning human experimentation and were approved by the local ethics committee of the Lausanne University. The 3–4 min of resting-state eyes-closed EEG data were collected in a semi-dark room with a low level of environmental noise while each subject was sitting in a comfortable chair. The resting-state EEGs were recorded with the 128-channel Geodesic Sensor Net (EGI, USA) with all the electrode impedances kept under 30 kO. The recordings were made with vertex reference using a low-pass filter set to 100 Hz. The signals were digitized at a 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 at 50 Hz), re-referenced against the common average reference (CAR), and segmented into non-overlapping epochs using the NS3 software (EGI, USA). Artifacts in all channels were edited off-line: first, automatically, based on an absolute voltage threshold (100 mV) and on a transition threshold (50 mV), and then by thorough visual inspection, which allowed us to identify and reject epochs or channels with moderate muscle artifacts not reaching threshold values. We also excluded from further analysis the sensors that recorded artifactual EEG in at least one subject. Finally, 101 sensors were used for further computation. Data were inspected in 1 s epochs and the number of artifact-free epochs entered into the analysis was 185751 for the patients, and 195745 for the control subjects. We have previously reported some aspects of this dataset, including the whole-head dysconnection maps [22], hypofrontality [23] of alpha rhythm, and asymmetry of functional connectivity [24]. 2.2. Constructing brain functional networks The connectivity matrices of brain networks were extracted from EEG time-series. In order to reveal the functional

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connectivity taking into account both direct and indirect links, one should consider partial and unpartial correlations [19]. We calculated the Pearson correlation coefficient for all possible pairs between 101 artifact-free sensors. More precisely, the Pearson correlation coefficient between sensor i and j is obtained as covði,jÞ rij ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðiÞvarðjÞ

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 partial correlation matrices. The correlation between sensor i and j partialized to the group of sensors k is obtained as rij rjk rik rijk ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Þð1r 2 Þ ð1rik jk

ð2Þ

where rij is the unpartialized correlation between i and j, rik is the correlation 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 sensor locations. The first-order partial correlation coefficient has been proposed for constructing brain networks [19]. In other words, the partial correlation between two sensors would be the minimum of 100 correlation values: one unpartialized correlation and 99 correlation values partialized to each of the 99 remaining sensors. In this way, for each epoch in each subject, two 101  101 weighted correlation matrices were obtained: partialized and unpartialized. As the partial and unpartial correlations were computed for all epochs, they were then averaged over all artifact-free epochs for each subject. These averaged correlations were used to construct the corresponding functional networks pooled in the groups of SZ patients and control subjects. A precise way of computing partial correlation is to consider all possible choices and then to select the minimum among the obtained correlation values. In other words, to compute the correlation between two sensor locations, one should first compute the unpartialized correlation (zero-order correlation), all first-order correlations (correlations partialized to single sensors), all second-order correlations (those partialized to any combination of two sensors), all thirdorder correlations (those partialized to any combination of three sensors), and so on. Then, the minimum among these values is the true correlation between the two sensors. For many networks (especially if one analyzes a number of subjects each of who is represented by many EEG epochs), the computation of such a measure is an expensive task. We limited the computations to firstorder partial correlations. The next step was to binarize the weighted matrices. It resulted in binary adjacency matrices, with elements equal to 1, if the absolute correlation value exceeds a threshold TH, or 0, if it does not. The choice of the threshold TH has significant influence on the constructed connectivity structures such that fixing TH low values generates densely connected networks, whereas networks based on large TH values are sparse. To analyze the network properties as a function of TH, the correlation matrices were thresholded at different TH values.

2.3. Network measures The binary adjacency matrices were summarized in a few neurobiologically meaningful network metrics [2]. Let us show the binary adjacency matrix as A ¼(aij), where aij indicates the corresponding element in the (i,j) entry of A. A simple measure for the network is node degree that is defined by the sum of the links connecting a node. More precisely, the degree of node i is

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defined as X aij ki ¼

ð3Þ

j

Alternative nodal information for the original weighted network is node-strength. Let us consider a weighted network W¼(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 the links tipping to it X si ¼ wij ð4Þ j

2.3.1. Small-worldness index measuring functional segregation and integration Brain’s ability in functional segregation and integration is important for binding and information processing. Functional segregation is the ability in information processing in a specialized manner. A possible measure for quantifying the functional segregation is the clustering coefficient defined as [7,25] P 1 X i,j aij aik ajk C¼ ð5Þ N k kk ðkk 1Þ 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 frequent graph theory measure used for this purpose is the global efficiency defined as [7] X1 1 E¼ ð6Þ NðN1Þ i,j lij 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 nor regular but somewhere in between [25,26]. They are indeed small-worlds. A measure, called small-worldness, has been proposed to capture the network ability of segregation and integration [27], which estimates its clustering coefficient and efficiency compared to those of a number of properly random networks: Small-worldness ¼

C E Crandom Erandom

ð7Þ

where Crandom and Erandom are the average clustering coefficient and efficiency in the corresponding Erdos–Renyi graph [28] having the same number of nodes and edges as compared to the original graph. For each case, we created 10 randomized networks with the same degree distribution and computed their metrics by averaging over these 10 realizations. 2.3.2. Modularity index Another metric for measuring the ability of a network for segregation is its modularity, which is observed in many realworld networks. To capture the degree of modularity in the network with predetermined M modules, the following index has been proposed [29]: 2 0 12 3 X6 X 7 Q¼ eij A 5 ð8Þ 4eii @ iAM

jAM

where the network is fully partitioned into M non-overlapping modules (clusters), and eij represents the proportion of all links connecting nodes in module i with those in module j. The modularity index is computed by estimating the optimal modular structure for a given network [2,29].

2.3.3. Measures of centrality: In order to take into account the significance (centrality) of network elements, i.e. nodes and edges, their centrality can be considered [30]. Let us denote the edge between the nodes i and j by eij. Edge-betweenness centrality (load or traffic) rij of the network is defined as

rij ¼

X Gpu ðeij Þ pau

Gpu

ð9Þ

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 paths making 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 use of node i (except those between the ith node with the other nodes) [30]. More precisely

Oi ¼

X Gjk ðiÞ jaiak

Gjk

ð10Þ

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 the node i. 2.3.4. Measures of resiliency Networks may undergo random and/or intentional failures in their components, and their resiliency against such a failure is of high importance for their proper functioning. If the performance of the network is associated to its efficiency, the vulnerability of a node would be the amount of drop in the performance when the node is removed along all tipping edges from the network. More precisely Vi ¼

EEi E

ð11Þ

where E is the efficiency of the network and Ei is the efficiency of the network after the removal of the nodes i. A measure for the network vulnerability is the maximum vulnerability for all its nodes V ¼ maxi Vi ;

i ¼ 1,2,. . .,N

ð12Þ

Another measure for resiliency of networks is based on the degree–degree correlation, i.e. assortativity of the network. Many real-world networks show assortative or disassortative behavior. In assortative networks, nodes with high degree tend to connect to other nodes with high degree, whereas in disassortative networks, nodes with high degree tend to be linked to those with low degree [31]. In order to calculate the degree correlation, one may use the Pearson correlation of the degrees at both ends of the edges of the network [32] P P ð1=MÞ j 4 i ki kj aij ½ð1=MÞ j 4 i ð1=2Þðki þ kj Þaij 2 r¼ ð13Þ P P ð1=MÞ j 4 i ð1=2Þðk2i þk2j Þaij ½ð1=MÞ j 4 i ð1=2Þðki þ kj Þaij 2 where M is the total number of the edges of the network and ki is the degree of node i. If r 40, the network is assortative, whereas ro0 indicates a disassortative network. For r ¼0 there is no correlation between the node-degrees. The assortative networks are likely to consist of mutually coupled high-degree nodes and to be resilient against random failures. In contrast, the disassortative networks are likely to have vulnerable high-degree nodes. 2.3.5. Measure of synchronizability Various brain disorders have been linked to abnormalities in brain synchronization [33]. The eigenratio of the Laplacian matrix of the connection graph has been proposed as a measure for its

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synchronizability [34]. The Laplacian of a graph A ¼(aij) is obtained as L¼D–A, where D ¼(dii) is a diagonal matrix with node-degrees in the diagonal entries. Let us consider the eigenvalues of the Laplacian matrix as li, ordered as 0 ¼ l1 r l2 ryr lN, where N is the number of nodes in the network. The eigenratio lN/l2 is a measure for the degree of synchronization properties of the network and the smaller the eigenratio of a network the better its synchronizability [34]. 2.4. Statistical assessments The assessment for statistical significance of any possible differences between the graph metrics of brain networks in SZ patients and normal controls was performed through Wilcoxon’s ranksum test. The tests were carried out separately for any values of the threshold. The P-values are not corrected for multiple comparisons. All the computations were performed in MatLab. We used some available tools, i.e. BGL toolbox [35] and connectivity toolbox [36].

3. Application The computations resulted in two 101  101 weighted connection matrices for each subject; one based on unpartial crosscorrelations and another one based on partial correlations. As the unpartial correlation values were compared between SZ and control groups, the percentage of sensor pairs with significantly different correlation values (Po0.05, Wilcoxon’s ranksum test) varied from 8.73% in beta band to 4.04% in delta band (Fig. 1). The percentage of sensor pairs, where the partial correlations were significantly different between SZ patients and controls (Po 0.05, Wilcoxon’s ranksum test), varied from 7.72% in beta band to 5.03% in delta band (Fig. 1). In order to obtain a better spatial pattern on the changes we computed the node-strength that is the sum of the weights of edges linking a node to others (Fig. 2). The strength-maps were not the same for partial and unpartial correlations. The difference maps (SZ patients vs. controls) showed the patchy changes of strength in SZ patients when partial correlations were used to

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construct the networks. These changes were represented either by singletons or by small clusters of sensors. By contrast, there was a clear pattern of changes for unpartial correlations in alpha, beta, and gamma bands. We found the clusters of sensors over the right frontal and left parietal regions (alpha band) and over the right frontal continued to the central region (gamma band), showing decreased values of node-strength in the SZ patients compared to the controls. Node-strengths based on unpartial correlations in beta band were characterized with increased values 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, we binarized the weighted correlation matrices by thresholding them. The binary adjacency matrices in SZ patients and controls were then compared for their graph metrics. Functional networks of SZ brains constructed through partial correlations showed no significant changes in the small-worldness index as compared to those of normal controls (Fig. 3). However, there were significant differences for the networks based on unpartial correlations. In particular, we found that for the high-value TH threshold, the small-worldness was different in alpha and beta bands. The modularity metric is used for measuring the segregation properties of a network. The modularity index of unpartial networks was significantly higher in SZ than in controls (P o0.05, Wilcoxon’s ranksum test) for a large range of THs in the beta band (Fig. 4). Except for a single TH value in the gamma band with significant difference for unpartial case, we found no changes in the modularity (Fig. 4). Therefore, the beta band can serve as a marker for abnormal modular structure in SZ functional networks. Resiliency of brain networks belongs to those properties that may be altered by SZ. We considered two markers that are indirectly related to the resilient behavior of a network: vulnerability and assortativity. The vulnerability index showed little changes for the networks based on partial correlations. For the case of unpartial correlations, there were a few changes especially for the two clusters of TH values in middle ranges in gamma band (Fig. 5), where the vulnerability of functional networks in the SZ patients was lower than in controls (Po0.05, Wilcoxon’s ranksum test). The pattern of changes in the assortativity, i.e., degree–degree correlation, in delta band was almost the same for partial and

Fig. 1. Differences in the cross-correlation matrices in SZ patients compared to controls. The graphs show the significant differences for the partial and unpartial crosscorrelation matrices (101  101 matrices) at P o 0.05 (Wilcoxon’s ranksum test) in SZ patients vs. normal controls. The significantly different correlations are in black, whereas nonsignificant entries are left white. The analysis was carried out in different frequency bands, including delta (1–3 Hz), theta (3–7 Hz), alpha (7–13 Hz), beta (13–30 Hz), and gamma (30–70 Hz).

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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 (P o 0.05, 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 smallworldness 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 smallworldness index is significantly different between SZ and normal groups (Po 0.05, 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.)

unpartial case; SZ networks showed decreased assortativity for a cluster of high values of the threshold (Fig. 6). It is worth mentioning that the unpartial correlation values between a pair of sensors are often much larger than the corresponding partial correlation value. The assortativity of brain networks was increased for a cluster of low threshold values in alpha band for unpartial and in beta band for partial case. We also calculated the centrality measures (node and edge-betweenness centrality measures); however, SZ patients showed no significant differences as compared to control in these centrality measures.

Finally, we tested the brain networks for their synchronization properties. Fig. 7 shows the synchronizability index, i.e. the eigenratio of the Laplacian matrix of the connection graph for the SZ patients and controls. The networks based on partial correlations have altered synchronizability only in the beta band and for some high values of the threshold. However, for the networks based on unpartial correlations, decreased synchronizability, i.e. increased eigenratio, is characteristic in the theta, alpha, beta, and gamma bands. We found a cluster of intermediate threshold values (with a larger range in the alpha and gamma

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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.

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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.

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

4. Discussion Recent advancements in statistical methods for analyzing complex networks have influenced studies in brain network organization. It has been shown that large-scale brain networks constructed through recordings such as EEG, MEG, FMRI, and DTI show attributes such as small-world property, modularity, and scale-free degree distribution [1,36–40]. Graph theory analysis of brain signals may give useful information on the mechanisms that various brain disorders influence the brain structural and functional organization. SZ is one of those disorders that has been shown to alter a number of metrics of brain functional networks [4,14–18]. The abnormality in the cotico-cortical connectivity has been widely reported in SZ [37–40]. A necessary link between abnormal circuitry and basic SZ symptoms is functional connectivity. However, the changes in the anatomical and functional connectivity in SZ are not in the same direction [41]. While the anatomical connectivity assessed by DTI showed nearly uniform decrease in SZ, the functional connectivity captured through fMRI showed both increased (for some connections such as cingulate and thalamus) and decreased (for some connections such as middle temporal gyrus) regimes [41]. The application of statespaced based synchronization measure (S-estimator) on the data used for the present study has revealed the coexistence of hypoand hyper-synchronization clusters in SZ [22]. The abnormal functional connectivity and synchronization in SZ has also been revealed by other studies (reviewed in Ref. [42]). Following current views by ‘‘functional connectivity’’ we understand cooperation between distributed neural assemblies in the brain. Common ways of assessing the cooperation among cortical networks are measuring their synchronization, correlation, or coherency. Graph theory analysis is another method providing a global picture of the functional connectivity and cooperation among distributed brain regions [1,36–40]. In the present study we applied the graph theory techniques on the EEGs of SZ patients and compared their statistics with those of normal control subjects. The observed wide-spread morphological abnormalities in SZ such as enlarged ventricles (reviewed in

Ref. [43]), decreased cortical volume or thickness coupled with increased cell packing density [44–46], and reduced clustering of neurons [40], suggest the dysconnectivity model of SZ [47]. The dysconnection hypothesis suggests anomalous structural integrity and/or functional connectivity in SZ [47]. The association between anatomical and functional connectivity in the brain signifies a challenging issue in neuroscience research. Segregation and integration in the brain have been proposed as two potential principles linking these different modes of brain connectivity [48]. The interplay of segregation and integration in brain networks may cause information binding resulting in the generation of information that is simultaneously highly spread and highly mixed. These two principles, i.e. functional segregation and integration of brain networks, play important roles in information processing and proper functionality of the brain. The modularity index is one of those measures frequently used for characterizing the segregation properties of complex networks and the brain networks have been shown to have a modular structure [49,50]. We found the beta band as a marker of abnormal modularity of brain networks in SZ patients. The small-worldness is another measure revealing the functional segregation and integration in the networks [51–53]. In our data, this measure was significantly different in SZ patients as compared to controls for a cluster of high valued thresholds in alpha and beta bands. Previous fMRI studies in SZ patients and healthy controls also showed disturbed topological properties in the brain functional networks in patients, such as a lower strength and degree of connectivity, a lower efficiency, a lower clustering coefficient, and, hence, disrupted small-worldness [15]. Therefore, abnormal small-worldness is associated with partial disorganization of brain networks in SZ-affected brain. Networks may undergo random and/or intentional failures in their components. Many biological networks have shown resilient behavior against random failures and a number of measures such as assortativity and vulnerability are important in characterizing such a resilient behavior. The pattern of changes for these measures in SZ patients reveals the alternation in the resiliency of brain functional networks in SZ. Synchronization is believed to play an important role in information processing in the brain at both macroscopic and cellular levels [26,54]. Our results showing a pattern of significant differences in the synchronization properties of brain networks in SZ is of high importance, since a number of previous reports have

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shown that the synchronization among the cortical regions is altered in SZ [22,33,55–59]. In particular, our previous analysis on different aspects of the dataset reported here showed a synchronization landscape in SZ characterized by synchronization changes in centro-parietal sensors over the left hemisphere, a large cluster over the right hemisphere, and a midline cluster of locations over the centro-parieto-occipital region [22]. The analysis reported here looks at the synchronizability from different perspective and considers the synchronization properties of the brain networks rather than looking for a synchronous pattern in the original EEG signals. We found that, in general, the differences between SZ patients and control subjects in terms of graph metrics are more pronounced in the cases where unpartial correlations are used as compared to partial correlations. The unpartial correlation gives an estimate of the (direct or indirect) relationship between two variables. Because of low computational cost, it can be readily used to analyze large-scale complex networks such as brain networks 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 connectivity. The methods, therefore, provide different information on the system under study and may complement each other. Unpartial correlation analysis detects any kind of linear functional connectivity between two sites, which can be direct or indirect through other sites. Whereas, partial correlation analysis tries to minimize detecting the functional connectivity resulted by indirect links.

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of SZ brain networks extracted through fMRI has also been previously reported [60]. We also tested assortativity of brain networks. Functional MRI-based brain networks have shown increased assortativity in SZ patients [14]. Our data showed both increased and decreased assortativity, which was band- and method-specific. The changes in the centrality measures have also been reported for SZ brains [17]; however, we failed to show any significant difference in node and edge centrality measures in our data. There are many works reporting abnormalities in neural oscillation and synchronization in SZ (reviewed in Ref. [42]). However, we could not find any prior work studying the synchronization properties of brain networks in SZ. Our data showed SZ-specific wide-spread decreased synchronizability in theta, alpha, beta, and gamma bands. Temporal synchronization is important in information binding and cortical computing in the brain [54]. Decreased level of synchronizability in SZ may be related to the dysfunction of cortical networks in this illness. This study has several limitations, implying that our results need to be considered as preliminary requiring further validation. The studied group was relatively small and needs to be repeated in larger diverse groups. The study also needs to be performed through other imaging techniques such as fMRI, DTI, and/or MEG. In addition to graph theoretical tools, which enable the analysis of the network’s topological features, other analysis approaches can also be done in parallel. For example, approaches focusing on the three-dimensional structure of brain networks including morphometric methods (such as those for measuring wiring length or volume [61]) can be employed.

5. Summary A possible approach to study functional connectivity is to model it via the graph theory techniques and to analyze the graph properties of the networks. Given that different brain states respond to different measures, complementary information on the usefulness of functional brain networks can be obtained by combining the networks based on partial and unpartial correlations. Here we studied multichannel resting-state EEGs of fourteen SZ patients and fourteen matched healthy control subjects in terms of their graph properties. First, the pair-wise partial and unpartial cross-correlations were computed for all noise- and artifact-free EEG channels. The average correlation matrices were then used to extract the connection graph of the brain in SZ patients and normal subjects. Unpartial networks of SZ patients showed a distinct pattern in terms of node-strengths. The clusters of sensors with decreased strength were characteristic for the alpha and gamma bands, while those with increased strength were found in the beta band. Then, we thresholded the correlation matrices at different values and computed graph metrics for the constructed networks. We considered three classes of measures among various metrics available in graph theory: those related to the segregation/ integration, resiliency, and synchronizability of networks. The small-worldness and modularity were among those measures representing the segregation/integration properties of the brain networks. For high threshold values, the small-worldness of SZ brains was significantly higher in alpha band, whereas it was lower in beta band. Abnormal small-worldness has been reported in SZ patients through fMRI [14,15] and low-resolution EEG [4,15] studies. We found the beta band as a marker for abnormalities in the modularity of brain networks in SZ. The SZ-specific reduction of modularity in beta band was in agreement with the results obtained through fMRI [60]. Brain networks may undergo failures in their components. We found that the networks of SZ brains in gamma band are less vulnerable as compared to normal controls. Increased robustness

Conflict of interest statement None declared.

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