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Journal of Integrative Neuroscience, Vol. 10, No. 2 (2011) 213–232 c Imperial College Press DOI: 10.1142/S0219635211002725
CONSTRUCTING BRAIN FUNCTIONAL NETWORKS FROM EEG: PARTIAL AND UNPARTIAL CORRELATIONS MAHDI JALILI∗,§ and MARIA G. KNYAZEVA†,‡ ∗
Department of Computer Engineering Sharif University of Technology Azadi Avenue, Tehran, Iran
†
Department of Clinical Neuroscience and Department of Radiology Centre Hospitalier Universitaire Vaudois (CHUV) Lausanne, Switzerland ‡
University of Lausanne, Lausanne, Switzerland §
[email protected] Received 3 March 2011 Accepted 10 May 2011
We consider electroencephalograms (EEGs) of healthy individuals and compare the properties of the brain functional networks found through two methods: unpartialized and partialized cross-correlations. The networks obtained by partial correlations are fundamentally different from those constructed through unpartial correlations in terms of graph metrics. In particular, they have completely different connection efficiency, clustering coefficient, assortativity, degree variability, and synchronization properties. Unpartial correlations are simple to compute and they can be easily applied to large-scale systems, yet they cannot prevent the prediction of non-direct edges. In contrast, partial correlations, which are often expensive to compute, reduce predicting such edges. We suggest combining these alternative methods in order to have complementary information on brain functional networks. Keywords: Brain functional networks; EEG; synchronization; partial cross-correlation; graph theory measures.
1. Introduction The analysis of networks through graph theory have been extensively studied in recent years and the field is growing at a high rate [1]. A graph is a collection of nodes and connecting edges that can be directed or undirected, weighted or unweighted. Most strikingly, one of the systems where graph theory can be applied is the human brain — a complex system capable of integrating separately processed information from different sources [2–5]. Therefore, the brain can be regarded as a § Corresponding
author.
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dynamical system consisting of many individual units interacting within a graph [5]. An important question in neuroscience is how various aspects of cognition depend upon the integration of activity from distributed brain regions [6]. The brain networks can be studied at the microscale level as excitable systems containing a number of neurons with excitatory and/or inhibitory connections in between [7–9]. For the analysis of the whole-brain connectivity network, one should use fMRI, diffusion imaging, MEG, or EEG techniques to extract the large-scale functional and anatomical brain connectivity networks. Characterization of the anatomical and functional connectivity structure of the brain at the macroscale level is crucial because it will increase our knowledge of how functional states emerge from the brain structural properties and provide new insights into the influence of brain disorders on the structure and function of the brain [10–13]. Graph theoretical analysis of brain activity reliably models brain functional networks [14]. A few neurobiologically meaningful graph theoretic measures can characterize the properties of the brain networks [2, 3, 15–18]. The character of nodes and links in the brain networks depends upon the nature of its functional activity and the methods of its recording and analysis. For example, the nodes based on EEG signals (used in our analysis) can be individual sensors or groups of them, and the dependencies between the time series of these nodes represent their functional connectedness [19–21]. However, scalp EEG suffers from the volume conductance problem that limits the interpretation of the results obtained based on such signals. In order to minimize such an effect, alternatively, a graph theoretical analysis can be performed in the source space of EEG [22], where connectivity is studied at the level of reconstructed sources that are considered individual nodes. The links may be weighted or unweighted, directed or undirected. Binary links represent the absence or presence of connection, while weighted links denote the strength of the connections. As the structure of the brain functional network is extracted, the next step is to represent it by a number of meaningful measures. To this end, measures such as characteristic path length, efficiency of connections, clustering coefficient, modularity, node and edge centrality, assortativity, and synchronizability are often used [1, 23]. Bivariate measures such as cross-correlation, coherence, and synchronization likelihood are conventionally used to extract meaningful relations out of EEG, MEG, or fMRI time series [14, 18, 20, 24]. In particular, to estimate a functional relationship between two brain regions, e.g., between two EEG sensor locations, bivariate measures are usually applied independent of other regions, i.e., in an unpartialized manner. The alternative way of computing the relationship between two regions is to consider other regions as covariates, i.e., to apply partial measures [25, 26]. We consider EEGs obtained from a group of healthy individuals and show that the graphs obtained through these two methods, i.e., partial and unpartial cross-correlations, have fundamentally different topological properties.
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2. Materials and Methods 2.1. EEG recording Fifteen healthy control subjects (mean age 33.2 ± 15; 6 males) without known neurological or psychiatric illness or trauma were selected from our EEG database. All participants 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 University of Lausanne (Lausanne, Switzerland), where the EEG recordings were made. With their eyes closed, EEGs were collected in a semi-dark room with a low level of environmental noise while each subject was sitting in a comfortable chair. The data were recorded with the 128-channel Geodesic Sensor Net (EGI, USA) at the sampling frequency of 500 Hz. Since the sensors from the outer ring of the sensor net were excluded from the analysis because of low quality signal, only 111 sensors were used for computations (see [12, 13] for the detailed description of methods). The EEGs were filtered (FIR, band-pass of 1–50 Hz), re-referenced against the common average reference, and segmented into non-overlapping 1-s epochs using the NS3 software.
2.2. Constructing brain functional network structure The EEG time series were used to construct the connectivity matrix of the brain functional networks. In order to discover to which extent two sensor locations are coupled, one can use measures such as cross-correlation or coherence for linear dependence [14, 24] and synchronization likelihood for nonlinear relation [18, 20]. The first step is to calculate the weighted 111 × 111 correlation matrix (there are 111 sensor locations). We used the Pearson product momentum correlation coefficient as an index of the interdependence of the time series of two sensor locations. However, the correlation values between two sensors may result from the indirect relationship caused by sequential pathways, common sources, or sinks (Fig. 1). Partial correlation may prevent a false prediction of direct links between two sensors. To this end, a partial cross-correlation coefficient quantifies the correlation value between two sensors when conditioned by one or several other sensors. Let us consider the correlation between sensors i and j partialized to the group of sensors k. The Pearson correlation coefficient between the time series of i and j can be obtained as rij =
cov(i, j) var(i)var(j)
,
(1)
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Fig. 1. Sample network for a numerical assessment of the superiority of partial correlation over unpartial one. Consider a simulation of dynamical systems (Colpitts oscillators here) in the above network. There are indirect links between some nodes through other nodes, e.g., between nodes 1 and 3 through node 2. These indirect links result in a dependence between the time series of the nodes (e.g., nodes 1 and 3), and thus, unpartial cross-correlation wrongly identifies a link that does not exist in the real network. The usage of partial cross-correlation may exclude such false links.
where cov(i, j) is the covariance between nodes i and j, and var(i) is the variance of node i. The partial correlation is computed as rij |k =
rij − rjk rik 2 )(1 − r 2 ) (1 − rik jk
,
(2)
where rij is the unpartialized correlation between i and j, rik is the correlation between i and k, and rjk is the correlation between j and k. Note that k can be a group consisting of several sensor locations. 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 any two sensors), all third-order 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. This methodology has been applied considering the partial correlations of up to the first [27] and second-order [26, 28] for inferring the gene regulatory networks. Indeed, the reason behind the partial correlation is to avoid predicting nondirect functional links. However, a non-direct link may pass through various nodes. For example, the non-direct link between nodes A and B might follow through A → C → B, A → C → D → B, or A → C → D → K → H → J → B. To account for all of these non-direct connections, one should consider all the possibilities while doing a
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partial correlation analysis. For many networks (especially if one analyses a number of subjects, each of whom is represented by several EEG epochs), the computation of such a measure is an expensive task. We limited the computations to first-order partial correlations. Recently, partial correlations has been used for extracting effective connectivity from fMRI datasets [29]. Also, (N −2)-order partial correlations, where N is the number of nodes in the network, have been applied to structural and functional MRI data [30, 31]. Note that in the (N −2)-order partial correlations, the correlation between any two nodes is calculated considering the data of all of the remaining N −2 nodes in a single matrix as the covariate i.e., a single partial correlation value, is obtained. However, in the (N −2)-order partial correlation, the intra-connections in the N −2 nodes cannot be captured, even though one or more of these intra-connections cause indirect links between the parent nodes. As we already mentioned, to account for all possible indirect interactions, partial correlations of higher orders should also be calculated; however, we proceeded by calculating the first-order partial correlation due to the high computational complexity of higherorder partial correlations. Here, the partial correlation between two sensors was a minimum of 110 correlation values: one unpartialized correlation and 109 correlation values partialized to each of the 109 remaining sensors. In this way, for each epoch in each subject, two 111 × 111 weighted correlation matrices were obtained: partialized and unpartialized. By averaging them over the artefact-free epochs, we computed the average weighted correlation matrices for each subject. Since negative and positive correlation values similarly refer to the strength of functional connections, we considered the absolute values of correlations as synchronization measures scaling from zero for non-synchronized time series to one for completely synchronized time series. This is also the case for other synchronization measures such as phase synchrony, coherence and synchronization likelihood (see for example [18, 21, 30, 32–34]). The next step was to binarize the weighted matrices. It resulted in binary adjacency matrices, whose elements were 1, if the absolute correlation value exceeded a threshold TH, or 0, if it did not (Fig. 2). The adjacency matrix for each subject was an unweighted graph A = [aij ] comprising N = 111 nodes connected by E undirected edges. We defined the total number of edges in the graph, E, divided by the maximum possible number of edges, N (N − 1)/2, as (normalized) network cost [15, 35]. Different choices for the threshold TH have significant influence on the topological properties of the resulting brain functional networks. Considering low values for TH would generate densely connected networks with a large number of edges E corresponding to a high cost. On the other hand, networks based on large values of TH are sparse, have a small number of edges, and hence a low cost. Another difficulty is the thresholding of networks based on partialized and unpartialized correlations. Partial correlations are much smaller compared to the corresponding unpartial ones and have completely different distributions (Fig. 3). We refer to the
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Fig. 2. Construction of brain networks from EEG signals. The first step is to calculate the weighted cross-correlation matrix, with rows and columns representing nodes. Then, the network is reduced to a binary form, i.e., the entries of the binary connectivity matrix were set to 1, if they were greater than the threshold in the original weighted connectivity matrix; otherwise they were set to 0. The last step is to compute the graph metrics such as efficiency of connections, clustering coefficient, modularity, small-worldness, centrality measures, assortativity, and synchronizability measures.
networks extracted through partial and unpartial correlations as partialized and unpartialized networks, respectively. Therefore, instead of comparing the topological properties of the partialized and unpartialized networks at different threshold values, we performed the comparison as a function of the network cost. In particular, the partialized and unpartialized networks were thresholded at different TH values in a way that allowed them to have the same cost, i.e., the same number of edges E [15]. Since the topological properties of the resulting network might be dependent on a particular value of the cost, we thresholded the correlation matrices repeatedly over a range of costs and obtained the graph measurements. This procedure allowed us to compare various network properties as a function of network cost. It is worth mentioning that fixing the cost and binarizing the correlation matrix for this value may create edges based on weak correlations. To this end, the cost values should be varied over a wide range to deal with this problem as well as possible. As the partial and unpartial correlations were computed for each epoch, they were then averaged over all artefact-free epochs for each subject. These averaged
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Fig. 3. Histograms of partial and unpartial correlation values. The plots show histograms of (a) partial and (b) unpartial correlation coefficients. Note that partial and unpartial correlation values have completely different distributions, with partial correlations being much smaller compared to unpartial correlations.
correlations were used to construct the two corresponding functional networks per subject for all the cost values. Furthermore, they were statistically assessed to compare the properties of the networks based on partial correlations with those obtained through unpartial correlations. 2.3. Graph theoretical measures As the structure of the brain functional network is extracted, it should be expressed by a number of measures characterizing global and local brain connectivity [3]. A basic graph theory measure, which indicates the number of connection links of a node, is the degree of a node. The degree of node i is computed as: aij , (3) ki = j
where aij is the entry of the adjacency matrix A in (i, j) location. Nodes with higher degrees are hubs that have been found to exist in brain networks [24, 36, 37]. 2.3.1. Measures of functional segregation and integration Functional segregation is the brain’s ability to process information in a specialized manner in each of its interconnected groups of regions. There are a number of measures quantifying such groups within the graph. We considered the clustering coefficient and the modularity index. The clustering coefficient is calculated by
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counting all triangular connections existing in the graph and dividing that number by all the theoretically possible triangular connections [1, 38]: 1 i,j aij aik ajk . (4) C= N kk (kk − 1) k
Many real-world networks show a modular structure. To capture the degree of modularity in a graph with predetermined M modules, the following index has been proposed [39]: 2 eii − eij , (5) Q= i∈M
j∈M
where the graph 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 graph [3, 39]. Functional integration is the ability of the brain to combine information processed in distributed brain regions. The frequently used measure for functional integration is the characteristic path length or global efficiency [1]. The following metric has been suggested for representing the global efficiency of wirings in the graph [40]: 1 1 , (6) Eglobal = N (N − 1) lij i,j
where lij is the length of the shortest path between nodes i and j. The average of the shortest path lengths is called characteristic path length L of the graph. Watts and Strogatz showed in their influential paper that many real-world networks have neither random nor regular structure [38]. A measure, called smallworldness, has been proposed to capture the network ability of segregation and integration [41], which estimates its clustering coefficient and characteristic path length compared to those of a number of properly random networks: Small-worldness =
C/Crandom , L/Lrandom
(7)
where Crandom and Lrandom are the average clustering coefficient and characteristic path length in the randomized networks. We randomized the networks in such a way that they would have the same degree distribution as the original brain functional network [42]. For each case, we created 10 randomized networks and computed their metrics by averaging over these 10 realizations. 2.3.2. Measures of centrality In graph theory, there is a term indicating the centrality of a node or an edge, which shows its contribution to the interactions between the nodes of the network
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by assuming that these interactions follow the shortest paths between the nodes [43]. Let us denote the edge between nodes i and j by eij . Edge-betweenness centrality (load or traffic) ρij of the network is defined by [43] as Γpu (eij ) , (8) ρij = Γpu p=u
where Γpu is the number of shortest paths between nodes p and u in the graph, and Γpu (eij ) is the number of shortest paths making use of the edge eij . In a similar way, one can define node-betweenness centrality Ωi as [43] Γjk (i) , (9) Ωi = Γjk j=i=k
where Γjk is the number of shortest paths between nodes j and k, and Γjk (i) is the number of shortest paths making use of the node i. 2.3.3. Measures of resiliency Measures of network resiliency determine the features that reflect network vulnerability to failure. It has been shown that degree-heterogeneous networks are resilient against random failures [44, 45]. We considered the variance of the node degrees to be a measure of the heterogeneity of the networks, which also reflects the resiliency properties of the network. Many real-world networks show assortative or disassortative behavior. To this end, a measure called the assortativity coefficient has been proposed, which is the correlation coefficient between the degree of all nodes on the two ends of a link [46]. The assortativity r of a network is defined as
2 1 1 1 k k a − (k + k )a i j ij i j ij j>i j>i 2 E E (10) r=
2 , 1 2 1 1 2 )a − 1 (k + k (k + k )a j ij j>i 2 i j>i 2 i j ij E E where E is the total number of the edges of the network. If r > 0, the network is assortative, whereas r < 0 indicates a disassortative network. 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. 2.3.4. Measures of synchronizability It has been shown that the eigenvalues of the Laplacian matrix of the connection graph are important for determining the network synchronization properties [47]. Let us consider the eigenvalues of the Laplacian matrix as λi , ordered as 0 = λ1 ≤ λ2 ≤ · · · ≤ λN , in which λ1 = 0 is associated with the synchronized manifold (synchronization cannot be defined in disconnected networks where λ2 = 0). The master-stability-function formalism proposes considering either the eigenratio λN λ2
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or the inverse of algebraic connectivity 1/λ2 as a measure of network synchronizability [47]; the smaller the value of these parameters, the better the synchronization properties of a network. In general, these measures of synchronizability do not have the same behavior [48], and thus we considered both. 2.4. Numerical assessment In order to compare the real neural network properties extracted through partial and unpartial correlation analysis, their functional connectivity in terms of graph theory metrics should be known in advance. However, the existing models of functional networks have biased through imaging and analysis methods. Even in cases when anatomical connectivity is known, linking it to functional connectivity is not straightforward. To avoid uncertainties associated with real neural networks, a numerical test was performed on the time series obtained from a network with a known adjacency matrix. To this end, we considered the simulation of coupled chaotic Colpitts oscillators, which generate irregular sine-like signals such as the EEG [49]. The dynamics of the oscillator at node i is given by d g −yi − 1) + z ] + ε x [α(e = aij (xj − xi ) i i dt Q(1 − k) j∈Ni dX g d −y = F (X) = (11) yi = [(1 − α)(e i − 1) + zi ] dt dt Qk d zi = Qk(1 − k) [xi + yj ] − 1 zi dt g Q where X = (x, y, z) is the state vector and g, Q, α, and k are parameters. A = (aij ) is the (binary) adjacency matrix, ε is the uniform coupling strength, and Ni is the set of the neighbors of node i. In the simulations, k = 0.5 and ε = 0.5 for all the oscillators, while different g, Q, and α have been chosen for each oscillator, with values in the intervals [4.0361, 4.6385], [1.2833, 1.4829], and [0.9004, 0.9889], respectively. The differential Eq. (11) were iterated, starting from random initial conditions, using the Heun algorithm [50]. Neglecting a number of first iterations in order to eliminate transients, unpartial and partial cross-correlations were obtained using Eqs. (1) and (2), respectively. All the computations were performed in MatLab. The differential equations of the coupled Colpitts oscillators were numerically integrated using the Heun algorithm in MatLab [50]. 3. Results Numerical simulation was performed on Colpitts oscillators coupled through the network shown in Fig. 1. Since the connections in the nervous system are directed, we considered a sample-directed network. However, as the time series was produced from the coupled oscillators and correlation analysis was performed on them, the
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Table 1. Cross-correlation between nodes of the sample network shown in Fig. 1. The table shows the absolute value of unpartial (partial) cross-correlation between nodes of the sample network of Fig. 1. The time series was obtained by numerically solving chaotic Colpitts oscillators coupled through the network of Fig. 1. Nodes
1
2
3
4
5
6
7
8
9
10
1
—
0.60 (0.31)
0.42 (0.02)
0.62 (0.66)
0.34 (0.04)
0.33 (0.02)
0.33 (0.01)
0.34 (0.03)
0.42 (0.04)
0.33 (0.01)
2
0.60 (0.31)
—
0.70 (0.34)
0.04 (0.01)
0.55 (0.02)
0.58 (0.07)
0.57 (0.09)
0.54 (0.1)
0.68 (0.08)
0.56 (0.07)
3
0.42 (0.02)
0.70 (0.34)
—
0.02 (0.01)
0.63 (0.06)
0.61 (0.29)
0.58 (0.1)
0.54 (0.3)
0.99 (0.93)
0.59 (0.1)
4
0.62 (0.66)
0.040 (0.01)
0.02 (0.01)
—
0.05 (0.02)
0.05 (0.01)
0.06 (0.01)
0.01 (0.00)
0.02 (0.00)
0.06 (0.01)
5
0.34 (0.04)
0.55 (0.02)
0.63 (0.06)
0.05 (0.02)
—
0.96 (0.86)
0.85 (0.09)
0.42 (0.09)
0.69 (0.29)
0.90 (0.53)
6
0.33 (0.02)
0.58 (0.07)
0.61 (0.29)
0.05 (0.01)
0.96 (0.86)
—
0.95 (0.4)
0.41 (0.07)
0.69 (0.07)
0.97 (0.65)
7
0.33 (0.01)
0.57 (0.09)
0.58 (0.1)
0.06 (0.01)
0.85 (0.09)
0.95 (0.4)
—
0.40 (0.1)
0.65 (0.08)
0.98 (0.76)
8
0.34 (0.03)
0.54 (0.1)
0.54 (0.3)
0.01 (0.00)
0.42 (0.09)
0.41 (0.07)
0.40 (0.1)
—
0.53 (0.22)
0.37 (0.06)
9
0.42 (0.04)
0.68 (0.08)
0.99 (0.93)
0.02 (0.00)
0.69 (0.29)
0.69 (0.07)
0.65 (0.08)
0.53 (0.22)
—
0.67 (0.23)
10
0.33 (0.01)
0.56 (0.07)
0.59 (0.1)
0.06 (0.01)
0.90 (0.53)
0.97 (0.65)
0.98 (0.76)
0.37 (0.06)
0.67 (0.23)
—
structure of the network was identified as an undirected network. Partial and unpartial cross-correlations were calculated for the time series of variable x of oscillators (Table 1). These cross-correlation values were then used to reconstruct the network structure. Figure 4 shows the number of over-identified connections, i.e., those identified in the reconstructed network while they were missing in the original network, as a function of the thresholds for partial and unpartial cases. It can be easily seen that the networks based on partial correlations coincide with the original network in Fig. 1 across a wider range of threshold values compared to the networks constructed from unpartial correlations. Also, the networks based on the unpartial correlations over-identify the links for many significant threshold values. Using first-order partial correlation, many of these indirect links were excluded, e.g., the link between nodes 1 and 3. However, there were still some wrongly identified links between the nodes (e.g., between nodes 6 and 10), which can be excluded by considering higher order partial correlations. Therefore, partial correlations minimize the prediction of links that do not exist in the real system. It is worth mentioning that typically the genuine network structure is not known; however, this example shows that the network structure obtained through partial correlations is closer to the real structure than that based on unpartial correlations.
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Fig. 4. Number of overidentified links using partial and unpartial correlations. The plot shows the number of over identified links as a function of the threshold in the networks based on partial (black line) and unpartial (red line) correlation values. The correlation coefficients were calculated from the time series of variable x of the Colpitts oscillators coupled over the sample network shown in Fig. 1. The networks based on partial correlations coincide with the original network shown in Fig. 1 for a larger range of threshold values compared to those based on unpartial correlations.
The results of real EEG time series obtained from 15 healthy subjects (see “Methods”) are shown in Figs. 5–8. Figure 5 shows the measures of segregation and integration in functional brain networks. The values are plotted as a function of network cost. As can be seen from the figure, brain functional networks constructed through unpartial correlations have significantly higher efficiency, cost efficiency, and clustering coefficient than those extracted based on partial correlations (P < 0.001, Wilcoxon’s rank sum test). Thus, not partializing the cross-correlation artificially increases the efficiency and clustering coefficient of a network. Moreover, for the values of the network cost less than 0.2, the small-worldness index is underestimated (P < 0.001, Wilcoxon’s rank sum test), while the modularity is overestimated for some small and large values of the cost (P < 0.001, Wilcoxon’s rank sum test). The centrality measurements for brain functional networks extracted using the partialized and unpartialized correlations are shown in Fig. 6. The picture shows the mean node-betweenness centrality averaged over 111 nodes as well as the mean edgebetweenness centrality. The edge-betweenness centrality in the networks depends on which of the two strategies is used (P < 0.001, Wilcoxon’s rank sum test) only if the networks are dense, i.e., the network cost is large. However, the mean nodebetweenness centrality is significantly different (P < 0.001, Wilcoxon’s rank sum test) for a large range of network costs.
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Fig. 5. Measures of functional segregation and integration of brain networks extracted by means of partial and unpartial correlation matrices. The graphs show the mean values of (a) efficiency, (b) clustering coefficient, (c) small-worldness, and (d) modularity index for 15 healthy subjects as a function of network cost — that is, the number of links divided by the total number of possible links in the network. The blue dots represent the cost values, where the value of the corresponding network metric is significantly different in partial and unpartial cases (P < 0.001, Wilcoxon’s rank sum test; uncorrected for multiple comparison across multiple cost values). The shaded regions represent the standard deviations of the variables. Other designations are in Fig. 4.
Degree variability and assortativity are significantly overestimated (P < 0.001, Wilcoxon’s rank sum test) in brain functional networks constructed through unpartial correlations (Fig. 7). The synchronizability of the brain functional networks using two different synchronizability measures is shown as a function of the network cost in Fig. 8. As the network cost increases, the number of the links increases and the communication between the nodes is facilitated. As a result, the nodes can coordinate their behavior better and the synchronizability of the network is enhanced, i.e., synchronizability measures decrease. However, the pattern is completely different for the networks based on partial correlations compared to networks based on unpartial correlations. Those constructed through partial correlations show significantly better synchronization properties than those obtained through unpartial
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Fig. 6. Measures of the centrality of the brain networks. Panel (a) shows mean node-betweenness centrality and panel (b) shows mean edge-betweenness centrality as a function of network cost. Other designations are in Fig. 5.
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Fig. 7. Measure of resiliency of the brain networks. Panel (a) demonstrates the variance of nodedegrees and panel (b) shows the assortativity measurement, i.e., the degree–degree correlation coefficient as a function of network cost. Other designations are in Fig. 5.
correlations (P < 0.001, Wilcoxon’s rank sum test). Note that the comparison is based on the same number of links (i.e., the same network cost) for both cases. 4. Discussion Studies in brain network organization have significantly benefited from recent developments in statistical methods for analyzing networks. Large-scale brain networks,
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Fig. 8. Measure of the synchronizability of brain networks. Panel (a) shows the eigenratio defined as the largest eigenvalue of the Laplacian over the second smallest eigenvalue (algebraic connectivity) and panel (b) shows the inverse of algebraic connectivity as a function of network cost. Other designations are in Fig. 5.
comprising anatomically and functionally distinct regions and inter-regional pathways, show attributes such as small-world property, modularity, and highly connected hubs. The first step in studying the network properties of the brain is to construct the connectivity matrix. A common technique for this purpose is to use methods such as cross-correlation, coherence, or other synchronization measures estimating functional connectivity. However, a functional link between two brain regions might be a result of links with other brain regions. Thus, another way of predicting a link between any two regions of the brain is to consider other relevant regions as a covariate through partial coherence, correlation, or synchronization. In this paper, we propose to simultaneously use partial and unpartial correlations to discover the topology of brain functional networks. Functional brain networks obtained through partial correlation showed topologically different properties than those obtained through unpartial correlation. The networks constructed through unpartial correlations had higher efficiency, cost efficiency, and clustering coefficient than those extracted from partial correlations. These measures are important for characterizing integration and segregation in the brain. The partial and unpartial networks also showed some differences in the smallworldness and the node-betweenness centrality. Degree variability and assortativity have significant influences on the resiliency of networks. These parameters were overestimated in functional brain networks constructed through unpartial correlations. Hub nodes, essential for coordinating brain functions due to their connectivity with numerous brain regions [51], have been shown to exist in functional brain networks [24, 36, 37]. The vulnerability of a network is highly dependent on the structure of its hub nodes [52], and
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such hub nodes may explain the pattern of vulnerability in brain disorders such as Alzheimer’s disease [53]. For example, the superior frontal and parietal regions are among the most vulnerable cortical territories that also have a central role in functional connectivity [54]. The network synchronizability was also different in the partial and unpartial functional brain networks. Synchronization is believed to play an important role in information processing in the brain at both macroscopic and cellular levels [55–60]. Neuronal temporal synchronization is a fundamental process in cortical computation [61]. Brain oscillations that are ubiquitous phenomena in all brain areas get into synchrony and consequently allow an implementation of the whole range of brain functions [8, 61]. Finally, although we considered sensor space provided by potential scalp EEG, the inferences attained in this work hold true for the source EEG and high-resolution EEG (e.g., surface Laplacian), which are preferable for better localization of the brain networks, and can be generalized across different models of EEG activity. 5. Conclusion Graph theoretical analysis of anatomical and functional brain networks is increasingly used in neuroscience. The first step in characterizing properties of brain networks is to extract the structure of the networks. Then, a number of graph theoretic measures such as connection efficiency, clustering coefficient, modularity, degree distribution, assortativity, and synchronizability are employed to describe the properties of the networks. Methods such as cross-correlation, coherence, and synchronization are frequently used for predicting the connection links between brain regions. Taking into account that the functional dependence between the signals recorded from two brain regions can result from the indirect link between these regions through other brain regions, the accurate way of extracting the functional link between any two brain regions is to consider other relevant regions as covariates by calculating partial correlation, coherence, or synchronization measures. Graph theoretical metrics of functional brain networks are often used for characterizing the mechanism by which various disorders affect brain functionality. Here we provide evidence that networks constructed through partial and unpartial correlation matrices of EEG signals are dissimilar in terms of various network metrics. The unpartial correlation, e.g., the Pearson correlation coefficient, 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 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. We suggest using them jointly in order to comprehensively characterize brain networks, including both direct and indirect functional connections.
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Acknowledgments We thank Prof. R. Poznanski for the thoughtful review of our paper. This work was partly supported by Swiss National Foundation Grant No 320030-127538/1. References [1] Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU, Complex networks: Structure and dynamics, Physics Reports 424:175–308, 2006. [2] Bullmore E, Sporns O, Complex brain networks: Graph theoretical analysis of structural and functional systems, Nat Rev Neurosci 10:186–198, 2009. [3] Rubinov M, Sporns O, Complex network measures of brain connectivity: Uses and interpretations, Neuroimage 52:1059–1069, 2010. [4] Sporns O, Tononi G, Kotter R, The human connectome: A structural description of the human brain, PLoS Comput Biol 1:0245–0251, 2005. [5] Sporns O, Networks of the Brain, MIT Press, 2011. [6] Varela F, Lachaux J-P, Rodriguez E, Martinerie J, The brainweb: Phase synchronization and large-scale integration, Nat Rev Neurosci 2:229–239, 2001. [7] Markram H, A network of tufted layer 5 pyramidal neurons, Cereb Cortex 7:523–533, 1997. [8] Buzsaki G, Rhythms of the Brain, Oxford University Press, New York, 2006. [9] Bonifazi P, Goldin M, Picardo MA, Jorquera I, Cattani A, Bianconi G, Represa A, Ben-Ari Y, Cossart R, GABAergic hub neurons orchestrate synchrony in developing hippocampal networks, Science 326:1419–1424, 2009. [10] Sporns O, Chialvo DR, Kaiser M, Hilgetag CC, Organization, development and function of complex brain networks, Trends Cogn Sci 8:418–425, 2004. [11] Sporns O, Tononi G, Edelman GM, Theoretical neuroanatomy: Relating anatomical and functional connectivity in graphs and cortical connections matrices, Cereb Cortex 10:127–141, 2000. [12] Jalili M, Lavoie S, Deppen P, Meuli R, Do KQ, Cuenod M, Hasler M, De Feo O, Knyazeva MG, Dysconnection topography in schizophrenia with state-space analysis of EEG, PLoS ONE 2:e1059, 2007. [13] Knyazeva MG, Jalili M, Frackowiak RS, Rossetti AO, Psychogenic seizures and frontal disconnection: EEG synchronisation study, J Neurol Neurosurg Psychiatry 82: 505–511, 2011. [14] Deuker L, Bullmore ET, Smith M, Christensen S, Nathan PJ, Rockstroh B, Bassett DS, Reproducibility of graph metrics of human brain functional networks, Neuroimage 47:1460–1468, 2009. [15] Achard S, Bullmore E, Efficiency and cost of economical brain functional networks, PLoS Comput Biol 3:174–183, 2007. [16] Achard S, Salvador R, Whitcher B, Suckling J, Bullmore E, A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs, J Neurosci 26:63–72, 2006. [17] Zalesky A, Fornito A, Harding IH, Cocchi L, Y¨ ucel M, Pantelis C, Bullmore ET, Wholebrain anatomical networks: Does the choice of nodes matter? Neuroimage 50:970–983, 2010.
June 8, 2011
10:25
230
WSPC/S0219-6352
179-JIN
S0219635211002725
Jalili & Knyazeva
[18] Stam CJ, de Haan W, Daffertshofer A, Jones BF, Manshanden I, van Cappellen van Walsum AM, Montez T, Verbunt JP, de Munck JC, van Dijk BW, Berendse HW, Scheltens P, Graph theoretical analysis of magnetoencephalographic functional connectivity in Alzheimer’s disease, Brain 132:213–224, 2009. [19] Ferri R, Rundo F, Bruni O, Terzano MG, Stam CJ, Small-world network organization of functional connectivity of EEG slow-wave activity during sleep, Clin Neurophysiol 118:449–456, 2007. [20] Micheloyannis S, Pachou E, Stam CJ, Breakspear M, Bitsios P, Vourkas M, Erimaki S, Zervakis M, Small-world networks and disturbed functional connectivity in schizophrenia, Schizophr Res 87:60–66, 2006. [21] Stam CJ, Jones BF, Nolte G, Breakspear M, Scheltens P, Small-world networks and functional connectivity in Alzheimer’s disease, Cereb Cortex 17:92–99, 2006. [22] Schoffelen JM, Gross J, Source connectivity analysis with MEG and EEG, Hum Brain Mapp 30:1857–1865, 2009. [23] Costa LdF, Rodrigues FA, Travieso G, Villas Boas PR, Characterization of complex networks: A survey of measurements, Advances in Physics 56:167–242, 2007. [24] Eguiluz VM, Chialvo DR, Cecchi GA, Baliki M, Apkarian AV, Scale-free brain functional networks, Phys Rev Lett 94:018102, 2005. [25] Sun Y, Zhang H, Feng T, Qiu Y, Zhu Y, Tong S, Early cortical connective network relating to audiovisual stimulation by partial directed coherence analysis, IEEE Trans Biomed Eng 56:2721–2724, 2009. [26] Fuente Adl, Bing N, Hoeschele I, Mendes P, Discovery of meaningful associations in genomic data using partial correlation coefficients, Bioinformatics 20:3565–3574, 2004. [27] Magwene PM, Kim J, Estimating genomic coexpression networks using first-order conditional independence, Genome Biol 5:R100, 2004. [28] Keurentjes JJB, Fu J, de Vos CHR, Lommen A, Hall RD, Bino RJ, van der Plas LH, Jansen RC, Vreugdenhil D, Koornneef M, The genetics of plant metabolism, Nat Genet 38, 2006. [29] Marrelec G, Krainik A, Duffau H, Pelegrini-Issac M, Lehericy S, Doyon J, Benali H, Partial correlation for functional brain interactivity investigation in functional MRI, Neuroimage 32:228–237, 2006. [30] Bassett DS, Bullmore E, Verchinski BA, Mattay VS, Weinberger DR, MeyerLindenberg A, Hierarchical organization of human cortical networks in health and schizophrenia, J Neurosci 28:9239–9248, 2008. [31] Salvador R, Suckling J, Coleman MR, Pickard JD, Menon D, Bullmore E, Neurophysiological architecture of functional magnetic resonance images of human brain, Cereb Cortex 15:1332–1342, 2005. [32] Stam CJ, van Dijk BW, Synchronization likelihood: An unbiased measure of generalized synchronization in multivariate data sets, Physica D 163:236–251, 2002. [33] Carmeli C, Knyazeva M, Innocenti G, De Feo O, Assessment of EEG synchronization based on state-space analysis, Neuroimage 25:339–354, 2005. [34] Knyazeva MG, Jalili M, Brioschi A, Bourquin I, Fornari E, Hasler M, Meuli R, Maeder P, Ghika J, Topography of EEG multivariate phase synchronization in early Alzheimer’s disease, Neurobiol Aging 31:1132–1144, 2010.
June 8, 2011
10:25
WSPC/S0219-6352
179-JIN
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[35] Latora V, Marchiori M, Economic small-world behavior in weighted networks, Eur Phys J B 32:249–263, 2003. [36] Hagmann P, Cammoun L, Gigandet X, Meuli R, Honey CJ, Wedeen VJ, Sporns O, Mapping the structural core of human cerebral cortex, PLoS Biol 6:e159, 2008. [37] Hagmann P, Kurant M, Gigandet X, Thiran P, Wedeen VJ, Meuli R, Thiran JP, Mapping human whole-brain structural networks with diffusion MRI, PLoS ONE 2:e597, 2007. [38] Watts DJ, Strogatz SH, Collective dynamics of ‘small-world’ networks, Nature 393:440– 442, 1998. [39] Newman MEJ, Modularity and community structure in networks, Proc Natl Acad Sci USA 103:8577–8582, 2006. [40] Latora V, Marchiori M, Efficient behavior of small-world networks, Phys Rev Lett 87:198701, 2001. [41] Humphries MD, Gurney K, Network ‘small-world-ness’: A quantitative method for determining canonical network equivalence, PLoS ONE 3:e0002051, 2008. [42] Maslov S, Sneppen K, Specificity and stability in topology of protein networks, Science 296:910–913, 2002. [43] Freeman LC, Set of measures of centrality based on betweenness, Siociometry 40:35–41, 1977. [44] Albert R, Barabasi A-L, Statistical mechanics of complex networks, Rev Mod Phys 74:47–97, 2002. [45] Albert R, Jeong H, Barabasi A-L, Error and attack tolerance of complex networks, Nature 406:378–382, 2000. [46] Newman MEJ, Assortative mixing in networks, Phys Rev Lett 89:208701, 2002. [47] Pecora LM, Carroll TL, Master stability functions for synchronized coupled systems, Phys Rev Lett 80:2109–2112, 1998. [48] Jalili M, Ajdari Rad A, Synchronizability of dynamical networks: Different measures and coincidence, International Journal of Nonlinear Sciences and Numerical Simulation 10:1481–1489, 2009. [49] De Feo O, Maggio GM, Kennedy MP, The Colpitts oscillator: Families of periodic solutions and their bifurcations, Int J Bifurcation and Chaos 10:935–958, 2000. [50] Quarteroni A, Sacco R, Saleri F, Numerical Mathematics, Springer-Verlag, Berlin, Germany, 2000. [51] Cole MW, Pathak S, Schneider W, Identifying the brain’s most globally connected regions, Neuroimage 49:3132–3148, 2010. [52] Boccaletti S, Criado R, Pello J, Romance M, Vela-Perez M, Vulnerability and fall of efficiency in complex networks: A new approach with computational advantages, Int J Bifurcation and Chaos 19:727–735, 2009. [53] Buckner RL, Sepulcre J, Talukdar T, Krienen FM, Cortical hubs revealed by intrinsic functional connectivity: Mapping, assessment of stability, and relation to Alzheimer’s disease, J Neurosci 29:1860–1873, 2009. [54] Iturria-Medina Y, Sotero RC, Canales-Rodriguez EJ, Aleman-Gomez Y, Melie-Garcia L, Studying the human brain anatomical network via diffusion-weighted MRI and graph theory, Neuroimage 40:1064–1076, 2008. [55] Singer W, Neuronal synchrony: A versatile code for the definition of relations? Neuron 24:49–65, 1999.
June 8, 2011
10:25
232
WSPC/S0219-6352
179-JIN
S0219635211002725
Jalili & Knyazeva
[56] Uhhaas PJ, Singer W, Neural synchrony in brain disorders: Relevance for cognitive dysfunctions and pathophysiology, Neuron 52:155–168, 2006. [57] Neuenschwander S, Singer W, Long-range synchronization of oscillatory light responses in the cat retina and lateral geniculate nucleus, Nature 379:728–733, 1996. [58] Glass L, Synchronization and rhythmic processes in physiology, Nature 410:277–284, 2001. [59] Gray CM, Konig P, Engel AK, Singer W, Oscillatory responses in cat visual-cortex exhibit inter-columinar synchronization which reflects global stimulus properties, Nature 338:334–337, 1989. [60] Dzakpasu R, Zochowski M, Discriminating differing types of synchrony in neural systems, Physica D 208:115–122, 2005. [61] Fries P, Neuronal gamma-band synchronization as a fundamental process in cortical computation, Ann Rev Neurosci 32:209–224, 2009.
