Evaluation of the Brain Network Organization From EEG SignalsA Preliminary Evidence in Stroke Patient.код для вставкиСкачать
THE ANATOMICAL RECORD 292:2023–2031 (2009) Evaluation of the Brain Network Organization From EEG Signals: A Preliminary Evidence in Stroke Patient FABRIZIO DE VICO FALLANI,1,2 LAURA ASTOLFI,1,3 FEBO CINCOTTI,1 DONATELLA MATTIA,1 DARIA LA ROCCA,1 ELIRA MAKSUTI,1 SERENELLA SALINARI,3 FABIO BABILONI,1,2* BALAZS VEGSO,4 GYORGY KOZMANN,4 AND ZOLTAN NAGY4 1 Laboratory of ‘‘Neuroﬁsiopatologia Clinica,’’ IRCCS ‘‘Fondazione Santa Lucia,’’ Roma, Italy 2 Department of Human Physiology and Pharmacology, University ‘‘Sapienza,’’ Rome, Italy 3 Department of ‘‘Informatica e Sistemistica,’’ University ‘‘Sapienza,’’ Rome, Italy 4 Department of Information Systems, University of Pannonia, Veszprem, Hungary ABSTRACT Synchronous brain activity in motor cortex in perception or in complex cognitive processing has been the subject of several studies. The advanced analysis of cerebral electro-physiological activity during the course of planning (PRE) or execution of movement (EXE) in a high temporal resolution could reveal interesting information about the brain functional organization in patients following stroke damage. High-power (128 channels) electroencephalography registration was carried out on 8 healthy subjects and on a patient with stroke with capsular lacuna in the right hemisphere. For activation of motor cortex, the ﬁnger tapping paradigm was used. In this preliminary study, we tested a theoretical graph approach to characterize the task-related spectral coherence. All of the obtained brain functional networks were analyzed by the connectivity degree, the degree distribution, and efﬁciency parameters in the Theta, Alpha, Beta, and Gamma bands during the PRE and EXE intervals. All the brain networks were found to hold a regular and ordered topology. However, signiﬁcant differences (P < 0.01) emerged between the patient with stroke and the control subjects, independently of the neural processes related to the PRE or EXE periods. In the Beta (13–29 Hz) and Gamma (30–40 Hz) bands, the signiﬁcant (P < 0.01) decrease in globaland local-efﬁciency in the patient’s networks, reﬂected a lower capacity to integrate communication between distant brain regions and a lower tendency to be modular. This weak organization is principally due to the signiﬁcant (P < 0.01 Bonferroni corrected) increase in disconnected nodes together with the signiﬁcant increase in the links in some other crucial C 2009 Wiley-Liss, Inc. vertices. Anat Rec, 292:2023–2031, 2009. V Key words: cerebral disorder; functional connectivity; graph theory Sequential activation of parietal, premotor, and supplementary motor areas as well as subcortical extrapyramidal system and cerebellum before activation of motor cortex is well documented in different experimental, neuromorphological studies. In the clinical praxis, the low temporal resolution of widely used neuroimaging Grant sponsor: Hungarian Research Foundation; Grant number: NKFP 2/004/04 and OTKA K69240; Grant sponsor: COST EU Project NEUROMATH; Grant number: BM0601. *Correspondence to: Fabio Babiloni, Laboratorio Neuroﬁsiopatologia Clinica, IRCCS ‘‘Fondazione Santa Lucia,’’ Via Ardeatina, 306 I-00179 Rome, Italy. Fax: þ39 06 5150 1465. E-mail: email@example.com Received 18 February 2009; Accepted 10 June 2009 DOI 10.1002/ar.20965 Published online in Wiley InterScience (www.interscience.wiley. com). C 2009 WILEY-LISS, INC. V 2024 DE VICO FALLANI ET AL. technologies like photon emission tomography (PET), single photon emission computerized tomography (SPECT), or even functional magnetic resonance imaging (fMRI) do not allow to visualize activated loops or networks in the course of movement. Other technologies, like magnetoencephalography (MEG) or electroencephalography (EEG) with high temporal resolution offer new insight into mechanisms of movement-organization and execution in the brain. These methods could characterize compromised brain function in patients with paretic stroke. The functional connectivity networks estimated from brain-imaging technologies (MEG, fMRI, and EEG) can be investigated by using graph theory (Stam, 2004; Salvador et al., 2005; Eguiluz et al., 2005; Bartolomei et al., 2006; Micheloyannis et al., 2006; Achard and Bullmore, 2007; De Vico Fallani et al., 2008a). As a graph is a mathematical representation of a network essentially reduced to nodes and connections between them, the use of a graph-theoretical approach is potentially relevant and useful, as ﬁrst demonstrated on a set of anatomical brain networks (Strogatz, 2001; Sporns, 2002). In those studies, the authors employed two characteristic features, the average shortest path L and the clustering index C, to extract the global and local properties of the network structure, respectively (Watts and Strogatz, 1998). They found that anatomical brain networks exhibit many local connections (i.e., a high C) and few random long distance connections (i.e., a low L), characterizing a particular model that interpolates between a regular lattice and a random structure. Such a topological property of the network (designated as small-world) has a strong impact on neurosciences, as it is related to optimal architectures for information processing and signal transmission among different cerebral structures (Lago-Fernandez et al., 2000; Sporns, 2002). Advances in brain imaging technologies, such as MEG and EEG have shown that activity takes place in different parts of the human brain. In several domains of engagement, these areas of activity are disparate in geography yet analogous in context. In particular, several studies (Classen et al. 1998; Andres and Gerloff, 1999; Lachaux et al., 1999; Miltner et al., 1999; Ghilardi et al., 2000) have examined synchronous brain activity using EEG technology. Varela et al. (Lachaux et al., 1999; Thompson and Varela, 2001) and others (Rappelsberger et al., 1994; Singer, 1999) have shown the role that synchronous cerebral activity plays in higher cognitive function including associative memory, emotional tone, and motor planning (Rodriguez et al., 1999). Such a synchronous cooperation between two or more distant brain regions to achieve a particular result may be referred to as a form of functional connectivity, which can be treated as a complex network. In this study, we show results from a novel analysis of functional networks estimated from a set of scalp EEG signals in a stroke patient and in a group of healthy subjects during the performance of a self-paced motor task. This study wishes to ascertain if the cerebral lesion changed the functional organization of the brain network related to the performance of a self-paced movement of the hand. MATERIALS AND METHODS Experimental Design To test the method of EEG graph analysis, eight healthy male subjects (age 30.75 15.39 years) and one stroke male patient (age 65 years) were recruited. The stroke patient had poorly controlled hypertension. Besides his index lacunar stroke in the right capsula, he had multiple lacunas in the white matter in both hemispheres without focal neurological symptoms. The ﬁrst step in the recording procedure was to inform the subject of what participation in the study would entail. A consent form was then signed by the subjects. For the EEG data acquisition, subjects were comfortably seated on a reclining chair, in an electrically shielded, dimly lit room. They were asked to perform a rapid extension of their second ﬁnger with the right hand. This task was repeated every 10 sec in a self-paced manner and 60 repetitions (trials) were recorded, digitalized, and stored in a hard disk. EEG signals were recorded with a sampling frequency of 2,048 Hz from 128 scalp electrodes disposed according to the scheme in Fig. 1. After the artifact removal in the present study, we analyzed separately the EEG segments of the preparation (PRE) and the execution (EXE) of the ﬁnger movement. In particular, we considered 2 sec before the movement onset for the PRE period and 2 sec after the onset for the EXE period. Before the computation of the spectral properties, all the EEG signals were downsampled—through an antialiasing (low-pass) FIR ﬁlter to compensate for the ﬁlter’s delay—to 128 Hz to allow a more efﬁcient data processing, given that the frequencies of interest were below 50 Hz. Brain Functional Synchronization Brain functional connectivity is achieved through the computation of task-related coherence. Andres and Gerloff (1999) describe task-related coherence as the ‘‘analysis of systematic interregional correlations of oscillatory cortical activity.’’ As described earlier several studies have suggested that such interregional correlations are associated with conscious cognitive processing and active perception. Task-related coherence in the context of our study is essentially a measure of similarity that provides us with an estimation of the level of synchronicity between two or more regions of the brain. The estimation is based on a mathematical manipulation of electrical signals obtained from the regions of the brain under investigation. Spectral coherence is a function that operates in the frequency domain and generates a value between 0 and 1. Given two signals x and y, spectral coherence is calculated for a particular frequency f by taking the square of the cross-spectrum |Sxy(f)|2 and then dividing by the product of the two corresponding auto power spectra. SCxy ðf Þ ¼ Sxy ðf Þ2 Sxxðf ÞSyyðf Þ (1) In the present study, the spectral coherence (SC) is calculated by estimating the value of linear correlation between each possible pairs of EEG channels with respect to each single frequency bin. The spectral resolution was ﬁxed to 1 Hz, thus for each channel pair the spectral range was from 0 to 64 Hz. To compute the correlation of a pair of time series, each trial was considered as a unique not overlapped section and a ‘‘hanning’’ window of 2 sec was employed to smooth the noise due to the ﬁnite-size trials. FUNCTIONAL CONNECTIVITY DURING SIMPLE MOTOR ACTS 2025 Fig. 1. The EEG cap map of the active-two recording system from Biosemi. Afterward, the coherence values were stored opportunely in a matrix with 128 rows and 128 columns (i.e., 128 channels) for each PRE and EXE intervals and for each frequency bin (see Fig. 2). To study the level of synchronization in speciﬁc physiological frequency ranges, we considered a frequencyaveraged coherence map in the bands of interest Theta y (4–7 Hz), Alpha a (8–12 Hz), Beta b (13–29 Hz), and Gamma c (30–40 Hz). Network Analysis The theoretical representation of a network is the graph. A graph consists of a set of vertices (or nodes) and a set of edges (or connections) indicating the presence of some sort of interaction between the vertices. The adjacency matrix A contains the information about the connectivity structure of the graph. When a link connects two nodes i and j, the corresponding entry of the adjacency matrix is aij ¼ 1; otherwise aij ¼ 0. Node degree. The simplest attribute of a node is its connectivity degree, which is the total number of connec- Fig. 2. Schematic representation of the obtained spectral coherence connectivity maps. 2026 DE VICO FALLANI ET AL. tions with other vertices. The formulation of the degree index d(i) can be introduced as follows: dðiÞ ¼ X ai;j (2) j2V V is the set of the available nodes and aij indicates the presence of the arc connecting the point j to the point i. Thus, it represents the total amount of links connected to the vertex i. The node degree has an obvious functional interpretation because it gives an index of the importance of that node within the network. Degree distribution. To get information about the behavior of degree within the system, it is useful to introduce P(d), the fraction of vertices in the graph that have degree d. Equivalently, P(d) is the probability that a vertex chosen uniformly at random has degree d. A plot of P(d) for any given network can be constructed by making a histogram of the degrees of vertices. This histogram is the degree distribution for the graph and it allows better understanding of the degree allocation in the system. Efﬁciency. Two measures are frequently used to characterize the local and global structure of unweighted graphs: the average shortest path L and the clustering index C (Watts and Strogatz, 1998; Newman, 2003; Grigorov, 2005). The former measures the efﬁciency of the passage of information among the nodes and the latter indicates the tendency of the network to form highly connected clusters of vertices. Recently, a more general setup has been examined to investigate weighted networks (Boccaletti et al., 2006). In particular, Latora and Marchiori (2001) considered weighted networks and deﬁned the efﬁciency coefﬁcient e of the path between two vertices as the inverse of the shortest distance (dist) between the vertices (note that in weighted graphs the shortest path is not necessarily the path with the smallest number of edges). In the case where a path does not exist, the distance is inﬁnite and e ¼ 0. The average of all the pair-wise efﬁciencies eij is the global-efﬁciency Eg of the graph. Thus, global-efﬁciency can be deﬁned as: Eg ðAÞ ¼ X 1 1 NðN 1Þ i6¼j2V disti;j 1X Eg ðAi Þ N i2V between the ﬁrst neighbors of i when i is removed. Global- (Eg) and local-efﬁciency (El) were demonstrated to reﬂect the same properties of the inverse of the average shortest path 1/L and the clustering index C (Latora and Marchiori, 2003). Hence, the deﬁnition of small-world can be rephrased and generalized in terms of the efﬁciency indexes (Boccaletti et al., 2006; Stam and Reijneveld, 2007). Small-world networks have high Eg (i.e., high 1/L) and high El (i.e., high C). This new deﬁnition is attractive because it takes into account the full information contained in the weighted links of the graph and provides an elegant solution to handle disconnected vertices (Latora and Marchiori, 2001). Contrast with random graphs. Also the comparison with 100 random graphs has been addressed by comparing the original brain networks with connectivity patterns holding the same number of nodes and the same number of links arranged in a random fashion (Erdos and Renyi, 1960). This contrast has the power to evaluate statistically the distance of the real brain network from structures obtained in a casual way. (3) where N is the number of vertices composing the graph. As the efﬁciency e also applies to disconnected graphs, the local properties of the graph can be characterized by evaluating for every vertex i the efﬁciency coefﬁcients of Ai, which is the subgraph composed by the neighbors of the node i. The local-efﬁciency El is the average of all the subgraphs global-efﬁciencies: El ðAÞ ¼ Fig. 3. Functional networks of the stroke patient in the Alpha band during the PRE and EXE intervals of left side ﬁnger tapping. Each node represents a scalp electrode. Each link represents a signiﬁcant coherence between the signals of two electrodes. Yellow links represent connections common to both the intervals, whereas black links are the changing connections. (4) As the node i does not belong to the subgraph Ai, this measure reveals the level of fault tolerance of the system, showing how the communication is efﬁcient Connections ﬁltering. In the present study, we dealt with brain networks holding the same number of connections. This should be done when one wants to prevent that a different number of connections could bias the comparison of two conditions or subjects. For this reason, the resulting values of SC were ﬁltered deﬁning a threshold. An edge between two channels was drawn only if the corresponding coherence value is higher than the speciﬁed threshold. In particular, each coherence matrix was ﬁltered by maintaining only the strongest 585 links (0.07197 of connection density), which were subsequently converted to logical values (0, absence of link; 1, presence of link). See ‘‘Methodological considerations’’ paragraph, in the Discussion section, for more details. The resulting coherence maps, which represent the functional interactions between all the EEG signals, can 2027 FUNCTIONAL CONNECTIVITY DURING SIMPLE MOTOR ACTS TABLE 1. Statistical comparisons of global- and local-efﬁciency Theta Eg El Alpha Beta PRE EXE PRE EXE PRE 0.80 1.81 1.50 1.99 0.34 1.80 1.06 2.37 2.83 2.72a a Gamma EXE PRE EXE 2.01 3.42a 4.89 2.67a a 3.43a 3.42a Values are represented by the z-score. a Signiﬁcant difference between the stroke and control graphs (|Z| > 2.58, P < 0.01). be drawn through a set of vertices disposed in a circle. Figure 3 illustrates the connectivity networks in the Alpha band for the stroke patient during the PRE and EXE period. RESULTS The main theoretical graph indexes obtained from the brain networks of the stroke patient were contrasted statistically (Z-score) with the distribution of values obtained from the healthy population in the same frequency bands and conditions. The analysis of the network topology through the efﬁciency indexes revealed a signiﬁcant (P < 0.01) difference between the stroke patient and the control healthy subjects. Table 1 shows the obtained Z-values for Eg and El in all the frequency bands and conditions. The differences involve mainly the highest spectral contents (Beta and Gamma band). In those bands, both the global- and local-efﬁciency of the patient are statistically (P < 0.01) lower with respect to control subjects during both the PRE and EXE intervals. Only in the Beta band during the execution (EXE) was this difference less strong. Figure 4 shows the original Eg and El values in the Beta and Gamma bands and during both situations. The decrease in efﬁciency in the patient’s brain network is maintained either during the PRE or EXE periods. Eventually, all the estimated brain networks were contrasted with a distribution of 100 random graphs having the same number of nodes and links. The results demonstrate that all the brain networks present a statistically different topology when compared with random networks. In particular, the Z-test revealed a signiﬁcant (P < 0.01) lower Eg and a signiﬁcant (P < 0.01) higher El (data not shown here). The different structural properties observed in the patient’s brain are due to a different arrangement of the links within the functional network. The evaluation of how these connections are distributed within the network was addressed through the degree-distribution indexes. Figure 5 shows the obtained Z-values for P in the Beta and Gamma bands and during the PRE and EXE periods. The general outcome suggests in the patient’s brain network the presence of many nodes (Z < 0) with a lower degree, that is, d [ (1,25), than control subjects and few nodes (Z > 0) with a higher degree, d [ (26,55). Also, it can be noted how the number of disconnected nodes (d ¼ 0) is signiﬁcantly higher (P < 0.01, Bonferroni corrected for multiple comparisons). Moreover, a signiﬁcant increase (P < 0.01, Bonferroni corrected) of the nodes with 32 and 38 links can be observed in the Beta band during the PRE interval (Fig. 5a). In the Fig. 4. Comparison between the efﬁciency values of the stroke patient and the control subjects. (a) Contrast for the Eg indexes in the Beta band during the PRE and EXE intervals. The red bar represents the value of the stroke patient. The blue bar represents the mean value of the control subjects; the vertical line measures the standard deviation. The asterisk indicates a signiﬁcant difference (P < 0.01). (b) Contrast for the El indexes in the Beta band during the PRE and EXE intervals. (c) Contrast for the Eg indexes in the Gamma band during the PRE and EXE intervals. (d) Contrast for the El indexes in the Gamma band during the PRE and EXE intervals. same band, we can ﬁnd a signiﬁcant increase in the vertices (P < 0.01, Bonferroni corrected) with 26 and 40 connections (Fig. 5b). A signiﬁcant increase (P < 0.01, Bonferroni corrected) of the nodes with 28, 29, 33, and 41 links can be observed in the Gamma band during the PRE interval (Fig. 5c). In the same band, we can ﬁnd also a signiﬁcant increase in the vertices (P < 0.01, Bonferroni corrected) with 23 and 29 connections (Fig. 5d). To identify the main nodes, that is, the electrodes on the scalp, which are responsible for the observed structural change in the brain functional networks of the stroke patient, we evaluated the degree indexes of each vertex. Figure 6 shows the obtained d values for the stroke patient in the Beta and Gamma bands and during the PRE and EXE intervals. Each degree value is represented through a sphere located on the schematic map of the EEG electrodes cap at the respective sensor position. The size of the sphere measures the role of the electrode in the brain functional network. The larger spheres indicate those electrodes in the stroke patient’s network whose number of links was signiﬁcantly higher (P < 0.01 Bonferroni corrected) than control subjects. Thus, according to the degree distribution ﬁndings the main nodes in the Beta band during the PRE period are 2028 DE VICO FALLANI ET AL. Fig. 5. Statistical contrast (Z-test) of the degree distributions of the stroke patient and control subjects. (a) comparison for the P indexes in the Beta band during the PRE period. On x-axes, the degree values, on y-axes the Z-values of the P indexes. The numbers at the line peaks indicate those connections that responsible of a signiﬁcant dif- ference (P < 0.01, Bonferroni corrected) between the brain network of the patient and control subjects. (b) Comparison for the P indexes in the Beta band during the EXE interval. (c) Comparison for the P indexes in the Gamma band during the PRE period. (d) Comparison for the P indexes in the Gamma band during the EXE interval. those with 32 and 38 links, that is, A18, B17, D17, and D29 (Fig. 6a); the main vertices in the Beta band during the EXE interval are those with 26 and 40 connections, that is, A29, C1, C29, D12, D28, and D29 (Fig. 6b); the main nodes in the Gamma band during the PRE period are those with 28, 29, 33, and 41 links, that is, A7, B1, B7, B17, B18, C12, C17, C29, D1, and D17 (Fig. 6c); the main vertices in the Gamma band during the EXE interval are those with 23 and 29 connections, that is, B23, B29, C7, D8, and D23 (Fig. 6d). In the present work, we characterized the functional networks through a theoretical graph approach in a stroke patient and in eight healthy subjects during a ﬁnger-tapping task from scalp EEG signals. All the estimated networks were found to be very unlikely from random graphs. In particular, the brain functional networks presented a more regular and ordered structure, as revealed by the signiﬁcant (P < 0.01) higher values of local-efﬁciency and lower values of global-efﬁciency. Brain networks have been widely demonstrated as differing from random graphs in several studies (Stam, 2004; Salvador et al., 2005; Eguiluz et al., 2005; Bartolomei et al,. 2006; Micheloyannis et al., 2006; Achard and Bullmore, 2007; De Vico Fallani et al., 2008b); however, this aspect does not represent the focus of this article. The theoretical graph approach we used for analysis of 128 channels EEG recording aimed at evaluating the level of degeneration in the brain functional network of a stroke patient. Results suggest a signiﬁcant change in the structural properties of the brain network in the stroke patient either in the PRE or in the EXE interval of the motor act. This means that the network topology in the patient’s brain could be generally affected by the stroke lesion independently of the neural processes DISCUSSION The possibility to adopt a mathematical approach improves the capability of detecting the topological features from real complex networks. The topology of a network describes the organization and the architecture of the links within the system. In the neuroscience, this aspect has a strong impact as the arrangement of the functional links between different brain sites could affect the level of information processing and signal synchronization. In this respect, the graph theory may facilitate the analysis of the functional connectivity patterns estimated from actual neuroimaging technologies. FUNCTIONAL CONNECTIVITY DURING SIMPLE MOTOR ACTS 2029 Fig. 6. Patterns of degree values for the stroke patient. Each degree value is represented through a sphere located on the schematic map of the EEG electrodes cap at the respective sensor position. The darker is the color, the higher is the degree value according to the colorbar. The larger spheres indicate those electrodes in the stroke patient’s network whose number of links was signiﬁcantly higher (P < 0.01 Bonferroni corrected) than control subjects. (a) Degree values in the Beta band during the PRE interval. (b) Degree values in the Beta band during the EXE period. (c) Degree values in the Gamma band during the PRE interval. (d) Degree values in the Gamma band during the EXE period. related to preparation or execution of the movement. These changes are particularly prominent in the Beta band (13–29 Hz), which is already known to be involved in motor tasks (Pfurtsheller and Lopes da Silva, 1999), but also in the Gamma (30–40 Hz) band. This would demonstrate that this type of cerebral damage would affect the functional network mainly in the highest spectral contents. In particular, the signiﬁcant (P < 0.01) decrease in global- and local-efﬁciency in the patient’s networks reﬂects a lower capacity to integrate the communication between distant brain regions and a lower tendency to be modular. This weak organization is principally due to the different arrangement of the functional links within the network, as the number of connections was the same in the stroke patient and in the healthy subjects. The signiﬁcant (P < 0.01 Bonferroni corrected) increase of disconnected nodes, that is, electrodes, together with the signiﬁcant (P < 0.01 Bonferroni corrected) increase of the links in some other crucial vertices, is thought to be the main aspect responsible for the observed structural and organizational degeneration. This change indicates that the overall connectivity in the patient’s network is ruled by a lower number of brain regions. A possible interpretation of this preliminary evidence is that the capsular lacunas could affect the rapid oscillating activities (13–40 Hz) of the standard cortical areas involved during the motor task by inhibiting their degree of connectivity. The subsequent increase in connectivity in a limited number of brain regions does not seem sufﬁcient to avoid a drastic reduction in the information propagation in the functional networks of the patient’s brain. Methodological considerations The spectral coherence, the method we used in this study, is one of the simplest measures to detect functional connectivity between signals. It is straightforward and it does not need restrictive a-priori hypotheses. However, despite its intuitive nature it only gives undirected links between electrodes. No information can be achieved about the direction of the functional relationship. Other interesting methods, based on multivariate autoregressive models (MVAR) models (Sameshima and Baccala, 1999; Kaminski et al., 2001), could give the direction of the information, but there is a restrictive 2030 DE VICO FALLANI ET AL. relation between the number of the elements among which it is possible estimate the connectivity and the available data. Eventually, it should be very hard to have reliable estimates with the MVAR model on 128 electrodes. Furthermore, a serious limitation of extracranial EEG coherence measures is contamination by volume conduction through the tissues separating sources and electrodes (Urbano et al., 1997, 1998; Babiloni et al., 2001; Astolﬁ et al., 2005, 2007; De Vico Fallani et al., 2007). However, as the main effect seems to be a coherence elevation in the highest frequencies >40 Hz (Ramesh et al., 2007), the estimates in the present study should be relatively stable, as we considered EEG spectral contents up to 40 Hz. The choice of the ‘‘optimum’’ connection-density of the estimated networks was surely the most favorable condition for the signiﬁcance of the network structure indexes (De Vico Fallani et al., in press). In fact, at this connection-density the efﬁciency indexes describing the structural properties of the network are maximally separated and independent (data not shown here), thus giving a more robust analysis. This is not trivial as arbitrary thresholding methods could still lead to highly connected networks or vice versa, to very sparse networks. This possibility dramatically affects the independence and the separation of the efﬁciency indexes (i.e., in a full connected graph both the global- and local-efﬁciency are equal to 1, in an empty graph they are both equal to 0), which could give misleading results across different subjects or tasks. Eventually, it is worth to mention that previous studies showed that age affects the structure of the healthy brain functional network (Achard and Bullmore, 2007) obtained from fMRI signals during resting state conditions. Even if in the present study there is a sensible difference between the age of the stroke patient and the mean age of the healthy subjects, we would like to state that we are dealing with functional networks obtained from a totally different brain imaging technique (i.e., EEG) and that the experimental conditions deviate actually from a simple resting state. For these reasons, we would not be able to assess exactly the role of the age in the present study, which mainly focuses on the possible structural changes in the brain functional network of the stroke patient. ACKNOWLEDGEMENTS Dr. Fabrizio De Vico Fallani whishes to thank Ing. Alessandro Tabarrini for his support in the last computational steps. LITERATURE CITED Achard S, Bullmore E. 2007. Efﬁciency and cost of economical brain functional networks. PloS Comp Biol 3:e17. Andres FG, Gerloff C. 1999. Coherence of sequential movements and motor learning. J Clin Neurophsiol 16:520–527. Astolﬁ L, Cincotti F, Babiloni C, Carducci F, Basilisco A, Rossigni PM, Salinari S, Mattia D, Cerutti S, Ben Dayan D, Ding L, Ni Y, He B, Babiloni F. 2005. Assessing cortical functional connectivity by linear inverse estimation and directed transfer function: simulations and application to real data. Clin Neurophysiol 116:920–932. Astolﬁ L, De Vico Fallani F, Cincotti F, Mattia D, Marciani MG, Bufalari S, Salinari S, Colosimo A, Ding L, Edgar JC, Heller W, Miller GA, He B, Babiloni F. 2007. Imaging functional brain connectivity patterns from high-resolution EEG and fMRI via graph theory. Psychophysology 44:880–893. Babiloni C, Babiloni F, Carducci F, Cincotti F, Rosciarelli F, Rossini PM, Arendt-Nielsen L, Chen ACN. 2001. Mapping of early and late human somatosensory evoked brain potentials to phasic galvanic painful stimulation. Human Brain Mapp 12:168–179. Bartolomei F, Bosma I, Klein M, Baayen JC, Reijneveld JC, Postma TJ, Heimans JJ, van Dijk BW, de Munck JC, de Jongh A, Cover KS, Stam CJ. 2006. Disturbed functional connectivity in brain tumour patients: evaluation by graph analysis of synchronization matrices. Clin Neurophysiol 117:2039–2049. Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU. 2006. Complex networks: structure and dynamics. Phys Reports 424: 175–308. Classen J, Gerloff C, Honda M, Hallett M. 1998. Integrative visuomotor behavior is associated with interregionally coherent oscillations in human brain. J Neurophysiol 79:1567–1573. De Vico Fallani F, Astolﬁ L, Cincotti F, Mattia D, Tocci A, Marciani MG, Colosimo A, Salinari S, Gao S, Cichocki A, Babiloni F. 2007. Extracting information from cortical connectivity patterns estimated from high resolution EEG recordings: a theoretical graph approach. Brain Topogr 19:125–136. De Vico Fallani F, Astolﬁ L, Cincotti F, Mattia D, Tocci A, Salinari S, Marciani MG, Witte H, Colosimo A, Babiloni F. 2008b. Brain network analysis from high resolution EEG recordings by the application of theoretical graph indexes. IEEE Trans Neural System Rehabilit Eng 16:442–452. De Vico Fallani F, Baluch F, Astolﬁ L, Subramanian D, Zouridakis G, Babiloni. Structural organization of functional networks from EEG signals during motor learning tasks. Int J Bifurcat Chaos. De Vico Fallani F, Latora V, Astolﬁ L, Cincotti F, Mattia D, Marciani MG, Salinari S, Colosimo A and Babiloni F. 2008a. Persistent patterns of interconnection in time-varying cortical networks estimated from high-resolution EEG recordings in humans during a simple motor act. J Phys A. Eguiluz VM, Chialvo DR, Cecchi GA, Baliki M, Apkarian AV 2005. Scale-free brain functional networks. Phys Rev Lett 94:018102. Erdos P, Renyi A. 1960. On the evolution of random graphs. Publ Math Inst Hung Acad Sci 5:17–61. Ghilardi M, Ghez C, Dhawan V, Moeller J, Mentis M, Nakamura T, Antonini A, Eidelberg D. 2000. Patterns of regional brain activation associated with different forms of motor learning. Brain Res 871:127–145. Grigorov MG. 2005. Global properties of biological networks. Drug Discov Today 10:365–372. Lachaux J, Rodriguez E, Martinerie J, Varela FJ. 1999. Measuring phase synchrony in brain signals. Hum Brain Mapp 8:194–208. Lago-Fernandez LF, Huerta R, Corbacho F, Siguenza JA. 2000. Fast response and temporal coherent oscillations in small-world networks. Phys Rev Lett 84:2758–2761. Latora V, Marchiori M. 2001. Efﬁcient behaviour of small-world networks. Phys Rev Lett 87:198701. Latora V, Marchiori M. 2003. Economic small-world behaviour in weighted networks. Eur Phys J B 32:249–263. Kaminski M, Ding M, Truccolo WA, Bressler S. 2001. Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of signiﬁcance. Biol Cybern 85:145–157. Micheloyannis S, Pachou E, Stam CJ, Vourkas M, Erimaki S, Tsirka V. 2006. Using graph theoretical analysis of multi channel EEG to evaluate the neural efﬁciency hypothesis. Neurosci Lett 402:273–277. Miltner HRW, Braun C, Mathias A, Witte H, Taub E. 1999. Coherence of c-band EEG activity as a basis for associative learning. Nature 397:434–436. Newman MEJ. 2003. The structure and function of complex networks. SIAM Rev 45:167–256. Pfurtsheller G, Lopes da Silva FH. 1999. Event-related EEG/EMG synchronizations and desynchronization: basic principles. Clin Neurophysiol 110:1842–1857. FUNCTIONAL CONNECTIVITY DURING SIMPLE MOTOR ACTS Srinivasan R, Winter WR, Ding J, Nunez PL. 2007. EEG and MEG coherence: measures of functional connectivity at distinct spatial scales of neocortical dynamics. J Neurosci Methods 166:41–52. Rappelsberger P, Pfurtscheller G, Filz O. 1994. Calculation of eventrelated coherence-a new method to study short-lasting coupling between brain areas. Brain Topogr 7:121–127. Rodriguez E, George N, Lachaux J, Martinerie J, Renault B, Varela FJ. 1999. Perception’s shadow: long-distance synchronization of human brain activity. Nature 397:430–433. Salvador R, Suckling J, Coleman MR, Pickard JD, Menon D, Bullmore E. 2005. Neurophysiological architecture of functional magnetic resonance images of human brain. Cereb Cortex 15:1332–1342. Sameshima K, Baccala LA. 1999 Using partial directed coherence to describe neuronal ensemble interactions. J Neurosci Methods 94:93–103. Singer W. 1999. Striving for coherence. Nature 397:391–393. Sporns O. 2002. Graph theory methods for the analysis of neural connectivity patterns In: Kötter R, editor. Neuroscience databases. A practical guide. Boston: Kluwer. p 171–186. 2031 Stam CJ. 2004. Functional connectivity patterns of human magnetoencephalographic recordings: a ‘small-world’ network? Neurosci Lett 35. Stam CJ, Reijneveld JC. 2007. Graph theoretical analysis of complex networks in the brain. Nonlinear Biomed Phys 1:3. Strogatz SH. 2001. Exploring complex networks. Nature 410:268–276. Thompson E, Varela FJ. 2001. Radical embodiment: neural dynamics and consciousness. Trends Cogn Sci 5:418–425. Urbano A, Babiloni F, Babiloni C, Ambrosini A, Onorati P, Rossini PM. 1997. Human short-latency cortical responses to somatosensory stimulation. A high resolution study. Neuro Report 8:3239– 3243. Urbano A, Babiloni C, Carducci F, Fattorini L, Onorati P, Babiloni F. 1998. Dynamic functional coupling of high resolution EEG potentials related to unilateral internally triggered one-digit movements. Electroencephalogr Clin Neurophysiol 106:477– 487. Watts DJ, Strogatz SH. 1998. Collective dynamics of ‘‘small-world’’ networks. Nature 393:440–442.