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## Graph analysis and visualization for brain function characterization using EEG data (2010)

Venue: | J. Healthcare Eng |

Citations: | 2 - 1 self |

### Citations

3316 |
Collective dynamics of “small-world” networks
- Watts, Strogatz
- 1998
(Show Context)
Citation Context ...vertices within its neighborhood divided by the number of links that could possibly K n v v V = ∈ ∑1 deg( ) Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 439 exist between them. For an undirected graph, if a vertex vi has degree ki, edges could exist among the vertices within its neighborhood, thus (2) This measure is 1 if every neighbor connected to vi is also connected to every other vertex within the neighborhood, and 0 if no vertex that is connected to vi connects to any other vertex that is connected to vi. The clustering coefficient for a graph is given by Watts and Strogatz [43] as the average of the clustering coefficient for each vertex, (3) and is a measure of the tendency of graph vertices to form local clusters. For example, in the graph of Figure 1, in order to compute the cluster coefficient for vertex STG.L, we first determine the other vertices to which it is directly connected. These neighbors are vertices STG.R, MTG.R, MTG.L and SMG.L. Then we determine how many edges exist in the set of neighbors. In this case, the edges among neighbors of vertex STG.L are 3 edges: {STG.R, MTG.R}, {MTG.L, MTG.R}, {MTG.L, SMG.L}. Next, we determine how many edges could hav... |

3047 | Choosability in graphs
- Erdős, Rubin, et al.
- 1979
(Show Context)
Citation Context ...n between their elements that are neither totally regular nor totally random. Uncovering the hidden regularities and organizational principles of brain networks often requires comparison with a null model network that has similar statistical properties. Well established network models which have been extensively used as null models are the Erdös-Rényi random graph, the small world, the scale free, and the geometric random graph models. These are briefly mentioned below: • Erdös-Rényi (random graph) model: Erdös and Rényi introduced the random graph model and initiated a large area of research [65, 66, 67, 68]. There are two closely related variants of the Erdös–Rényi random graph model. In the G(n, m) σ σ st st v( ) C v n n v B st stt V v ss V v ( ) ( )( ) ( ) \{ , }\{ } = − − ∈∈ ∑1 1 2 σ σ∑ d v u d v uG G ( , ) ( , ) .=+∞ ⇒ = 1 0 C v n d v uEf Gu V v ( ) ( , )\{ } = − ∈ ∑1 1 1 Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 443 model, a graph is chosen uniformly at random from the collection of all graphs which have n vertices and m edges. In the G(n, q) model, a graph is thought to be constructed by connecting vertices randomly. Each edge is included in the graph with pro... |

2383 | The diameter of random graphs
- Bollobás
- 1981
(Show Context)
Citation Context ...n between their elements that are neither totally regular nor totally random. Uncovering the hidden regularities and organizational principles of brain networks often requires comparison with a null model network that has similar statistical properties. Well established network models which have been extensively used as null models are the Erdös-Rényi random graph, the small world, the scale free, and the geometric random graph models. These are briefly mentioned below: • Erdös-Rényi (random graph) model: Erdös and Rényi introduced the random graph model and initiated a large area of research [65, 66, 67, 68]. There are two closely related variants of the Erdös–Rényi random graph model. In the G(n, m) σ σ st st v( ) C v n n v B st stt V v ss V v ( ) ( )( ) ( ) \{ , }\{ } = − − ∈∈ ∑1 1 2 σ σ∑ d v u d v uG G ( , ) ( , ) .=+∞ ⇒ = 1 0 C v n d v uEf Gu V v ( ) ( , )\{ } = − ∈ ∑1 1 1 Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 443 model, a graph is chosen uniformly at random from the collection of all graphs which have n vertices and m edges. In the G(n, q) model, a graph is thought to be constructed by connecting vertices randomly. Each edge is included in the graph with pro... |

2141 | Statistical mechanics of complex networks
- Albert, Barabási
- 2002
(Show Context)
Citation Context ...v) of a vertex v which for an undirected graph is simply defined as the degree deg(v) of v. Degree centrality is normalized to range [0, 1] by dividing deg(v) with the maximum possible degree (n – 1). (8) In directed networks, two variants of the degree centrality may be appropriate: the in-degree centrality C–D(v) = deg−(v)/(n−1) and the out-degree centrality C+D (v) = deg+(v)/(n−1). Degree is the most fundamental network measure and most other measures are linked to vertex degree. The degree sequence is argued to reflect some fundamental aspects of natural, social and technological networks [59]. Manke et al. [60] advocate the view that the degree can be considered as correlation of underlying dynamical properties, such as the stability of a dynamic process to random perturbations. In the C v v nD ( ) deg( )= −1 442 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data graph of Figure 1 we have deg(PrCG.R) = 11, deg(SPL.R) = 11 and deg(SFG.R) = 11. The corresponding normalized centralities are CD(PrCG.R) = 0.25, CD(SPL.R) = 0.25 and CD(SFG.R) = 0.25. Shortest-Path Efficiency Latora and Marchiori [61, 62] defined the efficiency efvu in the communication b... |

1859 |
Investigating causal relations by econometric models and cross-spectral methods
- Granger
- 1969
(Show Context)
Citation Context ...ross spectral density function Sx y( f ), which is simply derived via the FFT of the cross-correlation, normalized by their individual autospectral density functions. Hence, MSC is calculated using the Welch’s method as: (11) where <·> indicates window averaging. The estimated MSC for a given frequency f ranges between 0 (no coupling) and 1 (maximum linear interdependence). Partial Directed Coherence (PDC) The main advantage of this linear method is that it is able to derive additional information on the “driver and response” relationship between observations. The concept of Granger-causality [81] is based on the commonsense idea that causes precede their effects in time and is formulated in terms of predictability. In a linear framework, Granger-causality is commonly evaluated by fitting Vector Autoregressive Models. Suppose that a set of n simultaneously observed time series is adequately represented by a Vector Autoregressive Model of order p (MVAR(p)): (12) where is the coefficient matrix at time lag k, and w(t) = [w1(t),…,wn(t)]T is the vector of model innovations having zero mean and covariance matrix Σw. The autoregressive coefficients aij(k), i, j = 1,…, n represent the A k n n... |

869 |
Network biology: understanding the cell’s functional organization
- Barabasi, Oltvai
- 2004
(Show Context)
Citation Context ... − 2 2 22 eg( ) deg( )) { , } u v u v E +( )∈∑ 2 d v u d v uG G ( , ) ( , ) .=+∞ ⇒ = 1 0 E n n d u vf G u v V u v = − ∈ ≠∑ 1 1 1 ( ) ( , ), , d u v G ( , ) =+∞d u v M n G ( , ) = >>d u v G ( , ) =+∞ Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 441 Many social networks are assortative since vertices having many connections tend to connect with other highly connected vertices [50]. On the other hand, most technological and biological networks are disassortative since they have the property that vertices with high degree are preferably connected with ones of low degree [51]. Exceptions are the protein contact networks and the brain functional networks which are assortative [52, 53]. Disassortative mixing observed in certain biological networks (metabolic signaling pathways network, and gene regulatory network) is conjectured to be responsible for decreasing the likelihood of crosstalk between different functional modules of the cell, and increasing the overall robustness of a network by localizing effects of deleterious perturbations [54]. On the other hand, from computational studies, it has been observed that information can be easily transferred through assor... |

670 |
Exploring complex networks
- Strogatz
- 2001
(Show Context)
Citation Context ...em, although each interaction depends on time, space, cognitive task and many other intrinsic details. Especially during the last decade, it became evident that brain functional networks as well as brain anatomical networks are characterized by the same topological properties that are present in most real networks, as for instance relatively small characteristic path lengths (average length over all shortest paths between each pair of vertices), high clustering coefficients, fat tailed shapes in the degree distributions, degree correlations, and the presence of motifs and community structures [2, 19]. Brain anatomical and functional networks are neither totally regular nor entirely random. Uncovering the hidden regularities and organizational principles of brain networks often requires comparison with a null model network that has similar statistical properties. A well-fitting network model that reproduces the network structure and/or the laws through which the network has emerged can enable us to understand the underlying processes and to predict the structure and behavior of the brain. Thus far, four null model networks have been considered: the Erdös-Rényi random graph, the small world... |

609 |
Complex brain networks: graph theoretical analysis of structural and functional systems.
- Bullmore, Sporns
- 2009
(Show Context)
Citation Context ...tion; brain dynamics; brain functional networks; graph measures; random graph models 1. INTRODUCTION The brain is a complex dynamical system in which information is continuously processed and transferred to other interconnected regions to make up a functional network [1–5]. Functional networks are thought to provide the physiological basis for *Corresponding author. Address: BMI lab, Institute of Computer Science (ICS), Foundation for Research and Technology (FORTH), Heraklion 71110, Greece. E-mail address: sakkalis@ics.forth.gr (V. Sakkalis). information processing and mental representations [6, 7], and have been studied across different conditions of rest [8, 9, 10] or cognitive load [11]. Studies with detailed electroencephalography (EEG) and magnetoencephalography (MEG) signals have revealed local synchronization patterns and cortico-cortical interactions involved in several cognitive functions [12], with composite subtasks being triggered within different brain regions by unitary brain sources that subsequently synchronize to complete the task. Thus, the dynamics of interaction among different EEG/ MEG channels may be used for indexing neural synchrony of such local or distant brain... |

543 | A Faster Algorithm for Betweenness Centrality
- Brandes
(Show Context)
Citation Context ...g normalized centralities are CD(PrCG.R) = 0.25, CD(SPL.R) = 0.25 and CD(SFG.R) = 0.25. Shortest-Path Efficiency Latora and Marchiori [61, 62] defined the efficiency efvu in the communication between vertices v and u to be inversely proportional to the shortest distance, 1/dG (v, u). Then the average efficiency of vertex v is given by: (9) Note that eqn. (9) can also be used for disconnected graphs. If some vertices v and u are not connected, they do not contribute to CEf (v) Shortest-Path Betweenness Centrality The shortest-path betweenness centrality CB(v) of a vertex v ∈ V is defined to be [63, 64] (10) where σst is the number of shortest (s, t)-paths, and let σst(v) be the number of shortest (s, t)-paths passing through some vertex v other than s, t. The relative numbers are interpreted as the extent to which vertex v controls the communication between vertices s and t. A vertex is central, if it is between many pairs of other vertices. The definition of betweenness applies to disconnected graphs without modification. 2.4. Network Models The basic idea behind using graphs as the first step in the study of a brain network is that measuring some basic properties of a complex network can ... |

543 |
Random geometric graphs
- Penrose
- 2003
(Show Context)
Citation Context ...roperty, like random graphs, with high clustering coefficient like lattices. This network model is also called small-world. In brain network literature, the term small-world refers to this model (viz. networks with small diameter and high clustering coefficient). Both brain functional and anatomical networks are small-world [47, 69]. • Geometric random model: A geometric graph G(V, ρ) with radius ρ is a graph where points in a metric space correspond to vertices, and two vertices are adjacent if the distance between them is at most ρ. More details about geometric random graphs can be found in [78]. When the position of vertices in a (possibly high dimensional) Euclidean space is important, the random geometric graph is a good candidate null model. Computational experiments have revealed close matches between key topological properties of Protein-Protein Interaction networks and geometric random graph models [79]. Oikonomou showed that brain functional networks estimated at sensor space from schizophrenia and epilepsy data also share key topological properties with geometric random graphs [80]. Recent articles illustrate that there may be evidence for small world networks in characteriz... |

527 |
Lethality and centrality in protein networks
- Jeong, Mason, et al.
(Show Context)
Citation Context ... scale and that there is a small number of very highly connected vertices called hubs. Recent interest in scalefree networks started in 1999 by Barabási, et al. [70, 71] who proposed a mechanism to explain the appearance of the power-law distribution in a stochastic growth model in which new vertices are added continuously and they preferentially attach to existing vertices with probability proportional to the degree of the target vertex. Many real world networks have power-law degree distributions, such as the World Wide Web [72], the metabolic reaction networks [73] and the protein networks [74]. There is no consensus in scientific community on whether the degree distribution of brain function or anatomical networks follows either a power law [69, 75] or a power law with exponential cut-off [76, 77]. • Small-world model: Many real world networks exhibit what is called the smallworld property, viz. the condition that most vertices can be reached from every other vertex through short paths. Erdös-Rényi and scale-free networks also have the small-world property. The network model published by Watts and Strogatz [43] combines the small-world property, like random graphs, with high cluste... |

512 |
The brainweb: Phase synchronization and large-scale integration
- Varela, Lachaux, et al.
- 2001
(Show Context)
Citation Context ...tion; brain dynamics; brain functional networks; graph measures; random graph models 1. INTRODUCTION The brain is a complex dynamical system in which information is continuously processed and transferred to other interconnected regions to make up a functional network [1–5]. Functional networks are thought to provide the physiological basis for *Corresponding author. Address: BMI lab, Institute of Computer Science (ICS), Foundation for Research and Technology (FORTH), Heraklion 71110, Greece. E-mail address: sakkalis@ics.forth.gr (V. Sakkalis). information processing and mental representations [6, 7], and have been studied across different conditions of rest [8, 9, 10] or cognitive load [11]. Studies with detailed electroencephalography (EEG) and magnetoencephalography (MEG) signals have revealed local synchronization patterns and cortico-cortical interactions involved in several cognitive functions [12], with composite subtasks being triggered within different brain regions by unitary brain sources that subsequently synchronize to complete the task. Thus, the dynamics of interaction among different EEG/ MEG channels may be used for indexing neural synchrony of such local or distant brain... |

407 | Speci and stability in topology of protein networks
- Maslov, Sneppen
(Show Context)
Citation Context ...e disassortative since they have the property that vertices with high degree are preferably connected with ones of low degree [51]. Exceptions are the protein contact networks and the brain functional networks which are assortative [52, 53]. Disassortative mixing observed in certain biological networks (metabolic signaling pathways network, and gene regulatory network) is conjectured to be responsible for decreasing the likelihood of crosstalk between different functional modules of the cell, and increasing the overall robustness of a network by localizing effects of deleterious perturbations [54]. On the other hand, from computational studies, it has been observed that information can be easily transferred through assortative networks as compared to that in disassortative networks [48, 49, 55]. Note that the network of Figure 1 is assortative having an assortativity coefficient of 0.083. 2.3. Centrality Measures Characterizing local properties of networks is important both in theory and practice since at the local scale, we can detect which vertices are the most relevant to the organization and functioning of a network. These local measures are commonly named centrality measures (or c... |

388 |
Efficient behavior of smallworld networks,”
- Latora, Marchiori
- 2001
(Show Context)
Citation Context ... aspects of natural, social and technological networks [59]. Manke et al. [60] advocate the view that the degree can be considered as correlation of underlying dynamical properties, such as the stability of a dynamic process to random perturbations. In the C v v nD ( ) deg( )= −1 442 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data graph of Figure 1 we have deg(PrCG.R) = 11, deg(SPL.R) = 11 and deg(SFG.R) = 11. The corresponding normalized centralities are CD(PrCG.R) = 0.25, CD(SPL.R) = 0.25 and CD(SFG.R) = 0.25. Shortest-Path Efficiency Latora and Marchiori [61, 62] defined the efficiency efvu in the communication between vertices v and u to be inversely proportional to the shortest distance, 1/dG (v, u). Then the average efficiency of vertex v is given by: (9) Note that eqn. (9) can also be used for disconnected graphs. If some vertices v and u are not connected, they do not contribute to CEf (v) Shortest-Path Betweenness Centrality The shortest-path betweenness centrality CB(v) of a vertex v ∈ V is defined to be [63, 64] (10) where σst is the number of shortest (s, t)-paths, and let σst(v) be the number of shortest (s, t)-paths passing through some ver... |

358 | Mean-field theory for scale-free random networks
- Barabasi, Jeong
- 1999
(Show Context)
Citation Context ...ce they have, among other properties, low clustering coefficient whereas both anatomical and functional networks have high clustering coefficient [7, 47, 69]. • Scale-free model: The scale-free network model is characterized by its degree distribution, viz. the probability distribution of the degrees over the whole network, following a power law. The power law implies that the degree distribution of these networks has no characteristic scale and that there is a small number of very highly connected vertices called hubs. Recent interest in scalefree networks started in 1999 by Barabási, et al. [70, 71] who proposed a mechanism to explain the appearance of the power-law distribution in a stochastic growth model in which new vertices are added continuously and they preferentially attach to existing vertices with probability proportional to the degree of the target vertex. Many real world networks have power-law degree distributions, such as the World Wide Web [72], the metabolic reaction networks [73] and the protein networks [74]. There is no consensus in scientific community on whether the degree distribution of brain function or anatomical networks follows either a power law [69, 75] or a ... |

261 |
Spontaneous fluctuations in brain activity observed with functional magnetic resonance imaging
- Fox, E
- 2007
(Show Context)
Citation Context ...andom graph models 1. INTRODUCTION The brain is a complex dynamical system in which information is continuously processed and transferred to other interconnected regions to make up a functional network [1–5]. Functional networks are thought to provide the physiological basis for *Corresponding author. Address: BMI lab, Institute of Computer Science (ICS), Foundation for Research and Technology (FORTH), Heraklion 71110, Greece. E-mail address: sakkalis@ics.forth.gr (V. Sakkalis). information processing and mental representations [6, 7], and have been studied across different conditions of rest [8, 9, 10] or cognitive load [11]. Studies with detailed electroencephalography (EEG) and magnetoencephalography (MEG) signals have revealed local synchronization patterns and cortico-cortical interactions involved in several cognitive functions [12], with composite subtasks being triggered within different brain regions by unitary brain sources that subsequently synchronize to complete the task. Thus, the dynamics of interaction among different EEG/ MEG channels may be used for indexing neural synchrony of such local or distant brain sources [13]. Such sources act synchronously behaving similar to coup... |

223 |
A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs,”
- Achard, Salvador, et al.
- 2006
(Show Context)
Citation Context ...n the appearance of the power-law distribution in a stochastic growth model in which new vertices are added continuously and they preferentially attach to existing vertices with probability proportional to the degree of the target vertex. Many real world networks have power-law degree distributions, such as the World Wide Web [72], the metabolic reaction networks [73] and the protein networks [74]. There is no consensus in scientific community on whether the degree distribution of brain function or anatomical networks follows either a power law [69, 75] or a power law with exponential cut-off [76, 77]. • Small-world model: Many real world networks exhibit what is called the smallworld property, viz. the condition that most vertices can be reached from every other vertex through short paths. Erdös-Rényi and scale-free networks also have the small-world property. The network model published by Watts and Strogatz [43] combines the small-world property, like random graphs, with high clustering coefficient like lattices. This network model is also called small-world. In brain network literature, the term small-world refers to this model (viz. networks with small diameter and high clustering coe... |

221 | Mapping the structural core of human cerebral cortex.
- Hagmann, Cammoun, et al.
- 2008
(Show Context)
Citation Context ...rk of Figure 1 is assortative having an assortativity coefficient of 0.083. 2.3. Centrality Measures Characterizing local properties of networks is important both in theory and practice since at the local scale, we can detect which vertices are the most relevant to the organization and functioning of a network. These local measures are commonly named centrality measures (or centrality indices), and have proven of great value in analyzing the role played by individuals in social networks and in identifying essential proteins [56], keystone species [57], and functionally important brain regions [58]. There is no commonly accepted definition for centrality. Many authors introduced their own centrality, avoiding a strict definition for centrality in general. The intuition about a centrality measure is that it denotes an order of importance on the vertices or edges of a graph by assigning real values to them. As a minimal requirement, most centrality measures depend only on the structure of a graph. Degree Centrality The simplest and most widely studied centrality is the degree centrality CD(v) of a vertex v which for an undirected graph is simply defined as the degree deg(v) of v. Degree c... |

217 |
Why social networks are different from other types of networks
- MEJ, Park
- 2003
(Show Context)
Citation Context ...ng to a scalar vertex property is mixing according to vertex degree. r E u v u v u v E= − + ∈∑4 ||deg( )deg( ) (deg( ) deg( )){ , } {u v E u v E E u v , } { , } ||(deg( ) deg( ) ) (d ∈ ∈ ∑ ∑ ( ) + − 2 2 22 eg( ) deg( )) { , } u v u v E +( )∈∑ 2 d v u d v uG G ( , ) ( , ) .=+∞ ⇒ = 1 0 E n n d u vf G u v V u v = − ∈ ≠∑ 1 1 1 ( ) ( , ), , d u v G ( , ) =+∞d u v M n G ( , ) = >>d u v G ( , ) =+∞ Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 441 Many social networks are assortative since vertices having many connections tend to connect with other highly connected vertices [50]. On the other hand, most technological and biological networks are disassortative since they have the property that vertices with high degree are preferably connected with ones of low degree [51]. Exceptions are the protein contact networks and the brain functional networks which are assortative [52, 53]. Disassortative mixing observed in certain biological networks (metabolic signaling pathways network, and gene regulatory network) is conjectured to be responsible for decreasing the likelihood of crosstalk between different functional modules of the cell, and increasing the overall robustnes... |

211 | Functional and Effective Connectivity in Neuroimaging: A Synthesis. - Friston - 1994 |

183 | Organization, development and function of complex brain networks
- Sporns, Chialvo, et al.
- 2004
(Show Context)
Citation Context ...em, although each interaction depends on time, space, cognitive task and many other intrinsic details. Especially during the last decade, it became evident that brain functional networks as well as brain anatomical networks are characterized by the same topological properties that are present in most real networks, as for instance relatively small characteristic path lengths (average length over all shortest paths between each pair of vertices), high clustering coefficients, fat tailed shapes in the degree distributions, degree correlations, and the presence of motifs and community structures [2, 19]. Brain anatomical and functional networks are neither totally regular nor entirely random. Uncovering the hidden regularities and organizational principles of brain networks often requires comparison with a null model network that has similar statistical properties. A well-fitting network model that reproduces the network structure and/or the laws through which the network has emerged can enable us to understand the underlying processes and to predict the structure and behavior of the brain. Thus far, four null model networks have been considered: the Erdös-Rényi random graph, the small world... |

168 | Efficiency and cost of economical brain functional networks,”
- Achard, Bullmore
- 2007
(Show Context)
Citation Context ...s the arithmetic mean of the shortest path distances. Note that eqn. (6) can also be used for disconnected graphs. If some vertices v and u are not connected, they do not contribute to Ef Network efficiency quantifies the effectiveness of information flow within brain networks. He et al. showed that in multiple sclerosis, the efficiency of the anatomical network is reduced in a manner proportional to the extent of total white matter lesions [46]. Achard and Bullmore found that the resting state functional networks provide high efficiency of information processing for low anatomical connection [47]. Efficiency was reduced in older people and the detrimental effects of age to efficiency were localized to frontal and temporal cortical and subcortical regions [47]. The efficiency of graph in Figure 1 is 0.407. Assortativity An important network feature is the similarity between properties of adjacent network vertices. Newman [48, 49] proposed a measure to quantify the degree of similarity (dissimilarity) between adjacent vertices in a network using assortative mixing, which is given by correlation between the degrees of every pair of adjacent vertices. The assortativity coefficient for an ... |

138 |
Partial directed coherence: a new concept in neural structure determination
- A, Sameshima
- 2001
(Show Context)
Citation Context ...e the task. Thus, the dynamics of interaction among different EEG/ MEG channels may be used for indexing neural synchrony of such local or distant brain sources [13]. Such sources act synchronously behaving similar to coupled oscillators [14] and their interactions can be measured using pair-wise linear (cross-coherence or phasecoherence) [16] or nonlinear dynamics and models [15, 16, 17]. Furthermore, the causality of the functional coupling of such oscillatory activities can be assessed with partially directed coherence, which reveals the direction of statistically significant relationships [18]. Synchronization can be evaluated not only on the actual recordings on the scalp electrodes but also on independent components. The later are derived from linear un-mixing transforms and are free from volume conduction effects [15, 16]. Networks are modeled by graphs which consist of a set of vertices and a set of pair of vertices called edges (Figure 1). In this respect, graph theory offers a unique perspective and a common framework for studying interactions between local and remote cortical areas, where areas correspond to vertices and interactions to edges. These interactions can be estim... |

130 |
editors. Network Analysis: Methodological Foundations
- Brandes, Erlebach
- 2005
(Show Context)
Citation Context ...w many edges could have existed between the neighbors. In this case, there could be 6 edges (one edge for every pair of vertices STG.R, MTG.R, MTG.L, SMG.Y ). The clustering coefficient of vertex STG.L is the ratio of 3/6 = 0.5. The clustering coefficient of the graph of Figure 1 is 0.299. Eccentricity, Radius and Diameter of a Graph There are several measures defined in terms of distance, such as the eccentricity of a vertex, the radius, and the diameter of a graph. The eccentricity ε (v) of a vertex v in a connected graph G is the maximum graph distance between v and any other vertex u of G [44, 45] (4) For a disconnected graph, all vertices are defined to have infinite eccentricity. The radius and the diameter of a graph are the minimum and the maximum eccentricity of any vertex in the graph, respectively. The diameter represents the greatest distance between any two vertices. Average Shortest Path Length The average shortest path length (5)L n n d u v Gu v V u v = − ∈ ≠∑ 1 1( ) ( , ) , , ε( ) |max{ ( , ) |} |v d u v u V G = ∈ C n C v v V = ∈ ∑1 ( ) C v k k e v v v v N v i i i j k j k i ( ) ( ) { { , } |, ( )}= − = ∈2 1 k k i i ( )− 1 2 440 Graph Analysis and Visualization for Brain Fun... |

115 |
EEG dynamics in patients with Alzheimer’s disease. Clinical Neurophysiology.
- Jeong
- 2004
(Show Context)
Citation Context ... subjects with neuropathologies such as 436 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data epilepsy, Alzheimer’s disease, autism, Parkinson’s disease and schizophrenia [20, 21, 22, 23, 24]. All these diseases have been associated with abnormal neural synchronization, and as a result, with functional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a disturbance in connectivity between different brain regions. In particular, there is reduction in both local and long-r... |

111 | Small-world anatomical networks in the human brain revealed by cortical thickness from MRI.
- He, Chen, et al.
- 2007
(Show Context)
Citation Context ... from multivariate neurophysiological signals (EEG, MEG, electrocorticography (ECoG)) and/or haemodynamic response images (functional magnetic resonance imaging (fMRI)). Then a network is formed by corresponding brain areas or channels to vertices and by considering an edge between two vertices if and only if the estimated interdependence is above a threshold (Figure 1). The next step in the analysis is to measure some statistics to characterize the network. Using the network characterization, one can draw conclusions on the effect of illnesses or of cognitive loads on functional connectivity [9, 28, 41, 42]. Figure 2 outlines a basic signal preprocessing and analysis schema. 438 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data EEG/MEG/ECoG data Artifacts removal (Cardiac artifacts, eye movements, etc.) Sources estimation (optional) Bandpass filtering Synchronization measures – similarity matrices Adjacency functions – networks Analysis and visualization of networks Relate network concepts to external information Figure 2. Sample outline of neurophysiological signals analysis. A graph G = (V, E ) is a pair of vertices V = {v1, v2, …, vn} and edges E = {e1, e2, …... |

107 | Economic small-world behavior in weighted networks
- Latora, Marchiori
(Show Context)
Citation Context ... aspects of natural, social and technological networks [59]. Manke et al. [60] advocate the view that the degree can be considered as correlation of underlying dynamical properties, such as the stability of a dynamic process to random perturbations. In the C v v nD ( ) deg( )= −1 442 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data graph of Figure 1 we have deg(PrCG.R) = 11, deg(SPL.R) = 11 and deg(SFG.R) = 11. The corresponding normalized centralities are CD(PrCG.R) = 0.25, CD(SPL.R) = 0.25 and CD(SFG.R) = 0.25. Shortest-Path Efficiency Latora and Marchiori [61, 62] defined the efficiency efvu in the communication between vertices v and u to be inversely proportional to the shortest distance, 1/dG (v, u). Then the average efficiency of vertex v is given by: (9) Note that eqn. (9) can also be used for disconnected graphs. If some vertices v and u are not connected, they do not contribute to CEf (v) Shortest-Path Betweenness Centrality The shortest-path betweenness centrality CB(v) of a vertex v ∈ V is defined to be [63, 64] (10) where σst is the number of shortest (s, t)-paths, and let σst(v) be the number of shortest (s, t)-paths passing through some ver... |

101 |
Network structure of cerebral cortex shapes functional connectivity on multiple time scales,
- Honey, Kötter, et al.
- 2007
(Show Context)
Citation Context ...andom graph models 1. INTRODUCTION The brain is a complex dynamical system in which information is continuously processed and transferred to other interconnected regions to make up a functional network [1–5]. Functional networks are thought to provide the physiological basis for *Corresponding author. Address: BMI lab, Institute of Computer Science (ICS), Foundation for Research and Technology (FORTH), Heraklion 71110, Greece. E-mail address: sakkalis@ics.forth.gr (V. Sakkalis). information processing and mental representations [6, 7], and have been studied across different conditions of rest [8, 9, 10] or cognitive load [11]. Studies with detailed electroencephalography (EEG) and magnetoencephalography (MEG) signals have revealed local synchronization patterns and cortico-cortical interactions involved in several cognitive functions [12], with composite subtasks being triggered within different brain regions by unitary brain sources that subsequently synchronize to complete the task. Thus, the dynamics of interaction among different EEG/ MEG channels may be used for indexing neural synchrony of such local or distant brain sources [13]. Such sources act synchronously behaving similar to coup... |

97 |
Pathological synchronization in Parkinson's disease: networks, models and treatments,
- Hammond, Bergman, et al.
- 2007
(Show Context)
Citation Context ...nctional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a disturbance in connectivity between different brain regions. In particular, there is reduction in both local and long-range synchronization [34, 35, 36, 37]. Additionally, a number of studies demonstrate that there is a strong negative association between the characteristic path length (the average shortest path lengths between each pair of vertices) of the resting-state brain functional network and the intelligence quotient (IQ), sugge... |

95 | Disrupted small-world networks in schizophrenia.
- Liu, Liang, et al.
- 2008
(Show Context)
Citation Context ...ons associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a disturbance in connectivity between different brain regions. In particular, there is reduction in both local and long-range synchronization [34, 35, 36, 37]. Additionally, a number of studies demonstrate that there is a strong negative association between the characteristic path length (the average shortest path lengths between each pair of vertices) of the resting-state brain functional network and the intelligence quotient (IQ), suggesting that human intellectual Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 437 PCU.L PHG.R PHG.L OP.L OP.R IFG.L SOG.L MFG.R SOG.R PoCG.L PoCG.R SFG.R SFG.L PCU.R CUN.R SPL.R ANG.L ANG.R LOFG.R LOFG.L LOTG.R PrCG.R PrCG.L ITG.L ITG.R STG.R STG.L SMG.R LING.R SMG.LMTG.L LING.L SPL.L MdFG.L MFG.L MTG.R C... |

95 |
A set of measures of centrality based upon betweenness
- Freeman
- 1977
(Show Context)
Citation Context ...g normalized centralities are CD(PrCG.R) = 0.25, CD(SPL.R) = 0.25 and CD(SFG.R) = 0.25. Shortest-Path Efficiency Latora and Marchiori [61, 62] defined the efficiency efvu in the communication between vertices v and u to be inversely proportional to the shortest distance, 1/dG (v, u). Then the average efficiency of vertex v is given by: (9) Note that eqn. (9) can also be used for disconnected graphs. If some vertices v and u are not connected, they do not contribute to CEf (v) Shortest-Path Betweenness Centrality The shortest-path betweenness centrality CB(v) of a vertex v ∈ V is defined to be [63, 64] (10) where σst is the number of shortest (s, t)-paths, and let σst(v) be the number of shortest (s, t)-paths passing through some vertex v other than s, t. The relative numbers are interpreted as the extent to which vertex v controls the communication between vertices s and t. A vertex is central, if it is between many pairs of other vertices. The definition of betweenness applies to disconnected graphs without modification. 2.4. Network Models The basic idea behind using graphs as the first step in the study of a brain network is that measuring some basic properties of a complex network can ... |

94 | Functional and anatomical cortical underconnectivity in autism: Evidence from an FMRI study of an executive function task and corpus callosum morphometry.
- ust, Cherkassky, et al.
- 2007
(Show Context)
Citation Context ...Alzheimer’s disease, autism, Parkinson’s disease and schizophrenia [20, 21, 22, 23, 24]. All these diseases have been associated with abnormal neural synchronization, and as a result, with functional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a disturbance in connectivity between different brain regions. In particular, there is reduction in both local and long-range synchronization [34, 35, 36, 37]. Additionally, a number of studies demonstrate that there is a strong negative association between t... |

94 | S.: Graph Drawing by Stress Majorization
- Gansner, Koren, et al.
- 2005
(Show Context)
Citation Context ... (indicating complete synchronization). Although, in general, , this measure is not suitable to infer driver response relationships. Accordingly, we used the mean value , as a Robust Interdependence Measure (RIM) between X and Y. 3. BRAIN NETWORK VISUALIZATION AND APPLICATION DOMAINS Brain functional networks can be practically visualized by assigning to vertices the coordinates of the corresponding channels as described in section 2.1. Additionally, in order to show the network structure, one can use network visualization techniques such as the stress majorization technique of Gansner et al. [86] and the binary stress model of Koren et al. [87]. Apart from real world network visualization and graph analysis cases, there are some initial but indicative examples of possible translation of graph research outcomes in the clinical practice. More specifically, considering that the so called “disconnectivity syndromes” represent many pathological and neuropsychological diseases, based on their functional impairment symptoms, one can infer directly the usability and significance of network analysis and visualization tools based on graph theory. This section briefly discusses applications of g... |

92 |
Intrinsic functional architecture in the anaesthetized monkey brain. Nature 447
- Vincent, Patel, et al.
- 2007
(Show Context)
Citation Context ...andom graph models 1. INTRODUCTION The brain is a complex dynamical system in which information is continuously processed and transferred to other interconnected regions to make up a functional network [1–5]. Functional networks are thought to provide the physiological basis for *Corresponding author. Address: BMI lab, Institute of Computer Science (ICS), Foundation for Research and Technology (FORTH), Heraklion 71110, Greece. E-mail address: sakkalis@ics.forth.gr (V. Sakkalis). information processing and mental representations [6, 7], and have been studied across different conditions of rest [8, 9, 10] or cognitive load [11]. Studies with detailed electroencephalography (EEG) and magnetoencephalography (MEG) signals have revealed local synchronization patterns and cortico-cortical interactions involved in several cognitive functions [12], with composite subtasks being triggered within different brain regions by unitary brain sources that subsequently synchronize to complete the task. Thus, the dynamics of interaction among different EEG/ MEG channels may be used for indexing neural synchrony of such local or distant brain sources [13]. Such sources act synchronously behaving similar to coup... |

92 |
Breakdown in cortical effective connectivity during midazolam-induced loss of consciousness.
- Ferrarelli, Massimini, et al.
- 2010
(Show Context)
Citation Context ... from multivariate neurophysiological signals (EEG, MEG, electrocorticography (ECoG)) and/or haemodynamic response images (functional magnetic resonance imaging (fMRI)). Then a network is formed by corresponding brain areas or channels to vertices and by considering an edge between two vertices if and only if the estimated interdependence is above a threshold (Figure 1). The next step in the analysis is to measure some statistics to characterize the network. Using the network characterization, one can draw conclusions on the effect of illnesses or of cognitive loads on functional connectivity [9, 28, 41, 42]. Figure 2 outlines a basic signal preprocessing and analysis schema. 438 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data EEG/MEG/ECoG data Artifacts removal (Cardiac artifacts, eye movements, etc.) Sources estimation (optional) Bandpass filtering Synchronization measures – similarity matrices Adjacency functions – networks Analysis and visualization of networks Relate network concepts to external information Figure 2. Sample outline of neurophysiological signals analysis. A graph G = (V, E ) is a pair of vertices V = {v1, v2, …, vn} and edges E = {e1, e2, …... |

91 | The Elusive Concept of Brain Connectivity, - Horwitz - 2003 |

91 | The importance of bottlenecks in protein networks: Correlation with gene essentiality and expression dynamics
- Yu, Kim, et al.
- 2007
(Show Context)
Citation Context ...d to that in disassortative networks [48, 49, 55]. Note that the network of Figure 1 is assortative having an assortativity coefficient of 0.083. 2.3. Centrality Measures Characterizing local properties of networks is important both in theory and practice since at the local scale, we can detect which vertices are the most relevant to the organization and functioning of a network. These local measures are commonly named centrality measures (or centrality indices), and have proven of great value in analyzing the role played by individuals in social networks and in identifying essential proteins [56], keystone species [57], and functionally important brain regions [58]. There is no commonly accepted definition for centrality. Many authors introduced their own centrality, avoiding a strict definition for centrality in general. The intuition about a centrality measure is that it denotes an order of importance on the vertices or edges of a graph by assigning real values to them. As a minimal requirement, most centrality measures depend only on the structure of a graph. Degree Centrality The simplest and most widely studied centrality is the degree centrality CD(v) of a vertex v which for an ... |

80 |
A robust method for detecting interdependences: application to intracranially recorded EEG
- Arnhold, Grassberger, et al.
- 1999
(Show Context)
Citation Context ...ructed domain [84]. A physiological time series such as the EEG appears to have more than the single degree of freedom represented just by plotting the voltage as a function of time. To free up some of these unknown parameters, a standard technique is to map the scalar time series to a vector-valued one in a higher dimensional space Rm, thereby giving it an extension in space as well as time. Hence, one may measure how neighborhoods (recurrences) in state space located in one attractor map to each other. This idea turned out to be the most robust and reliable way of assessing the extent of GS [85]. First, we reconstruct delay vectors out of our time series: and (20) where i = 1…N ′, N′ = N − (m−1)τ and m, τ are the embedding dimension and time lag, respectively. Let ri, j and si, j, j = 1, …, k, denote the time indices of the k nearest neighbors of xi and yi, respectively. For each xi, the squared mean Euclidean distance to its k neighbors is defined as: (21) The Y-conditioned squared mean Euclidean distance is defined by replacing the nearest neighbors by the equal time partners of the closest neighbors of yi. If the set of reconstructed vectors (point cloud xi) has an average squared... |

77 |
The application of graph theoretical analysis to complex networks in the brain
- Reijneveld, Berendse
- 2007
(Show Context)
Citation Context ...hat are further utilized in describing the functional connectivity of a brain network, discussed in sections 2.2 and 2.3. Section 2.4 covers network models, and section 2.5 briefly reviews the most robust and popular linear and nonlinear methods for quantifying synchronization between time series pointing to references for further detailed information. Section 3 presents real case application paradigms, while section 4 concludes this paper. 2. METHODS 2.1. Preliminaries A recent trend in brain functional connectivity analysis is to model the interdependencies among brain signals with networks [7, 39]. Interdependence among different brain areas is estimated from multivariate neurophysiological signals (EEG, MEG, electrocorticography (ECoG)) and/or haemodynamic response images (functional magnetic resonance imaging (fMRI)). Then a network is formed by corresponding brain areas or channels to vertices and by considering an edge between two vertices if and only if the estimated interdependence is above a threshold (Figure 1). The next step in the analysis is to measure some statistics to characterize the network. Using the network characterization, one can draw conclusions on the effect of i... |

74 | Small-world networks and functional connectivity in Alzheimer’s disease,”
- Stam, Jones, et al.
- 2007
(Show Context)
Citation Context ... subjects with neuropathologies such as 436 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data epilepsy, Alzheimer’s disease, autism, Parkinson’s disease and schizophrenia [20, 21, 22, 23, 24]. All these diseases have been associated with abnormal neural synchronization, and as a result, with functional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a disturbance in connectivity between different brain regions. In particular, there is reduction in both local and long-r... |

73 |
Adaptive reconfiguration of fractal small-world human brain functional networks.
- Bassett, Meyer-Lindenberg, et al.
- 1951
(Show Context)
Citation Context ...CTION The brain is a complex dynamical system in which information is continuously processed and transferred to other interconnected regions to make up a functional network [1–5]. Functional networks are thought to provide the physiological basis for *Corresponding author. Address: BMI lab, Institute of Computer Science (ICS), Foundation for Research and Technology (FORTH), Heraklion 71110, Greece. E-mail address: sakkalis@ics.forth.gr (V. Sakkalis). information processing and mental representations [6, 7], and have been studied across different conditions of rest [8, 9, 10] or cognitive load [11]. Studies with detailed electroencephalography (EEG) and magnetoencephalography (MEG) signals have revealed local synchronization patterns and cortico-cortical interactions involved in several cognitive functions [12], with composite subtasks being triggered within different brain regions by unitary brain sources that subsequently synchronize to complete the task. Thus, the dynamics of interaction among different EEG/ MEG channels may be used for indexing neural synchrony of such local or distant brain sources [13]. Such sources act synchronously behaving similar to coupled oscillators [14] an... |

67 |
Rapoport A: Connectivity of random nets
- Solomonoff
- 1951
(Show Context)
Citation Context ...n between their elements that are neither totally regular nor totally random. Uncovering the hidden regularities and organizational principles of brain networks often requires comparison with a null model network that has similar statistical properties. Well established network models which have been extensively used as null models are the Erdös-Rényi random graph, the small world, the scale free, and the geometric random graph models. These are briefly mentioned below: • Erdös-Rényi (random graph) model: Erdös and Rényi introduced the random graph model and initiated a large area of research [65, 66, 67, 68]. There are two closely related variants of the Erdös–Rényi random graph model. In the G(n, m) σ σ st st v( ) C v n n v B st stt V v ss V v ( ) ( )( ) ( ) \{ , }\{ } = − − ∈∈ ∑1 1 2 σ σ∑ d v u d v uG G ( , ) ( , ) .=+∞ ⇒ = 1 0 C v n d v uEf Gu V v ( ) ( , )\{ } = − ∈ ∑1 1 1 Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 443 model, a graph is chosen uniformly at random from the collection of all graphs which have n vertices and m edges. In the G(n, q) model, a graph is thought to be constructed by connecting vertices randomly. Each edge is included in the graph with pro... |

67 |
Structural insights into aberrant topological patterns of large-scale cortical networks in Alzheimer’s disease.
- He, Chen, et al.
- 2008
(Show Context)
Citation Context ...rly one driving the late alpha. The early alpha associated with attention drives a delta component related to cognition, which in turn drives the late alpha component that is also related to memory operations; this relationship in Figure 6 is observed as node influence from 5 to 7 to 4 in the left network and from 6 to 19 to 24 in the right network. Similar tendencies maybe easily identified using directed graphs based on directed coupling methods such as PDC. Apart from the presented application domains, several studies have been conducted focusing on other pathologies like Alzheimer Disease [93], Epilepsy [94] and Parkinson’s disease [95]. 4. CONCLUSIONS Our brain is a complex network in which information is continuously processed and transported between spatially distributed but functionally linked regions [1]. This ongoing integration of information enables us to evaluate the world around us and to respond quickly and flexibly to complex situations. A snapshot of a subject’s dynamic brain functional network can be estimated from high resolution EEG measurements. Then this network can be characterized and analyzed using well formed concepts mainly based on graph theory and statistic... |

65 |
Connectivity and complexity: the relationship between neuroanatomy and brain dynamics
- Sporns, Tononi, et al.
(Show Context)
Citation Context ...efinition for centrality. Many authors introduced their own centrality, avoiding a strict definition for centrality in general. The intuition about a centrality measure is that it denotes an order of importance on the vertices or edges of a graph by assigning real values to them. As a minimal requirement, most centrality measures depend only on the structure of a graph. Degree Centrality The simplest and most widely studied centrality is the degree centrality CD(v) of a vertex v which for an undirected graph is simply defined as the degree deg(v) of v. Degree centrality is normalized to range [0, 1] by dividing deg(v) with the maximum possible degree (n – 1). (8) In directed networks, two variants of the degree centrality may be appropriate: the in-degree centrality C–D(v) = deg−(v)/(n−1) and the out-degree centrality C+D (v) = deg+(v)/(n−1). Degree is the most fundamental network measure and most other measures are linked to vertex degree. The degree sequence is argued to reflect some fundamental aspects of natural, social and technological networks [59]. Manke et al. [60] advocate the view that the degree can be considered as correlation of underlying dynamical properties, such as the ... |

65 |
Resting state cortical connectivity reflected in EEG coherence in individuals with autism. Biological Psychiatry.
- Murias, SJ, et al.
- 2007
(Show Context)
Citation Context ...Alzheimer’s disease, autism, Parkinson’s disease and schizophrenia [20, 21, 22, 23, 24]. All these diseases have been associated with abnormal neural synchronization, and as a result, with functional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a disturbance in connectivity between different brain regions. In particular, there is reduction in both local and long-range synchronization [34, 35, 36, 37]. Additionally, a number of studies demonstrate that there is a strong negative association between t... |

65 |
A review of diffusion tensor imaging studies in schizophrenia.
- Kubicki, McCarley
- 2007
(Show Context)
Citation Context ...ons associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a disturbance in connectivity between different brain regions. In particular, there is reduction in both local and long-range synchronization [34, 35, 36, 37]. Additionally, a number of studies demonstrate that there is a strong negative association between the characteristic path length (the average shortest path lengths between each pair of vertices) of the resting-state brain functional network and the intelligence quotient (IQ), suggesting that human intellectual Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 437 PCU.L PHG.R PHG.L OP.L OP.R IFG.L SOG.L MFG.R SOG.R PoCG.L PoCG.R SFG.R SFG.L PCU.R CUN.R SPL.R ANG.L ANG.R LOFG.R LOFG.L LOTG.R PrCG.R PrCG.L ITG.L ITG.R STG.R STG.L SMG.R LING.R SMG.LMTG.L LING.L SPL.L MdFG.L MFG.L MTG.R C... |

57 |
Disordered connectivity in the autistic brain: Challenges for the ‘new psychophysiology’,
- Rippon, Brock, et al.
- 2007
(Show Context)
Citation Context ...Alzheimer’s disease, autism, Parkinson’s disease and schizophrenia [20, 21, 22, 23, 24]. All these diseases have been associated with abnormal neural synchronization, and as a result, with functional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a disturbance in connectivity between different brain regions. In particular, there is reduction in both local and long-range synchronization [34, 35, 36, 37]. Additionally, a number of studies demonstrate that there is a strong negative association between t... |

52 |
Neurodegenerative diseases target large-scale human brain networks,”
- Seeley, Crawford, et al.
- 2009
(Show Context)
Citation Context ...ered: the Erdös-Rényi random graph, the small world, the scale free, and the geometric random graph models. The information contained in a network can be summarized with graph measures. Graph measures have been applied to topological analysis of brain functional networks, and many of them have been shown to reflect disease and statistically significant differences between healthy subjects and subjects with neuropathologies such as 436 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data epilepsy, Alzheimer’s disease, autism, Parkinson’s disease and schizophrenia [20, 21, 22, 23, 24]. All these diseases have been associated with abnormal neural synchronization, and as a result, with functional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing ... |

51 |
On random graphs. Publicationes Mathematicae 6:290–297
- Erdös, Rényi
- 1959
(Show Context)
Citation Context |

49 | Phase synchronization: from theory to data analysis
- Pikovsky, Schafer, et al.
- 2001
(Show Context)
Citation Context ...new PDC is defined as [82] (14) where A – (λ) = I − A(λ) and σ 2i refers to the variance of the innovation processes wi (t). ranges between 0 (indicating independence) and 1 (indicating maximum coherence). Phase Locking Value (PLV) Phase synchronization presents a different approach in analyzing the possible nonlinear interdependencies of the EEG signal and focuses on the phases of the signals. The Phase Locking Value (PLV) is one of the most used robust phase coupling measures. It assumes that two dynamic systems may have their phases synchronized even if their amplitudes are zero correlated [83]. The phase synchronization (PS) is defined as the locking of the phases associated to each signal: (15) In order to estimate the instantaneous phase of our signal, we transform it using the Hilbert transform (HT), whereby the analytical signal H(t) is computed as: (16) where x~(t) is the HT of x(t), defined as: (17)x t PV x t t t dt( ) ( )= ′ − ′ ′ −∞ ∞ ∫ 1 π H t x t ix t( ) ( ) ( )= + φ φ x y t t const( ) ( )− = π λi j← ( ) π λ σ λ σ λ λ i j i ij m mj mj H m p A A A ← = = ∑ ( ) ( ) ( )* ( ) 1 1 2 1 i = −1. A A ek i k k p ( )λ πλ= − = ∑ 2 1 446 Graph Analysis and Visualization for Brain Fu... |

47 |
Eccentricity and centrality in networks.
- Hage, Harary
- 1995
(Show Context)
Citation Context ...w many edges could have existed between the neighbors. In this case, there could be 6 edges (one edge for every pair of vertices STG.R, MTG.R, MTG.L, SMG.Y ). The clustering coefficient of vertex STG.L is the ratio of 3/6 = 0.5. The clustering coefficient of the graph of Figure 1 is 0.299. Eccentricity, Radius and Diameter of a Graph There are several measures defined in terms of distance, such as the eccentricity of a vertex, the radius, and the diameter of a graph. The eccentricity ε (v) of a vertex v in a connected graph G is the maximum graph distance between v and any other vertex u of G [44, 45] (4) For a disconnected graph, all vertices are defined to have infinite eccentricity. The radius and the diameter of a graph are the minimum and the maximum eccentricity of any vertex in the graph, respectively. The diameter represents the greatest distance between any two vertices. Average Shortest Path Length The average shortest path length (5)L n n d u v Gu v V u v = − ∈ ≠∑ 1 1( ) ( , ) , , ε( ) |max{ ( , ) |} |v d u v u V G = ∈ C n C v v V = ∈ ∑1 ( ) C v k k e v v v v N v i i i j k j k i ( ) ( ) { { , } |, ( )}= − = ∈2 1 k k i i ( )− 1 2 440 Graph Analysis and Visualization for Brain Fun... |

45 |
Neuronal Substrates of Sleep and Epilepsy,
- Steriade
- 1993
(Show Context)
Citation Context ... them have been shown to reflect disease and statistically significant differences between healthy subjects and subjects with neuropathologies such as 436 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data epilepsy, Alzheimer’s disease, autism, Parkinson’s disease and schizophrenia [20, 21, 22, 23, 24]. All these diseases have been associated with abnormal neural synchronization, and as a result, with functional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a dist... |

44 |
Small-world and scale-free organization of voxel-based resting-state functional connectivity in the human brain,”
- Heuvel, Stam, et al.
- 2008
(Show Context)
Citation Context ...eering · Vol. 1 · No. 3 · 2010 443 model, a graph is chosen uniformly at random from the collection of all graphs which have n vertices and m edges. In the G(n, q) model, a graph is thought to be constructed by connecting vertices randomly. Each edge is included in the graph with probability q, with the presence or absence of any two distinct edges in the graph being independent. The random graph model fails to adequately represent brain networks since they have, among other properties, low clustering coefficient whereas both anatomical and functional networks have high clustering coefficient [7, 47, 69]. • Scale-free model: The scale-free network model is characterized by its degree distribution, viz. the probability distribution of the degrees over the whole network, following a power law. The power law implies that the degree distribution of these networks has no characteristic scale and that there is a small number of very highly connected vertices called hubs. Recent interest in scalefree networks started in 1999 by Barabási, et al. [70, 71] who proposed a mechanism to explain the appearance of the power-law distribution in a stochastic growth model in which new vertices are added contin... |

43 |
Efficiency of functional brain networks and intellectual performance,
- Heuvel, Stam, et al.
- 2009
(Show Context)
Citation Context ... SPL.L MdFG.L MFG.L MTG.R CING.L LOTG.L MOFG.R MdFG.R MOFG.L MOTG.L MOTG.R IFG.R IOG.R Figure 1. Large-scale anatomical network in the human brain that consists of 45 areas [46]. Two areas are considered anatomically connected if they show statistically significant correlations in cortical thickness measurements from magnetic resonance images. This network is modeled as a graph which consists of vertices (yellow rectangles) and edges (line segments between pairs of vertices). performance is likely to be related to how efficiently our brain integrates information between multiple brain regions [38]. The purpose of this paper is to review the most robust linear and nonlinear synchronization measures for constructing functional networks including measures for comparing and charactering networks. It also stresses the significance of network measures as potential biomarkers of brain pathophysiology. This paper proceeds as follows: Section 2.1 provides an introductory review of basic graph concepts and indices that are further utilized in describing the functional connectivity of a brain network, discussed in sections 2.2 and 2.3. Section 2.4 covers network models, and section 2.5 briefly re... |

37 | Small-world properties of nonlinear brain activity in schizophrenia.
- Rubinov, Knock, et al.
- 2009
(Show Context)
Citation Context ...ons associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a disturbance in connectivity between different brain regions. In particular, there is reduction in both local and long-range synchronization [34, 35, 36, 37]. Additionally, a number of studies demonstrate that there is a strong negative association between the characteristic path length (the average shortest path lengths between each pair of vertices) of the resting-state brain functional network and the intelligence quotient (IQ), suggesting that human intellectual Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 437 PCU.L PHG.R PHG.L OP.L OP.R IFG.L SOG.L MFG.R SOG.R PoCG.L PoCG.R SFG.R SFG.L PCU.R CUN.R SPL.R ANG.L ANG.R LOFG.R LOFG.L LOTG.R PrCG.R PrCG.L ITG.L ITG.R STG.R STG.L SMG.R LING.R SMG.LMTG.L LING.L SPL.L MdFG.L MFG.L MTG.R C... |

35 |
Graph theoretical analysis of magnetoencephalographic functional connectivity in Alzheimer’s disease,”
- Stam, Haan, et al.
- 2009
(Show Context)
Citation Context ... subjects with neuropathologies such as 436 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data epilepsy, Alzheimer’s disease, autism, Parkinson’s disease and schizophrenia [20, 21, 22, 23, 24]. All these diseases have been associated with abnormal neural synchronization, and as a result, with functional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a disturbance in connectivity between different brain regions. In particular, there is reduction in both local and long-r... |

29 |
Emergence of scaling in random networks, Science,
- Barabási, Albert
- 1999
(Show Context)
Citation Context ...ce they have, among other properties, low clustering coefficient whereas both anatomical and functional networks have high clustering coefficient [7, 47, 69]. • Scale-free model: The scale-free network model is characterized by its degree distribution, viz. the probability distribution of the degrees over the whole network, following a power law. The power law implies that the degree distribution of these networks has no characteristic scale and that there is a small number of very highly connected vertices called hubs. Recent interest in scalefree networks started in 1999 by Barabási, et al. [70, 71] who proposed a mechanism to explain the appearance of the power-law distribution in a stochastic growth model in which new vertices are added continuously and they preferentially attach to existing vertices with probability proportional to the degree of the target vertex. Many real world networks have power-law degree distributions, such as the World Wide Web [72], the metabolic reaction networks [73] and the protein networks [74]. There is no consensus in scientific community on whether the degree distribution of brain function or anatomical networks follows either a power law [69, 75] or a ... |

24 | Fitting a geometric graph to a protein-protein interaction network, Bioinformatics,
- Higham, Rasajski, et al.
- 2008
(Show Context)
Citation Context ...orld [47, 69]. • Geometric random model: A geometric graph G(V, ρ) with radius ρ is a graph where points in a metric space correspond to vertices, and two vertices are adjacent if the distance between them is at most ρ. More details about geometric random graphs can be found in [78]. When the position of vertices in a (possibly high dimensional) Euclidean space is important, the random geometric graph is a good candidate null model. Computational experiments have revealed close matches between key topological properties of Protein-Protein Interaction networks and geometric random graph models [79]. Oikonomou showed that brain functional networks estimated at sensor space from schizophrenia and epilepsy data also share key topological properties with geometric random graphs [80]. Recent articles illustrate that there may be evidence for small world networks in characterizing certain brain pathologies like the Alzheimer disease [28]. 2.5. Interdependence Measures Before analyzing graph metrics, the graph itself has to be estimated from data such as EEG. In brain functional networks, each vertex is identified with a brain area and each edge corresponds to statistical dependence in the act... |

22 |
Induced gamma-band activity and human brain function, The Neuroscientist,
- Kaiser, Lutzenberger
- 2003
(Show Context)
Citation Context ...ovide the physiological basis for *Corresponding author. Address: BMI lab, Institute of Computer Science (ICS), Foundation for Research and Technology (FORTH), Heraklion 71110, Greece. E-mail address: sakkalis@ics.forth.gr (V. Sakkalis). information processing and mental representations [6, 7], and have been studied across different conditions of rest [8, 9, 10] or cognitive load [11]. Studies with detailed electroencephalography (EEG) and magnetoencephalography (MEG) signals have revealed local synchronization patterns and cortico-cortical interactions involved in several cognitive functions [12], with composite subtasks being triggered within different brain regions by unitary brain sources that subsequently synchronize to complete the task. Thus, the dynamics of interaction among different EEG/ MEG channels may be used for indexing neural synchrony of such local or distant brain sources [13]. Such sources act synchronously behaving similar to coupled oscillators [14] and their interactions can be measured using pair-wise linear (cross-coherence or phasecoherence) [16] or nonlinear dynamics and models [15, 16, 17]. Furthermore, the causality of the functional coupling of such oscilla... |

22 |
Impaired small-world efficiency in structural cortical networks in multiple sclerosis associated with white matter lesion load.
- He, Dagher, et al.
- 2009
(Show Context)
Citation Context ...ortest path lengths between each pair of vertices) of the resting-state brain functional network and the intelligence quotient (IQ), suggesting that human intellectual Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 437 PCU.L PHG.R PHG.L OP.L OP.R IFG.L SOG.L MFG.R SOG.R PoCG.L PoCG.R SFG.R SFG.L PCU.R CUN.R SPL.R ANG.L ANG.R LOFG.R LOFG.L LOTG.R PrCG.R PrCG.L ITG.L ITG.R STG.R STG.L SMG.R LING.R SMG.LMTG.L LING.L SPL.L MdFG.L MFG.L MTG.R CING.L LOTG.L MOFG.R MdFG.R MOFG.L MOTG.L MOTG.R IFG.R IOG.R Figure 1. Large-scale anatomical network in the human brain that consists of 45 areas [46]. Two areas are considered anatomically connected if they show statistically significant correlations in cortical thickness measurements from magnetic resonance images. This network is modeled as a graph which consists of vertices (yellow rectangles) and edges (line segments between pairs of vertices). performance is likely to be related to how efficiently our brain integrates information between multiple brain regions [38]. The purpose of this paper is to review the most robust linear and nonlinear synchronization measures for constructing functional networks including measures for comparing ... |

21 |
Studying the human brain anatomical network via diffusion-weighted MRI and graph theory,
- Iturria-Medina, Sotero, et al.
- 2008
(Show Context)
Citation Context ...n the appearance of the power-law distribution in a stochastic growth model in which new vertices are added continuously and they preferentially attach to existing vertices with probability proportional to the degree of the target vertex. Many real world networks have power-law degree distributions, such as the World Wide Web [72], the metabolic reaction networks [73] and the protein networks [74]. There is no consensus in scientific community on whether the degree distribution of brain function or anatomical networks follows either a power law [69, 75] or a power law with exponential cut-off [76, 77]. • Small-world model: Many real world networks exhibit what is called the smallworld property, viz. the condition that most vertices can be reached from every other vertex through short paths. Erdös-Rényi and scale-free networks also have the small-world property. The network model published by Watts and Strogatz [43] combines the small-world property, like random graphs, with high clustering coefficient like lattices. This network model is also called small-world. In brain network literature, the term small-world refers to this model (viz. networks with small diameter and high clustering coe... |

21 |
Evolving functional network properties and synchronizability during human epileptic seizures,”
- Schindler, Bialonski, et al.
- 2008
(Show Context)
Citation Context ... the late alpha. The early alpha associated with attention drives a delta component related to cognition, which in turn drives the late alpha component that is also related to memory operations; this relationship in Figure 6 is observed as node influence from 5 to 7 to 4 in the left network and from 6 to 19 to 24 in the right network. Similar tendencies maybe easily identified using directed graphs based on directed coupling methods such as PDC. Apart from the presented application domains, several studies have been conducted focusing on other pathologies like Alzheimer Disease [93], Epilepsy [94] and Parkinson’s disease [95]. 4. CONCLUSIONS Our brain is a complex network in which information is continuously processed and transported between spatially distributed but functionally linked regions [1]. This ongoing integration of information enables us to evaluate the world around us and to respond quickly and flexibly to complex situations. A snapshot of a subject’s dynamic brain functional network can be estimated from high resolution EEG measurements. Then this network can be characterized and analyzed using well formed concepts mainly based on graph theory and statistical physics. Our... |

17 |
Small-world networks and disturbed functional connectivity in schizophrenia.
- Micheloyannis, Vourkas, et al.
- 2009
(Show Context)
Citation Context |

15 |
Measuring Phase Synchronization of Superimposed Signals, Physical Review Letters,
- Meinecke, Ziehe, et al.
- 2005
(Show Context)
Citation Context ...ad [11]. Studies with detailed electroencephalography (EEG) and magnetoencephalography (MEG) signals have revealed local synchronization patterns and cortico-cortical interactions involved in several cognitive functions [12], with composite subtasks being triggered within different brain regions by unitary brain sources that subsequently synchronize to complete the task. Thus, the dynamics of interaction among different EEG/ MEG channels may be used for indexing neural synchrony of such local or distant brain sources [13]. Such sources act synchronously behaving similar to coupled oscillators [14] and their interactions can be measured using pair-wise linear (cross-coherence or phasecoherence) [16] or nonlinear dynamics and models [15, 16, 17]. Furthermore, the causality of the functional coupling of such oscillatory activities can be assessed with partially directed coherence, which reveals the direction of statistically significant relationships [18]. Synchronization can be evaluated not only on the actual recordings on the scalp electrodes but also on independent components. The later are derived from linear un-mixing transforms and are free from volume conduction effects [15, 16]. ... |

15 |
Scale-free brain functional networks, Physical Review Letters,
- Eguíluz, Chialvo, et al.
- 2005
(Show Context)
Citation Context ...t al. [70, 71] who proposed a mechanism to explain the appearance of the power-law distribution in a stochastic growth model in which new vertices are added continuously and they preferentially attach to existing vertices with probability proportional to the degree of the target vertex. Many real world networks have power-law degree distributions, such as the World Wide Web [72], the metabolic reaction networks [73] and the protein networks [74]. There is no consensus in scientific community on whether the degree distribution of brain function or anatomical networks follows either a power law [69, 75] or a power law with exponential cut-off [76, 77]. • Small-world model: Many real world networks exhibit what is called the smallworld property, viz. the condition that most vertices can be reached from every other vertex through short paths. Erdös-Rényi and scale-free networks also have the small-world property. The network model published by Watts and Strogatz [43] combines the small-world property, like random graphs, with high clustering coefficient like lattices. This network model is also called small-world. In brain network literature, the term small-world refers to this model (viz. net... |

13 |
Reshufling scale-free networks: From random to assortative, Physical Review E,
- Xulvi-Brunet, Sokolov
- 2004
(Show Context)
Citation Context ...ctional networks which are assortative [52, 53]. Disassortative mixing observed in certain biological networks (metabolic signaling pathways network, and gene regulatory network) is conjectured to be responsible for decreasing the likelihood of crosstalk between different functional modules of the cell, and increasing the overall robustness of a network by localizing effects of deleterious perturbations [54]. On the other hand, from computational studies, it has been observed that information can be easily transferred through assortative networks as compared to that in disassortative networks [48, 49, 55]. Note that the network of Figure 1 is assortative having an assortativity coefficient of 0.083. 2.3. Centrality Measures Characterizing local properties of networks is important both in theory and practice since at the local scale, we can detect which vertices are the most relevant to the organization and functioning of a network. These local measures are commonly named centrality measures (or centrality indices), and have proven of great value in analyzing the role played by individuals in social networks and in identifying essential proteins [56], keystone species [57], and functionally imp... |

11 |
Epileptic seizure disorders,
- Niedermeyer
- 2005
(Show Context)
Citation Context ... them have been shown to reflect disease and statistically significant differences between healthy subjects and subjects with neuropathologies such as 436 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data epilepsy, Alzheimer’s disease, autism, Parkinson’s disease and schizophrenia [20, 21, 22, 23, 24]. All these diseases have been associated with abnormal neural synchronization, and as a result, with functional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing in Parkinson’s disease with excessive synchrony at low beta frequencies in basal ganglia-cortical loops [33]. Concerning schizophrenia, there is a growing body of evidence that the clinical symptoms and cognition dysfunctions observed in schizophrenia are caused by a dist... |

11 |
Assortative mixing in networks. Physical Review Letters,
- Newman
- 2002
(Show Context)
Citation Context ...ncy of the anatomical network is reduced in a manner proportional to the extent of total white matter lesions [46]. Achard and Bullmore found that the resting state functional networks provide high efficiency of information processing for low anatomical connection [47]. Efficiency was reduced in older people and the detrimental effects of age to efficiency were localized to frontal and temporal cortical and subcortical regions [47]. The efficiency of graph in Figure 1 is 0.407. Assortativity An important network feature is the similarity between properties of adjacent network vertices. Newman [48, 49] proposed a measure to quantify the degree of similarity (dissimilarity) between adjacent vertices in a network using assortative mixing, which is given by correlation between the degrees of every pair of adjacent vertices. The assortativity coefficient for an undirected graph is defined as the (sample) Pearson product-moment correlation coefficient written in a symmetrical form: (7) where E denotes the number of edges. The assortativity coefficient r lies between –1 and 1, whereby r = 1 means perfect assortativeness, r = –1 means perfect disassortativeness, and r = 0 means no assortativenes... |

11 |
Comparison of the small-world topology between anatomical and functional connectivity in the human brain, Physica A,
- Park, Kim, et al.
- 2008
(Show Context)
Citation Context ... n d u vf G u v V u v = − ∈ ≠∑ 1 1 1 ( ) ( , ), , d u v G ( , ) =+∞d u v M n G ( , ) = >>d u v G ( , ) =+∞ Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 441 Many social networks are assortative since vertices having many connections tend to connect with other highly connected vertices [50]. On the other hand, most technological and biological networks are disassortative since they have the property that vertices with high degree are preferably connected with ones of low degree [51]. Exceptions are the protein contact networks and the brain functional networks which are assortative [52, 53]. Disassortative mixing observed in certain biological networks (metabolic signaling pathways network, and gene regulatory network) is conjectured to be responsible for decreasing the likelihood of crosstalk between different functional modules of the cell, and increasing the overall robustness of a network by localizing effects of deleterious perturbations [54]. On the other hand, from computational studies, it has been observed that information can be easily transferred through assortative networks as compared to that in disassortative networks [48, 49, 55]. Note that the network of Figure 1... |

11 |
Graph structure in the web, Computer Networks,
- Broder, Kumar, et al.
- 2000
(Show Context)
Citation Context ...that the degree distribution of these networks has no characteristic scale and that there is a small number of very highly connected vertices called hubs. Recent interest in scalefree networks started in 1999 by Barabási, et al. [70, 71] who proposed a mechanism to explain the appearance of the power-law distribution in a stochastic growth model in which new vertices are added continuously and they preferentially attach to existing vertices with probability proportional to the degree of the target vertex. Many real world networks have power-law degree distributions, such as the World Wide Web [72], the metabolic reaction networks [73] and the protein networks [74]. There is no consensus in scientific community on whether the degree distribution of brain function or anatomical networks follows either a power law [69, 75] or a power law with exponential cut-off [76, 77]. • Small-world model: Many real world networks exhibit what is called the smallworld property, viz. the condition that most vertices can be reached from every other vertex through short paths. Erdös-Rényi and scale-free networks also have the small-world property. The network model published by Watts and Strogatz [43] com... |

10 |
Generalized partial directed coherence,
- Baccalá, Sameshima
- 2007
(Show Context)
Citation Context ...toregressive coefficients aij(k), i, j = 1,…, n represent the A k n n nn a k a k a k a k = 11 1 1 ( ) ( ) ( ) ( ) x x w( ) ( ) ( )t A t k t k k p = − + = ∑ 1 x( ) ( ) , , ( )t x t x t N T = [ ]1 … γ xy xy xx yy f S f S f S f ( ) ( ) ( ) ( ) = 2 Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 445 linear interaction effect of xj(t-k) onto xi(t). In order to provide a frequency domain description of Granger-causality, Baccala and Sameshima [18] introduced the concept of Partial Directed Coherence (PDC) which has recently been generalized to the new PDC [82] as follows: Let (13) be the Fourier transform of the coefficient matrices, where λ is the normalized frequency in the interval [–0.5, 0.5] and Then the new PDC is defined as [82] (14) where A – (λ) = I − A(λ) and σ 2i refers to the variance of the innovation processes wi (t). ranges between 0 (indicating independence) and 1 (indicating maximum coherence). Phase Locking Value (PLV) Phase synchronization presents a different approach in analyzing the possible nonlinear interdependencies of the EEG signal and focuses on the phases of the signals. The Phase Locking Value (PLV) is one of the most ... |

9 | Functional connectivity in the brain-is it an elusive concept, Neuroscience & Biobehavioral Reviews, - Fingelkurts, Fingelkurts, et al. - 2005 |

9 |
Quantifying positional importance in food webs: a comparison of centrality indices
- Jordán, Benedek, et al.
- 2007
(Show Context)
Citation Context ...tive networks [48, 49, 55]. Note that the network of Figure 1 is assortative having an assortativity coefficient of 0.083. 2.3. Centrality Measures Characterizing local properties of networks is important both in theory and practice since at the local scale, we can detect which vertices are the most relevant to the organization and functioning of a network. These local measures are commonly named centrality measures (or centrality indices), and have proven of great value in analyzing the role played by individuals in social networks and in identifying essential proteins [56], keystone species [57], and functionally important brain regions [58]. There is no commonly accepted definition for centrality. Many authors introduced their own centrality, avoiding a strict definition for centrality in general. The intuition about a centrality measure is that it denotes an order of importance on the vertices or edges of a graph by assigning real values to them. As a minimal requirement, most centrality measures depend only on the structure of a graph. Degree Centrality The simplest and most widely studied centrality is the degree centrality CD(v) of a vertex v which for an undirected graph is sim... |

8 |
The integrated analysis of metabolic and protein interaction networks reveals novel molecular organizing principles, BMC System Biology,
- Durek, Walther
- 2008
(Show Context)
Citation Context ... n d u vf G u v V u v = − ∈ ≠∑ 1 1 1 ( ) ( , ), , d u v G ( , ) =+∞d u v M n G ( , ) = >>d u v G ( , ) =+∞ Journal of Healthcare Engineering · Vol. 1 · No. 3 · 2010 441 Many social networks are assortative since vertices having many connections tend to connect with other highly connected vertices [50]. On the other hand, most technological and biological networks are disassortative since they have the property that vertices with high degree are preferably connected with ones of low degree [51]. Exceptions are the protein contact networks and the brain functional networks which are assortative [52, 53]. Disassortative mixing observed in certain biological networks (metabolic signaling pathways network, and gene regulatory network) is conjectured to be responsible for decreasing the likelihood of crosstalk between different functional modules of the cell, and increasing the overall robustness of a network by localizing effects of deleterious perturbations [54]. On the other hand, from computational studies, it has been observed that information can be easily transferred through assortative networks as compared to that in disassortative networks [48, 49, 55]. Note that the network of Figure 1... |

8 |
Learning partially directed functional networks from meta-analysis imaging data, NeuroImage,
- Neumann, Fox, et al.
- 2010
(Show Context)
Citation Context ...oup. Rather recently, there is an interest in analyzing motor tasks using graphs in order to identify the involvement of different brain lobes. Such applications are central in the Brain-Computer Interface (BCI) field. Such visualizations allow us to select the most relevant sensors to take into account in the BCI classification tasks (i.e., focusing at those regions that act like hubs). Figure 3 depicts such networks using the PDC interdependence method applied at the mu frequency band [88]. Not only the interdependences but also the directionality of the coupling is illustrated using arrows [89]. Apart from the visualization itself, one is able to calculate and use the connectivity graphs to extract network properties and utilize them as biomarkers for certain pathologies such as alcoholism (Figure 4). In this case, synchronization was calculated using magnitude squared coherence and the standard 10/20. International montage along with an additional 41 sites as depicted in Figure 4 were used. Each subject was exposed to randomized pictures of objects chosen from the 1980 Snodgrass and Vanderwart picture set presented on a white background at the centre of a computer monitor and was r... |

7 | Working memory in schizophrenia: An EEG study using power spectrum and coherence analysis to estimate cortical activation and network behavior, Human Brain Mapping, - Pachou, Vourkas, et al. - 2008 |

7 |
Timesignificant wavelet coherence for the evaluation of schizophrenic brain activity using a graph theory approach,
- Sakkalis, Oikonomou, et al.
- 2006
(Show Context)
Citation Context ...f Healthcare Engineering · Vol. 1 · No. 3 · 2010 451 Healthy (a) (b) Patient Figure 5. (a) A “healthy” brain network during a working memory task appears to have higher average degree K and clustering coefficient C, and lower average shortest path length L values compared to the “schizophrenic” one (b). These disturbances are more prominent for the connections of the frontal lobes as well as the temporal lobes. brain activity of schizophrenic populations than healthy ones. More direct connections among different functional lobes are needed when coping with mental tasks of increased difficulty [90]. Finally, as mentioned in the introduction one can also study neural interdependencies based on its independent components (ICs). Instead of directly measuring the synchronization using the actual EEG traces, ICs are first obtained and then identified based on their spatial and frequency properties using ICA decomposition in a concatenated trials scheme for each channel. Such approaches are most commonly used in analysing cognitive brain function. In the case of an auditory working memory paradigm (oddball experiment), induced responses are attributed to oscillatory bursts from local or dista... |

6 |
Mixing patterns in networks,” Physical Review E.
- Newman
- 2003
(Show Context)
Citation Context ...ncy of the anatomical network is reduced in a manner proportional to the extent of total white matter lesions [46]. Achard and Bullmore found that the resting state functional networks provide high efficiency of information processing for low anatomical connection [47]. Efficiency was reduced in older people and the detrimental effects of age to efficiency were localized to frontal and temporal cortical and subcortical regions [47]. The efficiency of graph in Figure 1 is 0.407. Assortativity An important network feature is the similarity between properties of adjacent network vertices. Newman [48, 49] proposed a measure to quantify the degree of similarity (dissimilarity) between adjacent vertices in a network using assortative mixing, which is given by correlation between the degrees of every pair of adjacent vertices. The assortativity coefficient for an undirected graph is defined as the (sample) Pearson product-moment correlation coefficient written in a symmetrical form: (7) where E denotes the number of edges. The assortativity coefficient r lies between –1 and 1, whereby r = 1 means perfect assortativeness, r = –1 means perfect disassortativeness, and r = 0 means no assortativenes... |

6 |
An entropic characterization of protein interaction networks and cellular robustness,
- Manke, Demetrius, et al.
- 2006
(Show Context)
Citation Context ...ich for an undirected graph is simply defined as the degree deg(v) of v. Degree centrality is normalized to range [0, 1] by dividing deg(v) with the maximum possible degree (n – 1). (8) In directed networks, two variants of the degree centrality may be appropriate: the in-degree centrality C–D(v) = deg−(v)/(n−1) and the out-degree centrality C+D (v) = deg+(v)/(n−1). Degree is the most fundamental network measure and most other measures are linked to vertex degree. The degree sequence is argued to reflect some fundamental aspects of natural, social and technological networks [59]. Manke et al. [60] advocate the view that the degree can be considered as correlation of underlying dynamical properties, such as the stability of a dynamic process to random perturbations. In the C v v nD ( ) deg( )= −1 442 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data graph of Figure 1 we have deg(PrCG.R) = 11, deg(SPL.R) = 11 and deg(SFG.R) = 11. The corresponding normalized centralities are CD(PrCG.R) = 0.25, CD(SPL.R) = 0.25 and CD(SFG.R) = 0.25. Shortest-Path Efficiency Latora and Marchiori [61, 62] defined the efficiency efvu in the communication between vertices v a... |

5 |
Dynamic Functional Connectivity, Current Opinion in Neurobiology,
- Ioannides
- 2007
(Show Context)
Citation Context ...erns and cortico-cortical interactions involved in several cognitive functions [12], with composite subtasks being triggered within different brain regions by unitary brain sources that subsequently synchronize to complete the task. Thus, the dynamics of interaction among different EEG/ MEG channels may be used for indexing neural synchrony of such local or distant brain sources [13]. Such sources act synchronously behaving similar to coupled oscillators [14] and their interactions can be measured using pair-wise linear (cross-coherence or phasecoherence) [16] or nonlinear dynamics and models [15, 16, 17]. Furthermore, the causality of the functional coupling of such oscillatory activities can be assessed with partially directed coherence, which reveals the direction of statistically significant relationships [18]. Synchronization can be evaluated not only on the actual recordings on the scalp electrodes but also on independent components. The later are derived from linear un-mixing transforms and are free from volume conduction effects [15, 16]. Networks are modeled by graphs which consist of a set of vertices and a set of pair of vertices called edges (Figure 1). In this respect, graph theor... |

4 |
New perspectives in pharmacoelectroencephalography,
- Fingelkurts, Fingelkurts, et al.
- 2005
(Show Context)
Citation Context ...ered: the Erdös-Rényi random graph, the small world, the scale free, and the geometric random graph models. The information contained in a network can be summarized with graph measures. Graph measures have been applied to topological analysis of brain functional networks, and many of them have been shown to reflect disease and statistically significant differences between healthy subjects and subjects with neuropathologies such as 436 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data epilepsy, Alzheimer’s disease, autism, Parkinson’s disease and schizophrenia [20, 21, 22, 23, 24]. All these diseases have been associated with abnormal neural synchronization, and as a result, with functional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing ... |

4 |
Optimal brain network synchrony visualization: Application in an alcoholism paradigm,
- Sakkalis, Tsiaras, et al.
- 2007
(Show Context)
Citation Context ...ered: the Erdös-Rényi random graph, the small world, the scale free, and the geometric random graph models. The information contained in a network can be summarized with graph measures. Graph measures have been applied to topological analysis of brain functional networks, and many of them have been shown to reflect disease and statistically significant differences between healthy subjects and subjects with neuropathologies such as 436 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data epilepsy, Alzheimer’s disease, autism, Parkinson’s disease and schizophrenia [20, 21, 22, 23, 24]. All these diseases have been associated with abnormal neural synchronization, and as a result, with functional networks that systematically differ from those of control subjects. Epilepsy has been associated with too high and too extended neural synchronization [25, 26]. Patients with Alzheimer’s disease show reduced synchronization in the alpha and beta frequency bands [27, 28, 29]. Cognitive dysfunctions associated with autism are explained with reduced functional connectivity and neural synchronization [30, 31, 32]. There are an increasing amount of data linking impaired motor processing ... |

4 |
Effects of normal aging on event-related desynchronization/synchronizatio during a memory task, Neuroscience Letters,
- Karrasch, Laine, et al.
- 2004
(Show Context)
Citation Context ...ng cognitive processes may be successfully assessed using connectivity measures applied on independent components, which reflect distinct spatial patterns of activity. The results suggest increased phase locked activity most prominently in the delta/ theta band, while alpha is also apparent in measures of non phase-locked activity [91]. Figure 6 depicts numbered ICs. Even though the control group involves a large variation in age, it has been demonstrated that processes related to working memory have been found to weaken with normal aging, but in general follow the same patterns of activation [92]. This is also verified in Figure 6 showing two similar networks of subjects of different ages. The strong interaction in the alpha band is indicated by directed lightblue lines, whereas the weak influence of the theta band from the alpha band is indicated with dashed lines. The independent components identified relate to the alpha, theta and delta bands on the basis of their (major) frequency activity. The delta components (related to cognitive processing) strongly relate to the alpha components. 452 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data Alpha Del... |

4 |
Dopaminergic modulation of cortico-cortical functional connectivity in Parkinson’s disease: An MEG study,
- Stoffers, Bosboom, et al.
- 2008
(Show Context)
Citation Context ...pha associated with attention drives a delta component related to cognition, which in turn drives the late alpha component that is also related to memory operations; this relationship in Figure 6 is observed as node influence from 5 to 7 to 4 in the left network and from 6 to 19 to 24 in the right network. Similar tendencies maybe easily identified using directed graphs based on directed coupling methods such as PDC. Apart from the presented application domains, several studies have been conducted focusing on other pathologies like Alzheimer Disease [93], Epilepsy [94] and Parkinson’s disease [95]. 4. CONCLUSIONS Our brain is a complex network in which information is continuously processed and transported between spatially distributed but functionally linked regions [1]. This ongoing integration of information enables us to evaluate the world around us and to respond quickly and flexibly to complex situations. A snapshot of a subject’s dynamic brain functional network can be estimated from high resolution EEG measurements. Then this network can be characterized and analyzed using well formed concepts mainly based on graph theory and statistical physics. Our research has been facilitate... |

3 |
Neural synchrony in brain review disorders: Relevance for cognitive dysfunctions and pathophysiology,
- Uhlhaas, Singer
- 2006
(Show Context)
Citation Context |

2 |
Assessment of linear and nonlinear synchronization measures for analyzing EEG in a mild epileptic paradigm,
- Sakkalis, Giurcaneanu, et al.
- 2009
(Show Context)
Citation Context ...erns and cortico-cortical interactions involved in several cognitive functions [12], with composite subtasks being triggered within different brain regions by unitary brain sources that subsequently synchronize to complete the task. Thus, the dynamics of interaction among different EEG/ MEG channels may be used for indexing neural synchrony of such local or distant brain sources [13]. Such sources act synchronously behaving similar to coupled oscillators [14] and their interactions can be measured using pair-wise linear (cross-coherence or phasecoherence) [16] or nonlinear dynamics and models [15, 16, 17]. Furthermore, the causality of the functional coupling of such oscillatory activities can be assessed with partially directed coherence, which reveals the direction of statistically significant relationships [18]. Synchronization can be evaluated not only on the actual recordings on the scalp electrodes but also on independent components. The later are derived from linear un-mixing transforms and are free from volume conduction effects [15, 16]. Networks are modeled by graphs which consist of a set of vertices and a set of pair of vertices called edges (Figure 1). In this respect, graph theor... |

1 |
The large scale organization of metabolic networks, Letters to Nature,
- Jeong, Tombor, et al.
- 2000
(Show Context)
Citation Context ...networks has no characteristic scale and that there is a small number of very highly connected vertices called hubs. Recent interest in scalefree networks started in 1999 by Barabási, et al. [70, 71] who proposed a mechanism to explain the appearance of the power-law distribution in a stochastic growth model in which new vertices are added continuously and they preferentially attach to existing vertices with probability proportional to the degree of the target vertex. Many real world networks have power-law degree distributions, such as the World Wide Web [72], the metabolic reaction networks [73] and the protein networks [74]. There is no consensus in scientific community on whether the degree distribution of brain function or anatomical networks follows either a power law [69, 75] or a power law with exponential cut-off [76, 77]. • Small-world model: Many real world networks exhibit what is called the smallworld property, viz. the condition that most vertices can be reached from every other vertex through short paths. Erdös-Rényi and scale-free networks also have the small-world property. The network model published by Watts and Strogatz [43] combines the small-world property, like r... |

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InBrAiN: An Interactive Tool for Brain Analysis and Visualization.
- Oikonomou
- 2007
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Citation Context ...distance between them is at most ρ. More details about geometric random graphs can be found in [78]. When the position of vertices in a (possibly high dimensional) Euclidean space is important, the random geometric graph is a good candidate null model. Computational experiments have revealed close matches between key topological properties of Protein-Protein Interaction networks and geometric random graph models [79]. Oikonomou showed that brain functional networks estimated at sensor space from schizophrenia and epilepsy data also share key topological properties with geometric random graphs [80]. Recent articles illustrate that there may be evidence for small world networks in characterizing certain brain pathologies like the Alzheimer disease [28]. 2.5. Interdependence Measures Before analyzing graph metrics, the graph itself has to be estimated from data such as EEG. In brain functional networks, each vertex is identified with a brain area and each edge corresponds to statistical dependence in the activities of two brain areas. We 444 Graph Analysis and Visualization for Brain Function Characterization Using EEG Data assume here that the activity of a brain area is measured in a sc... |

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Graph Analysis and Visualization for Brain Function Characterization Using EEG Data
- Sauer, Yorke, et al.
- 1991
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Citation Context ...the sampling period and N is the sample number of each signal. PLV takes values within the [0, 1] space, where 1 indicates perfect phase synchronization and 0 indicates lack of synchronization. Nonlinear synchronization (state-space approach) Finally, another group of synchronization measures are based on the assumption that neurons are highly nonlinear devices, which in some cases show chaotic behavior. Such measures belong to the generalized synchronization (GS) concept and are based on analyzing the interdependence between the amplitudes of the signals in a state-space reconstructed domain [84]. A physiological time series such as the EEG appears to have more than the single degree of freedom represented just by plotting the voltage as a function of time. To free up some of these unknown parameters, a standard technique is to map the scalar time series to a vector-valued one in a higher dimensional space Rm, thereby giving it an extension in space as well as time. Hence, one may measure how neighborhoods (recurrences) in state space located in one attractor map to each other. This idea turned out to be the most robust and reliable way of assessing the extent of GS [85]. First, we re... |

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The binary stress model for graph drawing. In: Tollis,
- Koren, Civril
- 2008
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Citation Context ... in general, , this measure is not suitable to infer driver response relationships. Accordingly, we used the mean value , as a Robust Interdependence Measure (RIM) between X and Y. 3. BRAIN NETWORK VISUALIZATION AND APPLICATION DOMAINS Brain functional networks can be practically visualized by assigning to vertices the coordinates of the corresponding channels as described in section 2.1. Additionally, in order to show the network structure, one can use network visualization techniques such as the stress majorization technique of Gansner et al. [86] and the binary stress model of Koren et al. [87]. Apart from real world network visualization and graph analysis cases, there are some initial but indicative examples of possible translation of graph research outcomes in the clinical practice. More specifically, considering that the so called “disconnectivity syndromes” represent many pathological and neuropsychological diseases, based on their functional impairment symptoms, one can infer directly the usability and significance of network analysis and visualization tools based on graph theory. This section briefly discusses applications of graph visualization and analysis in cognitive and ... |

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Brainnetvis: Analysis and visualization of brain functional networks,
- Tsiaras, Andreou, et al.
- 2009
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Citation Context ...f graph visualization and analysis in cognitive and clinical domains, referring to published works of our group. Rather recently, there is an interest in analyzing motor tasks using graphs in order to identify the involvement of different brain lobes. Such applications are central in the Brain-Computer Interface (BCI) field. Such visualizations allow us to select the most relevant sensors to take into account in the BCI classification tasks (i.e., focusing at those regions that act like hubs). Figure 3 depicts such networks using the PDC interdependence method applied at the mu frequency band [88]. Not only the interdependences but also the directionality of the coupling is illustrated using arrows [89]. Apart from the visualization itself, one is able to calculate and use the connectivity graphs to extract network properties and utilize them as biomarkers for certain pathologies such as alcoholism (Figure 4). In this case, synchronization was calculated using magnitude squared coherence and the standard 10/20. International montage along with an additional 41 sites as depicted in Figure 4 were used. Each subject was exposed to randomized pictures of objects chosen from the 1980 Snodgr... |

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Activity detection and causal interaction analysis among independent EEG components from memory related tasks,
- Michalopoulos, Sakkalis, et al.
- 2009
(Show Context)
Citation Context ...study neural interdependencies based on its independent components (ICs). Instead of directly measuring the synchronization using the actual EEG traces, ICs are first obtained and then identified based on their spatial and frequency properties using ICA decomposition in a concatenated trials scheme for each channel. Such approaches are most commonly used in analysing cognitive brain function. In the case of an auditory working memory paradigm (oddball experiment), induced responses are attributed to oscillatory bursts from local or distant neural assemblies with variable latency and frequency [91]. The functional coupling and role of independent components are investigated through the concept of PDC method. The EEG signals used in this work are selected from two representative subjects out of 9 healthy participants (age 37–74), who had no history of neurological or psychiatric disorder. This study indicates that functional connectivity during cognitive processes may be successfully assessed using connectivity measures applied on independent components, which reflect distinct spatial patterns of activity. The results suggest increased phase locked activity most prominently in the delta/... |