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A graph-theoretical approach in brain functional networks. Possible implications in EEG studies

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PROCEEDINGS Open Access
A graph-theoretical approach in brain functionalnetworks. Possible implications in EEG studies
Fabrizio De Vico Fallani
1,2
, Luciano da Fontoura Costa
5
, Francisco Aparecido Rodriguez
5
, Laura Astolfi
1,3
,Giovanni Vecchiato
1,2
, Jlenia Toppi
1,3
, Gianluca Borghini
1
, Febo Cincotti
1
, Donatella Mattia
1
, Serenella Salinari
3
,Roberto Isabella
4
, Fabio Babiloni
1,2*
From
Consciousness and its Measures: Joint Workshop for COST Actions NeuroMath and ConsciousnessLimassol, Cyprus. 29 November
–
1 December 2009
Abstract
Background:
Recently, it was realized that the functional connectivity networks estimated from actual brain-imaging technologies (MEG, fMRI and EEG) can be analyzed by means of the graph theory, that is a mathematicalrepresentation of a network, which is essentially reduced to nodes and connections between them.
Methods:
We used high-resolution EEG technology to enhance the poor spatial information of the EEG activity onthe scalp and it gives a measure of the electrical activity on the cortical surface. Afterwards, we used the Directed Transfer Function (DTF) that is a multivariate spectral measure for the estimation of the directional influencesbetween any given pair of channels in a multivariate dataset. Finally, a graph theoretical approach was used tomodel the brain networks as graphs. These methods were used to analyze the structure of cortical connectivityduring the attempt to move a paralyzed limb in a group (N=5) of spinal cord injured patients and during themovement execution in a group (N=5) of healthy subjects.
Results:
Analysis performed on the cortical networks estimated from the group of normal and SCI patientsrevealed that both groups present few nodes with a high out-degree value (i.e. outgoing links). This property isvalid in the networks estimated for all the frequency bands investigated. In particular, cingulate motor areas (CMAs)ROIs act as
‘‘
hubs
’’
for the out
ﬂ
ow of information in both groups, SCI and healthy. Results also suggest that spinalcord injuries affect the functional architecture of the cortical network sub-serving the volition of motor acts mainlyin its local feature property.In particular, a higher local efficiency
E
l
can be observed in the SCI patients for three frequency bands, theta (3-6Hz), alpha (7-12 Hz) and beta (13-29 Hz).By taking into account all the possible pathways between different ROI couples, we were able to separate clearlythe network properties of the SCI group from the CTRL group. In particular, we report a sort of compensatorymechanism in the SCI patients for the Theta (3-6 Hz) frequency band, indicating a higher level of
“
activation
”
Ω
within the cortical network during the motor task. The activation index is directly related to diffusion, a type of dynamics that underlies several biological systems including possible spreading of neuronal activation across sev-eral cortical regions.
Conclusions:
The present study aims at demonstrating the possible applications of graph theoretical approachesin the analyses of brain functional connectivity from EEG signals. In particular, the methodological aspects of the i)cortical activity from scalp EEG signals, ii) functional connectivity estimations iii) graph theoretical indexes areemphasized in the present paper to show their impact in a real application.
* Correspondence: fabio.babiloni@uniroma1.it
1
IRCCS
“
Fondazione Santa Lucia
”
, Rome, Italy
Fallani
et al
.
Nonlinear Biomedical Physics
2010,
4
(Suppl 1):S8http://www.nonlinearbiomedphys.com/content/4/S1/S8
© 2010 Babiloni et al; licensee BioMed Central Ltd. This is an open access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the srcinal work is properly cited.
Background
Over the last decade, there has been a growing interestin the detection of the functional connectivity in thebrain from different neuroelectromagnetic and hemo-dynamic signals recorded by several neuro-imagingdevices such as the functional Magnetic ResonanceImaging (fMRI) scanner, electroencephalography (EEG)and magnetoencephalography (MEG) apparatus. Many methods have been proposed and discussed in the lit-erature with the aim of estimating the functional rela-tionships among different cerebral structures [1-5].However, the necessity of an objective comprehensionof the network composed by the functional links of different brain regions is assuming an essential role inthe Neuroscience. The extraction of salient characteris-tics from brain connectivity patterns is an open chal-lenging topic, since often the estimated cerebralnetworks have a relative complex structure. Recently, itwas realized that the functional connectivity networksestimated from actual brain-imaging technologies(MEG, fMRI and EEG) can be analyzed by means of the graph theory [6,7]. In those studies, the authorshave evaluated two characteristic measures, the averageshortest path L and the clustering coefficient C, toextract respectively the global and local properties of the network structure. They have found that anatomi-cal brain networks exhibit a high degree to whichnodes tend to cluster together (i.e. a high C) and arelatively short distance between all the nodes (i.e. alow L). These values identify a particular model thatinterpolate between a regular lattice and a randomstructure. Such a model has been designated as
“
small-world
”
network in analogy with the concept of thesmall-world phenomenon observed more than 30 yearsago in social systems [8]. In a similar way, many typesof functional brain networks have been analyzedaccording to this mathematical approach. In the func-tional brain connectivity context, these properties havebeen demonstrated to reflect an optimal architecturefor the information processing and propagation amongthe involved cerebral structures. However, the perfor-mance of cognitive and motor tasks as well as the pre-sence of neural diseases has been demonstrated toaffect such a small-world topology, as revealed by thesignificant changes of L and C [9-11].The small-world concept in a complex network isstrictly related to the length of the shortest pathswithin the network, which is given by the smallestnumber of edges needed to go from a starting vertex
i
to a target node
j
[12]. However, shortest paths justrepresent one possible way in which two nodes in thenetwork can communicate and other existing pathwaysshould be generally taken into account to characterizethe connectivity pattern. In particular, by neglectingthe longer pathways important information is lostabout the alternative trails that could connect any twonodes in a network. This information appears strictly related to the concepts of
“
redundancy
”
and
“
robust-ness
”
, critical resources for the survival of many biolo-gical systems as they provide reliable function despitethe death of individual elements. Indeed, the presenceof more than one path between two nodes in thegraph tends to increase the interaction between them,while enhancing the resilience to damages. In particu-lar, the human brain is supposed to exhibit a highlevel of alternative anatomical and functional pathwaysbetween adjacent regions and sites. This type of orga-nization would allow the brain to reshape its physiolo-gic mechanisms in order to compensate the criticalconsequences of possible diseases [13].Recently, an interesting methodology
–
the superedgesapproach - has been proposed [14] in physics to obtaina detailed analysis of networks considering the conceptof generalized connectivity. This approach allows char-acterizing the networks properties by taking intoaccount all the possible paths between pairs of nodes.In order to illustrate the potential of the graph theore-tical approach in the brain functional network analysis,we report the results obtained with a set of high-resolu-tion
EEG
signals from spinal cord injured patients andcontrol subjects during the preparation of an intendedmotor act.
Methods
Cortical activity estimation
High-resolution EEG technology involves the use of alarger number of scalp electrodes (64-256). In addition,high-resolution EEG uses realistic MRI-constructed sub- ject head models and spatial de-convolution estimations,which are commonly computed by solving a linearinverse problem based on boundary-element mathe-matics [15,16]. In the present applications, the corticalactivity was estimated from EEG recordings by using arealistic head model, whose cortical surface consisted of about 5000 triangles disposed uniformly.Each triangle represents the electrical dipole of a parti-cular neuronal population and the estimation of its cur-rent density was computed by solving the linear inverseproblem (see following paragraphs). In this way, the elec-trical activity in different Regions Of Interest (ROIs) canbe obtained by averaging the current density of the var-ious dipoles within the considered cortical area.
Fallani
et al
.
Nonlinear Biomedical Physics
2010,
4
(Suppl 1):S8http://www.nonlinearbiomedphys.com/content/4/S1/S8Page 2 of 13
Head models and regions of interest
In order to estimate cortical activity from conventionalEEG scalp recordings, realistic head models reconstructedfrom T1-weighted MRIs are employed. Scalp, skull anddura mater compartments are segmented from MRIs andtessellated with about 5000 triangles. Then, the corticalregions of interest (ROIs) are drawn by a neuroradiologiston the computer-based cortical reconstruction of the indi- vidual head model by following a Brodmann
’
s mappingcriterion.
Estimation of cortical source current density
The solution of the following linear system:
Ax
=
b
+
n
(1)provides an estimation of the dipole source configura-tion
x
which generates the measured EEG potential dis-tribution
b
. The system includes also the measurementnoise
n
, assumed to be normally distributed.
A
is thelead field matrix, where each
j-th
column describes thepotential distribution generated on the scalp electrodesby the
j-th
unitary dipole. The current density solution vector
ξ
of Eq. 1 was obtained as:
= − +
( )
arg min
x
Ax b x
M N
2 2 2
(2)where
M
,
N
are matrices associated to the metrics of data and source space, respectively;
l
is a regularizationparameter; ||
…
||
M
represent the M-norm of the dataspace
b
and ||
…
||
N
the N-norm of the solutions space
x
. The formula 2 represents a minimization problemalso known as
linear inverse
problem.As a metric of the data space the identity matrix isgenerally employed. However, the metric in the sourcespace can be opportunely modified when hemodynamicinformation is available from recorded fMRI data. Thisaspect can notably improve the localization of thesource activity. An estimate of the signed magnitude of the dipolar moment for each one of the 5000 corticaldipoles was then obtained for each time point. Theinstantaneous average of all the dipoles
’
magnitudewithin a particular ROI was used to deal with the aver-age activity in that ROI during the whole time intervalof the experimental task. Figure 1 illustrates the effect of the linear inverse problem
’
s solution. From a scalppotential distribution one can estimate accurately thesrcinal cortical potential.
Functional connectivity estimation
Many EEG and/or MEG frequency-based methods thathave been proposed in recent years for assessment of thedirectional influence of one signal on another are basedmainly on the Granger theory of causality. Granger the-ory mathematically defines what a
“
causal
”
relationbetween two signals is. According to this theory, anobserved time series x(n) is said to cause another series y (n) if the knowledge of x(n)
’
s past significantly improvesprediction of y(n); this relation between time series is notnecessarily reciprocal, i.e., x(n) may cause y(n) without y (n) causing x(n). This lack of reciprocity allows the eva-luation of the direction of information flow betweenstructures. Kaminski and Blinowska [3] proposed a multi- variate spectral measure, called the Directed TransferFunction (DTF), which can be used to determine thedirectional influences between any given pair of channelsin a multivariate dataset. DTF is an estimator that simul-taneously characterizes the direction and spectral proper-ties of the interaction between brain signals and requiresonly one multivariate autoregressive (MVAR) model tobe estimated simultaneously from all the time series. Theadvantages of MVAR modeling of multichannel EEG sig-nals in order to compute efficient connectivity estimateshave recently been stressed [17-19].
MultiVariate AutoRegressive models
The approach based on multivariate autoregressivemodels (MVAR) can simultaneously model a wholeset of signals. Let
X
be a set of estimated cortical timeseries:
x x t x t x t
N
=
[ ( ), ( ),... ( )]
1 2
(2)where
t
refers to time and
N
is the number of corticalareas considered. Given an MVAR process which is anadequate description of the data set
X
:
Λ
( ) ( ) ( )
k X t k E t
kp
− =
=
∑
0
(3)where
X(t)
is the data vector in time;
E(t)=[e
1
(t),
…
,e
N
]
is a vector of multivariate zero-mean uncorrelatedwhite noise processes;
Λ
(1),
Λ
(2),
…
Λ
(p)
are the
NxN
matrices of model coefficients (
Λ
(0)=I);
and
p
is themodel order. The
p
order is chosen by means of theAkaike Information Criteria (AIC) for MVAR processes.In order to investigate the spectral properties of theexamined process, the Eq. (3) is transformed into thefrequency domain:
Λ
( ) ( ) ( )
f X f E f
=
(4)
Fallani
et al
.
Nonlinear Biomedical Physics
2010,
4
(Suppl 1):S8http://www.nonlinearbiomedphys.com/content/4/S1/S8Page 3 of 13
where:
Λ Λ
∆
( ) ( )
f k e
j f tkkp
=
−=
∑
20
(5)and
∆
t
is the temporal interval between two samples.Eq. (4) can then be rewritten as:
X f f E f H f E f
( ) ( ) ( ) ( ) ( )
= =
−
Λ
1
(6)
H(f)
is the transfer matrix of the system, whose ele-ment
H
ij
represents the connection between the
j-th
input and the
i-th
output of the system.
Directed Transfer Function
The Directed Transfer Function, representing the cau-sal influence of the cortical waveform estimated in the
j-th
ROI on that estimated in the
i-th
ROI is defined interms of elements of the transfer matrix
H
, is:
ij ij
f H f
2 2
( ) ( )
=
(7)In order to compare the results obtained for corticalwaveforms with different power spectra, normalizationcan be performed by dividing each estimated DTF by the squared sums of all elements of the relevant row,thus obtaining the so-called normalized DTF:
ijijimm N
f H f H f
2221
( )( )( )
=
=
∑
(8)where N indicates the number of ROIs,
g
2ij
(f)expresses the ratio of influence of the cortical waveformestimated in the
j-th
ROI on the cortical waveform esti-mated in the
i-th
ROI, with respect to the influence of all the estimated cortical waveforms. Normalized DTF values are in the interval [0 1], and the normalizationcondition:
inn N
f
21
1( )
=
=
∑
(9)is applied.Figure 2 shows a schematic representation of the func-tional connectivity estimation from a set of high-resolu-tion EEG signals to the cortical network.
Graph theory
A graph is an abstract representation of a network. Itconsists of a set of vertices (or nodes) and a set of edges (or connections) indicating the presence of someof interaction between the vertices. The adjacency matrix
W
contains the information about the
Figure 1
Electrical activity estimation in the Brodmann area 7 from the scalp measurement in the parietal sensor P3.
Fallani
et al
.
Nonlinear Biomedical Physics
2010,
4
(Suppl 1):S8http://www.nonlinearbiomedphys.com/content/4/S1/S8Page 4 of 13
connectivity structure of the graph. When a weightedand directed edge exists from the node
i
to
j
, the cor-responding entry of the adjacency matrix is
W
ij
≠
0;otherwise
W
ij
= 0.
Node strength
The simplest attribute of a node is its connectivity degree, which is the total number of connections withother vertices. In a weighted graph, the natural generali-zation of the degree of a node
i
is the node strength ornode weight or weighted-degree. This quantity has tobe split into in-strength
s
in
and out-strength
s
out
, whendirected relationships are being considered. Thestrength index integrates the information of the links
’
number (degrees) with the connections
’
weight, thusrepresenting the total amount of outgoing intensity from a node or incident intensity into it. The formula-tion of the in-strength index
s
in
can be introduced asfollows:
s i w
in ij j V
( )
=
∈
∑
(10)It represents the whole functional flow incoming tothe vertex
i
.
V
is the set of the available nodes and
w
ij
isthe weight of the particular arc from the point
j
to thepoint
i
. In a similar way, for the out-strength:
s i w
out ji j V
( )
=
∈
∑
(11)It represents the whole functional flow outgoing fromthe vertex
i
.
Strength distributions
For a weighted graph, the arithmetical average of all thenodes
’
strengths
<s>
only gives little information aboutthe distributions of the links intensity within the system.Hence, it is useful to introduce
R(s)
as the fraction of
Figure 2
From a set of cortical time series the MVAR method estimates in the frequency domain a functional connectivity pattern that can bemodeled by means of a graph.
Fallani
et al
.
Nonlinear Biomedical Physics
2010,
4
(Suppl 1):S8http://www.nonlinearbiomedphys.com/content/4/S1/S8Page 5 of 13

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