IEEE Signal Processing - May 2018 - 87

The phase-lag index (PLI) [25] quantifies the asymmetry of
the phase difference, rendering it insensitive to shared signals
at zero phase lag:

constraints, these techniques are not described in this article.
Readers may refer to [21] for a review.

PLI = G sign{(t) H .

From the previous step, and whatever the connectivity method being used (functional or effective), an R # R adjacency
matrix is produced. This matrix represents the pairwise connections between all of the ROIs. An example of a functional
connectivity matrix for R = 68 is presented in Figure 2(c). To
retain significant interactions, these matrices are usually
thresholded (e.g., keeping only the top 10% of connections)
to distinguish real functional connections from spurious ones.
A variety of thresholding methods are available, but none is
free of bias. It is then prudent to perform studies across different threshold values (in addition to using alternative strategies) to ensure that the obtained findings are robust to this
methodological factor. In the context of EEG, other techniques are also available to test the significance of interactions, such as the use of surrogate data analysis. Readers can
check [31] for a complete overview of most network-related
methodological issues.
Interestingly, this R # R adjacency matrix can be characterized and quantified using network measures derived
from graph theory. Graph theory is a branch of mathematics
focused on the analysis of systems consisting of interconnected elements. Such a system can be represented as a graph in
which nodes (or vertices) are connected by edges (or links). In
the context of brain networks, the nodes represent the brain
regions and the edges reflect the functional and/or effective
connections. Once nodes and edges are defined, network topological properties can be studied by graph-theory metrics. As
illustrated in Figure 1, these quantitative metrics can be used
to characterize the normal brain network architecture during
rest or during cognitive functions. They can also be used in a
clinical perspective, such as the localization of epileptic zones
[Figure 2(d), right] or the development of neuromarkers for
other brain disorders [Figure 2(d), left].
A simple graph can be represented by G = (V, E) where V
is the set of nodes and E is the set of edges. In the weighted
undirected graph, each node can be identified by an integer
value i = 1, 2, ..., N, and an edge can be identified by (i, j),
which represents the connection going from node I to node
j, to which a weight A ij can be associated. We now briefly
describe some of the main network measures, illustrated in
Figure 3.
■ The degree d denotes the total number of links connected
to a given node [Figure 3(a)].
■ The clustering coefficient C reflects the tendency of a network to form topologically local circuits [Figure 3(b)]. For
a given node i with degree d, the local C i is defined as

Another method used in MEG/EEG source connectivity is AEC. It consists of estimating the amplitude correlation
between signals using the linear correlations (or partial correlations) of the envelopes of filtered signals. The envelopes of the
signals can be computed using the Hilbert transform [26]. The
r 2, PLV, PLI, and AEC values range from zero (independent
signals) to one (fully correlated/synchronized signals).
The aforementioned functional connectivity methods consider only the degree of coupling. In contrast, effective connectivity methods aim to estimate the causality (in the sense
of Granger causality) or the directionality of coupling between
the signals. Several techniques have been proposed based on
the multivariate autoregressive model (MVAR), such as the
directed transfer function (DTF) and partial directed coherence (PDC) [27].
As an example, we describe here the method based on the
parametric representation of multichannel time series, which
is widely used to study causal brain interactions. For signals
X (t) with M dimensions, the MVAR with order p can be
defined as
p

X (t) = / A (i) X (t - i) + f (t),
i =1

where f (t) denotes the additive noise and A (i) are the model
coefficients (M # M). This time-domain representation can be
transformed into a frequency-domain model:
X ( f ) = A -1 ( f ) f ( f ) = H ( f ) f( f ),
where H ( f ) is the transfer function and A ( f ) is the Fourier
transform of the coefficients. Using MVAR coefficients, the
PDC estimator, characterized by the outflow from channel j
to channel i at frequency f, is defined as
PDC 2ij =

A 2ij ( f )
k

,

/ A 2rj ( f )

r =1

and the DTF estimator, which describes the causal influence
of channel j on channel i at frequency f, is defined as
DTF ij2 ( f ) =

H ij ( f )
k

/

2

H ir ( f )

.
2

r =1

Other methods are also available to compute the effective
connectivity. They are based on a directionality index derived
from nonlinear regression analysis [28], on the transfer entropy
[29], or on the combination of effective connectivity with neural mass models identified from time series. This latter method
is known as dynamic causal modeling [30]. Due to space

Network measures

Ci =

2L i ,
d i (d i - 1)

where L i denotes the number of links between the d i
neighbors of node i. C i varies between zero and one,
and it is considered the main graph metric of information

IEEE Signal Processing Magazine

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May 2018

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87



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