IEEE Signal Processing - May 2018 - 86
the computation is more difficult, as the difference between the
signal and baseline is very small. A long EEG segment is traditionally used to estimate the C matrix. When the noise can be
assumed as spatially uniform across all channel sites, then C = I.
Another popular inverse solution is beamforming. Here,
the beamformer filter extracts the components of a signal with
some specific spatial features. More particularly, it allows for
scanning each source location and for retaining a signal contribution that originates from that spatial location, while it rejects
any contribution stemming from other locations. The weights
in matrix W (which correspond to each specific source location) are therefore estimated one by one from the data. The data
covariance matrix C is used for this purpose. One of the most
widely used beamformers is the linearly constrained minimum variance (LCMV) [19], which makes use of the following
weight estimation for the source placed at a given location:
-1
W beamformer = 6(G T .C -1) .G@ . (G T .C -1) .
The two previously described methods belong to a wide set of
signal processing methods aimed at solving the EEG inverse
problem, i.e., estimating matrix W, from which the dynamics
of the brain sources can be reconstructed using (1). This estimation is usually done on high-resolution surface mesh (e.g., 8,000
or 15,000 vertex). However, this number of reconstructed sources is too great to perform the second step of the connectivity
analysis. Therefore, in practice, spatially closed brain sources
are clustered based on a set of R predefined ROIs, with R chosen as small with respect to the number of estimated sources.
To define ROIs, many anatomical and/or functional atlases
are available, such as the Desikan-Killiany atlas, with 68 ROIs,
used in the illustrative example in Figure 1(b), and the Destrieux atlas, with 148 ROIs. This procedure leads to R regional time series R (t), each one representing the average brain
activity generated by one of the R predefined brain regions.
Note that the 3-D surface of neocortical patches is folded. To
avoid activity cancellation due to the opposite direction of the
dipole sources, the averaging is performed on the absolute
value of the dipole moments. Averaging the time series across
ROIs is a simple way to produce a single time-series representative of the activity of a given extended brain source (ROI).
Note that the absolute value transformation is a bit anecdotal.
It accounts for calculation errors that affect an extremely small
number of sources that are flipped to the dominant (and correct) direction before averaging. Nevertheless, there certainly
exist some other approaches to estimate the activity associated
with a given ROI, e.g., the use of a dimensionality reduction
technique, such as principal component analysis.
Functional and effective connectivity
Once the R(t) time series are reconstructed, the statistical couplings between these regional time series can be estimated.
When the estimated quantity is related only to the degree of
coupling, the method is referred to as functional connectivity.
When the objective is to estimate directionality in this coupling or causality between considered time series, the method
86
is referred to as effective connectivity. Both functional and
effective connectivity methods have been the topic of intensive research over the past two decades, and many metrics are
now available (a review is in [21]).
Concerning functional connectivity, the most widely used
approaches in the EEG context are those based on linear/
nonlinear correlation, the coherence function, PS, mutual
information, and amplitude envelope correlation (AEC) (see
[22] for a review and [23] for comparative studies). A key
issue is performance, and, in regard to this, whatever the
context (cognitive research or clinical application), each
method has its own advantages and limitations, and there
is no consensus about one standard approach that would
outperform the others. In this section, we present three main
families of methods: linear correlation, PS, and AEC, as
they represent the most-used techniques in the context of EEG
source connectivity.
The cross-correlation coefficient (r 2xy) is one of the oldest
and probably the most classical measure of interdependence
between two time series. Conceptually very close to the socalled Pearson's correlation coefficient in statistics, it is a measure of the linear correlation between two signals, x and y,
possibly delayed by x:
rxy2 (x) =
cov 2 (x (t), y (t + x))
,
(v x (t) v y (t +x)) 2
(2)
where v and cov denote the standard deviation and the covariance, respectively. Starting from (2), the metrics r 2xy classically used to characterize the coupling between x and y are
given by
r 2xy = max [r 2xy (x)]
- x max 1 x 1 x max,
where x max denotes the maximum time shift between the
two signals.
The second family of methods is PS. It is well known that
the phases of two time series can be synchronized, even if their
amplitudes are independent. The general principle of PS is to
detect the presence of a phase locking between two systems
defined as
{ (t) = U x (t) - U y (t) # C,
where U x (t), U y (t) are the unwrapped phases of the signals
x and y at time bins t, and C is a constant. The first step is to
extract the instantaneous phase of each signal. Two different
techniques can be used: the Hilbert transform and the wavelet
transform. Both approaches produce relatively close results.
The second step is the definition of a metric that measures
the synchronization degree between the estimated phases.
Several measures have been proposed to measure the PS
between two signals.
The phase-locking value (PLV) [24] is defined as
PLV = G e i{ (t) H ,
where G$H denotes the average over time and trials.
IEEE Signal Processing Magazine
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May 2018
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