Modeling sparse connectivity between underlying brain sources for EEG/MEG

1Modeling sparse connectivity between
underlying brain sources for EEG/MEG
Stefan Haufe, Ryota Tomioka, Guido Nolte, Klaus-Robert Mu¨ller and Motoaki Kawanabe
Abstract—We propose a novel technique to assess functional
brain connectivity in EEG/MEG signals. Our method, called
Sparsely-Connected Sources Analysis (SCSA), can overcome the
problem of volume conduction by modeling neural data innova-
tively with the following ingredients: (a) the EEG is assumed to
be a linear mixture of correlated sources following a multivariate
autoregressive (MVAR) model, (b) the demixing is estimated
jointly with the source MVAR parameters, (c) overfitting is
avoided by using the Group Lasso penalty. This approach allows
to extract the appropriate level cross-talk between the extracted
sources and in this manner we obtain a sparse data-driven model
of functional connectivity. We demonstrate the usefulness of
SCSA with simulated data, and compare to a number of existing
algorithms with excellent results.
I. INTRODUCTION
A. Functional brain connectivity
The analysis of neural connectivity plays a crucial role
for understanding the general functioning of the brain. In
the past two decades such analysis has become possible
thanks to tremendous progress that has been made in the
fields of neuroimaging and mathematical modeling. Today, a
multiplicity of imaging modalities exists, allowing to monitor
brain dynamics at different spatial and temporal scales.
Given multiple simultaneously-recorded time-series reflect-
ing neural activity in different brain regions, a functional (task-
related) connection (sometimes also called information flow or
(causal) interaction in this paper) between two regions is com-
monly inferred, if a significant time-lagged influence between
the corresponding time-series is found. Different measures
have been proposed for quantifying this influence, most of
them being formulated either in terms of the cross-spectrum
(e.g., coherence, phase slope index [1]) or an autoregressive
models (e.g., Granger causality [2], directed transfer function
[3], partial directed coherence [4], [5]).
B. Volume conduction problem in EEG and MEG
In electroencephalography (EEG) and magnetoencephalog-
raphy (MEG), sensors are placed outside the head and the
problem of volume conduction arises. That is, rather than
measuring activity of only one brain site, each sensor captures
a linear superposition of signals from all over the brain. This
mixing introduces instantaneous correlations in the data, which
can cause traditional analyses to detect spurious connectivity
[6].
S. Haufe and K.-R. Mu¨ller are with the Berlin Institute of Technology,
Germany.
R. Tomioka is with the University of Tokyo, Japan.
G. Nolte and M. Kawanabe are with Fraunhofer Institute FIRST, Berlin,
Germany.
C. Existing source connectivity analyses
Only recently, methods have been brought up, which qualify
for EEG/MEG connectivity analysis, since they account for
volume conduction effects. These methods can roughly be
divided as follows.
One type of methods aims at providing meaningful connec-
tivity estimates between sensors. The idea here is, that only the
real part of the cross-spectrum and related quantities is affected
by instantaneous effects. Thus, by using only the imaginary
part, many traditional coupling measures can be made robust
against volume-conduction [1], [6].
Another group of methods attempts to invert the mixing
process in order to apply standard measures to the obtained
source estimates. These methods can be further divided into
(i) source-localization approaches (where sources are obtained
as solutions to the EEG/MEG inverse problem), (ii) methods
using statistical assumptions, and (iii) combined methods. The
first approach is pursued, for example, in [7], [8]. Methods
in the second category can be appealing, since they avoid
finding an explicit inversion of the physical forward model.
Instead, both the sources and the (de-)mixing transformation
are estimated. To make such decomposition unique, assump-
tions have to be formulated, the choice of which is not so
straightforward. We will now briefly review some possibilities
for such assumptions.
Principal component analysis (PCA) and independent com-
ponent analysis (ICA) are the most prominent linear decom-
position techniques for multivariate data. Unfortunately, these
methods contradict either with the goal of EEG/MEG connec-
tivity analysis (assumption of independent sources in ICA1) or
even with the physics underlying EEG/MEG generation (as-
sumption of orthogonal loadings in PCA). Nevertheless, both
concepts have been successfully used in more sophisticated
ways to find meaningful EEG/MEG decompositions.
For example, an interesting use of ICA is proposed in [10].
The authors of this paper do not assume independence of the
source traces, but rather argue that this property holds for the
residuals of an MVAR model if no instantaneous correlations
in the data exist. Hence, in their MVARICA approach they
apply ICA to the residuals of a sensor-space MVAR model.
In this work, we first propose a single-step procedure to
estimate all parameters (i.e. the mixing matrix and MVAR
coefficients) of the linear mixing model of MVAR sources
[10] based on temporal-domain convolutive ICA, instead of
the combination of MVAR parameter fitting and demixing by
instantaneous ICA. Furthermore, the approach enables us to
1Although, under some circumstances this approach can be justified [9].
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2integrate a sparsity assumption on brain connectivity, i.e. on
interactions between underlying brain sources via the Group
Lasso penalty. The additional sparsity prior can avoid over-
fitting in practical applications and yields more interpretable
estimators of brain connectivity. We remark that it is hard to
incorporate such sparsity priors in MVARICA, since MVAR
is fit to the sensor signals where interactions (i.e. MVAR
coefficients) are not at all sparse due to the volume conduction.
The remainder of the paper is organized as follows. In
Section II, our procedure will be explained step by step.
The correlated source model assumed in this paper will be
defined in II-B. The identification procedure called connected
sources analysis (CSA) based on the convolutive ICA will be
introduced (II-C) and followed by its sparse version, sparse
connected sources analysis (SCSA) with the Group Lasso prior
(II-D). The relations of our methods with existing approaches
such as MVARICA [10] and CICAAR [11] will be elucidated
in detail (II-E). Finally, the optimization algorithms for CSA
and SCSA will be explained (II-F). We implemented two
versions for SCSA, one based on L-BFGS and the other by EM
algorithm which is slower, but numerically more stable. The
next section III will provide our experimental results on simu-
lated data sequences emulating realistic EEG recordings. The
plausibility of our correlated source model will be discussed
with future research directions in the context of computational
neuroscience (Section IV), before the concluding remarks
(Section V).
II. CONNECTED SOURCES ANALYSIS WITH SPARSITY
PRIOR
A. MVAR for modeling causal interactions
Autoregressive (AR) models are frequently used to define
directed “Granger-causal” relations between time-series. The
original procedure by Granger involves the comparison of two
models for predicting a time series zi, containing either past
values of zi and zj , or zi only [2]. If involvement of zj leads
to a lower prediction error, (Granger-causal) information flow
from zj to zi is inferred. Since this may lead to spurious
detection of causality if both zi and zj are driven by a common
confounder z∗, it is advisable to include the set {z1, . . . , zM}\
{zi, zj} of all other observable time series in both models.
It has been pointed out, that pairwise analysis can be
replaced by fitting one multivariate autoregressive (MVAR)
model to the whole dataset, and that Granger-causal inference
can be performed based on the estimated MVAR model
coefficients (e.g., [5], [12]). Several connectivity measures are
derived from the MVAR coefficents [3], [4], but probably the
following definition is most straightforward from Granger’s
argument that the cause should always precede the effect. We
say that time series zi has a causal influence on time series
zj if the present and past of the combined time series zi and
zj can better predict the future of zj than the present and
past of zj alone. In the bivariate case this is equivalent to
saying that for at least one p ∈ {1, . . . , P}, the coefficient
H(p)ji corresponding to the interaction between zj and zi at the
pth time-lag is nonzero (significantly different from zero). In
the multivariate case, Granger causality also includes indirect
causes not contained in non-vanishing H(p)ji .
B. Correlated sources model
In this paper we propose a method for demixing the
EEG/MEG signal into causally interacting sources. We start
from the same model as in [10]: the sensor measurement is
assumed to be generated as a linear instantaneous mixture of
sources, which follow an MVAR model
x(t) = Ms(t) (1)
s(t) =
P∑
p=1
H(p)s(t− p) + ε(t) . (2)
Here, x(t) is the EEG/MEG signal at time t, M is a mixing
matrix representing the volume conduction effect, s(t) is the
demixed (source) signal. The sources at time t are modeled as
a linear combination of their P past values plus an innovation
term ε(t), according to an MVAR model with coefficient
matrices H(p). In the standard MVAR analysis, the innova-
tion ε(t) is a temporally- and spatially-uncorrelated Gaussian
sequence. In contrast, we assume here that it is i.i.d. in time
and the components are subject to non-Gaussian distributions
in order to apply blind source separation (BSS) techniques
based on higher-order statistics [10], [11].
For simplicity, we deal with the case that the numbers of
sensors and sources are equal and the mixing matrix M is
invertible. When there exist less sources than sensors, the
problem falls into the current setting after being preprocessed
by PCA [10]. Under our model assumptions, the innovation
sequence can be obtained by a finite impulse response (FIR)
filtering of the observation, i.e.
ε(t) = M−1x(t)−
P∑
p=1
H(p)M−1x(t− p) (3)
=
P∑
p=0
W (p)x(t− p) , (4)
where the filter coefficients are determined by the mixing
matrix M and the MVAR parameters {H(p)} as
W (p) =
{
M−1 p = 0
−H(p)M−1 p > 0
. (5)
Thanks to the non-Gaussianity assumption on the innovation
ε(t), we can use BSS techniques based on higher-order
statistics to identify the inverse filter {W (p)}. Since we
would like to impose sparse connectivity as a plausible prior
information later on, it is preferable to apply temporal-domain
convolutive ICA algorithms. The obtained FIR coefficients
{W (p)} directly identify the mixing matrix M and the MVAR
model of the same order P .
C. Identification by convolutive ICA
We use temporal-domain convolutive ICA for inferring
volume conduction effects and causal interactions between
extracted brain signals. The model parameters can be identified

3based on the mild assumptions that the innovations are non-
Gaussian and (spatially and temporally) independent. For EEG
and MEG data, a super-Gaussian is prefered to a sub-Gaussian
distribution, assuming that ongoing activity of brain networks
is triggered by spontaneous local bursts. We here adopt the
super-Gaussian sech-distribution that was proposed in [11].
The Likelihood of the data under the model is then
p({x(t)}Tt=P+1|{W
(p)})
= |W (0)|T−P
T∏
t=P+1
D∏
d=1
1
pi
sech (εd(t)) , (6)
where ε(t) = M−1x(t)−
∑P
p=1H
(p)M−1x(t− p). The cost
function to be minimized is the negative log-Likelihood
L({W (p)}) = (P − T ) log |W (0)|
−
T∑
t=P+1
D∑
d=1
log
(
1
pi
sech (εd(t))
)
. (7)
The solution of Eq. ((7)) leads to the estimators of the mixing
matrix M and the MVAR coefficients {H(p)} via Eq. ((5)). We
will call this procedure Connected Sources Analysis (CSA).
We remark that the temporal-domain algorithm of convolu-
tive ICA has obvious indeterminacy due to permutations and
sign flips. However, once we fix a rule to chose one from all
candidates, the cost function can be considered as convex.
D. Sparse connectivity as regularization
In practice, we usually have to consider a long-range lag P
to explain temporal structures of data sequences. However,
this causes too many parameters to be estimated reliably.
Maximum-Likelihood estimation may easily lead to overfit-
ting, especially if T is small. For this reason, it is advisable to
adopt a regularization scheme. Several authors have suggested
that the complexity of MVAR models can be reduced by
shrinking MVAR coefficients towards zero. In [12] and [13],
MVAR-based functional brain connectivity is estimated from
functional magnetic resonance imaging (fMRI) recordings us-
ing an `1-norm based (Lasso) penalty, which has the property
of shrinking some coefficients exactly to zero. In [5] it is
pointed out, that, by using a so-called Group Lasso penalty,
whole connections between time-series can be pruned at once.
In this approach, all coefficients H(p)ij , p = 1, . . . , P modeling
the information flow from si to sj are grouped together and
can only be pruned jointly. From the practical standpoint such
sparsification is very appealing, since fewer connections are
much easier to interpret. But assuming sparse connectivity in
fMRI data might also be justified from a neurophysiological
point of view, since under appropriate experimental conditions
only a few macroscopic brain areas are expected to show
significant interaction. This reasoning also applies to EEG and
MEG data.
We note that, besides the penalty-based approach, other
strategies for obtaining sparse connectivity graphs exist. For
example, post-hoc sparsification can be achieved for dense
estimators by means of statistical testing [5], [14]. However,
due to the compelling built-in regularization, we here adopt
Group Lasso sparsification.
Before applying our regularization to the cost function of
the correlated sources model, it is important to note that
the sparsity assumption is only reasonable for the MVAR
coefficients {H(p)}, but not for the W (p) matrices which
combine MVAR coefficients and the instantaneous demixing.
Hence, in order to apply sparsifying regularization, one has to
split the parameters into demixing and MVAR parts again, as
in the original model Eq. ((1)). Since the offdiagonal elements
{H(p)} correspond to interaction between sources, we propose
to put a Group Lasso penalty on them analogously to [5]. I.e.,
we penalize the sum of the `2-norms of each of the groups
{H(p)df }, d 6= f .
Let B := M−1(= W (0)), s(t) = Bx(t) and s˜(t) =
∑P
p=1H
(p)s(t− p). The regularized cost function is
LSCSA(B, {H(p)})
= (P − T ) log |B|+ λ
∑
d6=f
∥
∥
∥
∥
(
H(1)df , . . . ,H
(P )
df
)>
∥
∥
∥
∥
2
−
T∑
t=P+1
D∑
d=1
log
(
1
pi
sech (sd(t)− s˜d(t))
)
, (8)
λ being a positive constant. The solution to Eq. ((8)) for a
choice of λ is called the Sparsely-Connected Sources Analysis
(SCSA) estimate.
E. Relation to other methods
The proposed method extends previously suggested MVAR-
based sparse causal discovery approaches [5], [12] by a linear
demixing, which is appropriate for EEG/MEG connectiv-
ity analysis. Although the correlated sources model Eq. (1)
leads to an MVAR model of the observation sequence [10],
sparsity of the coefficients cannot be expected after mixing
by volume conduction effects. Our method compares with
MVARICA [10], which uses the same model Eq. (1), but
estimates its parameters differently. More precisely, the authors
of MVARICA suggest to first fit an MVAR model in sensor-
space. The demixing can then be obtained by performing
instantaneous ICA on the MVAR innovations, i.e., a dedicated
contrast function (Infomax) is used to model independence
of the innovations. The obtained sources follow an MVAR
model with time-lagged effects (interactions), but ideally no
instantaneous correlations (as caused by volume conduction).
It also turns out that the model Eq. (1) is very similar to
the convolutive ICA (cICA) [11], [15]–[17] model. The only
difference is that Eq. (1) employs a FIR filter to extract the
innovations, while an infinite response filter (IIR) is usually
used in the cICA literature (see, e.g., [11]). This discrepancy
is explained by the different philosophies that are associated
with both methods. While in our approach the innovations ε(t)
arise as residuals of a finite-length source-MVAR model, cICA
understands them as sources of a finite-length convolutional
mixture. Nevertheless, our unregularized cost function can
be regarded as a maximum-Likelihood approach to an IIR
version of convolutive ICA. This leads us also to a new view
of convolutive ICA as performing an instantaneous demixing
into correlated sources. Hence, it is possible to conduct source
connectivity analysis using cICA (see Fig. 1 for illustration).

4Compared to MVARICA and time-domain implementations
of convolutive ICA such as CICAAR [11], our formulation has
the advantage that sparse connectivity can easily be modeled
by an additional penalty. This is not possible for CICAAR,
because CICAAR only indirectly estimates the MVAR co-
efficients through their inverse filters. However, these are
generally nonsparse, even if the true connectivity structure
is sparse. Inverting the inverse coefficients is also generally
not possible (recall, that convolutive ICA is equivalent to
an infinite-length source-MVAR model). It is furthermore not
possible to introduce a sparse regularization for MVARICA,
since this method carries out the MVAR-estimation step in
sensor-space, where no sparsity can be assumed.
By variation of the regularization parameter, our method
is able to interpolate between a fully-correlated source model
(comparable to convolutive ICA) and a model which allows no
cross-talk between sources. Interestingly, the latter extreme can
be seen as a variant of traditional instantaneous ICA, in which
independence is measured in terms of mutual predictability
with a Granger-type criterion.
( )p{ }H
( )p{ }Wε x
s M
(IIR)
(FIR)
( )p M−1{ }HMM−1
ε (IIR)
Mε
(FIR)
(ARfit)(ICA)
x
(a) (b)
( )pW{ }
( )pA{ }ε (FIR)(IIR) x
(c)
Fig. 1. Relations between (a) SCSA, (b) MVARICA and (c) CICAAR.
All approaches assume a non-Gaussian innovation sequence ε. SCSA and
MVARICA fit an IIR model to the observed sequence x, while CICAAR
assumes an FIR filter for it. Therefore, in SCSA and MVARICA the inverse
filter from x to the innovation ε is an FIR. MVARICA is a two step approach
with AR fitting to the observed sequence x and spartial demixing of the
innovation Mε obtained in the first step. On the other hand, SCSA is a one-
step approach which compute the inverse FIR filter by convolutive ICA. We
remark that the AR fitting in MVARICA relies only on the second order
statistics, which may cause the performance drops compared to CSA.
F. Optimization
1) CSA: The gradient of the unregularized cost function
Eq. (7) is obtained as
∂L
∂W (p)d
= δ(p)
(
(P − T )W (p)
−>
ed
)
+
T∑
t=P+1
tanh
(
P∑
p=0
W (p)d
>
x(t− p)
)
x(t− p) , (9)
where W (p)d := W
(p)>ed, i.e. the d-th column vector of
W (p)
>
.
We plug the gradient into a limited memory Broyden-
Fletcher-Goldfarb-Shanno (L-BFGS) optimizer [18]2 and ob-
serve that the algorithm always converges to the global op-
timum, while only the signs and order of the components
may depend on the initialization. We use W (0) = I and
W (p) = 0, p = 1, . . . , P as a default initializer.
2) SCSA via a modified L-BFGS algorithm: Using sparse
regularization, two additional difficulties emerge compared
to the unregularized cost function. First, using the factor-
ization Eq. (5) the guaranteed convergence to the minimum
observed for CSA is unlikely to be retained. Furthermore,
the function Eq. (8) is not differentiable, when one of the
terms ‖(H(1)df , . . . ,H
(P )
df )
>‖2, d 6= f becomes zero, which is
expected to be the case at the optimum.
For tackling these difficulties we here propose to use a
modified version of the L-BFGS algorithm, which allows joint
nonlinear optimization of B and {H(p)}, while taking special
care of the nondifferentiability of the regularizer. The gradient
of Eq. (8) is obtained as
∂LSCSA
∂H(p)df
= −
T∑
t=P+1
tanh (sd(t)− s˜d(t)) sf (t− p)
+ λ
H(p)df∥
∥
∥
∥
(
H(1)df , . . . ,H
(P )
df
)>
∥
∥
∥
∥
2
(10)
and
∂LSCSA
∂Bd
= (P − T )B−>ed
+
T∑
t=P+1
D∑
d=1



tanh (sd(t)− s˜d(t))
×
(
x(t)−
P∑
p=1
xd(t− p)H
(p)
d
)}
.(11)
Our modified L-BFGS algorithm checks before
each gradient evaluation, whether some terms
‖(H(1)df , . . . ,H
(P )
df )
>‖2, d 6= f are already (close to) zero.
If any of the terms equals zero, the gradient is not defined
uniquely but as a set (subdifferential). Nevertheless it is
straightforward to compute the element of the subdifferential
with minimum norm, whose sign inversion is always a
descent direction. Care must be taken because in practice we
would not find any of the above terms exactly equal to zero.
Thus we truncate the elements of H corresponding to the
terms with small norms below some threshold to zero before
computing the minimum norm subgradient. If the minimum
is indeed attained at the truncated point, the minimum norm
subgradient will be zero. Otherwise the subgradient will
drive the solution out of zero. Further care must be taken in
practice to prevent the solution from oscillating in and out of
some zero.
2We use an implementation by Naoaki Okazaki,
http://www.chokkan.org/software/liblbfgs/.

5We find that, using the outlined optimization procedure,
sparse solutions can be found in shorter time, if the solution of
the unregularized cost function is used as the initializer. The
starting point can be obtained using the inverse transformation
of Eq. (5), which is given by
B = W (0) (12)
H(p) = −W (p)B−1, p > 0 . (13)
3) SCSA via an EM algorithm: Using joint optimization of
B and {H(p)}, the heuristic pruning of connections might in
some cases lead to suboptimal solutions regarding the com-
posite cost function. For this reason, we present an alternative
optimization scheme, which does not require any heuristic
step. The idea here is to alternate between the estimation of
both unknowns. Doing so can be justified as an application of
the Expectation Maximization (EM) algorithm (see [19]).
Estimation of B given {H(p)} (here called E-step) amounts
to solving an unconstrained nonlinear optimization problem.
Importantly, this problem is also convex, in contrast to the joint
approach to SCSA parameter fitting. The convexity follows
from the concavity of log |X| and log(sech(ax)) for constant a
(and from the fact that the sum of convex functions is convex.).
The great advantage of convex problems is, that they feature a
unique (local and global) minimum. In our case, the objective
is smooth, so the minimum is guaranteed to be found by the
L-BFGS algorithm, making use of the gradient in Eq. (11).
Optimization with respect to {H(p)} for fixed B (M-step)
is more involved, since the nondifferentiable Group Lasso
regularizer remains. Smooth optimization methods like L-
BFGS are unlikely to find the exact solution here. However,
this problem is not as difficult as the joint optimization
problem, since it is convex. This can be seen from the fact
that it is composed of a sum of − log(sech(ax)) terms (loss
function) and the Group Lasso term (regularizer), which is
a sum of `2-norms and thus convex. Hence we can solve
this problem using the Dual Augmented Lagrangian (DAL)
procedure [20], which has recently been introduced as a
method for minimizing arbitrary convex loss functions with
additional Lasso or Group Lasso penalties. Application of
DAL requires the loss function and its gradient, the convex
conjugate (Legendre transform) of the loss function, as well as
gradient and Hessian of the conjugate loss. Let s(t) = Bx(t)
be the demixed sources and s˜(t) =
∑P
p=1H
(p)s(t− p) be
their autoregressive approximations. The loss function in terms
of s˜ is defined as
LM(s˜) = −
T∑
t=P+1
D∑
d=1
log
(
1
pi
sech (s˜d(t)− sd(t))
)
. (14)
The gradient is
∂LM
∂s˜d(t)
= tanh(s˜d(t)− sd(t)) . (15)
Let ad(t) (d = 1, . . . , D, t = P + 1, . . . , T ) denote the
dual variables associated with the Legendre transform. The
conjugate loss function is defined on the interval [−1, 1] and
evaluates to
DM(a)
=
T∑
t=P+1
D∑
d=1
sup
s˜d(t)
(
ad(t)s˜d(t)− log
sech (s˜d(t)− sd(t))
pi
)
=
T∑
t=P+1
D∑
d=1
(
1− ad(t)
2
log
1− ad(t)
2
+
1 + ad(t)
2
log
1 + ad(t)
2
− ad(t)sd(t) + log
2
pi
)
.
(16)
The gradient of the conjugate loss is given by
∂DM(a)
∂ad(t)
=
1
2
log
1 + ad(t)
1− ad(t)
− sd(t) . (17)
The Hessian is diagonal with elements
∂2DM(a)
∂ad(t)2
=
1
2(1− a2d(t))
. (18)
Having defined the E- and M-steps, we have turned a
nonconvex estimation problem into a sequence of two convex
problems, which can both be solved exactly. A final estimate
of the model parameters can now be obtained by alternating
between E- and M-steps until convergence.
G. Treating source autocorrelations
Diagonal parts of the MVAR matrices {H(p)} model the
sources’ autocorrelation and should preferably not be pruned.
However, in some cases numerical stability can be increased
if these variables are also penalized, especially if D and P
are large. For this reason, we use a slight variation of the cost
function Eq. (8) in practice, which includes
∥
∥
∥
∥
(
H(1)11 , . . . ,H
(P )
11 , . . . ,H
(1)
DD, . . . ,H
(P )
DD
)>
∥
∥
∥
∥
2
(19)
as an additional penalty term. The augmented objective func-
tion can be minimized using the techniques presented in
Section II-F.
III. PERFORMANCE UNDER REALISTIC CONDITIONS
We conducted the following simulations in order to assess
the performance of the proposed source connectivity analysis
compared to those of existing approaches.
A. Data generation
We simulated seven time-series (pseudo-sources) of length
N = 2000 according to an MVAR model of order P = 4.
Seven out of the forty-two possible interactions were modeled
by allowing the corresponding offdiagonal MVAR coefficients
H(p)df , d 6= f, 1 ≤ p ≤ P to be nonzero. The innovations were
drawn from the sech-distribution (Note that the assumption of
non-Gaussianity is crucial for recovering mixed sources.).
The pseudo-sources were mapped to 118 EEG channels
using the theoretical spread of seven randomly placed dipoles.