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# Large brain effective network from EEG/MEG data and dMR information

Brahim Belaoucha, Théodore Papadopoulo

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Brahim Belaoucha, Théodore Papadopoulo. Large brain effective network from EEG/MEG data and dMR information. PRNI 2017 - 7th International Workshop on Pattern Recognition in NeuroImaging, Jun 2017, Toronto, Canada. hal-01533445

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# Large brain effective network from EEG/MEG data and dMR information

Brahim Belaoucha

Université Côte d'Azur

Inria

France

Théodore Papadopoulo

Université Côte d'Azur

Inria

France

**Abstract**—Over the past 30 years, neuroimaging has become a predominant technique. One might envision that over the next years it will play a major role in disclosing the brain's functional interactions. In this work, we use information coming from diffusion magnetic resonance imaging (dMRI) to reconstruct effective brain network from two functional modalities: electroencephalography (EEG) and magnetencephalography (MEG).

## I. INTRODUCTION

MEG and EEG imaging techniques record the brain cortical activity in real time. Localizing active regions from these measurements, in the source distributed model, involves solving an ill-posed inverse problem. Prior on the source space must be set to obtain a unique solution.

Minimum norm estimate (MNE) is one of the most used inverse solvers because it leads to a linear inverse operator [1, 2]. It results to widely extended active regions which are, biologically, not plausible. Among other solvers, we can find the minimum current estimate (MCE). It provides a focal reconstruction but is non-smooth in time because the sparse prior is applied at each time sample independently. Mixed-norm estimate (MxNE) [3] was introduced to overcome this limitation. It provides spatial focality, by applying $l_1$-norm on the source space, and temporal smoothness, by using $l_2$-norm on time courses.

Functional connectivity analysis has been traditionally implemented at the sensor level, but lately, a number of studies have begun to use source-space analysis. It has the potential of providing more accurate information regarding regions' functional in-

teractions [4, 5, 6]. Some of the functional measures (e.g. Directed Transfer Function (DTF) and Partial Directed Coherence (PDC)) can be derived from multivariate autoregressive model (MAR) which is more than a combination of purely temporal and purely spatial correlation, as it also takes into account the time-lagged correlation between sources.

In [4], the authors constrain the dynamics of the sources by a MAR model. They assume that there exist only one time-lag, which depends on the length of anatomical connections, between a pair of regions. In [6], the authors use a variant of MxNE constrained with a MAR model of order 1 to reconstruct regions' interactions. In both works, complex time courses can not be reconstructed because of limits in the degree of freedom of the model. In this work, we generalize the work of [6] by constraining the dynamics of the cortical regions with higher MAR model whose matrix elements are constrained by the anatomical connectivity and neighbors. We call it iterative source and dynamics reconstruction of order p (iSDR(p)). We test and compared our method with real and synthetic data.

## II. METHOD

Under the quasi-static approximation of the Maxwell's equations, the EEG/MEG data $M_t$ is a linear superposition of the source signals $J_t$ at the same time $t$:

$$M_t = G J_t + \varepsilon_t, \quad (1)$$

where $G \in \mathbb{R}^{m \times n}$ ($m$: size of sensor array, $n$: size of source space) is the lead-field matrix and
<figure><img src="image_3.png" /><figcaption>Fig. 1: In (a), mean, over 100 simulations, of reconstruction error $E_r$ of MxNE and iSDR(p=2) with different initialization of A and SNR levels. (b) Estimated initial values of $S_1$ and $S_2$ using iSDR(p = 2). (c) Eigenvalues of the companion matrix. The results are shown at 3 different SNR. Black dots represent ground truth values.</figcaption></figure>

$\epsilon_t$ is $\mathcal{N}(0, \Sigma_c)$. To reduce the problem complexity, we assume a constant activation per cortical region. This reduces the size of the lead-field to $m \times K$, where K is the number of cortical regions. We use the mutual nearest neighbor parcellation algorithm presented in [7, 8] to divide the cortical surface into functional regions from dMRI. The source dynamics is constant during a time window T and it follows MAR model of order p:

$$J_t = \sum_{i=1}^{p} A_i J_{t-i} + \omega_t \quad (2)$$

where $\omega_t$ is $\mathcal{N}(0, \Sigma_s)$, p is a positive integer that defines the order of the MAR model, $A_i$ is a $K \times K$ matrix which relates sources' current values to its pasts. The only nonzero effective interactions in A corresponds to either structurally connected or neighboring regions. The behavior of the MAR model is obtained from the companion matrix $\Phi$:

$$\Phi = \begin{bmatrix} A_1 & A_2 & \cdots & A_{p-1} & A_p \\ I & & & & \\ & \ddots & & & \\ & & I & & \end{bmatrix} \quad (3)$$

where I is the $K \times K$ identity matrix. The MAR model is stable (stationary) if all eigenvalues of $\Phi$ are inside the unit circle and nonstationary if it has eigenvalues on the unit circle. The model is unstable if at least one eigenvalue is outside the unit circle. The sources estimates J are obtained by minimizing the following functional given a MAR model:

$$U(J) = \|M_v - \mathbf{G}_d J_v\|_2^2 + \lambda \|J\|_{21} \quad (4)$$

where $M_v = [M_{p+1}, \cdots, M_T]$, $J_v = \text{vec}(J) = \text{vec}([J_1, \cdots, J_{T-1}])$ and $\|J\|_{21} = \sum_i \sqrt{\sum_j J(i, j)^2}$ i.e. $l_2$-norm over time and $l_1$-norm over space.

$$\mathbf{G}_d = \begin{bmatrix} G_1 & G_2 & \cdots & G_p \\ \vdots & \vdots & \ddots & \vdots \\ G_1 & \cdots & G_{p-1} & G_p \end{bmatrix} \quad (5)$$

where $G_i = GA_i$. The MAR elements are obtained by fitting the source estimates to the MAR model:

$$\underset{A_1, \cdots, A_p}{\operatorname{argmin}} \sum_{t=p+1}^{T} \left\| J_t - \sum_{i=1}^{p} A_i J_{t-i} \right\|_{2}^{2} \quad (6)$$

We solve iteratively Eq 4 and 6 until $\|J^k - J^{k-1}\| < \epsilon$ where $J^k$ is the reconstructed sources at iteration k.

### III. DATA

MEG and EEG were measured using 306 system (102 magnetometers, 204 planar gradiometers), and 70 EEG channels recorded EEG data simultaneously. Stimuli were presented in six 7.5 min runs. The face stimuli contain two sets of 300 gray scale photographs, half from unfamiliar people (unknown to the participants) and the remaining from famous people. In a third condition, 150 photographs of scrambled faces are obtained from either famous or unfamiliar people. The reader is referred to [9] for more details.

In this work, we are interested in localizing face recognition areas. For this reason, we use only the measurement acquired when using photos of famous people subtracted to the ones obtained when
using scrambled faces. Low pass filter of cut-off
frequency 35 Hz was used to smooth the data. The
MEG/EEG forward problem, lead field matrix G, is
obtained using OpenMEEG [10, 11].

IV. RESULTS AND DISCUSSION

A. *Simulation*

We use a random lead-field matrix of size
20 × 500 and whose elements are obtained from
a normal distribution N(0, 1). Each region is
connected randomly to 4 different regions. We
activate 50 random pairs of regions, one pair
at a time, 100 times at different noise levels
(signal-to-noise ratio (SNR) = {15, 10, 5} dB)
using the following simple MAR model: [ S₁ S₂ ]ₜ =
[ 0.43 0.24 [ S₁ S₂ ]ₜ₋₁ + [ 0.47 0.12 [ S₁ S₂ ]ₜ₋₂ , with
J₀(S₁) = 6.15, J₀(S₂) = -3.64, J₁(S₁) =
- J₁(S₂) = 5nA i.e. Jₜ = A₁Jₜ₋₁ + A₂Jₜ₋₂.
We estimate the sources and their interactions by
initializing A₂ and A₁ with different values to
have an insight about the effect of the MAR's
initialization on the resulting estimate. We use the
following combinations: A₂ = βI and A₁ = αI
with (α, β) ∈ {(1, 0), (¾, ¼), (½, ½)}.
The results
are presented in Fig (1). Fig (1a) represents the re-
construction error computed as ||Jgt - Jr||₂, where
Jgt and Jr are the simulated ground truth and re-
constructed activation, using MxNE and iSDR(p=2)
respectively. The estimated initial values of S₁ and
S₂ obtained from iSDR(p=2) are shown in Fig (1b).
Fig (1c) represents the nonzero eigenvalues of the
estimated companion matrix Φ.

iSDR(p) outperforms MxNE especially in low
SNR. For (α, β) = (1, 0), iSDR favors the initial
values (J<sub>t=0</sub>) to be close to 0, see blue dots in Fig
(1b), due to initializing A<sub>2</sub> = 0 i.e no interactions
between t and t − 2. Setting β to nonzero improves
both the temporal reconstruction of signals and
their dynamics (eigenvalues of Φ). iSDR could
reconstruct accurately two eigenvalues, right side
of the plane in Fig (1c). The remaining two are
affected by the choice of (α, β). The results are less
dispersed when (α, β) = (1/2, 1/2) (see left side of Fig
(1c)).

<figure><img src="image_4.png" /><figcaption>Fig. 2: Reconstructed activation in the right fusiform
gyrus (RFF) of 7 subjects, left panel, and left FF
(RFF), of 2 subjects using iSDR(p = 2). The temporal
courses are normalized with their l₂-norm for subjects
comparison.</figcaption></figure>

B. Real data

iSDR(p=2) is applied to MEG data of 9 subjects
from [9]. In [12], the authors show that subjects may
have right or bilateral fusiform gyrus (FF) activation
if they are right or left handed, respectively. In Fig
2, we show the estimated temporal course of the
right/left FF (R/LFF). In our work, seven subjects
show higher activation in the RFF, Fig (2) left, while
only two show higher activation in LFF, Fig (2)
right. Due to page limitation, we show only the
effective connectivity of Subject 1, Fig (3).

Several works [9, 4] show that activation in the
FF between 170 and 200 ms is observed when
performing face recognition task which coincides
with what we observe in this work. In the ma-
jority of subjects, regions in the superior posterior
temporal lobe, medial parietal, temporal pole (TP),
orbitofrontal (OF) are found to be active. This
coincides with the findings of other works with
the same dataset [9, 4]. We have found interactions
between TP and frontal lobe (FL), left and right FL,
inferior occipital and temporal gyrus with OF.

We applied our method only with MAR model of
order 2. Our approach provides functionally plau-
sible results when tested with MEG data provided
by [9].

V. CONCLUSION

In this work, we proposed a source reconstruction
method constrained by multivariate autoregressive
model from functional imaging EEG/MEG data.
<figure><img src="image_5.png" /><figcaption>Fig. 3: Effective network of A₁, left, and A₂, right, obtained from iSDR(p=2) and MEG data [9] during face recognition task. Red spheres represent center of active cortical regions. Edges represent uni/bidirectional interactions. Self-interactions are not shown for visualization clarity. Edge's color represents its strength.</figcaption></figure>

Application to real data of other mental tasks must be investigated and compared to physiological literature to validate our method. Information criterion can be used to obtain the order of MAR model.

## ACKNOWLEDGMENT

This work has received funding from ERC under the European Union's Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665: CoBCoM) and French ANR contract Vibrations. We thank Dr. Daniel Wakeman and Prof. Richard Henson for providing the data.

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