## Section: Application Domains

### Inverse source problems in EEG

Participants : Laurent Baratchart, Juliette Leblond, Jean-Paul Marmorat, Christos Papageorgakis, Nicolas Schnitzler.

This work is conducted in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena EPI.

Solving overdetermined Cauchy problems for the Laplace equation on a
spherical layer (in 3-D) in order to extrapolate
incomplete data (see Section 3.2.1) is
a necessary
ingredient of the team's approach to inverse source problems, in particular
for applications to EEG, see [7]. Indeed, the latter involves propagating the
initial conditions through several layers of different conductivities,
from the boundary shell
down to the center of the domain where the
singularities (*i.e.* the sources) lie.
Once propagated
to the innermost sphere, it turns out that traces of the
boundary data on 2-D cross sections coincide
with analytic functions with branched singularities
in the slicing plane [6],
[41]. The singularities are
related to the actual location of the sources, namely their moduli
reach in turn a
maximum when the plane contains one of the sources. Hence we are
back to the 2-D framework of Section 3.3.3,
and recovering these singularities
can be performed *via* best rational approximation.
The goal is to produce a fast and sufficiently accurate
initial guess on the number
and location of the sources in order to run heavier
descent algorithms on the direct problem, which are more precise but
computationally costly and often
fail to converge if not properly initialized. Our belief
is that such a localization process can add a geometric, valuable piece of
information to the standard temporal analysis of EEG signal records.

Numerical experiments obtained with our software FindSources3D give very good results on simulated data and we are now engaged in the process of handling real experimental data (see Sections 3.4.2 and 5.1), in collaboration with the Athena team at Inria Sophia Antipolis, neuroscience teams in partner-hospitals (la Timone, Marseille), and the BESA company (Munich).