Section: Application Domains
Inverse source problems in EEG
Participants : Laurent Baratchart, Kateryna Bashtova, Juliette Leblond.
This work is done in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena Project-Team, and Jean-Paul Marmorat (Centre de mathématiques appliquées - CMA, École des Mines de Paris).
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 since the latter involves propagating the initial conditions through several layers of different conductivities, from the boundary down to the center of the domain where the singularities (i.e. the sources) lie. Then, once propagated to the innermost sphere, it turns out that that traces of the boundary data on 2-D cross sections (disks) coincide with analytic functions in the slicing plane, that has branched singularities inside the disk  . These singularities are related to the actual location of the sources (namely, they reach in turn a maximum in modulus when the plane contains one of the sources). Hence, we are back to the 2-D framework of Section 3.3.3 where approximately recovering these singularities can be performed using best rational approximation. The goal is to produce a fast but already good enough 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.
Numerical experiments give very good results on simulated data and we are now engaged in the process of handling real experimental magneto-encephalographic data, see also Sections 5.6 and 6.1 , in collaboration with the Athena team at Inria Sophia Antipolis, neuroscience teams in partner-hospitals (la Timone, Marseille), and the BESA company (Munich).