Expected duration: 4-6 months (with regular remuneration).

Location: INRIA Sophia Antipolis, BP 93, 06902 Sophia-Antipolis Cedex, France.

Advisor: Juliette Leblond, Team FACTAS,

Email: juliette.leblond@inria.fr,

web page: http://www-sop.inria.fr/members/Juliette.Leblond/

**Context**

Electroencephalography (EEG) and magnetoencephalography (MEG) are among the non

invasive imaging techniques used in medical engineering for functional or clinical brain

exploration. Electrical currents occurring in the brain produce an electrical potential and

a magnetic field, that are recorded at a finite number of pointwise sensors located on or

above the scalp. From these measures, we approach the inverse problem of localizing in

the brain the primary currents (sources) which have produced the records, as described

in [5] for the EEG inverse source problem (see also [6]).

The underlying model consists in a pair of (elliptic) partial differential equations relating

the electric and magnetic fields, more precisely the electric potential and the magnetic

field, to the unknown source term. They are obtained from Maxwell’s equation under

quasi-static assumptions and describe the forward model and the forward operator which,

to the source term, associates the measurements. Existence and uniqueness of solutions

to the PDE in suitable classes of functions can be established, see [7].

The associated inverse problems are as follows. Given (pointwise) values of the electric

potential at electrodes on the upper part of the scalp, and of the normal component of the

magnetic field at MEG sensors (SQUID) located above the scalp, recover the source term

that generated them. These inverse problems, that we want to consider either jointly or

separately, are described by integral equations related to the above PDEs. They are ill-

posed: solutions could be non unique, and the stability of these problems is not granted.

**Goals**

The topic of this internship is to study silent sources terms for both EEG and MEG

inverse problems. Silent sources are those among the source terms that produce either

a vanishing electric potential or a vanishing magnetic field outside the brain. They have

been “characterized” in Sobolev spaces for EEG in spherical domains [3], and quite a few

silent source terms for MEG are described in [7]. They are responsible for non-uniqueness

of solutions to the inverse problems, since two source distributions that differ by a silent

source term can not be distinguished one from the other from values of the produced

electric potential / magnetic field outside the brain. Note that silent sources distribu-

tions (measures) for a class of Poisson-Laplace equation similar to that involved in the

EEG model with sources distributed on cortical patches are described in [4]. Also, silent

magnetic sources related to inverse source problems in paleomagnetism are considered in

[2, 1].

Both bibliographical, theoretical and numerical studies of silent sources for EEG and MEG

will be addressed during the internship. Using available softwares / simulators for the

forward problems, numerical computations will be performed to exhibit the corresponding

behaviour of the fields. This will be done for a spherical model of the head, either

1homogeneous with piecewise constant conductivities on consecutive spherical layers (brain,

skull, scalp).

If times permits, “ almost” silent sources will be studied as well, that consist in source

term producing “small” electrical potential / magnetic field outside the head.

Moreover, for dipolar source terms in spherical head models, numerical solutions to the

corresponding inverse problems can be computed as well. Indeed, a dedicated software

FindSources3D (http://www-sop.inria.fr/apics/FindSources3D) is being developed

at INRIA, in collaboration with the CMA, Ecole des Mines Paristech, Sophia Antipolis,

that solves the inverse EEG and MEG problems for spherical head models and pointwise

dipolar brain sources. This development is pursued in close contatcs with colleagues from

the Institut de Neurosciences des Systèmes (INS), Université Aix-Marseille, France.

**Candidate profile**

• Second year of Master degree or Engineers School (PFE).

• Strong background in applied mathematics.

• Good knowledge of physics, algorithms and numerical analysis.

• Involvement in numerical simulation (Matlab) and in applications.

The internship may be followed by a PhD thesis, co-advised with our partners at INS, on

related topics of common interest.

**References**

[1] L. Baratchart, S. Chevillard, J. Leblond, Silent and equivalent magnetic distribu-

tions on thin plates. In Harmonic Analysis, Function Theory, Operator Theory,

and their Applications, Theta Series in Advanced Mathematics, 2017, The Theta

Foundation.

[2] L. Baratchart, D. P. Hardin, E. A. Lima, E. B. Saff, B. P. Weiss. Characterizing

kernels of operators related to thin-plate magnetizations via generalizations of

Hodge decompositions. In Inverse Problems, 29(1), 2013.

[3] L. Baratchart, J. Leblond, Silent electrical sources in domains of R 3 , Unpublished

notes, 2012.

[4] L. Baratchart, C. Villalobos, D. P. Hardin, M. C. Northington, E. B. Saff. Inverse

potential problems for divergence of measures with total variation regularization,

to appear.

[5] M. Clerc, J. Leblond, J.-P. Marmorat, T. Papadopoulo, Source localization using

rational approximation on plane sections. Inverse Problems, 28(5):055018, 2012.

[6] J. C. Mosher, R. M. Leahy, Recursive MUSIC: A Framework for EEG and MEG

Source Localization. IEEE Trans. Biomedical Engineering, 45(11), 1998.

[7] J. Sarvas, Basic mathematical and electromagnetic concepts of the biomagnetic

inverse problem, Phys. Med. Biol., 32(1), 1987.