Expected duration: 4-6 months (with regular remuneration).
Location: INRIA Sophia Antipolis, BP 93, 06902 Sophia-Antipolis Cedex, France.
Advisor: Juliette Leblond, Team FACTAS,
web page: http://www-sop.inria.fr/members/Juliette.Leblond/
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  for the EEG inverse source problem (see also ).
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 .
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.
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 , and quite a few
silent source terms for MEG are described in . 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 . Also, silent
magnetic sources related to inverse source problems in paleomagnetism are considered in
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,
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.
• 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.
 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
 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.
 L. Baratchart, J. Leblond, Silent electrical sources in domains of R 3 , Unpublished
 L. Baratchart, C. Villalobos, D. P. Hardin, M. C. Northington, E. B. Saff. Inverse
potential problems for divergence of measures with total variation regularization,
 M. Clerc, J. Leblond, J.-P. Marmorat, T. Papadopoulo, Source localization using
rational approximation on plane sections. Inverse Problems, 28(5):055018, 2012.
 J. C. Mosher, R. M. Leahy, Recursive MUSIC: A Framework for EEG and MEG
Source Localization. IEEE Trans. Biomedical Engineering, 45(11), 1998.
 J. Sarvas, Basic mathematical and electromagnetic concepts of the biomagnetic
inverse problem, Phys. Med. Biol., 32(1), 1987.