Section: New Results

Forward and Inverse Problems

Source localization using rational approximation on plane sections

Participants : Maureen Clerc, Théodore Papadopoulo, Juliette Leblond [Apics Project-Team, Inria, Sophia Antipolis, Méditerranée, France] , Jean-Paul Marmorat [CMA, Ecole des Mines Paristech, Sophia Antipolis, France] .

In functional neuroimaging, a crucial problem is to localize active sources within the brain non-invasively, from knowledge of electromagnetic measurements outside the head. Identification of point sources from boundary measurements is an ill-posed inverse problem. In the case of electroencephalography (EEG), measurements are only available at electrode positions, the number of sources is not known in advance and the medium within the head is inhomogeneous. This work presents a new method for EEG source localization, based on rational approximation techniques in the complex plane. The method is used in the context of a nested sphere head model, in combination with a cortical mapping procedure. Results on simulated data prove the applicability of the method in the context of realistic measurement configurations.

This work has been published in the journal Inverse Problems [14] .

The adjoint method of OpenMEEG for EEG and MEG with large source space

Participants : Maureen Clerc, Théodore Papadopoulo, Alexandre Gramfort [Telecom Paristech] , Emmanuel Olivi [Forner member of the Athena Project-Team] .

In EEG or MEG, a lead field is the linear operator which associates unitary dipolar sources to the resulting set of sensor measurements. In practise, the source space often includes over 10 000 dipoles, which sometimes causes memory problems. The adjoint approach considers the forward problem from the viewpoint of sensors instead of sources: this drops down the number of linear systems to solve by two orders of magnitude. The adjoint approach is here proposed in the context of the Boundary Element Method, and its implementation is provided by the OpenMEEG library.

This work was presented at the BIOMAG conference [38] .

Comparison of Boundary Element and Finite Element Approaches to the EEG Forward Problem

Participants : Maureen Clerc, Carsten Wolters [Institute for Biomagnetism and Biosignal Analysis, University of Münster] , Johannes Vorwerk [Institute for Biomagnetism and Biosignal Analysis, University of Münster] , Martin Burger [Institut für Numerische und Angewandte Mathematik, Fachbereich Mathematik und Informatik, Westfälische Wilhelms Universität (WWU) Münster] , Jan de Munck [Vrije Universiteit Medical Centre (VUMC), The Netherlands] .

The accurate simulation of the electric fields evoked by neural activity is crucial for solving the inverse problem of EEG. Nowadays, boundary element methods (BEM) are frequently applied to achieve this goal, usually relying on the simpli-fication of approximating the human head by three nested compartments with isotropic conductivities (skin, skull, brain). Here, including the highly-conducting cerebrospinal fluid (CSF) is a difficult task due to the complex geometrical structure of the CSF, demanding a high number of additional nodes for an accurate modeling and thus a strongly increased computational effort. Though, CSF conductivity is well-known and nearly not varying inter-individually and its significant influence on EEG forward simulation has been shown. The CSF can be included at negligible computational costs when applying finite element (FE) forward approaches. In this study we compare the accuracy and performance of state-of-the-art BE and FE approaches in both artificial and realistic three layer head models, showing that all approaches lead to high numerical accuracies. Furthermore, we demonstrate the significant influence of modeling the CSF compartment as disregarding this compartment leads to model errors that lie clearly above the observed numerical errors.

A book chapter on BEM and FEM models has been published in the Handbook for Neural Activity Measurement [40] . The comparison was presented at the BIOMAG conference [19] .

Domain Decomposition to handle versatile conductivity models

Participants : Maureen Clerc, Théodore Papadopoulo, Emmanuel Olivi [Forner member of the Athena Project-Team] .

Source localization from external data such EEG or MEG, requires a good understanding of the electromagnetic behavior of the patient head. Several models can been used, representing more or less complex geometrical shapes, and conductivity profiles. Different numerical methods allow to cope with different types of models: the Finite Element Method (FEM) can handle very general conductivity models, whereas the Boundary Element Method (BEM) is limited to piecewise constant conductivity. On the other hand, it is easier with BEM than with FEM to accurately represent sources in isotropic media. Thanks to domain decomposition, we propose to solve a EEG forward problem using BEM where the sources are (the brain) and FEM for other tissues (with notably inhomogeneities in the skull).

This work was presented at the BIOMAG conference [37] .