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Section: New Results

Brain functional imaging using MEG/EEG

EEG forward problem

OpenMEEG software library

Participants : Maureen Clerc, Emmanuel Olivi, Alexandre Gramfort, Théodore Papadopoulo.

This work was partly supported by the Regional Council of Provence Alpes Cote d'Azur and the ANR ViMAGINE.

To recover the sources giving rise to electro- and magnetoencephalography in individual measurements, realistic physiological modeling is required, and accurate numerical solutions must be computed. The OpenMEEG software library solves the electromagnetic forward problem in the quasistatic regime, for head models with piecewise constant conductivity. The core of OpenMEEG consists of the symmetric Boundary Element Method, which is based on an extended Green Representation theorem. OpenMEEG is able to provide lead fields for four different electromagnetic forward problems: Electroencephalography (EEG), Magnetoencephalography (MEG), Electrical Impedance Tomography (EIT), and intracranial electric potentials (IPs). OpenMEEG is open source and multiplatform. It can be used from Python and Matlab in conjunction with toolboxes that solve the inverse problem; its integration within FieldTrip is operational since release 2.0.

Some new developments have concerned the organization of the computations to compute the lead fields. A lead-field matrix is obtained by concatenating the forward fields computed for thousands of sources characterized by their positions, orientations and strengths. A line of this lead-field matrix represents the physical quantity (potential for EEG, or some components of the magnetic field for MEG) at a sensor for each source. The number of sources largely exceeds the number of sensors (up to 256 electrodes for EEG, and less than 600 squids for MEG). When solving the forward problem with a BEM (Boundary Element Method), the lead-field matrix is generally computed column-by-column, i.e. source by source, which represents nsources resolutions of the forward problem. Using the adjoint operator of the forward problem, one can reduce the computations to sensors resolutions. Some previous works [72] , [73] have used similar techniques for efficient computation of the lead-fields using finite element methods. The adjoint method [69] generalizes the Helmholtz reciprocity theorem and here is proposed its implementation using the BEM provided by the open-source software OpenMEEG

This work has been published in [15] and [36] .

Conductivity calibration for the EEG forward problem

Participants : Maureen Clerc, Emmanuel Olivi, Alexandre Gramfort, Théodore Papadopoulo, Jean-Michel Badier [INSERM U751, La Timone, Marseille] , Martine Gavaret [INSERM U751, La Timone, Marseille] , Laurent Koessler [CRAN Nancy] .

This work was partly supported by the Regional Council of Provence Alpes Cote d'Azur and the ANR ViMAGINE.

Bioelectric phenomena at low temporal frequency can be described by the electrostatic Poisson equation

divσV=divJ p ,

where J p are primary current sources and σV the Ohmic current. Appropriate boundary conditions (b.c.) must be set on the domain boundary, typically imposing the potential (Dirichlet b.c.) or the normal current (Neumann b.c.).

Solving this electrostatics equation for the electric potential is a problem common to different fields such as electroencephalograhy (EEG), electrocardiography (ECG), functional electrical stimulation. The main difficulty of the model concerns the conductivity field, which is not homogeneous, and whose values depend on the tissue type. Although the tissue structure can be revealed by imaging methods such as CT, Magnetic Resonance Imaging, diffusion Magnetic Resonance Imaging, conductivity values must nevertheless be assigned to the different tissues. In order to calibrate conductivity, injected current Electrical Impedance Tomography (EIT) can be used: it consists of injecting current on the outer boundary, and measuring the associated electric potential.

The thrust of OpenMEEG is to propose accurate forward problems, in several instances, including electro- and magneto-encephalography (EEG-MEG) and Electrical Impedance Tomography (EIT). OpenMEEG allows to compute the electric potential and magnetic fields due to boundary current injection or to sources within the compartments.

We apply this methodology to conductivity calibration in the context of high-density EEG pre-surgical epileptic exploration. High-density EEG measurements may be used to solve the inverse problem of source localization, in order to localize the foci of the epileptic activity within the brain. The solution of the inverse problem relies on a forward problem, linking sources to measurements. In turn, this forward problem is dependent on the conductivity of the head tissues.

Prior work has shown that the scalp-to-skull conductivity ratio is a sensitive parameter for source localization, because it has an influence on the depth of the estimated dipoles  [68] . There have been several studies demonstrating the feasibility of injected-current Electrical Impedance Tomography to calibrate the scalp-to-skull conductivity ratio  [57] , [49] . We are currently collaborating with our partners from La Timone in Marseille in a clinical assessment of the use of injected current Electrical Impedance Tomography to calibrate the scalp-to-skull conductivity ratio.

This work has been presented at the international conference on Electrical Impedance Tomography, see [27] and [28] .

Coupling numerical methods for the forward problem

Participants : Maureen Clerc, Emmanuel Olivi, Théodore Papadopoulo, Mariette Yvinec [Geometrica Project-Team, INRIA Sophia Antipolis Méditerranée] .

This work was partly supported by the ANR grant ViMAGINE.

Electroencephalography (EEG) and magnetoencephalography (MEG) are two modalities that aim at measuring brain activity. 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, BEM is more capable than FEM to accurately represent sources in isotropic media.

Using a domain decomposition approach, we propose to independently use BEM or FEM in different sub-domains. In the EEG forward problem considered, the BEM is limited to the domain where the sources lie (the brain) while the other tissues are handled with the FEM. This leads to an accurate description of the sources while allowing for inhomogeneous and anisotropic conductivity. The proposed method is first validated against analytical solutions in multi-sphere models. Results of the forward problem are presented for a four-layer realistic head-model incorporating a burr-hole in the skull. Convergence of the iterative algorithm is analyzed numerically. The domain decomposition framework provides a way of taking the best advantage of both methods, thus significantly improving the accuracy in the resolution of the forward EEG problem, as well as time and memory consumption.

This work is part of Emmanuel Olivi's PhD thesis [11] , and a journal article is under submission.

White matter anisotropy

Participants : Maureen Clerc, Emmanuel Olivi, Alexandre Gramfort, Théodore Papadopoulo.

This work was partly supported by the ANR grant ViMAGINE.

Conductivity of tissues in the vicinity of the sources is especially influential on the MEG and EEG forward fields. Those tissues include white matter, whose conductivity is anisotropic because of its fiber structure. While white matter anisotropy can be measured thanks to Diffusion-Weighted MRI, it is rarely incorporated in MEG and EEG head models. Boundary Element Methods can only deal with piecewise constant conductivities, therefore ruling out white matter anisotropy that has a complex structure, and Finite Element Methods have been developed to deal with anisotropic conductivity, but require very fine meshes, thus huge linear systems. We have extended the BEM framework to incorporate white matter anisotropy by treating anisotropic conductivity as a perturbation of an isotropic one. With this extension we have been able to compute the influence of a fiber within the brain white matter on the electric potential, and to validate the restults by comparison with a Finite Element Method.

This work has been published in [34] .

Inverse problems in MEG and EEG

Rational Approximations

Participants : Maureen Clerc, Théodore Papadopoulo, Juliette Leblond [APICS Project Team, INRIA Sophia Antipolis Méditerranée] , Jean-Paul Marmorat [Centre de Mathématiques Appliquées, Ecole des Mines] .

In functional neuroimaging, a crucial problem is to localize active sources within the brain non-invasively, from the knowledge of the 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. We pursue our ongoing work on 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 submitted to a journal and published as a Research Report in [38] .

Brain Computer Interfaces

New features for Motor Imagery

Participants : Maureen Clerc, Joan Fruitet, Théodore Papadopoulo, Eoin Thomas.

This work was partly supported by ANR grant CoAdapt.

Our goal is to build a training free BCI based on beta rebound detection and discrimination during the first stage of use, while the learning of the conventional sensorimotor rhythms is done. We show in this preliminary study that it is possible to use the beta rebound to discriminate, real and imagined, right versus left hand movement with either no or very little training.

This work has been published in [14] .

A bandit algorithm for exploring mental imagery

Participants : Maureen Clerc, Joan Fruitet, Alexandra Carpentier [Sequel Project Team, INRIA Lille Nord Europe] , Rémi Munos [Sequel Project Team, INRIA Lille Nord Europe] .

This work was partly supported by ANR grant Co-Adapt.

This study presents a new procedure to automatically select a discriminant motor task for an asynchronous brain-controlled button. This type of control pertains to Brain Computer Interfaces (BCI). When using sensorimotor rythms in a BCI, several motor tasks, such as moving the right or left hand, the feet or the tongue, can be considered as candidates for the control. This report presents a method to select as fast as possible the most promising task. We develop for this purpose an adaptive algorithm UCB-classif based on the stochastic bandit theory and build an EEG experiment to test our method. By not wasting time on inefficient tasks, our algorithm can focus on the most promising ones, resulting in a faster task selection and a more efficient use of the BCI training session. This leads to better classi cation rates for a xed time budget, compared to a standard task selection.

This work has been published in [39] and is currently under submission to a journal.