The study of brain mechanisms that underlie cognition (perception, motivity, language and memory) is a research field commonly referred to as cognitive neuroscience. In this domain, an important issue for further understanding of the brain is to establish the connection between function and localization, i.e. to find what parts of the brain are involved in the processing of a particular task. The first results of localization come from neuropsychology, by studying behavioral changes due to a well localized cerebral lesion, and electro-physiology technologies, that consist in measuring neurons electrical activity by implanting electrodes in the brain (mostly on animals). Nowadays, advances in the brain function localization are strongly linked to the use of several neuroimaging techniques that allow to study the human brain in an almost noninvasive way, and so to perform experiments on many subjects, even healthy .

Structural neuroimaging technologies appeared at the beginning of the seventies with x-ray Computed Tomography (CT), then developed with the arrival of MRI in the eighties. These methods generate a contrasted 3D image of the brain anatomy, and in particular allow to identify the localization and the extension of a cerebral lesion. Within the framework of cognitive neuroscience research, structural imaging brings elements to interpret behavioral observations in neuropsychology. By determining to which lesions corresponds a given cognitive deficit, it is possible to establish that the damaged cerebral region is involved in the underlying mechanism.

Functional neuroimaging goes beyond the simple anatomic image and aims at characterizing the brain in action. Its classical use in cognitive neuroscience consists in making a subject execute a task and in measuring the cerebral activity correlated to this task. Depending on the functional imaging technology, it is possible to find more or less precisely which region of the brain was activated and at which moment. We can distinguish two groups among these techniques :

• Metabolic methods
These methods measure a change in metabolism linked to cerebral activity. The most known is functional Magnetic Resonance Imaging (fMRI), that allows to measure the blood oxygenation level in the brain. In regions that consume energy, oxygenated blood flow increases and so fMRI allows to localize regions where a high cerebral activity is taking place. The fMRI modality generates a precise 3d image, but the acquisition time is of about one second : a temporal series of fMR images do not have a sufficient temporal resolution to follow in detail the dynamics of cognitive processes (duration of about 10 ms). We can also cite the Positron Emission Tomography (PET) that measures blood flow changes by injection of a radioactive label. This technique is less and less used in cognitive neuroscience because it has a temporal resolution even more limited than fMRI.

• Direct measures of the electrical activity
The most known technique, electroencephalography (EEG), is in fact the very first noninvasive neuroimaging method. Its realization by neurologist Hans Berger dates back to 1929. Its principle consists in measuring the electric potential at the surface of the patient’s head, thanks to a helmet with electrodes. Apart from all other artificial electrical sources, the electric potential measured is supposed to be only generated by brain electrical activity. Its advantage compared to metabolic methods is that the temporal resolution is only limited by the measurement electronics, and typically the signal is measured with a sampling frequency of 1 kHz, much higher than the dynamics of cerebral activity. Magnetoencephalography (MEG) is a technique very similar to EEG, but it measures above the head surface the magnetic field instead of the electric potential. Its interest lies in the fact that, contrary to the electric potential, the magnetic field is not very distorted by its propagation through organic tissues (notably the skull).

Hence, thanks to their high temporal resolution, EEG and MEG theoretically offer the possibility to follow in detail the dynamics of cerebral activity processes. A characteristic of these measures yet limit their imaging capacity : they are surface measures that, as such, only give a crude information on the localization of electrical activity that generated the measured electromagnetic field. To be able to estimate the position of electromagnetic signal sources, one needs to have recourse to mathematical methods of signal analysis. This problem of source localization falls in a category of mathematical problems designated as inverse problems, that generally have the characteristic of being ill-posed : the number of measurements is insufficient to perfectly determine the positions of signal sources. Before solving this inverse problem, an initial modeling of the studied phenomenon is required, called forward problem. In the case of EEG and MEG, the studied phenomenon is the electromagnetic field propagation from a source in the brain to the sensors located at the head surface. The quality of the models of this phenomenon has a direct effect on the accuracy of the source localization methods. It is hence necessary to build head models that correspond as well as possible to the real electromagnetic field propagation in the organic tissues of the head.

• The electromagnetic field propagation is described by partial differential equations (PDE). In complex media with complicated geometries, such as realistic head models built from structural MRI of the subjects, solving these PDEs requires to use numerical methods which are generally computationally expensive, both in memory and in time. It hence limits the use of complicated geometries that describe precisely the head geometry. Most experimenters in EEG and MEG still use spherical descriptions of the head, for which quasianalytical solutions were developed.

• The electrical conductivities of the organic tissues of the head are not well known, nevertheless these properties affect the electromagnetic field propagation (especially the electric potential). For most head models, the conductivity values come from in vitro measurements, which does not account for the reality of tissues in their natural environment. For the head models to correspond as well as possible to reality, methods are developed for in vivo conductivity estimation.