Cortical source imaging: from the laplacian montage to sparse inverse solutions

Cortical source imaging plays an important role for the comprehension of the functional or pathological brain. It allows to relate the activation of particular cortical areas in response to a given cognitive stimuli, hence to study the co-activations of underlying functional networks. It is also helpful in identifying the location of pathological activities. Various methods of clinical investigation can be used, from imaging modalities (PET, MRI) to electroencephalography (EEG, SEEG, MEG). The electroencephalographic (EEG) measurements have the great advantage to yield very high temporal resolution in milliseconds while being a non-invasive technique often used in primary clinical investigation. However the identification of the activated sources from EEG recordings remains an extremely difficult task due to the low spatial resolution this modality provides, to the smearing effect of the skull, to propagation model errors, as well as spatial (location and size) and temporal (synchronization) properties of the sources. In this thesis, different path to the estimation of cortical activities based on the EEG have been explored. Simplest cortical imaging methods requires only the assumptions of the geometrical properties of the head. Second order derivation of the interpolated scalp recordings removes smearing effect of the skull and yields an approximate estimation of the dura potentials. Despite small computational burden and simplified models such estimation do not provide accurate information about the individual neural generators nor their spatio-temporal properties. To overcome this, more elaborated models are used to construct realistic forward model and, thus, localize the sources by inverting it. In addition to the difficulty of forward model construction, inversion step requires regularization and/or sparsity constrains. Although dozens of methods already exists in the literature, only few are designed for the non-stationary nature of the unknown number of the sources. We reconsider the problem of cortical source imaging using as less information as possible in addition to the electrical activities provided through the EEG scalp measurements. We have avoided statistical assumptions due to the poor amount of physiological considerations they are able to integrate, and we have rather focused on methods that rely on basic geometrical and physiological considerations. Resulting full rank estimator is at the root of a large family of interpolation-based Surface Laplacian (SL) methods, based on the assumption that the scalp map is made of a linear mixing of smooth basis functions produced by the underlying sources. In the second part of the thesis, we relax the full rank constraint by adopting a dipolar distributed model and we follow the assumption that only a few cortical sources are simultaneously active. Such hypothesis is particularly valid in e.g., epileptic context or in the case of cognitive tasks, where a limited number of sources are responsible of the visible activity on the EEG electrodes. To enforce the regularization as well as the sparsity of the solution, we take benefit of the temporal dimension of the data, and propose two combined data-driven spatio-temporal dictionaries. At first the temporal atoms are learned based on a principal component analysis. Finally we exploit a time-frequency decomposition of the data based on wavelets, being more robust to noise and well adapted to the non-stationary nature of the electrophysiologic data.