Multi-condition M/EEG inverse modeling with sparsity assumptions: how to estimate what is common and what is specific in multiple experimental conditions

M/EEG inverse modeling with distributed dipolar source models and penalizations with sparsity inducing norms (e.g. `1 with MCE [1], `0 with FOCUSS [2], `2-`1 [3]) offer a way to select a set of active dipoles. Indeed, sparsity inducing norms lead to solutions where most of the sources are set to zero and the remaining non zero sources form the set of estimated active dipoles. When running cognitive studies multiple experimental conditions are usually involved and cognitive hypothesis classically consist in quantifying the difference between these conditions. The problem is that when a sparse inverse solver is used independently for each experimental condition, it happens that the selection of dipolar sources is not consistent across conditions, thus limiting further analysis. Even if all conditions share a common dipolar source, due to noise, it can happen that such solvers do not select exactly the same dipole but two neighboring ones. To circumvent this limitation, we propose in this contribution to run the inverse computation with all the experimental conditions simultaneously. We use a penalization that achieves a joint selection of active dipoles while estimating two parts in the reconstructed current distributions: a part that is common to all the different conditions and a part that is specific to each condition. The penalization used in the inverse problem is based on groups of `2-`1 norms. The optimization is achieved with iterative least squares (iterative `2 Minimum Norm) making the solver tractable on large datasets. The method is illustrated on toy data and validated on synthetic MEG data reproducing activations appearing for somesthesic finger stimulations. We call our solver SMC (Sparse Multi-Condition). Keywords— MEG, EEG, Inverse Problem, Sparse prior, IRLS