  The estimation of EEG generating sources constitutes an Inverse Problem (IP)
in Neuroscience. This is an ill-posed problem, due to the non-uniqueness of the
solution, and many kinds of prior information have been used to constrain it. A
combination of smoothness (L2 norm-based) and sparseness (L1 norm-based)
constraints is a flexible approach that have been pursued by important examples
such as the Elastic Net (ENET) and mixed-norm (MXN) models. The former is used
to find solutions with a small number of smooth non-zero patches, while the
latter imposes sparseness and smoothness simultaneously along different
dimensions of the spatio-temporal matrix solutions. Both models have been
addressed within the penalized regression approach, where the regularization
parameters are selected heuristically, leading usually to non-optimal
solutions. The existing Bayesian formulation of ENET allows hyperparameter
learning, but using computationally intensive Monte Carlo/Expectation
Maximization methods. In this work we attempt to solve the EEG IP using a
Bayesian framework for models based on mixtures of L1/L2 norms penalization
functions (Laplace/Normal priors) such as ENET and MXN. We propose a Sparse
Bayesian Learning algorithm based on combining the Empirical Bayes and the
iterative coordinate descent procedures to estimate both the parameters and
hyperparameters. Using simple but realistic simulations we found that our
methods are able to recover complicated source setups more accurately and with
a more robust variable selection than the ENET and LASSO solutions using
classical algorithms. We also solve the EEG IP using data coming from a visual
attention experiment, finding more interpretable neurophysiological patterns
with our methods, as compared with other known methods such as LORETA, ENET and
LASSO FUSION using the classical regularization approach.
