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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 thesolution, and many kinds of prior information have been used to constrain it. Acombination of smoothness (L2 norm-based) and sparseness (L1 norm-based)constraints is a flexible approach that have been pursued by important examplessuch as the Elastic Net (ENET) and mixed-norm (MXN) models. The former is usedto find solutions with a small number of smooth non-zero patches, while thelatter imposes sparseness and smoothness simultaneously along differentdimensions of the spatio-temporal matrix solutions. Both models have beenaddressed within the penalized regression approach, where the regularizationparameters are selected heuristically, leading usually to non-optimalsolutions. The existing Bayesian formulation of ENET allows hyperparameterlearning, but using computationally intensive Monte Carlo/ExpectationMaximization methods. In this work we attempt to solve the EEG IP using aBayesian framework for models based on mixtures of L1/L2 norms penalizationfunctions (Laplace/Normal priors) such as ENET and MXN. We propose a SparseBayesian Learning algorithm based on combining the Empirical Bayes and theiterative coordinate descent procedures to estimate both the parameters andhyperparameters. Using simple but realistic simulations we found that ourmethods are able to recover complicated source setups more accurately and witha more robust variable selection than the ENET and LASSO solutions usingclassical algorithms. We also solve the EEG IP using data coming from a visualattention experiment, finding more interpretable neurophysiological patternswith our methods, as compared with other known methods such as LORETA, ENET andLASSO FUSION using the classical regularization approach.

Empirical bayes formulation of the elastic net and mixed-norm models: application to the eeg inverse problem