Statistical Approaches to the Inverse Problem
Magnetoencephalography (MEG) can be considered as one of the most powerful instruments for non invasive investigation of the brain functions (Hamalainen et al. 1993) Its good temporal resolution (sampling frequency can reach several thousands of Hertz) only comparable to the sampling frequency of electroencephalography (EEG) allows following the neural dynamics on amillisecond time scale. This implies that, with respect to other functional imaging modalities, MEG provides an enormous amount of information about the brain activity. On the other hand, localization of brain activity from MEG data requires to solve a highly ill posed inverse problem (Sarvas, 1987) exact reconstruction of the neural current is not possible even with noise free data, and the debate about the actual spatial resolution of MEG is still open. Despite this drawback, the appeal of MEG data is such that a large number of methods have been developed for localizing brain activity from MEG recordings. Starting from the middle Eighties, when the first MinimumNorm Estimate (Hamalainen Ilmoniemi, 1984) appeared, to the recent developments in spatio temporal regularization (Ou et al. 2009) state space models (Long et al. 2006; Sorrentino et al. 2009) and Markov Chain Monte Carlo (Jun et al. 2005) the attempts to insert enough a priori information to actually provide stable and reliable solutions have been rising rather than diminishing. The goal and the scope are clear: automatic and reliable source localization can be of practical use in several fields, possibly including clinical applications (pre surgical evaluation, foci localization in epilepsy) new technology development (e.g. brain computer interfaces) and scientific research (e.g. for analysis of large datasets) Despite having been presented in rather different frameworks and with largely different names and techniques, most methods appeared so far in the MEG inverse literature can be described rather straightforwardly in a statistical setting, which helps unifying and clarifying the assumptions and the limitations of the methods. In this chapter, we attempt a brief review of the most largely known methods, as well as of some of the most recently appeared ones, highlighting the statistical features. We first provide a brief description of the statistical approach to inverse problems in Section 2; then we describe the main features of the MEG inverse problem in Section 3; in Section 4 and 5 we review methods developed by other authors, while in Section 6 we discuss our work on Bayesian filtering for MEG. 5