An empirical Bayesian solution to the source reconstruction problem in EEG

Abstract

Distributed linear solutions of the EEG source localisation problem are used routinely. In contrast to discrete dipole equivalent models, distributed linear solutions do not assume a fixed number of active sources and rest on a discretised fully 3D representation of the electrical activity of the brain. The ensuing inverse problem is underdetermined and constraints or priors are required to ensure the uniqueness of the solution. In a Bayesian framework, the conditional expectation of the source distribution, given the data, is attained by carefully balancing the minimisation of the residuals induced by noise and the improbability of the estimates as determined by their priors. This balance is specified by hyperparameters that control the relative importance of fitting and conforming to various constraints. Here we formulate the conventional bWeighted Minimum NormQ (WMN) solution in terms of hierarchical linear models. An bExpectation-MaximisationQ (EM) algorithm is used to obtain a bRestricted Maximum LikelihoodQ (ReML) estimate of the hyperparameters, before estimating the bMaximum a PosterioriQ solution itself. This procedure can be considered a generalisation of previous work that encompasses multiple constraints. Our approach was compared with the bclassicQ WMN and Maximum Smoothness solutions, using a simplified 2D source model with synthetic noisy data. The ReML solution was assessed with four types of source location priors: no priors, accurate priors, inaccurate priors, and both accurate and inaccurate priors. The ReML approach proved useful as: (1) The regularisation (or influence of the a priori source covariance) increased as the noise level increased. (2) The localisation error (LE) was negligible when accurate location priors were used. (3) When accurate and inaccurate location priors were used simultaneously, the solution was not influenced by the inaccurate priors. The ReML solution was then applied to real somatosensory-evoked responses to illustrate the application in an empirical setting. D

Variational free energy and the Laplace approximation

“…This is a common situation in underdetermined problems. An important example is source reconstruction in electroencephalography, where the number of sources is much greater than the number of measurement channels (see Phillips et al, 2005, for an application that uses spm _ ReML.m in this context). In these cases one can form conditional estimates of the parameters using the matrix inversion lemma and again avoid inverting large (p × p) matrices.…”

Automatic fMRI‐guided MEG multidipole localization for visual responses

“…Often, because of difficulties in the true multimodal integration of MEG/ EEG and fMRI, the suggested solutions have been for instance direct comparison of the separate results [e.g., Ahlfors et al, 1999], using fMRI data as a basis to adjust the source variance parameters [Dale et al, 2000;Liu et al, 1998], utilization of the functional results for constraining the possible source positions [e.g., Korvenoja et al, 1999], or by directly seeding the fMRI locations and cortical orientations to be optimized with a suitable EEG source dipole model [Vanni et al, 2004a,b]. Lately, there has been great interest in Bayesian methods utilizing fMRI prior information, for instance, with distributed linear solutions of the MEG/EEG inverse problem [e.g., Dale et al, 2000;Phillips et al, 2005] and also on determining the relevance of the fMRI prior information included in the inverse solution [Daunizeau et al, 2005].…”

Variational Bayesian inversion of the equivalent current dipole model in EEG/MEG

“…The inversion of this model amounts to a nonlinear optimization problem, because the forward model is nonlinear in dipole location (Mosher et al, 1992). Recently, the source reconstruction problem has been addressed by placing many dipoles in brain space, and using constraints on the solution to make it unique; for example (Baillet and Garnero, 1997;Mattout et al, 2006;Phillips et al, 2005). This approach is attractive, because it produces images of brain activity comparable to other imaging modalities and it eschews subjective constraints on the inversion.…”