Systematic Regularization of Linear Inverse Solutions of the EEG Source Localization Problem

Abstract

Distributed linear solutions of the EEG source localization problem are used routinely. Here we describe an approach based on the weighted minimum norm method that imposes constraints using anatomical and physiological information derived from other imaging modalities to regularize the solution. In this approach the hyperparameters controlling the degree of regularization are estimated using restricted maximum likelihood (ReML). EEG data are always contaminated by noise, e.g., exogenous noise and background brain activity. The conditional expectation of the source distribution, given the data, is attained by carefully balancing the minimization 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 prior constraints. Here we introduce a systematic approach to this regularization problem, in the context of a linear observation model we have described previously. In this model, basis functions are extracted to reduce the solution space a priori in the spatial and temporal domains. The basis sets are motivated by knowledge of the evoked EEG response and information theory. In this paper we focus on an iterative "expectationmaximization" procedure to jointly estimate the conditional expectation of the source distribution and the ReML hyperparameters on which this solution rests. We used simulated data mixed with real EEG noise to explore the behavior of the approach with various source locations, priors, and noise levels. The results enabled us to conclude: (i) Solutions in the space of informed basis functions have a high face and construct validity, in relation to conventional analyses. (ii) The hyperparameters controlling the degree of regularization vary largely with source geometry and noise. The second conclusion speaks to the usefulness of using adaptative ReML hyperparameter estimates. © 2002 Elsevier Science (USA)

A mesostate-space model for EEG and MEG

“…PEB requires a hierarchical observation model where the parameters and hyperparameters at any particular level can be treated as priors on the level below. In the context of the EEG inverse problem, this modelling strategy has already been employed successfully in Phillips et al, 2002) for constraining the spatial deployment of distributed sources. These models parameterize the prior covariance of dipole dynamics with a linear mixture of a priori covariance components.…”

An empirical Bayesian solution to the source reconstruction problem in EEG

“…The major problem here is the introduction of multiple constraints and their appropriate weighting, while accounting for observation noise (Gonzalez Andino et al, 2001). In Phillips et al (2002b), we introduced a simple bRestricted Maximum LikelihoodQ (ReML) procedure to estimate a single hyperparameter, i.e., balance between fitting the data and conforming to the priors. Here we reformulate the WMN solution in terms of a hierarchical linear model.…”

Recursive penalized least squares solution for dynamical inverse problems of EEG generation

“…Inverse problems arising in the analysis of data obtained by Electrical Impedance Tomography (EIT) and Single Photon Emission Tomography (SPET) have been formulated as state estimation problems (Karjalainen et al 1997;Kaipio et al 1999;Vauhkonen et al 2001) , and the use of Kalman filtering and Kalman smoothing has been suggested for the purpose of obtaining estimates of the state. Phillips et al (2002) have suggested to introduce a temporal constraint into the EEG/MEG inverse problem by employing a time window and Gaussian kernels.…”