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An empirical Bayesian solution to the source reconstruction problem in EEG Phillips, Christophe ; ; et al in Neuroimage (2005), 24(4), 997-1011 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 ... [more ▼] 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 "Weighted Minimum Norm" (WMN) solution in terms of hierarchical linear models. An "Expectation-Maximisation" (EM) algorithm is used to obtain a "Restricted Maximum Likelihood" (ReML) estimate of the hyperparameters, before estimating the "Maximum a Posteriori" solution itself. This procedure can be considered a generalisation of previous work that encompasses multiple constraints. Our approach was compared with the "classic" 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. (C) 2004 Elsevier Inc. All rights reserved. [less ▲] Detailed reference viewed: 84 (3 ULg)Systematic regularization of linear inverse solutions of the EEG source localization problem Phillips, Christophe ; ; in NeuroImage (2002), 17(1), 287-301 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 ... [more ▼] 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 "expectation-maximization" 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: M 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. (C) 2002 Elsevier Science (USA). [less ▲] Detailed reference viewed: 29 (6 ULg)Anatomically informed basis functions for EEG source localization : Combining functional and anatomical constraints Phillips, Christophe ; ; in Neuroimage (2002), 16(3), 678-6951 Distributed linear solutions have frequently been used to solve the source localization problem in EEG. Here we introduce an approach based on the weighted minimum norm (WMN) method that imposes ... [more ▼] Distributed linear solutions have frequently been used to solve the source localization problem in EEG. Here we introduce an approach based on the weighted minimum norm (WMN) method that imposes constraints using anatomical and physiological information derived from other imaging modalities. The anatomical constraints are used to reduce the solution space a priori by modeling the spatial source distribution with a set of basis functions. These spatial basis functions are chosen in a principled way using information theory. The reduced problem is then solved with a classical WMN method. Further (functional) constraints can be introduced in the weighting of the solution using fMRI brain responses to augment spatial priors. We used simulated data to explore the behavior of the approach over a range of the model's hyperparameters. To assess the construct validity of our method we compared it with two established approaches to the source localization problem, a simple weighted minimum norm and a maximum smoothness (Loreta-like) solution. This involved simulations, using single and multiple sources that were analyzed under different levels of confidence in the priors. (C) 2002 Elsevier Science (USA). [less ▲] Detailed reference viewed: 40 (1 ULg) |
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