An empirical Bayesian solution to the source reconstruction problem in EEG

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 br...

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Vydané v:	NeuroImage (Orlando, Fla.) Ročník 24; číslo 4; s. 997 - 1011
Hlavní autori:	Phillips, Christophe, Mattout, Jeremie, Rugg, Michael D., Maquet, Pierre, Friston, Karl J.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	United States Elsevier Inc 15.02.2005 Elsevier Limited Academic Press Inc Elsevier Science
Predmet:	Algorithms Bayes Theorem Bayesian analysis Distributed solution EEG Electroencephalography - statistics & numerical data Evoked Potentials, Somatosensory - physiology Expectation-maximisation (EM) procedure Humans Image Processing, Computer-Assisted - statistics & numerical data Inverse problems Likelihood Functions Magnetic Resonance Imaging Neurosciences & behavior Neurosciences & comportement Restricted maximum likelihood (ReML) solution Sciences sociales & comportementales, psychologie Social & behavioral sciences, psychology Source reconstruction Standard deviation Source reconstruction Restricted maximum likelihood (ReML) solution Expectation-maximisation (EM) procedure EEG Distributed solution
ISSN:	1053-8119, 1095-9572, 1095-9572
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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 “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.
AbstractList	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. 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.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. 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.
Author	Maquet, Pierre Phillips, Christophe Rugg, Michael D. Mattout, Jeremie Friston, Karl J.
Author_xml	– sequence: 1 givenname: Christophe surname: Phillips fullname: Phillips, Christophe email: c.phillips@ulg.ac.be organization: Centre de Recherches du Cyclotron, B30, Université de Liège, Liège 4000, Belgium – sequence: 2 givenname: Jeremie surname: Mattout fullname: Mattout, Jeremie organization: Wellcome Department of Imaging Neuroscience, Institute of Neurology, University College London, London, UK – sequence: 3 givenname: Michael D. surname: Rugg fullname: Rugg, Michael D. organization: Center for the Neurobiology of Learning and Memory, University of California, Irvine, CA 92717, USA – sequence: 4 givenname: Pierre surname: Maquet fullname: Maquet, Pierre organization: Centre de Recherches du Cyclotron, B30, Université de Liège, Liège 4000, Belgium – sequence: 5 givenname: Karl J. surname: Friston fullname: Friston, Karl J. organization: Wellcome Department of Imaging Neuroscience, Institute of Neurology, University College London, London, UK
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Snippet	Distributed linear solutions of the EEG source localisation problem are used routinely. In contrast to discrete dipole equivalent models, distributed linear...
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SubjectTerms	Algorithms Bayes Theorem Bayesian analysis Distributed solution EEG Electroencephalography - statistics & numerical data Evoked Potentials, Somatosensory - physiology Expectation-maximisation (EM) procedure Humans Image Processing, Computer-Assisted - statistics & numerical data Inverse problems Likelihood Functions Magnetic Resonance Imaging Neurosciences & behavior Neurosciences & comportement Restricted maximum likelihood (ReML) solution Sciences sociales & comportementales, psychologie Social & behavioral sciences, psychology Source reconstruction Standard deviation
Title	An empirical Bayesian solution to the source reconstruction problem in EEG
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