MEG source localization under multiple constraints: An extended Bayesian framework

Abstract

To use Electroencephalography (EEG) and Magnetoencephalography (MEG) as functional brain 3D imaging techniques, identifiable distributed source models are required. The reconstruction of EEG/MEG sources rests on inverting these models and is ill-posed because the solution does not depend continuously on the data and there is no unique solution in the absence of prior information or constraints. We have described a general framework that can account for several priors in a common inverse solution. An empirical Bayesian framework based on hierarchical linear models was proposed for the analysis of functional neuroimaging data [Friston, K., Penny, W., Phillips, C., Kiebel, S., Hinton, G., Ashburner, J., 2002. Classical and Bayesian inference in neuroimaging: theory. NeuroImage 16, 465–483] and was evaluated recently in the context of EEG [Phillips, C., Mattout, J., Rugg, M.D., Maquet, P., Friston, K., 2005. An empirical Bayesian solution to the source reconstruction problem in EEG. NeuroImage 24, 997–1011]. The approach consists of estimating the expected source distribution and its conditional variance that is constrained by an empirically determined mixture of prior variance components. Estimation uses Expectation-Maximization (EM) to give the Restricted Maximum Likelihood (ReML) estimate of the variance components (in terms of hyperparameters) and the Maximum A Posteriori (MAP) estimate of the source parameters. In this paper, we extend the framework to compare different combinations of priors, using a second level of inference based on Bayesian model selection. Using Monte-Carlo simulations, ReML is first compared to a classic Weighted Minimum Norm (WMN) solution under a single constraint. Then, the ReML estimates are evaluated using various combinations of priors. Both standard criterion and ROC-based measures were used to assess localization and detection performance. The empirical Bayes approach proved useful as: (1) ReML was significantly better than WMN for single priors; (2) valid location priors improved ReML source localization; (3) invalid location priors did not significantly impair performance. Finally, we show how model selection, using the log-evidence, can be used to select the best combination of priors. This enables a global strategy for multiple prior-based regularization of the MEG/EEG source reconstruction.

Introduction

Magnetoencephalography (MEG) and Electroencephalography (EEG) both provide a non-invasive and instantaneous measure of whole brain activity. These measures reflect synchronous post-synaptic potentials of cortical populations of neurons (Nunez and Silberstein, 2000). Unfortunately, localizing those electromagnetic sources is an ill-posed inverse problem that, in the absence of constraints, does not admit a unique solution (Baillet et al., 2001). Consequently, deriving a realistic and unique solution rests on prior knowledge, in addition to the observed measurements.

Any source reconstruction approach is characterized by three components. The first relates to the definition of the solution space and a parametric representation of the sources. The second embodies the information about the physical and geometrical properties of the head. The latter is needed for modeling the propagation of the sources electromagnetic field through various tissues. Together, these two components constitute a generative or forward model of the MEG/EEG data that can also be used for data simulation (see Synthetic MEG data). Finally, given a forward model, the third component is an inverse operator which, according to some criterion, defines a unique source distribution. For instance, when based on a probabilistic approach (Baillet and Garnero, 1997, Schmidt et al., 1999, Phillips et al., 2002, Amblard et al., 2004), the unique inverse solution corresponds to the most likely solution according to a predefined criterion formulated in terms of the source probability distribution.

Two types of inverse method can be distinguished by their respective source models: the equivalent current dipole (ECD) and the distributed model (DM). Although other source models have been used, such as multipoles (Jerbi et al., 2004) or continuous current densities (Riera et al., 1998), both approaches usually rely upon a dipolar representation of cortical sources, which are parameterized in terms of location, orientation and intensity. An ECD models the activity of a large cortical area. MEG or EEG data are then explained by few ECDs (usually less than five). Distributed models consider a large number (typically ∼10,000) of dipoles distributed at fixed locations over the cortical surface. Although the underlying parametric models are the same, the parametrization of the solution space is very different, calling for different forward calculations as well as different inverse operators and solutions.

ECDs are fitted using iterative algorithms that estimate the source parameters in order to explain the data as accurately as possible. In the iterative process, the source parameters are modified to minimize the residual error (Scherg and von Cramon, 1986, Koles, 1998). The solution is very sensitive to the number of sources and initial parameters (dipole locations and orientations), which need to be specified a priori. Indeed, ECD models require non-linear optimization with the possibility of local minima. Moreover, determining the optimal number of ECD (model complexity) is a non-trivial issue (Waldorp et al., 2005); some simulation studies have shown that, even with the right number of sources, ECD approaches are less reliable than distributed ones, when dealing with more than one source (Yao and Dewald, 2005). Finally, unlike distributed methods, ECD models do not address the anatomical deployment of an active region.

In contradistinction to ECD approaches, a DM uses the subject's anatomy derived from high resolution anatomical Magnetic Resonance Images (MRI) (Dale and Sereno, 1993). The solution space and associated forward models can then be made as realistic as allowed by computational constraints and the precision of head tissue conductivity measures. Moreover, due to the use of fixed dipoles, the forward solution only needs to be computed once, prior to any inverse operation. The DM represents a highly under-determined but linear system (see Notation). This (general) linear model, although under-determined, is formally similar to those encountered in signal and image processing and can be treated in a Bayesian way, using priors to furnish a unique solution.

In this paper, we focus on distributed source models and explore the usefulness of Bayesian model selection for determining the best combination of constraints on the inverse solution. To establish the face validity of the ensuing model selection, we also evaluated performance using conventional criteria based on detection and localization error. To assess localization error, we used simulations with quite focal sources. It is possible that ECD models would have been better than the distributed models for these focal responses. In principle, one could use Bayesian model selection to disambiguate between distributed source and ECD models for the same data. Furthermore, the application of Bayesian model selection to ECD models provides a principled way of finding the optimum number of ECDs. We are currently exploring this in the context of dynamic causal models for ERPs. In this paper, we introduce model selection and illustrate it in the context of selecting constraints (as opposed to sources).

In the context of distributed approaches, priors based on mathematical, anatomical, physiological and functional heuristics have been considered (Hämäläinen and Llmoniemi, 1994, Pascual-Marqui et al., 1994, Gorodnitsky et al., 1995, Baillet and Garnero, 1997, Dale et al., 2000, Phillips et al., 2002, Mattout et al., 2003, Babiloni et al., 2004). Although these approaches involve different constraints and inverse criteria, they all obtain a unique solution by optimizing a goodness of fit term and a prior term in a carefully balanced way. Most can be framed in terms of a Weighted Minimum Norm criterion (WMN), which represents the classical and most popular distributed approach (see Classical regularization: single prior) (Hauk, 2004).

However, a critical outstanding issue lies in the relative weighting of the accuracy and regularization criteria upon which the solution depends. Usually, in the context of Tikhonov regularization or WMN solutions, this weighting is fixed arbitrarily, or by using the L-curve heuristic (see Classical regularization: single prior). The latter case, which we will refer to as the (classical) WMN, is limited because it can only accommodate a single constraint on the source parameters. This means that multiple constraints (e.g., spatial and temporal; Baillet and Garnero, 1997) have to be mixed into a single prior term, using ad hoc criteria.

In this paper, we generalize the WMN approach, using a hierarchical (general) linear model that embraces, under the assumption of Gaussian errors, multiple constraints specified in terms of variance components (see Empirical Bayes: multiple priors). These priors can be formulated in sensor or source space. The optimal weight associated with each constraint is estimated from the data following an empirical Bayesian approach and is computed iteratively using Expectation Maximization (EM) (Friston et al., 2002). These weights are equivalent to restricted maximum likelihood (ReML) estimates of the prior covariance components.

In a companion paper (Phillips et al., 2005), we addressed the face validity of empirical Bayes in this context. In the present paper, the proposed framework is applied to simulated event related field (ERF) data with realistic noise. Our investigation focused on the comparison between the ReML approach and the classical WMN and on the comparison between single vs. multiple priors when solving the MEG inverse problem. The main contribution of this paper is the introduction of a second level of inference using Bayesian model selection. Because each model is defined by its prior covariance component, we can compare different combinations of priors in a principled way. We will illustrate this by showing that model selection can identify invalid priors and point to the optimum number of valid or useful priors.

The paper is organized as follows. In the Method section, we review the classical weighted minimum norm approach and present the ReML scheme that enables a principled and unique incorporation of multiple priors. In the Application section, we describe the simulations we have used to compare quantitatively the classical and ReML inverse approaches. The priors have been chosen to emphasize the role of ReML in the context of multimodal integration. In addition to the conventional localization error criterion, two complementary evaluation procedures are introduced. The first refers to the notion of detection power and is based upon Receiver Operating Characteristic (ROC) curve analysis. The second is based on Bayesian model selection and the evidence for different models with different prior covariance components. The results are presented in the final section and commented in the Discussion.