Keep it simple: a case for using classical minimum norm estimation in the analysis of EEG and MEG data

Abstract

The present study aims at finding the optimal inverse solution for the bioelectromagnetic inverse problem in the absence of reliable a priori information about the generating sources. Three approaches to tackle this problem are compared theoretically: the maximum-likelihood approach, the minimum norm approach, and the resolution optimization approach. It is shown that in all three of these frameworks, it is possible to make use of the same kind of a priori information if available, and the same solutions are obtained if the same a priori information is implemented. In particular, they all yield the minimum norm pseudoinverse (MNP) in the complete absence of such information. This indicates that the properties of the MNP, and in particular, its limitations like the inability to localize sources in depth, are not specific to this method but are fundamental limitations of the recording modalities. The minimum norm solution provides the amount of information that is actually present in the data themselves, and is therefore optimally suited to investigate the general resolution and accuracy limits of EEG and MEG measurement configurations. Furthermore, this strongly suggests that the classical minimum norm solution is a valuable method whenever no reliable a priori information about source generators is available, that is, when complex cognitive tasks are employed or when very noisy data (e.g., single-trial data) are analyzed. For that purpose, an efficient and practical implementation of this method will be suggested and illustrated with simulations using a realistic head geometry.

Introduction

In recent years, considerable effort has been spent on improving the spatial resolution of source estimation procedures in the analysis of electroencephalographic (EEG) and magnetoencephalographic (MEG) data. One of the ambitious goals in this endeavor is to make the interpretation of the results comparable to metabolic imaging methods such as positron emission tomography (PET) and functional magnetic resonance imaging (fMRI), which in turn lack the temporal resolution to track fast perceptual and cognitive processes in the human brain on the millisecond scale. Distributed inverse solutions, mostly of the linear type, are widely used to estimate the neuronal current distributions underlying the measured EEG and MEG signals, especially in studies on higher cognitive brain function Dhond et al., 2003, Haan et al., 2000, Halgren et al., 2002, Hauk et al., 2001, Pulvermüller and Shtyrov, 2004, Rinne et al., 2000. In some cases, descriptions of methods to estimate the neuronal sources from EEG and MEG signal suggest that tomography-like procedures exist, if only the right mathematical assumptions are implemented into the corresponding algorithms Ioannides et al., 1995, Pascual-Marqui et al., 2002, Singh et al., 2002. However, it is usually implicitly or explicitly acknowledged that according to the Helmholtz principle, the bioelectromagnetic inverse problem has no unique solution (von Helmholtz, 1853). Whatever result is obtained with one method, there are still infinitely many other possible solutions equally compatible with the recorded signal.

This alone implies that the question “Which method yields the correct solution to the bioelectromagnetic inverse problem?” without further specifications is futile. EEG or MEG signals alone do not carry sufficient information to determine the precise spatial distribution of the underlying neuronal sources. This problem can be compared to the reconstruction of a three-dimensional object from its shadow: Only the shape of the object along a two-dimensional plane is given by the data, making it impossible to infer anything about its 3D structure by these data alone.

Source estimation from EEG and MEG data therefore requires reformulating the question. Two general strategies can be distinguished: (1) focusing on those solution parameters that can be estimated reliably from the data alone; (2) including a priori knowledge from other sources than the data under analysis, thus reducing the amount of parameters to be estimated to a tractable number. Applied to the example of object reconstruction from a shadow, an example for strategy 1 would be to ask “What are the maximum extensions of the object in the plane parallel to the projection screen?” An example for strategy 2 would be “I know that the shadow represents the profile of a face, and it's either Gérard Depardieu or Michael Jackson. So does the profile have a big nose?” Obviously, the information we get following strategy 1 is rather limited, but might be enough for a given purpose (e.g., if we want to push the object through our front door). Following strategy 2, we get very specific information, but if we overlooked the possibility that another big-nosed person might have been in front of the screen, our conclusion could be completely wrong.

The same logic applies to source estimation procedures: Either questions must be asked that explicitly take into account the limitations of the technique, or restrictive modeling assumptions must be introduced. The problem, however, is that this can still be done in various ways, leading to a large number of procedures among which the experimenter has to choose (see Baillet et al., 2001, Dale and Halgren, 2001, Fuchs et al., 1999, Grave de Peralta Menendez et al., 1997a, Ilmoniemi, 1993, Pascual-Marqui, 1999, Vrba and Robinson, 2001 for comparisons or overviews of methods). It is therefore essential to investigate what features of an estimated current distribution are determined by the data, and how much depends on the specific method and corresponding modeling assumptions.

In this work, three approaches often employed in the literature to tackle the inverse problem will be compared: (1) the statistical approach, which aims at finding the “most likely” solution compatible with the data and possibly further constraints; (2) the minimum norm approach, which aims at finding a solution that is compatible with the data and fulfills further constraints on the amplitudes and/or covariances between source strengths; (3) the resolution optimization approach, which aims at estimating the source components as independently as possible from each other. It will be shown that all of these approaches are able to implement the same amount of a priori information, and that if this information is the same, their solutions are also the same. Most important is the case where no a priori knowledge is available at all, and the question “What is determined by the data alone?” is asked. Obviously, the amount of information present in the data cannot vary with the procedure employed to extract it, and therefore one may hope that a clear result for this case can be found. Indeed, all three approaches converge on the same method in this specific case, namely the classical minimum norm solution. This is of particular importance because even though the amount of a priori knowledge from metabolic imaging techniques or neuropsychology is continuously growing, in many experiments, the number, extension, or approximate locations of the generators are not reliably known. Obvious examples for this case are data obtained with complex cognitive paradigms, noisy data from individual subjects, with a low number of trials or on a single-trial level.

Based on these findings, an implementation for the classical minimum norm method is introduced that takes into account its limitations as well as its strengths. It is suggested for standardized routine analysis of large and complex data sets, such as in EEG and MEG studies on higher cognitive function, or for analyzing data on a single-trial level.