Keep it simple: a case for using classical minimum norm estimation in the analysis of EEG and MEG data
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.
Section snippets
General
In the following, bold capital letters (like G) represent matrices, and bold small letters (like s) refer to column vectors. Gi. and G.i represent the ith row and column of the matrix G, respectively. The superscript “T” denotes the transposition of a vector or a matrix (e.g., GT or sT).
The relationship between a given source distribution inside the head and the electric potential or magnetic field measured at discrete points on or above the scalp surface is linear Geselowitz, 1967, Sarvas, 1987
Null-space approach
In the case where nothing is known about the source distribution to be estimated, one might want to determine that part of the source that is solely determined by the data. Formally, one would like to find a solution ŝ for whichthat is, which produces the recorded signal, but which also cannot be separated into parts ŝ1 and ŝ2 such that
In other words, ŝ shall not contain any nonvanishing part that taken by itself would not produce any measurable signal at all
Motivation
The classical minimum norm solution is obtained by all of the abovementioned approaches in the case of nonexisting a priori knowledge about the source to be estimated. It would therefore be applicable in many situations where minimum modeling assumptions are required, such as localization of complex cognition-related brain activity, analysis of continuous EEG and MEG, or analysis on a single-trial level. In the following, an implementation of the minimum norm method will be described that
Theory
The theoretical part of this paper compared three main strategies to tackle the bioelectromagnetic inverse problem: the statistical, the minimum norm, and the resolution optimization approach. The main results are:
- (1)
All three approaches can incorporate the same constraints on the source distribution.
- (2)
All three approaches allow the handling of noisy data, and can take into account different noise levels at different sensors or the covariances between them.
- (3)
In the case where there is no a priori
Acknowledgements
I would like to thank Rhodri Cusack for valuable comments on an earlier version of this manuscript.
References (53)
- et al.
Multimodal integration of high-resolution EEG and functional magnetic resonance imaging data: a simulation study
NeuroImage
(2003) - et al.
Spatiotemporal mapping of brain activity by integration of multiple imaging modalities
Curr. Opin. Neurobiol.
(2001) - et al.
Dynamic statistical parametric mapping: combining fMRI and MEG for high-resolution imaging of cortical activity
Neuron
(2000) - et al.
Spatiotemporal maps of past-tense verb inflection
NeuroImage
(2003) On bioelectric potentials in an inhomogeneous volume conductor
Biophys. J.
(1967)- et al.
N400-like magnetoencephalography responses modulated by semantic context, word frequency, and lexical class in sentences
NeuroImage
(2002) - et al.
Comparison of data transformation procedures to enhance topographical accuracy in time-series analysis of the human EEG
J. Neurosci. Methods
(2002) - et al.
Magnetic field tomography of cortical and deep processes: examples of “real-time mapping” of averaged and single trial MEG signals
Int. J. Psychophysiol.
(1995) - et al.
Localization of primary auditory cortex in humans by magnetoencephalography
NeuroImage
(2003) - et al.
Anatomically informed basis functions for EEG source localization: combining functional and anatomical constraints
NeuroImage
(2002)
Separate time behaviors of the temporal and frontal mismatch negativity sources
NeuroImage
Task-related changes in cortical synchronization are spatially coincident with the hemodynamic response
NeuroImage
Visualization of magnetoencephalographic data using minimum current estimates
NeuroImage
Signal processing in magnetoencephalography
Methods
Smooth reconstruction of cortical sources from EEG or MEG recordings
NeuroImage
The resolving power of gross earth data
Geophys. J. R. Astron. Soc.
Uniqueness in the inversion of inaccurate gross earth data
Philos. Trans. R. Soc. Lond., A
Electromagnetic brain mapping
IEEE Signal Process. Mag.
Statistical flattening of MEG beamformer images
Hum. Brain Mapp.
Linear inverse problems with discrete data: I. General formulation and singular system analysis
Inverse Problems
Linear inverse problems with discrete data: II. Stability and regularisation
Inverse Problems
High-resolution frequency–wavenumber spectrum analysis
Proc. I.E.E.E.
Probabilistic methods in a biomagnetic inverse problem
Inverse Problems
Inverse Problems in Astronomy: A Guide to Inversion Strategies for Remotely Sensed Data
Improved localization of cortical activity by combining EEG and MEG with MRI cortical surface reconstruction: a linear approach
J. Cogn. Neurosci.
Source analysis of event-related cortical activity during visuo-spatial attention
Cereb. Cortex
Cited by (245)
Individual deviations from normative electroencephalographic connectivity predict antidepressant response
2024, Journal of Affective DisordersEEG cortical network reveals the temporo-spatial mechanism of visual search
2023, Brain Research BulletinDevelopmental aspects of fear generalization – A MEG study on neurocognitive correlates in adolescents versus adults
2022, Developmental Cognitive Neuroscience