Review of methods for solving the EEG inverse problem

This paper reviews the class of instantaneous, 3D, discrete, linear solutions for the EEG inverse problem. Five different inverse methods are analyzed and compared: minimum norm, weighted minimum norm, Backus and Gilbert, weighted resolution optimization (WROP), and low resolution brain electromagnetic tomography (LORETA). The inverse methods are compared by testing localization errors in the estimation of single and multiple sources. These tests constitute the minimum necessary condition to be satisfied by any tomography. Of the five inverse solutions tested, only LORETA demonstrates the ability of correct localization in 3D space. The other four inverse solutions should not be used if the research aim is to localize the neuronal generators of EEG in a 3D brain. In this sense, minimum norm, weighted minimum norm, Backus and Gilbert, and WROP can be likened to x-rays, where depth information is totally lacking. For the sake of reproducible research, all the material and methods used in this part of the study, consisting of computer programs (source code and executables) and data, are available upon request to the author. In this way, all the results and conclusions can be checked, reproduced, and validated by the interested reader. In the final part of this paper, LORETA in the standard Talairach human brain is presented. This technique allows the quantitative neuroanatomical localization of neuronal electric activity. A computer program for LORETA in Talairach space is available upon request from the author. 1. Localization properties of instantaneous, 3D, discrete, linear solutions for the EEG inverse problem One of the primary concerns in electrophysiology is the non-invasive localization of the neuronal generators responsible for measured EEG phenomena. Methods for localization are termed inverse solutions. This review is limited to the class of instantaneous, 3D, discrete, linear solutions for the EEG inverse problem. In order for an inverse solution of this class to qualify as a true functional “tomography”, it must at least be capable of localizing sources with a minimum of localization error. If an inverse solution of this class is incapable of correct localization, then it has no worth as a tomography. Harsh as this criterion may seem, it is fair and objective, but most important of all, it is applicable to any proposed method. The main difficulty impeding the development of a “good” tomography for the generators of the EEG is determined by the physics nature of the problem: the measurements do not contain enough information about the generators. This gives rise to what is known as the non-uniqueness of the inverse solution. Therefore, from the outset, it can be stated that a perfect tomography can not exist. R.D. Pascual-Marqui. Review of Methods for Solving the EEG Inverse Problem. International Journal of Bioelectromagnetism 1999, Volume 1, Number 1, pp:75-86. Printed Issue ISSN 1457-7857, Internet Issue ISSN 14567865 (http://www.tut.fi/ijbem). Author’s version. Page 2 of 13 Despite this obstacle, the search for better tomographies goes on, as witnessed by the number of papers being published in this field (see, e.g., Koles (1998) for a recent review). From a more optimistic point of view, one might expect that whatever little information is contained in EEG measurements, it should suffice to allow for the existence of at least an “approximate” tomography. Such a tomography should be capable of recovering the “true” generators with an acceptable low level of distortion (i.e., of error). Historically, the first tomography published in this field was the minimum norm inverse solution of Hamalainen and Ilmoniemi (1984). The properties of this method for 2D solution spaces (i.e., sources restricted to a plane or to a spherical surface running parallel to the measurement surface) were promising. Two-dimensional images of estimated current density corresponding to ideal point sources were recovered with blurring, but with correct localization of activity maxima. However, this method is incapable of correct localization in 3D solution spaces, as was shown in PascualMarqui (1995). The greatest challenge in the development of EEG source localization tomographies is to extend the good localization properties of the 2D minimum norm solution to 3D solution spaces. This was achieved with LORETA (low resolution brain electromagnetic tomography) (Pascual-Marqui et al., 1994; Pascual-Marqui, 1995). All the properties of a tomography, including its quality in terms of localization capability, can be completely characterized by means of the model resolution matrix (Menke, 1984; Backus and Gilbert, 1968). This approach was used by Pascual-Marqui (1995) to compare three tomographies (inverse solutions) in terms of their localization errors. The first part of this paper contains a brief review of the theory of instantaneous, 3D, discrete, linear solutions for the EEG inverse problem. A methodology is presented for the fair, objective, and rigorous comparison of EEG-based tomographies. The main results presented here correspond to a comparison of five different tomographies taken from the published literature. Some important aspects of inverse solutions not included in this study, such as the effect of noisy measurements and the effect of the reference electrode for EEG measurements, were considered in detail elsewhere (Pascual-Marqui, 1995). Other methods of source localization, such as single or multiple dipole fitting are not the object of this review. 1.1 Material and methods