The definite non-uniqueness results for deterministic EEG and MEG data

Abstract

The solvability of the inverse problems of electroencephalography and magnetoencephalography has been studied extensively in the literature using a variety of models, including spherical and non-spherical geometries, homogeneous and inhomogeneous head models, and neuronal excitations involving the discrete and continuous distribution of dipoles. Among the important methods used are the methods based on spectral decompositions, physical arguments and integral representation techniques. Regarding the uniqueness of these inverse problems, a general result, independent of the geometry and the homogeneity of the conducting medium, has been obtained recently by the second author. This paper summarizes this result which appears to be mathematically definitive, in the sense that the geometry is arbitrary, the brain is surrounded by shells of varying conductivities, the neuronal current is arbitrary, the data are complete and the proofs are analytical. Furthermore, this paper includes a summary of the main steps of the proofs leading to the above result. In addition, it demonstrates the consistency of this general uniqueness result with all earlier results known in the literature.

On the Inverse MEG Problem with a 1-D Current Distribution

“…; ; , U r a = ∇ > B r r r r (4) Then, a series of calculations lead to the following expression for the magnetic potential [10] [11], …”

“…However, a com-plete quantitative characterization of what part of the current is possible to be identified was a topic of intense investigation during the last two decades and the main results can be found in [4]. Fokas proved that, independently of the geometry of the conductor, we cannot recover more than one out of the three functions that define the current, in the case of electroencephalography, and no more than two such functions in the case of magnetoencephalography.…”

EEG identification of a localized 1-D neuronal excitation

“…As far as the question of uniqueness of the solutions of these two inverse problem is concerned, that is the characterization of the class of currents that provide identical eclectic potentials on the head, and identical magnetic fluxes outside the head, the ultimate results have been obtain recently [3], [5]. The definitive result states that neither the EEG nor the MEG measurements can recover completely the primary neuronal current, and therefore no uniqueness for the inverse problems exists.…”

Inverse Modeling for MEG/EEG Data

“…This is a difficult mathematical issue, requiring the solution of a dynamical, highly ill-posed inverse problem. In particular, such ill-posedness implies that the neural configuration explaining the measurements is not unique (there is an infinite number of neural current distributions producing the same dataset, [9]) and this technical difficulty has inspired the adoption of many different strategies (some proposed by the inverse problems community, some others coming from the engineering framework) for the selection of the optimal neural constellation from the infinite set of possible solutions. The available algorithms are usually divided into two classes, based on the physical model used to represent the neural currents: distributed methods assume a continuous current distribution and solve a linear inverse problem consisting in recovering the dynamics of the local strength of the current density at each point of a computational grid introduced in the brain volume; dipolar methods introduce in the reconstruction procedure the information that the neural sources can be modeled as a set of a small number of point-like currents (current dipoles), whose parameters (position, orientation, strength) have to be recovered; assuming this dipolar model, the inverse problem is non linear since the measurements have a strongly non-linear dependence with respect to the unknown source locations.…”