A Meshfree Solver for the MEG Forward Problem

Abstract

Noninvasive estimation of brain activity via magnetoencephalography (MEG) involves an inverse problem whose solution requires an accurate and fast forward solver. To this end, we propose the Method of Fundamental Solutions (MFS) as a meshfree alternative to the Boundary Element Method (BEM). The solution of the MEG forward problem is obtained, via the Method of Particular Solutions (MPS), by numerically solving a boundary value problem for the electric scalar potential, derived from the quasi-stationary approximation of Maxwell's equations. The magnetic field is then computed by the Biot-Savart law. Numerical experiments have been carried out in a realistic single-shell head geometry. The proposed solver is compared with a state-of-the-art BEM solver. A good agreement and a reduced computational load show the attractiveness of the meshfree approach.Index Terms-Biomagnetics, magnetoencephalography (MEG), method of fundamental solutions (MFS), meshfree methods.

Detecting tri‐stability of 3D models with complex attractors via meshfree reconstruction of invariant manifolds of saddle points

“…In this section, we present a MLS method that achieves a polynomial reproduction approximation of any order d. In particular, we use a meshfree approach that allows to work with a large number of data and that is not influenced by the geometry of the domain. [26][27][28][29] For our applications, the set of data χ ¼ x 1 ; ⋯; x M f g ⊂R 2 is composed by the M scattered points of the separatrix manifold projected on the plane XY and the heights z i ji ¼ 1; ⋯; M f grepresent the set of the data values. The general idea of the MLS is to calculate the generating functions Φ i (y)=Φ(y,x i ) necessary for the construction of the approximant 30 :…”

“…In this section, we present a MLS method that achieves a polynomial reproduction approximation of any order d . In particular, we use a meshfree approach that allows to work with a large number of data and that is not influenced by the geometry of the domain …”

Meshfree and Particle Methods in Biomechanics: Prospects and Challenges

“…There is no doubt that the brain is one of the most delicate, complex, and important organs in Peng et al [204] recently studied the effect of head models and dipole source parameters on EEG fields using a point least squares (PLS) based meshless method. Similar to the EEG forward problem, the magnetoencephalography (MEG) forward problem, which involves computing the scalp potential and magnetic field distribution generated by a set of current sources and analyzing the complex activation patterns in the human brain, was studied using the SPH method, by Ala and co-workers [205,206] and the MFS method via the method of particular solutions (MOPS) [207]. In their studies, the three-layered and multilayered model was used, the magnetic field was computed by way of the Biot-Savart law and numerical experiments were carried out in a realistic single-shell head geometry.…”

On Inverse Problems for Characteristic Sources in Helmholtz Equations

“…Furthermore, this method has become a popular tool because of its applicability in several areas where we can cite, as an example, [7,8,19,22,23], including inverse problems.…”