Analysis of Dipolar Sources in the Solution of the Electroencephalographic Inverse Problem

Abstract

In this work, we propose a solution to the problem of identification of sources in the brain from measurements of the electrical potential, recorded on the scalp EEG (electroencephalogram), where boundary problems are used to model the skull, brain region, and scalp, solving the inverse problem from the EEG measurements, the so-called Electroencephalographic Inverse Problem (EIP), which is ill-posed in the Hadamard sense since the problem has numerical instability. We focus on the identification of volumetric dipolar sources of the EEG by constructing and modeling a simplification to reduce the multilayer conductive medium (two layers or regions Ω1 and Ω2) to a problem of a single layer of a homogeneous medium with a null Neumann condition on the boundary. For this simplification purpose, we consider the Cauchy problem to be solved at each time. We compare the results we obtained solving the multiple layers problem with those obtained by our simplification proposal. In both cases, we solve the direct and inverse problems for two different sources, as synthetic results for dipolar sources resembling epileptic foci, and a similar case with an external stimulus (intense light, skin stimuli, sleep problems, etc). For the inverse problem, we use the Tikhonov regularization method to handle its numerical instability. Additionally, we build an algorithm to solve both models (multiple layers problem and our simplification) in time, showing optimization of the problem when considering 128 divisions in the time interval [0,1] s, solving the inverse problem at each time (interval division) and comparing the recovered source with the initial one in the algorithm. We observed a significant decrease in the computation times when simplifying the numerical calculations, resulting in a decrease up to 50% in the execution times, between the EIP multilayer model and our simplification proposal, to a single layer homogeneous problem of a homogeneous medium, which translates into a numerical efficiency in this type of problem.

Keywords

• inverse problem
• electroencephalographic inverse problem
• bioelectrical source
• mathematical modeling
• boundary problem
• differential equation