A common formalism for the Integral formulations of the forward EEG problem

Abstract

Abstract-The forward electro-encephalography (EEG) problem involves finding a potential V from the Poisson equation ∇ · (σ∇V ) = f , in which f represents electrical sources in the brain, and σ the conductivity of the head tissues. In the piecewise constant conductivity head model, this can be accomplished by the Boundary Element Method (BEM) using a suitable integral formulation. Most previous work uses the same integral formulation, corresponding to a double-layer potential. In this article we present a conceptual framework based on a well-known theorem (Theorem 1) that characterizes harmonic functions defined on the complement of a bounded smooth surface. This theorem says that such harmonic functions are completely defined by their values and those of their normal derivatives on this surface. It allows us to cast the previous BEM approaches in a unified setting and to develop two new approaches corresponding to different ways of exploiting the same theorem. Specifically, we first present a dual approach which involves a single-layer potential. Then, we propose a symmetric formulation, which combines single and double-layer potentials, and which is new to the field of EEG, although it has been applied to other problems in electromagnetism. The three methods have been evaluated numerically using a spherical geometry with known analytical solution, and the symmetric formulation achieves a significantly higher accuracy than the alternative methods. Additionally, we present results with realistically shaped meshes. Beside providing a better understanding of the foundations of BEM methods, our approach appears to lead also to more efficient algorithms.

EEG extended source localization: Tensor-based vs. conventional methods

“…A realistic head model was built by segmenting the patient's MRI using the BrainVisa software (Rivière et al, 2003). The lead field matrix was then computed for a cortical mesh with 20003 vertices using Brainstorm (Tadel et al, 2011) and OpenMEEG (Gramfort et al, 2010;Kybic et al, 2005). In this case, each vertex of the mesh corresponded to one grid dipole.…”

Analysis of infant cortical synchrony is constrained by the number of recording electrodes and the recording montage

“…'true' neuronal) signals, we obtained virtual EEG recordings by forward modeling simulated time series in cortical parcellations of variable resolutions (20-200 parcels fully covering the cortical surface). As the basis for the forward model we used boundary element method (BEM; Geselowitz, 1967;Kybic et al, 2005) applied to 3D model based on an anatomical MRI of a healthy neonate.…”

“…1B) for the resulting three-shell head model (Fig. 2C) was computed using symmetric boundary element method (BEM) in the OpenMEEG software (Gramfort et al, 2010;Kybic et al, 2005). To validate the results obtained with this source model as well as to examine the impact that a fully gyrated cortical surface would have on the output results, a 'gyrated model' was also used.…”

The Symmetric BEM: Bringing in More Variables for Better Accuracy

“…Thus, three Boundary Element Methods can be derived within a unified setting: a BEM involving only single-layer potentials, a BEM involving only double-layer potentials, and a BEM combining single-and double-layer potentials [3]. We concentrate hereforth on the latter two BEMs.…”

“…This triggered a quest to improve the accuracy of Boundary Element Methods and led us to study the extended Green representation theorem [2]. We proposed a common formalism for the integral formulations of the forward EEG problem, and we derived three different Boundary Element Methods within the same framework [3]. In this paper we recall the mathematical background of Boundary Element Methods, and we present both the double-layer BEM, which is the most widespread method, and the Symmetric BEM, which is a new formulation.…”