The multipole approach for EEG forward modeling using the finite element method

Highlights

• We present a novel approach for EEG forward modeling using the finite element method.

• We evaluate this novel approach in both multi-layer sphere and realistic head models.

• The presented approach leads to an improved accuracy and stability for dipolar sources.

• We additionally demonstrate the possibility of modeling quadrupolar sources.

Abstract

For accurate EEG forward solutions, it is necessary to apply numerical methods that allow to take into account the realistic geometry of the subject’s head. A commonly used method to solve this task is the finite element method (FEM). Different approaches have been developed to obtain EEG forward solutions for dipolar sources with the FEM. The St. Venant approach is frequently applied, since its high numerical accuracy and stability as well as its computational efficiency was demonstrated in multiple comparison studies. In this manuscript, we propose a variation of the St. Venant approach, the multipole approach, to improve the numerical accuracy of the St. Venant approach even further and to allow for the simulation of additional source scenarios, such as quadrupolar sources. Exploiting the multipole expansion of electric fields, we demonstrate that the newly proposed multipole approach achieves even higher numerical accuracies than the St. Venant approach in both multi-layer sphere and realistic head models. Additionally, we exemplarily show that the multipole approach allows to not only simulate dipolar but also quadrupolar sources.

Introduction

Electroencephalography (EEG) is a frequently used tool to observe brain activity in both neuroscience and clinical applications, since it provides a unique time resolution in the millisecond range. In many of these applications, it is desirable to perform EEG source analysis, i.e., to reconstruct the active brain areas evoking a measured signal. To achieve this reconstruction it is necessary to simulate the EEG signal that is generated by activity in a certain region of the brain. This task is called the EEG forward problem (Brette and Destexhe, 2012).

The EEG forward problem can only be solved analytically in simple geometries, such as multi-layer sphere models (de Munck et al., 1988; de Munck and Peters, 1993). To realistically model the subject’s head, the use of numerical techniques such as boundary element methods (BEM, Gramfort et al., 2010), finite volume methods (FVM, Cook and Koles, 2006), finite difference methods (FDM, Vatta et al., 2009; Montes-Restrepo et al., 2014), or finite element methods (FEM, Wolters et al., 2007a) is necessary. For all these techniques, the major challenge is to deal with the strong singularity caused by the assumption of a dipolar source to represent brain activity, as it is common in EEG source analysis (de Munck et al., 1988; Hämäläinen et al., 1993; Sarvas, 1987).

Two classes of approaches to solve this problem when applying FEM exist: In the subtraction approach (Wolters et al., 2007b; Engwer et al., 2017), the dipolar source is subtracted from the original equation by using the analytical solution for an infinite, homogeneous volume conductor. Subsequently, a correction potential that accounts for the inhomogeneous conductivity distribution within the head is computed, and the EEG forward solution is obtained by summing up analytical solution and correction potential. In the direct approaches, such as St. Venant (Buchner et al., 1997), partial integration (Yan et al., 1991), and Whitney approach (Tanzer et al., 2005; Pursiainen et al., 2011), the dipole source is approximated by a discrete distribution of current sinks and sources placed on the vertices of the finite element mesh. Each of the approaches follows a different assumption to select these vertices and to determine the strength of the sinks and sources.

Whereas all of these approaches lead to accurate EEG forward simulations, comparison studies between subtraction and direct approaches have shown that the direct approaches have a much lower computational complexity (Bauer et al., 2015; Lew et al., 2009). Thus, a multitude of EEG forward solutions, as needed in many EEG source analysis approaches, can be obtained in a much shorter time. Among the direct approaches, the St. Venant approach was shown to lead to the most accurate results for sources with arbitrary positions and directions in simulation studies in both spherical and realistic head models. As a result, the St. Venant approach has been chosen as forward approach in a variety of simulation (e.g., Güllmar et al., 2010; Cho et al., 2015) and experimental studies (e.g., Aydin et al., 2014; Rullmann et al., 2009).

In this study, we propose to modify the formulation of the St. Venant approach based on the multipole expansion of electric fields. This new formulation leads to reduced numerical errors especially for eccentric sources, whereas the computational effort remains nearly unchanged. Furthermore, it allows to model quadrupolar sources, which is an important advantage over existing FEM approaches, since recent studies have shown that the inclusion of higher order sources may improve source localization for both EEG (Riera et al., 2012) and MEG (Jerbi et al., 2004, 2002; Mosher et al., 1999; Nolte and Curio, 1997). Besides, quadrupolar sources also occur in the diagnosis of spinal chord disorders (Tomori et al., 2010; Sumiya et al., 2017; Ishii et al., 2012).