The multipole approach for EEG forward modeling using the finite element method
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).
Section snippets
Theory
Assuming the quasi-static approximation of Maxwell’s equations (Brette and Destexhe, 2012), the EEG forward problem consists in finding the electric potential that solves the Poisson equationwhere is the head domain, the conductivity distribution of , and models the electric activity in the brain. In EEG source analysis, a common model for is the current dipole, . The current dipole describes an infinitesimally small current flow at
Implementation
We implemented the multipole approach in FieldTrip-SimBio (http://fieldtriptoolbox.org, Vorwerk et al., 2018), based on the already existing implementation of the St. Venant approach. Since it was shown in previous studies that placing monopoles in multiple conductive compartments leads to less accurate results for sources that are close to compartment interfaces, i.e., the gray matter/CSF surface in our study, we chose to exclude all monopole positions that are not in the same conductive
Sphere model studies
Firstly, we evaluated the multipole approach in comparison to the St. Venant approach in both hexahedral and tetrahedral sphere models. Table 3 shows the computation times of a single FE right-hand side for both approaches. We find no significant difference in computation times between multipole and St. Venant approach with an increase of less than 1% in both the tetrahedral and the hexahedral model.
Fig. 2 depicts the numerical accuracies of the multipole and the St. Venant approach in model
Discussion
The multipole approach proposed in this study improves the numerical accuracy and stability of FEM EEG forward solutions for both tetrahedral and hexahedral finite element meshes and achieves a constantly high accuracy in both spherical and realistic head models. The multipole approach therefore simplifies the use of the FEM in EEG source analysis, as there is no more need to strictly control the placement of sources to achieve optimal numerical accuracies, be it with regard to the position of
Conclusion
We have introduced the multipole approach for EEG forward solutions and demonstrated that it outperforms the established St. Venant approach for dipolar sources in spherical and realistic head models. The implementation of the multipole approach based on existing implementations of the St. Venant approach, such as in FieldTrip, is straight-forward. Therefore, it is a future goal to make the multipole approach available for an application in practice. Whereas the modeling of quadrupolar sources
Funding
This work was supported by the Austrian Wissenschaftsfonds (FWF), project I 3790-B27, the Deutsche Forschungsgemeinschaft (DFG), projects GR3179/3-1 and WO1425/3-1, 7-1, and by project WO1425/5-2 of the DFG Priority Program 1665.
References (46)
- et al.
Inverse localization of electric dipole current sources in finite element models of the human head
Electroencephalogr. Clin. Neurophysiol.
(1997) - et al.
An improved method for finite element mesh generation of geometrically complex structures with application to the skullbase
J. Biomech.
(1997) - et al.
Influence of the head model on EEG and MEG source connectivity analyses
Neuroimage
(2015) - et al.
Conductive neuromagnetic fields in the lumbar spinal canal
Clin. Neurophysiol.
(2012) - et al.
Localization of realistic cortical activity in MEG using current multipoles
Neuroimage
(2004) - et al.
Accuracy and run-time comparison for different potential approaches and iterative solvers in finite element method based EEG source analysis
Appl. Numer. Math.
(2009) - et al.
A realistic, accurate and fast source modeling approach for the EEG forward problem
Neuroimage
(2019) - et al.
On the calculation of magnetic fields based on multipole modeling of focal biological current sources
Biophys. J.
(1997) - et al.
EEG source analysis of epileptiform activity using a 1mm anisotropic hexahedra finite element head model
Neuroimage
(2009) - et al.
Diagnosis of incomplete conduction block of spinal cord from skin surface using spinal cord evoked magnetic fields
J. Orthop. Sci.
(2010)
A guideline for head volume conductor modeling in EEG and MEG
Neuroimage
Numerical approaches for dipole modeling in finite element method based source analysis
Int. Congr. Ser.
Combining EEG and MEG for the reconstruction of epileptic activity using a calibrated realistic volume conductor model
PLoS One
Comparison study for Whitney (Raviart-Thomas)-type source models in finite element method based EEG forward modeling
IEEE (Inst. Electr. Electron. Eng.) Trans. Biomed. Eng.
A finite element solution of the forward problem in EEG for multipolar sources
IEEE Trans. Neural Syst. Rehabil. Eng.
Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics
Handbook of Neural Activity Measurement
A high-resolution anisotropic finite-volume head model for EEG source analysis
Modeling of the human skull in EEG source analysis
Hum. Brain Mapp.
A fast method to compute the potential in the multisphere model
IEEE (Inst. Electr. Electron. Eng.) Trans. Biomed. Eng.
Mathematical dipoles are adequate to describe realistic generators of human brain activity
IEEE (Inst. Electr. Electron. Eng.) Trans. Biomed. Eng.
A discontinuous Galerkin method to solve the EEG forward problem using the subtraction approach
SIAM J. Sci. Comput.
OpenMEEG: opensource software for quasistatic bioelectromagnetics
Biomed. Eng. Online
Cited by (9)
Global sensitivity of EEG source analysis to tissue conductivity uncertainties
2024, Frontiers in Human NeuroscienceMultimodal study of the neural sources of error monitoring in adolescents and adults
2023, PsychophysiologyValidating EEG source imaging using intracranial electrical stimulation
2023, Brain Communications
- 1
The first two authors contributed equally to this work.