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The analytical subtraction approach for solving the forward problem in EEG

Leandro Beltrachini

TL;DR

The paper targets the EEG forward problem solved with the subtraction finite element method, addressing the challenge of singular dipole sources by deriving analytic expressions for all potential integrals. By employing volume coordinates and Gauss theorems, it reduces surface and volume integrals to closed-form 1D expressions, yielding the analytical subtraction (AS) approach. AS achieves higher accuracy than existing subtraction-based methods while maintaining similar computational cost to the lowest-order numerical schemes, enabling robust performance with highly eccentric sources and anisotropic head conductivities. The method is demonstrated across local, spherical, and real head models, and is made available via the MATLAB FEMEG toolbox, positioning AS as a practical gold standard for realistic EEG forward modeling.

Abstract

Objective: The subtraction approach is known for being a theoretically-rigorous and accurate technique for solving the forward problem in electroencephalography by means of the finite element method. One key aspect of this approach consists of computing integrals of singular kernels over the discretised domain, usually referred to as potential integrals. Several techniques have been proposed for dealing with such integrals, all of them approximating the results at the expense of reducing the accuracy of the solution. In this paper, we derive analytic formulas for the potential integrals, reducing approximation errors to a minimum. Approach: Based on volume coordinates and Gauss theorems, we obtained parametric expressions for all the element matrices needed in the formulation assuming first order basis functions defined on a tetrahedral mesh. This included solving potential integrals over triangles and tetrahedra, for which we found compact and efficient formulas. Main results: Comparison with numerical quadrature schemes allowed to test the advantages of the methodology proposed, which were found of great relevance for highly-eccentric sources, as those found in the somatosensory and visual cortices. Moreover, the availability of compact formulas allowed an efficient implementation of the technique, which resulted in similar computational cost than the simplest numerical scheme. Significance: The analytical subtraction approach is the optimal subtraction-based methodology with regard to accuracy. The computational cost is similar to that obtained with the lowest order numerical integration scheme, making it a competitive option in the field. The technique is highly relevant for improving electromagnetic source imaging results utilising individualised head models and anisotropic electric conductivity fields without imposing impractical mesh requirements.

The analytical subtraction approach for solving the forward problem in EEG

TL;DR

The paper targets the EEG forward problem solved with the subtraction finite element method, addressing the challenge of singular dipole sources by deriving analytic expressions for all potential integrals. By employing volume coordinates and Gauss theorems, it reduces surface and volume integrals to closed-form 1D expressions, yielding the analytical subtraction (AS) approach. AS achieves higher accuracy than existing subtraction-based methods while maintaining similar computational cost to the lowest-order numerical schemes, enabling robust performance with highly eccentric sources and anisotropic head conductivities. The method is demonstrated across local, spherical, and real head models, and is made available via the MATLAB FEMEG toolbox, positioning AS as a practical gold standard for realistic EEG forward modeling.

Abstract

Objective: The subtraction approach is known for being a theoretically-rigorous and accurate technique for solving the forward problem in electroencephalography by means of the finite element method. One key aspect of this approach consists of computing integrals of singular kernels over the discretised domain, usually referred to as potential integrals. Several techniques have been proposed for dealing with such integrals, all of them approximating the results at the expense of reducing the accuracy of the solution. In this paper, we derive analytic formulas for the potential integrals, reducing approximation errors to a minimum. Approach: Based on volume coordinates and Gauss theorems, we obtained parametric expressions for all the element matrices needed in the formulation assuming first order basis functions defined on a tetrahedral mesh. This included solving potential integrals over triangles and tetrahedra, for which we found compact and efficient formulas. Main results: Comparison with numerical quadrature schemes allowed to test the advantages of the methodology proposed, which were found of great relevance for highly-eccentric sources, as those found in the somatosensory and visual cortices. Moreover, the availability of compact formulas allowed an efficient implementation of the technique, which resulted in similar computational cost than the simplest numerical scheme. Significance: The analytical subtraction approach is the optimal subtraction-based methodology with regard to accuracy. The computational cost is similar to that obtained with the lowest order numerical integration scheme, making it a competitive option in the field. The technique is highly relevant for improving electromagnetic source imaging results utilising individualised head models and anisotropic electric conductivity fields without imposing impractical mesh requirements.

Paper Structure

This paper contains 20 sections, 2 theorems, 56 equations, 6 figures.

Table of Contents

  1. Introduction
  2. EEG forward problem
  3. Differential formulation
  4. FE discretisation
  5. Stiffness matrix
  6. Source vector
  7. Local coordinate system
  8. Gauss theorems
  9. Surface integral
  10. Volume integral
  11. Experiments
  12. Local errors
  13. Spherical head model
  14. Illustration in a real model
  15. Results
  16. ...and 5 more sections

Key Result

Proposition 1

Let $h:S\rightarrow \mathbb{R}$ be a function defined for any vector $\hbox{\boldmath $t$}$ belonging to the planar surface $S$. Then, the following relation holds

Figures (6)

  • Figure 1: Schematic representation of the local frames $\mathcal{L}$ and $\mathcal{L}_a$ (see Section \ref{['sec:4loc']}).
  • Figure 2: RE between the analytical and numerical element source vectors as a function of the normalised distance between the source and the element. Results are presented for the surface (a.) and volume (b.) element vectors and numerical integration orders $n$ equal to 2, 4, and 6 (with different markers).
  • Figure 3: Error measures (RE: top; RDM: centre; MAG: bottom) for the numerical solutions of the EEG-FP as a function of the distance to the next compartment considering tangentially-oriented sources and the model with 256k nodes. Results are presented for the AS and FS (with $n=2$ and $n=4$) methods (with different colours).
  • Figure 4: RE for different models and source locations. a. Percentile curves of the RE considering 100 sources located at 0.5mm to the next compartment using all models (with different colours) and the FS approach with $n=2$ (left), $n=4$ (centre), and the AS method (right). b. 90th percentile of the RE as a function of the distance to the next compartment and the number of nodes. Results considering the FS (with $n=2$ and $n=4$) and AS methods are displayed as in a.
  • Figure 5: Mean RE between the numerical approximations obtained with the AS and FS approaches as a function of $d/a$. Results are presented for all mesh discretisations (with different colours) and numerical integration orders $n=2$ and $n=4$ (with different markers). The black horizontal line indicates the $1\%$ threshold (see text).
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2