# The analytical subtraction approach for solving the forward problem in EEG

## TL;DR

This paper introduces an analytical method for calculating potential integrals in EEG forward problem solutions, significantly improving accuracy while maintaining computational efficiency, especially for complex source configurations.

## Contribution

The authors derive explicit formulas for potential integrals in the finite element method, reducing approximation errors and enabling efficient, accurate EEG forward modeling with complex head models.

## Key findings

- Analytic formulas outperform numerical quadrature in accuracy.
- Method is efficient for highly-eccentric sources in brain regions.
- Comparable computational cost to simple numerical schemes.

## 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.