# A Calderon Regularized Symmetric Formulation for the Electroencephalography Forward Problem

## TL;DR

This paper introduces a Calderon regularized symmetric formulation for the EEG forward problem, improving numerical stability and conditioning, especially with high discretization density and conductivity contrast.

## Contribution

It proposes a novel regularized symmetric formulation using Calderon identities, resulting in a stable, well-conditioned second kind operator for EEG forward modeling.

## Key findings

- The new formulation is stable under high discretization density.
- It maintains compatibility with existing EEG software.
- Numerical results validate improved stability and conditioning.

## Abstract

The symmetric formulation of the electroencephalography (EEG) forward problem is a well-known and widespread equation thanks to the high level of accuracy that it delivers. However, this equation is first kind in nature and gives rise to ill-conditioned problems when the discretization density or the brain conductivity contrast increases, resulting in numerical instabilities and increasingly slow solutions. This work addresses and solves this problem by proposing a new regularized symmetric formulation. The new scheme is obtained by leveraging on Calderon identities which allow to introduce a dual symmetric equation that, combined with the standard one, results in a second kind operator which is both stable and well-conditioned under all the above mentioned conditions. The new formulation presented here can be easily integrated into existing EEG imaging packages since it can be obtained with the same computational technology required by the standard symmetric formulation. The performance of the new scheme is substantiated by both theoretical developments and numerical results which corroborate the theory and show the practical impact of the new technique.