The Surface Laplacian Technique in EEG: Theory and Methods

TL;DR

This paper reviews surface Laplacian methods in EEG, comparing finite difference and spline techniques, and introduces new approximations and solutions to improve accuracy and computational efficiency for EEG analysis.

Contribution

It provides detailed mathematical derivations, addresses peripheral electrode approximation, and discusses regularization and computational strategies for surface Laplacian in EEG.

Findings

Finite difference method is simple but prone to discretization errors.

Spline method reduces noise but increases computational complexity.

Matrix representation enhances computational performance.

Abstract

This paper reviews the method of surface Laplacian differentiation to study EEG. We focus on topics that are helpful for a clear understanding of the underlying concepts and its efficient implementation, which is especially important for EEG researchers unfamiliar with the technique. The popular methods of finite difference and splines are reviewed in detail. The former has the advantage of simplicity and low computational cost, but its estimates are prone to a variety of errors due to discretization. The latter eliminates all issues related to discretization and incorporates a regularization mechanism to reduce spatial noise, but at the cost of increasing mathematical and computational complexity. These and several others issues deserving further development are highlighted, some of which we address to the extent possible. Here we develop a set of discrete approximations for Laplacian estimates at peripheral electrodes and a possible solution to the problem of multiple-frame regularization. We also provide the mathematical details of finite difference approximations that are missing in the literature, and discuss the problem of computational performance, which is particularly important in the context of EEG splines where data sets can be very large. Along this line, the matrix representation of the surface Laplacian operator is carefully discussed and some figures are given illustrating the advantages of this approach. In the final remarks, we briefly sketch a possible way to incorporate finite-size electrodes into Laplacian estimates that could guide further developments.