Efficient and accurate computation of the electric fields of excitable cells

Abstract

The numerical computation of the electric fields produced by excitable cells is important in many applications. Traditionally, a potential formulation was used. An integral formulation based on the differentiation of Green's theorem, which solves directly for the electric field, is presented herein. This is desirable because the electric field is proportional to current density, which can be calculated on the cell membrane. Fredholm equations of the second kind are produced, which are more appropriate than are those of the first kind (produced by formulations based on potential). Analytic formulae are presented to calculate the required matrix entries for zeroth order triangular elements that are generally used for field computations in boundary element methods. Results indicated that significantly more accurate answers may be obtained with significantly less computation by formulating the problem directly in terms of electric field as opposed to potential. This approach has the additional advantage that, for equal intracellular and extracellular conductivities, only one matrix must be generated, and no system of simultaneous equations must be solved; this drastically reduces storage and computation requirements. Examples are given to illustrate this technique and to compare the electric field formulation with the potential formulation.