High-order accurate difference schemes for the Hodgkin-Huxley equations

TL;DR

This paper introduces high-order accurate difference schemes using Summation-By-Parts operators for simulating potential propagation in neuronal branches governed by the Hodgkin-Huxley equations, achieving high accuracy and computational efficiency.

Contribution

The work is the first to demonstrate high accuracy for the Hodgkin-Huxley equations using high-order difference schemes with stability and well-posedness proofs.

Findings

High-order schemes improve CPU efficiency.

Effective handling of complex boundary conditions.

Demonstrated high accuracy in neuronal potential simulations.

Abstract

A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate difference schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin-Huxley equations. This work is the first demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial differential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.