Solving Hodgkin-Huxley equations using the compact difference scheme - somadendrite

TL;DR

This paper introduces a compact finite-difference scheme to solve Hodgkin-Huxley equations on dendrites, providing spectral-like accuracy with easier implementation, applicable to various PDEs in biological and non-biological systems.

Contribution

The paper demonstrates the effectiveness of a compact finite-difference scheme for Hodgkin-Huxley equations, offering an alternative to spectral methods with comparable accuracy and broader applicability.

Findings

The scheme reproduces spectral method results.

It is easier to implement than spectral methods.

Applicable to other PDEs in diverse systems.

Abstract

Dendrites have voltage-gated ion channels which aid in production of action potentials. Thus dendrites are not just passive conductors of information, but actively act on the incoming input. Here we assume Hodgkin-Huxley formulations of voltage-gated ion channels on the dendrite. These equations are normally solved by some form of central difference scheme or the spectral methods. We use a compact finite-difference scheme to solve these equations. This scheme gives spectral-like spatial resolution while being easier to solve than spectral methods. The scheme has shown to be able to reproduce the results from spectral methods. In this paper cylindrical dendrites are described. It may also be noted that the compact difference scheme can be used to solve any other PDE both in biological as well as non biological systems. It is increasingly used in studying turbulence in airflow.