Solving the Cable Equation Using a Compact Difference Scheme -- Passive Soma Dendrite

TL;DR

This paper introduces a compact finite difference scheme to solve the cable equation for dendritic voltage propagation, offering spectral-like accuracy and improved resolution over traditional methods, applicable to complex neuronal structures.

Contribution

It is the first application of a compact difference scheme to neuronal cable equations, enhancing accuracy and adaptability for complex dendritic morphologies.

Findings

Compact scheme provides spectral-like resolution.

Superior accuracy over central difference schemes.

Effective for complex dendritic models.

Abstract

Dendrites are extensions to the neuronal cell body in the brain which are posited in several functions ranging from electrical and chemical compartmentalization to coincident detection. Dendrites vary across cell types but one common feature they share is a branched structure. The cable equation is a partial differential equation that describes the evolution of voltage in the dendrite. A solution to this equation is normally found using finite difference schemes. Spectral methods have also been used to solve this equation with better accuracy. Here we report the solution to the cable equation using a compact finite difference scheme which gives spectral like resolution and can be more easily used with modifications to the cable equation like nonlinearity, branching and other morphological transforms. Widely used in the study of turbulent flow and wave propagation, this is the first time it is being used to study conduction in the brain. Here we discuss its usage in a passive, soma dendrite construct. The superior resolving power of this scheme compared to the central difference scheme becomes apparent with increasing complexity of the model.