A PML method for signal-propagation problems in axon

TL;DR

This paper develops a PML-based numerical method to model and simulate signal propagation in myelinated axons, providing theoretical guarantees and demonstrating efficiency through finite element experiments.

Contribution

It introduces a PML method for axon signal modeling, establishing well-posedness, exponential convergence, and validating with numerical experiments.

Findings

PML effectively truncates unbounded domains in axon models.

The method shows exponential convergence to the true solution.

Numerical results confirm the efficiency of the proposed approach.

Abstract

This work is focused on the modelling of signal propagations in myelinated axons to characterize the functions of the myelin sheath in the neural structure. Based on reasonable assumptions on the medium properties, we derive a two-dimensional neural-signaling model in cylindrical coordinates from the time-harmonic Maxwell's equations. The well-posedness of model is established upon Dirichlet boundary conditions at the two ends of the neural structure and the radiative condition in the radial direction of the structure. Using the perfectly matched layer (PML) method, we truncate the unbounded background medium and propose an approximate problem on the truncated domain. The well-posedness of the PML problem and the exponential convergence of the approximate solution to the exact solution are established. Numerical experiments based on finite element discretization are presented to demonstrate the theoretical results and the efficiency of our methods to simulate the signal propagation in axons.