Electro-Neutral Models for dynamic Poisson-Nernst-Planck System: 2D Case

TL;DR

This paper extends electro-neutral models for the Poisson-Nernst-Planck system to two dimensions, demonstrating computational efficiency and ease of analysis for ion transport problems with boundary layers, validated through numerical simulations.

Contribution

The paper develops and validates 2D electro-neutral models for PNP systems, simplifying computations and analysis compared to the original models, especially for multi-ion and complex boundary conditions.

Findings

EN models are computationally cheaper than PNP systems.

EN models accurately replicate ion transport dynamics.

Numerical validation shows efficiency and effectiveness of EN models.

Abstract

The Poisson-Nernst-Planck (PNP) system is a standard model for describing ion transport. In many applications, e.g., ions in biological tissues, the presence of thin boundary layers poses both modelling and computational challenges. In a previous paper, we derived simplified electro-neutral (EN) models in one dimensional space where the thin boundary layers are replaced by effective boundary conditions. In this paper, we extend our analysis to the two dimensional case where the EN model enjoys even greater advantages. First of all, it is much cheaper to solve the EN models numerically. Secondly, EN models are easier to deal with compared with the original PNP system, therefore it is also easier to derive macroscopic models for cellular structures using EN models. The multi-ion case with general boundary is considered, for a variety of boundary conditions including either Dirichlet or flux boundary conditions. Using systematic asymptotic analysis, we derive a variety of effective boundary conditions directly applicable to the EN system for the bulk region. To validate the EN models, numerical computations are carried out for both the EN and original PNP system, including the propagation of action potential for both myelinated and unmyelinated axons. Our results show that solving the EN models is much more efficient than the original PNP system.