Electroneutral models for a multidimensional dynamic Poisson-Nernst-Planck system

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 modeling 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 multidimensional case where the EN model enjoys even greater advantages. First, it is much cheaper to solve the EN models numerically. Second, 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 a 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.

Figure 1

Comparison between analytic bulk solution with numerical solution at $t=20$, with $\epsilon =0.05$. Dots represent the exact solutions of $p,n,\psi$, and solid lines are the approximate solutions of $c,\varphi$.

Figure 2

Comparison of concentrations: (a) $p(r,\theta ,t)$ from the PNP system, (b) $c(r,\theta ,t)$ from the EN model with present condition (12) at $t=0.5$.

Figure 3

Comparison of electric potentials: (a) $\psi (r,\theta ,t)$ from the PNP system, (b) $\varphi (r,\theta ,t)$ from the EN model with present condition (12) at $t=0.5$.

Figure 4

Comparison of concentrations: (a) $p(r,\theta ,t)$ from the PNP system, (b) $c(r,\theta ,t)$ from the EN model with present condition (12) at $t=0.5$.

Figure 5

Comparison of electric potentials: (a) $\psi (r,\theta ,t)$ from the PNP system, (b) $\varphi (r,\theta ,t)$ from the EN model with present condition (12) at $t=0.5$.

Figure 6

Sketch of the domain $\mathrm{\Omega }$.

Figure 7

Numerical results of original system (red) and electro-neutral (EN) model (blue) to generate the resting state: (a) dynamics of membrane potential $V_m$; (b) distribution of electric potential at $t=6$.

Figure 8

Numerical results of original system with $\mathrm{\Delta }t={10}^{-4}$ and EN model with three different time step sizes: (a) $\mathrm{\Delta }t={10}^{-4}$, (b) $\mathrm{\Delta }t=5\times {10}^{-4}$, (c) $\mathrm{\Delta }t={10}^{-3}$.

Figure 9

Numerical results of propagation of action potential for myelinated axon: (a) 3/4 myelinated, (b) 9/10 myelinated.