Poisson-Nernst-Planck equations in a ball

TL;DR

This paper analyzes the Poisson-Nernst-Planck equations in a spherical domain to understand ion distribution, electric potential, and escape rates, with applications to neuronal dendritic spines and electro-diffusion modeling.

Contribution

It provides a detailed analysis of steady states, potential distribution, and ion escape rates in a spherical geometry with Neumann boundary conditions, extending understanding of electro-diffusion in neurons.

Findings

Potential is maximal at the center of the sphere.

Steady state solutions depend on charge magnitude and dimension.

Escape rate of ions is derived for small absorbing windows.

Abstract

The Poisson Nernst-Planck equations for charge concentration and electric potential in a ball is a model of electro-diffusion of ions in the head of a neuronal dendritic spine. We study the relaxation and the steady state when an initial charge of ions is injected into the ball. The steady state equation is similar to the Liouville-Gelfand-Bratu-type equation with the difference that the boundary condition is Neumann, not Dirichlet and there a minus sign in the exponent of the exponential term. The entire boundary is impermeable to the ions and the electric field satisfies the compatibility condition of Poisson's equation. We construct a steady radial solution and find that the potential is maximal in the center and decreases toward the boundary. We study the limit of large charge in dimension 1,2 and 3. For the case of a small absorbing window in the sphere, we find the escape rate of an ion from the steady density.