Steady-State Poisson-Nernst-Planck Systems: Asymptotic expansions and applications to ion channels

Abstract

Important properties of ion channels can be described by a steady state Poisson-Nernst-Plank system for electrodiffusion. The solution to the PNP system gives a relation between the current and electric potential of the ions in the channel, called the I-V curve. In this thesis, we will discuss the matched asymptotic expansions method of solving a singularly perturbed system and apply this method to find an approximate solution to the steady-state Poisson-Nernst-Planck system. In general, for nonlinear systems, it is impossible to obtain any representations of solutions. Due to the presence of a singular parameter in the PNP system, we can treat the system as a singularly perturbed problem. This system with specific nonlinearity has special structures that are crucial for the explicit higher order asymptotic expansions of the solutions. Although the ion channel problem considers only one cell, the I-V relation obtained in this thesis is consistent with the cubic-like assumption of the I-V relation in the FitzHugh-Nagumo model for action potential involving a population of ion channels. However, applications of the results of this thesis to ion channels are limited, since we considered a simplified model with two species of ions and a zero permanent charge in the channel.