Asymptotic Expansions of I-V Relations via a Poisson–Nernst–Planck System

This is the published version, also available here: http://dx.doi.org/10.1137/070691322.We investigate higher order matched asymptotic expansions of a steady-state Poisson–Nernst–Planck (PNP) system with particular attention to the I-V relations of ion channels. Assuming that the Debye length is small relative to the diameter of the narrow channel, the PNP system can be viewed as a singularly perturbed system. Special structures of the zeroth order inner and outer systems make it possible to provide an explicit derivation of higher order terms in the asymptotic expansions. For the case of zero permanent charge, our results concerning the I-V relation for two oppositely charged ion species are (i) the first order correction to the zeroth order linear I-V relation is generally quadratic in V; (ii) when the electro-neutrality condition is enforced at both ends of the channel, there is NO first order correction, but the second order correction is cubic in V. Furthermore (Theoremsigmo), up to the second order, the cubic I-V relation has (except for a very degenerate case) three distinct real roots that correspond to the bistable structure in the FitzHugh–Nagumo simplification of the Hodgkin–Huxley model

Transport of solvated ions in nanopores: Asymptotic models and numerical study

Improved Poisson--Nernst--Planck systems taking into account finite ion size and solvation effects provide a more accurate model of electric double layers compared to the classical setting. We introduce and discuss several variants of such improved models. %Based on spatially fully resolved numerical models We study the effect of improved modeling in large aspect ratio nanopores. Moreover, we derive approximate asymptotic models for the improved Poisson--Nernst--Planck systems which can be reduced to one-dimensional systems. In a numerical study, we compare simulation results obtained from solution of the asymptotic 1D-models with those obtained by discretization of the full resolution models

Modeling ionic flow between small targets: insights from diffusion and electro-diffusion theory

The flow of ions through permeable channels causes voltage drop in physiological nanodomains such as synapses, dendrites and dendritic spines, and other protrusions. How the voltage changes around channels in these nanodomains has remained poorly studied. We focus this book chapter on summarizing recent efforts in computing the steady-state current, voltage and ionic concentration distributions based on the Poisson-Nernst-Planck equations as a model of electro-diffusion. We first consider the spatial distribution of an uncharged particle density and derive asymptotic formulas for the concentration difference by solving the Laplace's equation with mixed boundary conditions. We study a constant particles injection rate modeled by a Neumann flux condition at a channel represented by a small boundary target, while the injected particles can exit at one or several narrow patches. We then discuss the case of two species (positive and negative charges) and take into account motions due to both concentration and electrochemical gradients. The voltage resulting from charge interactions is calculated by solving the Poisson's equation. We show how deep an influx diffusion propagates inside a nanodomain, for populations of both uncharged and charged particles. We estimate the concentration and voltage changes in relations with geometrical parameters and quantify the impact of membrane curvature.Comment: 17 pages, 8 figures, 1 tabl