Numerical analysis of the Nernst-Planck-Poisson system

The Nernst-Planck equation is used to describe membrane transport phenomena in biological systems, such as nerve excitation and cell transport of ions. Unfortunately, solving this equation is difficult and some assumptions are required. There are two assumptions that are widely used: the “constant field”and the “electroneutrality” assumptions. Neither assumption can be justified, however, once the Poisson equation equation is taken into account (Nernst-Planck-Poisson system), because the results obtained from these assumptions do not satisfy the Poisson equation. It was shown that the two assumptions are limiting cases of certain dimensionless parameter which is related to the ration of the Debye length to membrane thickness. For the cases far from both limits, however, the equations should be solved numerically. The numerical solution of the Nernst-Planck-Poisson equation was used to obtain the following results: (1) The result of the constant field assumption is not in good agreement with the numerical result and the first order perturbation considerably improved the result. (2) The range of applicability of the constant field assumption is larger than expected from the analysis based on the perturbation expansion. Using the solution for the case in which no approximation can be applied, transition between the constant fiels and the electroneutrality is shown.