具有空间相互作用的Poisson-Nernst-Planck方程的多个稳态解的磁滞和线性稳定性分析,Physical Review E

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具有空间相互作用的Poisson-Nernst-Planck方程的多个稳态解的磁滞和线性稳定性分析
Physical Review E ( IF 2.2 ) Pub Date : 2020-11-06 , DOI: 10.1103/physreve.102.053301
Jie Ding , Hui Sun , Shenggao Zhou

在这项工作中，我们对具有Dirichlet边界条件的一类空间Poisson-Nernst-Planck方程的多个稳态解的线性稳定性进行数值研究，该方程适用于离子通道。通过数值发现多个稳态解，我们得到

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形的电流-电压和电流-浓度曲线，显示离子电导对电压和边界浓度的滞后响应，并具有记忆效应。提出了一个边值问题来定位分叉点并预测出分叉点附近的局部分叉图。

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形曲线。开发了用于线性稳定性分析的数值方法，以了解仅在数值上可用的稳态解的稳定性。提出了有限差分方案来解决涉及微分算子的导出特征值问题。线性稳定性分析表明

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形曲线具有两个不同电导水平的线性稳定分支和一个线性不稳定的中间分支，表现出经典的双稳态磁滞。正如线性稳定性分析中所预测的那样，过渡动力学是由与主导特征值模式相关的扰动引起的，从不稳定分支上的稳态解到稳定分支上的稳态解。进一步的数值测试表明，线性稳定性分析中提出的有限差分方案是二阶精确的。在这项工作中开发的数值方法可以用于研究一类与时间相关的问题的线性稳定性，这些问题与通过数字计算的稳态解有关。

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Hysteresis and linear stability analysis on multiple steady-state solutions to the Poisson-Nernst-Planck equations with steric interactions

In this work, we numerically study linear stability of multiple steady-state solutions to a type of steric Poisson-Nernst-Planck equations with Dirichlet boundary conditions, which are applicable to ion channels. With numerically found multiple steady-state solutions, we obtain

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-shaped current-voltage and current-concentration curves, showing hysteretic response of ion conductance to voltages and boundary concentrations with memory effects. Boundary value problems are proposed to locate bifurcation points and predict the local bifurcation diagram near bifurcation points on the

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-shaped curves. Numerical approaches for linear stability analysis are developed to understand the stability of the steady-state solutions that are only numerically available. Finite difference schemes are proposed to solve a derived eigenvalue problem involving differential operators. The linear stability analysis reveals that the

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-shaped curves have two linearly stable branches of different conductance levels and one linearly unstable intermediate branch, exhibiting classical bistable hysteresis. As predicted in the linear stability analysis transition dynamics, from a steady-state solution on the unstable branch to one on the stable branches, are led by perturbations associated to the mode of the dominant eigenvalue. Further numerical tests demonstrate that the finite difference schemes proposed in the linear stability analysis are second-order accurate. Numerical approaches developed in this work can be applied to study linear stability of a class of time-dependent problems around their steady-state solutions that are computed numerically.

更新日期：2020-11-06

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