SIAM Journal on Applied Mathematics, Volume 80, Issue 1, Page 312-337, January 2020.
We analyze the diffusion equation in a bounded interval with a stochastically gated interior barrier at the center of the domain. This represents a stochastically gated gap junction linking a pair of identical cells. Previous work has modeled the switching of the gate as a two-state Markov process and used the theory of diffusion in randomly switching environments to derive an expression for the effective permeability of the gap junction. In this paper we extend the analysis of gap junction permeability to the case of a gate with age-structured switching. The latter could reflect the existence of a set of hidden internal states such that the statistics of the non-Markovian two-state model matches the statistics of a higher-dimensional Markov process. Using a combination of the method of characteristics and transform methods, we solve the partial differential equations for the expectations of the stochastic concentration, conditioned on the state of the gate and after integrating out the residence time of the age-structured process. This allows us to determine the jump discontinuity of the concentration at the gap junction and thus the effective permeability. We then use stochastic analysis to show that the solution to the stochastic PDE is a certain statistic of a single Brownian particle diffusing in a stochastically fluctuating environment. In addition to providing a simple probabilistic interpretation of the stochastic PDE, this representation enables an efficient numerical approximation of the solution of the PDE by Monte Carlo simulations of a single diffusing particle. The latter is used to establish that our analytical results match those obtained from Monte Carlo simulations for a variety of age-structured distributions.

中文翻译：

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具有年龄结构的开关的缝隙结点的有效磁导率

SIAM应用数学杂志，第80卷，第1期，第312-337页，2020年1月。
我们在一个有限的区间内分析扩散方程，该区间带有一个随机选通的内部势垒，位于域的中心。这代表了连接一对相同单元的随机门控间隙连接。先前的工作已将门的切换建模为两态马尔可夫过程，并使用了随机切换环境中的扩散理论来推导间隙结的有效磁导率表达式。在本文中，我们将间隙结导磁率的分析扩展到具有年龄结构开关的门的情况。后者可以反映出一组隐藏的内部状态的存在，从而使非马尔可夫两态模型的统计信息与高维马尔可夫过程的统计信息相匹配。结合使用特征方法和变换方法，我们以门的状态为条件，并在对年龄结构化过程的停留时间进行积分后，求解了偏微分方程，以期望获得随机浓度。这使我们能够确定间隙连接处浓度的跳跃不连续性，从而确定有效渗透率。然后，我们使用随机分析表明，随机PDE的解是一个随机波动环境中单个布朗粒子扩散的一定统计量。除了提供随机PDE的简单概率解释外，该表示还可以通过单个扩散粒子的蒙特卡洛模拟对PDE解的数值进行有效的数值逼近。