A computational methodology to simulate the diffusion of ions from point sources (e.g., ion channels) is described. The outlined approach computes the ion concentration from a cluster of many ion channels at pre-specified locations as a function of time using the theory of propagation integrals. How the channels’ open/closed states evolve in time does not need to be known at the start of the simulation, but can be updated on-the-fly as the simulation goes along. The technique uses analytic formulas for the solutions of the diffusion equation for three common geometries: (1) ions diffusing from a membrane (planar symmetry); (2) ions diffusing into a narrow cleft for effective two-dimensional diffusion (cylindrical symmetry); and (3) ions diffusing into open space like the cytosol (spherical symmetry). Because these formulas are exact solutions valid for arbitrarily long timesteps, no spatial or time discretizations are necessary. The only discrete locations are where the ion concentration is computed, and the only discrete timesteps are when the channels’ open/closed states are updated. Beyond pure diffusion, the technique is generalized to the Excess Buffer Approximation of ion chelation to give an analytic solution of this approximation of the full reaction/diffusion system. Both the pure diffusion and the diffusion/buffering algorithms scale linearly with the number of channels and the number of ion concentration locations.

描述了模拟来自点源（例如，离子通道）的离子扩散的计算方法。概述的方法使用传播积分理论，根据时间，根据预定位置的许多离子通道簇计算离子浓度。在仿真开始时无需知道通道的打开/关闭状态如何随时间变化，但是可以随着仿真的进行实时更新。该技术使用解析公式来求解三种常见几何形状的扩散方程：（1）离子从膜扩散（平面对称）；（2）离子扩散到狭窄的裂缝中以实现有效的二维扩散（圆柱对称）；（3）离子像胞质溶胶一样扩散到开放空间中（球形对称）。因为这些公式对于任意长的时间步都是有效的精确解，所以不需要空间或时间离散化。唯一离散的位置是计算离子浓度的位置，唯一离散的时间步长是更新通道的打开/关闭状态时的时间。除了纯扩散以外，该技术还可以推广到离子螯合的过量缓冲液近似，以给出整个反应/扩散系统近似值的解析解。纯扩散和扩散/缓冲算法都与通道数和离子浓度位置数成线性比例关系。唯一的离散时间步是通道的打开/关闭状态何时更新。除了纯扩散以外，该技术还可以推广到离子螯合的过量缓冲液近似，以给出整个反应/扩散系统近似值的解析解。纯扩散和扩散/缓冲算法都与通道数和离子浓度位置数成线性比例关系。唯一的离散时间步是通道的打开/关闭状态何时更新。除了纯扩散以外，该技术还可以推广到离子螯合的过量缓冲液近似，以给出整个反应/扩散系统近似值的解析解。纯扩散和扩散/缓冲算法都与通道数和离子浓度位置数成线性比例关系。