Multiscale Modeling &Simulation, Volume 18, Issue 2, Page 887-915, January 2020.
Volume transmission is an important neural communication pathway in which neurons in one brain region influence the neurotransmitter concentration in the extracellular space of a distant brain region. In this paper, we apply asymptotic analysis to a stochastic partial differential equation model of volume transmission to calculate the neurotransmitter concentration in the extracellular space. Our model involves the diffusion equation in a three-dimensional domain with interior holes that randomly switch between being either sources or sinks. These holes model nerve varicosities that alternate between releasing and absorbing neurotransmitter according to when they fire action potentials. In the case that the holes are small, we compute analytically the first two nonzero terms in an asymptotic expansion of the average neurotransmitter concentration. The first term shows that the concentration is spatially constant to leading order and that this constant is independent of many details in the problem. Specifically, this constant first term is independent of the number and location of nerve varicosities, neural firing correlations, and the size and geometry of the extracellular space. The second term shows how these factors affect the concentration at second order. Interestingly, the second term is also spatially constant under some mild assumptions. We verify our asymptotic results by high-order numerical simulation using radial-basis-function--generated finite differences.

中文翻译：

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体积传输随机PDE模型的渐近与数值分析。

2020年1月，《多尺度建模与仿真》，第18卷，第2期，第887-915页。
体积传输是一种重要的神经通讯途径，其中一个大脑区域的神经元影响远处大脑区域的细胞外空间中神经递质的浓度。在本文中，我们将渐进分析应用于体积传递的随机偏微分方程模型，以计算细胞外空间中神经递质的浓度。我们的模型涉及具有内部孔的三维域中的扩散方程，内部孔在源或汇之间随机切换。这些孔模拟神经静脉曲张，根据它们何时触发动作电位在释放和吸收神经递质之间交替。在孔很小的情况下，我们以平均神经递质浓度的渐近展开形式分析计算前两个非零项。第一项表明，浓度在空间上恒定为前导，并且该常数与问题中的许多细节无关。具体来说，这个恒定的第一项与神经静脉曲张的数量和位置，神经放电相关性以及细胞外空间的大小和几何形状无关。第二项显示这些因素如何影响二阶浓度。有趣的是，在某些温和的假设下，第二项在空间上也是恒定的。我们使用径向基函数生成的有限差分通过高阶数值模拟验证了渐近结果。这个恒定的第一项与神经静脉曲张的数量和位置，神经放电相关性以及细胞外空间的大小和几何形状无关。第二项显示这些因素如何影响二阶浓度。有趣的是，在一些温和的假设下，第二项在空间上也是恒定的。我们使用径向基函数生成的有限差分通过高阶数值模拟验证了渐近结果。这个恒定的第一项与神经静脉曲张的数量和位置，神经放电相关性以及细胞外空间的大小和几何形状无关。第二项显示这些因素如何影响二阶浓度。有趣的是，在某些温和的假设下，第二项在空间上也是恒定的。我们使用径向基函数生成的有限差分通过高阶数值模拟验证了渐近结果。