Restricted Diffusion in Cellular Media: Application to Cortical Spreading Depression

细胞介质中受限离子扩散的研究是指由于具有可变渗透性的细胞膜的介入几何形状而改变离子在组织中的扩散通量。 建模、数学和计算工具是为具有复杂几何形状和连通性的生物组织（特别是大脑）中的广泛一类问题而开发的。 脑细胞微环境被视为一个多孔介质，使用体积平均，已被用来研究在宏观水平上的物质的扩散。 数学模型的推导来研究细胞膜的渗透性，脑组织，包括神经元和神经胶质细胞的几何和拓扑结构，以及神经胶质细胞和血管网络的连通性的复杂的大脑现象的影响。 这些模型保留了基本的数学特征的限制扩散的离子物种在生物组织中含有细胞，即耦合系统的扩散方程与非线性源项，和分析和计算技术的开发用于研究这些模型。 另外的机制，被认为是重要的，在某些应用中，数学和计算研究。 这些脑的研究包括分别处理神经元和神经胶质细胞的细胞内空间，以及将细胞外空间作为单独的隔室处理。 此外，本文还从理论上研究了模型方程中简并源的影响，并将其应用于研究脑细胞复杂几何结构引起的离子受限扩散对皮层扩散抑制（cortical spreading depression，CSD）的影响。 虽然生理学家已经研究了扩散性抑郁症（SD）超过65年，但SD波传播的精确机制仍然难以确定。 CSD与有先兆的偏头痛有关;然而，我们仍然不完全了解CSD中已知和假设的机制如何合谋产生这种神秘的现象。 一位著名的神经生理学家说：“无论我们测序了多少通道蛋白，识别了多少神经调质，构建了多少神经网络，如果我们不能解释扩散性抑郁症，我们就不了解大脑是如何工作的。CSD涉及的机制包括离子扩散、膜离子电流、渗透、空间缓冲对细胞外钾离子快速转运的影响以及血管树的影响。 CSD波的特征在于不同动物的许多不同脑结构的皮质中的大离子浓度变化。 要深入了解哺乳动物脑的功能，就必须对CSD波的机制有一个很好的了解。 研究CSD的研究人员早先开发的连续框架进行了扩展和推广，以进行更详细和仔细的研究，可能涉及实验CSD波传播的假定机制。 此外，作为桥接连续和单神经元方法建模CSD的一步，还研究了神经元网络中的电扩散。

Research on restricted ion diffusion in cellular media refers to the alteration of the diffusion flux of ions in tissue because of the intervening geometry of cellular membranes with variable permeability. Modeling, mathematical, and computational tools are developed for a broad class of problems in biological tissues that have complicated geometry and connectivity, specifically the brain. The brain-cell microenvironment is treated as a porous medium using volume averaging that has been used to study the diffusion of substances at the macroscopic level. Mathematical models are derived to study the effects on complex brain phenomena of permeability of cell membranes, the geometrical and topological structure of brain tissue including neurons and glial cells, and the connectivity of glial cells and of the vascular network. These models retain essential mathematical features of restricted diffusion of ionic species in biological tissue containing cells, namely coupled systems of diffusion equations with nonlinear source terms, and analytical and computational techniques are developed for studying these models. Additional mechanisms, believed to be important in certain applications, are studied mathematically and computationally. These studies of the brain include treating the intracellular spaces of neurons and glial cells separately and treating the extracellular space as a separate compartment. In addition, the influence of degenerate sources in the model equations is studied theoretically.A specific application of this research is to the effects of restricted diffusion of ions due to the complex geometry of cells in the brain on cortical spreading depression (CSD), a nonlinear chemical and electrical (slow) wave phenomenon in the cortices of different brain structures. Although physiologists have studied spreading depression (SD) for more than 65 years, identification of the precise mechanisms involved in the propagation of SD waves has remained elusive. CSD has been implicated in migraine with aura; however, we still do not fully understand how the known and postulated mechanisms that are involved in CSD conspire to produce this enigmatic phenomenon. A prominent neurophysiologist has said, "No matter how many channel proteins we sequence, how many neuromodulators we identify and how many neural networks we construct, if we cannot explain spreading depression, we do not understand how the brain works." Mechanisms involved in CSD include ionic diffusion, membrane ionic currents, osmosis, effect of spatial buffering on fast transport of extracellular potassium ions, and effects of the vascular tree. CSD waves are characterized by large ionic concentration changes in the cortex of many different brain structures in different animals. It is essential to have a good understanding of the mechanisms underlying CSD waves to develop a thorough understanding of how the mammalian brainfunctions. The continuum frameworks developed earlier by researchers studying CSD are extended and generalized in order to carry out more detailed and careful studies of the putative mechanisms that may be involved in experimental CSD wave propagation. Also, as a step towards bridging the continuum and single neuron approaches to modeling CSD, electrodiffusion in a network of neurons is also studied.