Restricted Diffusion in Cellular Media: Application to Cortical Spreading Depression

限制离子在细胞介质中扩散的研究是指由于细胞膜具有可变通透性的中间几何构型而改变了离子在组织中的扩散通量。建模、数学和计算工具是为生物组织中的一大类问题开发的，这些组织具有复杂的几何和连通性，特别是大脑。利用体积平均法将脑细胞微环境视为多孔介质，已被用于在宏观水平上研究物质的扩散。建立了数学模型，研究了细胞膜的通透性、包括神经元和神经胶质细胞在内的脑组织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对复杂脑现象的影响。这些模型保留了离子物种在含有细胞的生物组织中受限扩散的基本数学特征，即具有非线性源项的耦合扩散方程组，并发展了分析和计算技术来研究这些模型。从数学和计算两方面研究了在某些应用中被认为是重要的附加机构。这些对大脑的研究包括将神经元和神经胶质细胞的细胞内空间分开处理，以及将细胞外空间作为单独的隔间处理。此外，还从理论上研究了模型方程中简并源的影响。这项研究的具体应用是研究由于大脑中细胞的复杂几何形状而导致的离子限制扩散对皮层扩散抑制(CSD)的影响，CSD是不同大脑结构的皮质中的一种非线性化学和电慢波现象。虽然生理学家对扩散性抑郁(SD)的研究已经超过65年，但对SD波传播的确切机制的识别仍然难以捉摸。CSD与偏头痛的先兆有关；然而，我们仍然不完全了解CSD所涉及的已知和假定的机制如何共同产生这种神秘的现象。一位著名的神经生理学家曾说过：“无论我们对多少个通道蛋白进行了测序，我们确定了多少神经调节剂，构建了多少神经网络，如果我们不能解释传播抑郁症的原因，我们就无法理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、渗透作用、空间缓冲对细胞外钾离子快速运输的影响以及维管束的影响。CSD波的特点是在不同动物的许多不同脑结构的皮质中离子浓度变化很大。对CSD波的潜在机制有一个很好的了解是至关重要的，以发展对哺乳动物大脑如何运作的透彻了解。对早期研究CSD的研究人员建立的连续介质框架进行了扩展和推广，以便对实验CSD波传播中可能涉及的假设机制进行更详细和仔细的研究。此外，作为建立CSD模型的连续神经元和单个神经元方法之间的桥梁，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.