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.
