Diffusing Substances during Spreading Depolarization: Analytical Expressions for Propagation Speed, Triggering, and Concentration Time Courses

Abstract

Spreading depolarization (SD) is an important phenomenon in stroke and migraine. However, the processes underlying the propagation of SD are still poorly understood, and an elementary model that is both physiological and quantitative is lacking. We show that, during the onset and propagation of SD, the concentration time courses of excitatory substances such as potassium and glutamate can be described with a reaction-diffusion equation. This equation contains four physiological parameters: (1) a concentration threshold for excitation; (2) a release rate; (3) a removal rate; and (4) an effective diffusion constant. Solving this equation yields expressions for the propagation velocity, concentration time courses, and the minimum stimulus that can trigger SD. This framework allows for analyzing experimental results in terms of these four parameters. The derived time courses are validated with measurements of potassium in rat brain tissue.