Simplified Reaction-Diffusion Equations for Potassium and Calcium Ion Concentrations During Spreading Cortical Depression

Abstract

The reaction terms in the reaction-diffusion equations previously derived for the concentrations of potassium and calcium ions in brain structures during spreading depression are evaluated for various values of the ion concentrations. The null isoclines are estimated and the solitary wave trajectory plotted in the phase plane to find its relation to the isoclines and critical points of the kinetic equations. Four sets of simplified reaction terms are devised which have approximately the same dependence on the ion concentrations as those in the original system. The properties of the simplified equations are determined by numerical integration. Each system has a subthreshold response to a small local elevation of external K⁺ concentration. The first, third and fourth systems support solitary wave solutions with properties similar to those of the original model system. Strength-duration curves are obtained and the response to a sustained stimulus evaluated. Collisions between waves are observed to result in their mutual destruction.

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Notes on contributors

Henry C. Tuckwell

Joyce Laing works in the Department of Child and Family Psychiatry, Playfield House, Cupar, Fife, and is a Consultant Art Therapist to Psychiatric Hospitals and Prisons and Chairwoman of the Scottish Society of Art and Psychology.