Predictions and Properties of a Model of Potassium and Calcium Ion Movements During Spreading Cortical Depression

Abstract

A mathematical model for spreading depression, in which the movement of potassium ions and calcium ions is taken into account by means of a coupled system of reaction diffusion equations, is analysed. The effects of varying the parameters of the model are ascertained. These effects are analysed in terms of phase portraits. A critical point in the phase plane determines the minimum amplitude of the wave solutions. The position of this critical point relative to the trajectory on the solitary wave solution determines the shape of the wave profile. The model is found to predict wave profiles as observed experimentally. The model predicts a threshold K+ concentration which is close to the observed threshold of 10 mM. When a stimulus of elevated K+ is locally sustained, a train of SD waves emerges as in experiments where suprathreshold concentrations of potassium chloride are applied. Under some conditions it is impossible to elicit an SD wave. The relative values of the strengths of the pumps and other source terms determines the shape of the waves. A comparison of results from the model equations with data on TTX treated cat cortex yields satisfactory agreement between the amplitude and possibly the speed of the waves. For these small amplitude waves (less than about 25 mM) a linear relation between velocity and amplitude of the K+ wave is found. Action potentials are included in the model by inserting an extra source term in the potassium equation, this being done in an approximate way which obviates the use of the Hodgkin-Huxley equations. When this is done waves of large amplitude (up to 100 mM) are obtained. It was found that treating the ratio of the presynaptic to extracellular volume differently from the ratio of the postsynaptic volume to the extracellular volume plays an important role in determining the amplitude of the response to locally elevated K+ concentration. With the large amplitude waves a linear relationship is also found between velocity and amplitude. When the theoretical and experimental amplitude and time course of the K+ wave are matched at a depth of 0.5 mm in rat cortex, the velocity predicted by the model is 2.9 mm/minute which compares favorably with the experimental value of 3 mm/minute. In other comparisons with experiment, the velocity is in the range of experimental velocities except in one case, cat cerebellum, where the predicted velocity is much smaller than the observed velocity. It is suspected, however, that the experimentally reported velocity may be too high, as it is based on indirect evidence. The questions not addressed by the model are discussed as are the successful predictions and the difficulties expected in obtaining reliable estimates of the magnitudes of the various source and sink terms for the ions in any cortical structure.