Pattern Formation in Semi-Discrete Excitable Media

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Hastings 9703630 Semi-discrete models of excitable media consist of coupled systems of ordinary differential equations, with each unit describing the electrical kinetics in a single "cell". They are intermediate between continuous (pde) models and purely discrete models, or cellular automata. The investigator studies whether principles developed for cellular automata models apply to the more realistic semi-discrete case. He considers specific models developed from experimental data to give quantitative descriptions of neural behavior. A model of special interest is that of Morris and Lecar, describing voltage oscillations in barnacle muscle fiber. This model has different features from the well-known FitzHugh-Nagumo system, features that are found in a number of experimental settings and that, moreover, appear amenable to mathematical analysis. The investigator explores the role of various parameters in the system in the propagation of spatial patterns. Among the factors to be considered are excitation threshold, coupling strength, relative rates of fast and slow processes, spatial geometry of the medium, and the relative time spent in the excited and refractory states. The goal of this project is to develop an understanding of pattern formation in semi-discrete excitable media. The main intended application is to neurobiology, though excitable media are found in many biological and chemical systems. Isolated cells are called ``excitable'' if they exhibit a threshold phenomenon in response to a brief external stimulus, but cannot support continued oscillations on their own. A semi-discrete model describes collections of excitable neurons coupled by a linear or nonlinear diffusion mechanism. Examples include nerve, cardiac, and muscle tissue, supporting phenomena such as wave propagation down a myelinated axon, waves of electrical stimulation that sweep through heart muscles, and spreading cortical depression, a brain wave phenomenon.