Bursting Phenomena in Excitable Membranes

A generalized Hodgkin-Huxley model of excitable membranes is defined, and traveling wave solutions of the model are analyzed using singular perturbation methods in phase space. A complete classification determines whether a system exhibits finite wave train and periodic bursting behavior or only single pulse and regular periodic behavior. Qualitative properties of the bursts are deduced and used to suggest underlying membrane mechanisms. The conclusions shed new light on the mechanisms of bursting in the epileptogenic focus and Aplysia ganglia. While periodic bursting is shown to be possible in a large class of membranes, a membrane which satisfies a special additional condition is shown to embody an infinite-dimensional temporal code in the form of arbitrary sequences of bursts. Other examples exhibit nonuniqueness and chaos.