This award will support collaborative research visits between researchers at the U.S. National Institutes of Health under the direction of Dr. John Rinzel and researchers at the Universidad Nacional Autonoma de Mexico (UNAM) under the leadership of Dr. Humberto Carrillo. For over 30 years, mathematical models have been formulated and used to describe and interpret the electrical activity (action potentials) of various nerve, muscle, and secretory cells. However, long-time behavior often involves dynamics on two different time scales, e.g., action potentials (time scale of msec) and variation of second messengers such as intracellular calcium (tens of seconds). The researchers intend to study such behavior using models that include the short-time averaged influence of action potentials but not the detailed time courses of individual events. As a model problem, they will consider the phenomenon of bursting electrical oscillations. The approach will involve both numerical and analytical methods and, in particular, the mathematical technique of averaging. They will derive and analyze biophysically detailed, "averaged" differential equations for the slow dynamics of bursting. The formulation and study of an idealized, analytically tractable model with important generic properties will be useful in elucidating the qualitative character of all such models. The work will proceed simultaneously at both institutions with extensive computations done in Bethesda and rigorous justification of averaging in Mexico City. The approach will allow a biophysical description and interpretation. Furthermore, it will provide insight into the important role played by slow variables as well as into new and interesting mathematical questions related to the method of averaging.
