Return map characterizations of singular solutions for a model of bursting with two slow variables

Abstract

Various physiological systems display bursting electrical activity (BEA). There exist numerous three variable models to describe this behavior. However, four variables may be required to explain some qualitative features of BEA. In this dissertation a model with two slow and two fast variables is presented. For some parameter values the system has stable equilibria while for other values there exist bursting solutions. A singular construction of the latter solutions corresponds to the existence of a fixed point of a one dimensional map. The map is the composition of two maps derived from the slow-subsystem and averaged fast-subsystem. In a degenerate case this fixed point is determined. For non-degenerate cases numerical methods for calculating the maps will be presented.