Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models

We present a qualitative analysis of a generic model structure that can sim ulate the bursting and spiking dynamics of many biological cells. Four diff erent scenarios for the emergence of bursting are described. In this connec tion a number of theorems are stated concerning the relation between the ph ase portraits of the fast subsystem and the global behavior of the full mod el. It is emphasized that the onset of bursting involves the formation of a homoclinic orbit that travels along the route of the bursting oscillations and, hence, cannot be explained in terms of bifurcations in the fast subsy stem. In one of the scenarios, the bursting oscillations arise in a homocli nic bifurcation in which the one-dimensional (ID) stable manifold of a sadd le point becomes attracting to its whole 2D unstable manifold. This type of homoclinic bifurcation, and the complex behavior that it can produce, have not previously been examined in detail. We derive a 2D flow-defined map fo r this situation and show how the map transforms a disk-shaped cross-sectio n of the flow into an annulus. Preliminary investigations of the stable dyn amics of this map show that it produces an interesting cascade of alternati ng pitchfork and boundary collision bifurcations.