[For the entire collection see Zbl 0657.00005.]
We will present schematically our qualitative view of a number of different mechanisms for bursting. To understand these mechanisms we exploit the different time scales. We first identify the fast and slow subsystems. Then the fast dynamics are considered with the slow variables treated as parameters. A full description of the steady state and periodic solution sets to the fast subsystem yields the slow manifold; i.e., this step is essentially a global bifurcation analysis of the fast subsystem with the slow variables treated as parameters. We find several different bifurcation structures with which we identify and correlate features of different observed burst patterns.
For example, the various types of transition behavior between steady state and oscillation branches: subcritical Hopf bifurcation, large amplitude homoclinic orbits, and degenerate homoclinic orbits (which contact saddle-node singularities) lead to different spike frequency characteristics at the beginning or end of the active phase. To complete this lowest order approximation we then consider the flow determined by the slow dynamics on the branches of the slow manifold. By varying parameters of the slow dynamics one obtains a variety of burst patterns and other activity which correspond to various experimental findings.
Our formal analysis is essentially the first step in a systematic singular perturbation treatment of these complex oscillators. Perhaps our presentation will motivate analysts to formulate and consider rigorously questions suggested by the phenomena described here. Furthermore, we will discuss some of the explicit models for excitable membrane behavior and offer biophysical interpretations of the theoretical results.