Abstract

Bursting behavior refers to the slow periodic change between epochs of fast oscillatory behavior and epochs of quiescent behavior. Since the first geometric explanation of bursting in a neuronal model by Rinzel in 1985, there has been a proliferation of bursting types and several attempts to classify bursting systems in a mathematically meaningful way. I will review two of these approaches: The first by Bertram et al. gives a classification of bursters as slices through a bifurcation diagram. In the second Izhikevich uses the idea that two local or global bifurcations strung together lead to bursting to give a classification that contains 99 distinct bursting types. I will also suggest a third approach which was developed by M. Golubitsky, T. Kaper and myself and gives a classification of bursters in terms of paths in the unfolding space of singularities of vector fields. From this viewpoint there is an intrinsic, natural way to define the complexity of a burster and the problem has a local character making it easier to treat.