An Unfolding Theory Approach to Bursting in Fast-Slow Systems [ Back ]

Centre de Recerca Matemàtica
Kresimir Josic
Boston University


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.