Cross-currents between biology and mathematics on models of bursting

A great deal of work has gone into classifying bursting oscillations, periodic alternations of spiking and quiescence modeled by fast-slow systems. In such systems, one or more slow variables carry the fast variables through a sequence of bifurcations that mediate transitions between oscillations and steady states. The most rigorous approach is to characterize the bifurcations found in the neighborhood of a singularity. Fold/homoclinic bursting, along with most other burst types of interest, has been shown to occur near a singularity of codimension three by examining bifurcations of a cubic Liénard system. Modeling and biological considerations suggest that fold/homoclinic bursting should be found near fold/subHopf bursting, a more recently identified burst type whose codimension has not been determined yet. One would expect that fold/subHopf bursting has the same codimension as fold/homoclinic bursting, because models of these two burst types have very similar underlying bifurcation diagrams. However, we are unable to determine a codimension-three singularity that supports fold/subHopf bursting. Furthermore, we believe that it is not possible to find a codimension-three singularity that gives rise to all known types of bursting. Instead, we identify a three-dimensional slice that contains all known types of bursting in a partial unfolding of a doubly-degenerate Bodganov–Takens point, which has codimension four.