Multistability and Infinite Cycles in a Model of the Leech Heart Interneuron

We argue that the Lukyanov-Shilnikov bifurcation of a saddle-node periodic orbit with noncentral homoclinics explains the effect of bi-stability observed in a neuron model based on a HodgkinHuxley formalism. In this model the dominating regime, depending on the initial state, can be either spiking oscillations or weakly irregular bursting ones. It is also shown how the bifurcation of the blue-sky catastrophe may underlie a continuous transition between the above regimes. We exploit the features of this bifurcation (a) to explain transition from tonic spiking activity to periodic bursting regime; (b) to identify the principal bifurcational parameters in the system; (c) to control the temporal characteristics of bursting behavior via these parameters.