Asymptotic analysis of subcritical Hopf-homoclinic bifurcation

This paper discusses the mathematical analysis of a codimension two bifurca tion determined by the coincidence of a subcritical Hopf bifurcation with a homoclinic orbit of the Hopf equilibrium. Our work is motivated by our pre vious analysis of a Hodgkin-Huxley neuron model which possesses a subcritic al Hopf bifurcation (J. Guckenheimer, R. Harris-Warrick, J. Peck, A. Willms , J. Comput. Neurosci. 4 (1997) 257-277). In this model, the Hopf bifurcati on has the additional feature that trajectories beginning near the unstable manifold of the equilibrium point return to pass through a small neighborh ood of the equilibrium, that is, the Hopf bifurcation appears to be close t o a homoclinic bifurcation as well. This model of the lateral pyloric (LP) cell of the lobster stomatogastric ganglion was analyzed for its ability to explain the phenomenon of spike-frequency adaptation, in which the time in tervals between successive spikes grow longer until the cell eventually bec omes quiescent. The presence of a subcritical Hopf bifurcation in this mode l was the one identified mechanism for oscillatory trajectories to increase their period and finally collapse to a non-oscillatory solution. The analy sis presented here explains the apparent proximity of homoclinic and Hopf b ifurcations. We also develop an asymptotic theory for the scaling propertie s of the interspike intervals in a singularly perturbed system undergoing s ubcritical Hopf bifurcation that may be close to a codimension two subcriti cal Hopf-homoclinic bifurcation. (C) 2000 Elsevier Science B.V. All rights reserved.