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Codimension-Two Homoclinic Bifurcations Underlying Spike Adding in the Hindmarsh-Rose Burster

Abstract : The Hindmarsh-Rose model of neural action potential is revisited from the point of view of global bifurcation analysis, with the singular perturbation parameter held fixed. Of particular concern is a parameter regime where lobe-shaped regions of irregular bursting undergo a transition to stripeshaped regions of periodic bursting. The boundary of each stripe represents a fold bifurcation that causes a smooth spike adding transition where the number of spikes in each burst is increased by one. It is shown via numerical path-following that the lobe-to-stripe transition is organized by a sequence of codimension-one and -two homoclinic bifurcations. Specifically, each of a sequence of homoclinic bifurcation curves in the parameter plane is found to undergo a sharp turn, due to interaction between a two-dimensional unstable manifold and the one-dimensional slow manifold that persists from the singular limit. Local analysis using approximate Poincar'e maps shows that each turning point induces an inclination-flip bifurcation that gives birth to the fold curve that organizes the spike adding transition. Implications of this mechanism for other excitable systems are discussed.
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https://hal.inria.fr/hal-00765189
Contributor : Mathieu Desroches <>
Submitted on : Friday, December 14, 2012 - 11:49:54 AM
Last modification on : Thursday, January 7, 2021 - 3:26:19 PM

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Daniele Linaro, Alan Champneys, Mathieu Desroches, Marco Storace. Codimension-Two Homoclinic Bifurcations Underlying Spike Adding in the Hindmarsh-Rose Burster. SIAM Journal on Applied Dynamical Systems, Society for Industrial and Applied Mathematics, 2012, 11 (3), pp.939-962. ⟨10.1137/110848931⟩. ⟨hal-00765189⟩

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