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Title
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Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh-rose burster
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Author
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Abstract
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| 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 stripe-shaped 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 Poincare 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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Language
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English
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Source (journal)
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SIAM journal on applied dynamical systems. - Philadelphia, Pa
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Publication
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Philadelphia, Pa
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2012
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ISSN
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1536-0040
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DOI
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10.1137/110848931
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Volume/pages
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11
:3
(2012)
, p. 939-962
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ISI
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000309977400006
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Full text (Publisher's DOI)
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