Title
Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh-rose burster Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh-rose burster
Author
Faculty/Department
Faculty of Pharmaceutical, Biomedical and Veterinary Sciences . Biomedical Sciences
Publication type
article
Publication
Philadelphia, Pa ,
Subject
Mathematics
Physics
Source (journal)
SIAM journal on applied dynamical systems. - Philadelphia, Pa
Volume/pages
11(2012) :3 , p. 939-962
ISSN
1536-0040
ISI
000309977400006
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Affiliation
University of Antwerp
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 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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