期刊
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
卷 11, 期 3, 页码 939-962出版社
SIAM PUBLICATIONS
DOI: 10.1137/110848931
关键词
bifurcation analysis; codimension-two homoclinic degeneracies; Hindmarsh-Rose model; period adding
资金
- Research Foundation - Flanders (FWO) [12C9112N]
- EPSRC [EP/E032249/1]
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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