Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle

X. J. Wang

doi:10.1016/0167-2789(93)90286-A

Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle

X. J. Wang

Neural Science

Research output: Contribution to journal › Article › peer-review

Abstract

We present two hypotheses on the mathematical mechanism underlying bursting dynamics in a class of differential systems: (1) that the transition from continuous firing of spikes to bursting is caused by a crisis which destabilizes a chaotic state of continuous spiking; and (2) that the bursting corresponds to a homoclinicity to this unstable chaotic state. These propositions are supported by a numerical test on the Hindmarsh-Rose model, a prototype of its kind. We conclude by a unified view for three types of complex multi-modal oscillations: homoclinic systems, bursting, and the Pomeau-Manneville intermittency.

Original language	English (US)
Pages (from-to)	263-274
Number of pages	12
Journal	Physica D: Nonlinear Phenomena
Volume	62
Issue number	1-4
DOIs	https://doi.org/10.1016/0167-2789(93)90286-A
State	Published - Jan 30 1993

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Condensed Matter Physics
Applied Mathematics

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10.1016/0167-2789(93)90286-A

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title = "Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle",

abstract = "We present two hypotheses on the mathematical mechanism underlying bursting dynamics in a class of differential systems: (1) that the transition from continuous firing of spikes to bursting is caused by a crisis which destabilizes a chaotic state of continuous spiking; and (2) that the bursting corresponds to a homoclinicity to this unstable chaotic state. These propositions are supported by a numerical test on the Hindmarsh-Rose model, a prototype of its kind. We conclude by a unified view for three types of complex multi-modal oscillations: homoclinic systems, bursting, and the Pomeau-Manneville intermittency.",

author = "Wang, {X. J.}",

note = "Funding Information: It is a pleasuret o thank J. Rinzel, D. Terman and A. Sherman for explainingt o me their recent work, and for interestingd iscussionos n the subject treated here. This work is partly supported by the Office of Naval Research under the contractN o. N00014-90J-1194T. he numerical simulationsw ere carried out on the National Cancer Institute Advanced ScientificC omputing Laboratory.",

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AU - Wang, X. J.

N1 - Funding Information: It is a pleasuret o thank J. Rinzel, D. Terman and A. Sherman for explainingt o me their recent work, and for interestingd iscussionos n the subject treated here. This work is partly supported by the Office of Naval Research under the contractN o. N00014-90J-1194T. he numerical simulationsw ere carried out on the National Cancer Institute Advanced ScientificC omputing Laboratory.

PY - 1993/1/30

Y1 - 1993/1/30

N2 - We present two hypotheses on the mathematical mechanism underlying bursting dynamics in a class of differential systems: (1) that the transition from continuous firing of spikes to bursting is caused by a crisis which destabilizes a chaotic state of continuous spiking; and (2) that the bursting corresponds to a homoclinicity to this unstable chaotic state. These propositions are supported by a numerical test on the Hindmarsh-Rose model, a prototype of its kind. We conclude by a unified view for three types of complex multi-modal oscillations: homoclinic systems, bursting, and the Pomeau-Manneville intermittency.

AB - We present two hypotheses on the mathematical mechanism underlying bursting dynamics in a class of differential systems: (1) that the transition from continuous firing of spikes to bursting is caused by a crisis which destabilizes a chaotic state of continuous spiking; and (2) that the bursting corresponds to a homoclinicity to this unstable chaotic state. These propositions are supported by a numerical test on the Hindmarsh-Rose model, a prototype of its kind. We conclude by a unified view for three types of complex multi-modal oscillations: homoclinic systems, bursting, and the Pomeau-Manneville intermittency.

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Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle

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