HOPF BIFURCATION TO REPETITIVE ACTIVITY IN NERVE.

John Rinzel, James P. Keener

Research output: Contribution to journalArticle

Abstract

Various nerve axons and other excitable systems exhibit repetitive activity (e. g. , trains of propagated impulses) in response to a spatially localized, time-independent stimulus. If the stimulus is too weak, or in some cases too strong, one finds rather a spatially nonuniform, steady response which attenuates with distance from the input site. These stimulus-response properties are considered for a qualitative model of nerve conduction, the FitzHugh-Nagumo (parabolic partial differential) equations. For each stimulus amplitude there is a unique steady state solution. At critical stimulus values this steady solution loses stability and a branch of time periodic (spatially nonuniform) solutions appears via Hopf bifurcation. This is associated with the onset of repetitive activity. Analytic bifurcation formulae are derived and evaluated for the case of a cubic nonlinearity to determine regions in parameter space where the steady solution is unstable. The effect of stimulus forms, either voltage or current input, and either spatially localized or spatially uniform stimulus is interpreted physiologically.

Original languageEnglish (US)
Pages (from-to)907-922
Number of pages16
JournalSIAM Journal on Applied Mathematics
Volume43
Issue number4
StatePublished - Aug 1983

Fingerprint

Hopf bifurcation
Nerve
Hopf Bifurcation
Partial differential equations
Electric potential
Excitable Systems
FitzHugh-Nagumo
Parabolic Partial Differential Equations
Steady-state Solution
Conduction
Impulse
Critical value
Parameter Space
Branch
Bifurcation
Unstable
Voltage
Nonlinearity
Axons
Model

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

HOPF BIFURCATION TO REPETITIVE ACTIVITY IN NERVE. / Rinzel, John; Keener, James P.

In: SIAM Journal on Applied Mathematics, Vol. 43, No. 4, 08.1983, p. 907-922.

Research output: Contribution to journalArticle

Rinzel, John ; Keener, James P. / HOPF BIFURCATION TO REPETITIVE ACTIVITY IN NERVE. In: SIAM Journal on Applied Mathematics. 1983 ; Vol. 43, No. 4. pp. 907-922.
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