### 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 language | English (US) |
---|---|

Pages (from-to) | 907-922 |

Number of pages | 16 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 43 |

Issue number | 4 |

State | Published - Aug 1983 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*43*(4), 907-922.

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

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 43, no. 4, pp. 907-922.

}

TY - JOUR

T1 - HOPF BIFURCATION TO REPETITIVE ACTIVITY IN NERVE.

AU - Rinzel, John

AU - Keener, James P.

PY - 1983/8

Y1 - 1983/8

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=0020807585&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0020807585

VL - 43

SP - 907

EP - 922

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 4

ER -