### Abstract

A simplified FitzHugh Nagumo nerve conduction equation with known traveling wave solutions is considered. The spatial stability of these solutions is analyzed to determine which solutions should occur in signal transmission along such a nerve model. It is found that the slower of the two pulse solutions is unstable while the faster one is stable, so the faster one should occur. This agrees with conjectures which have been made about the solutions of other nerve conduction equations. Furthermore for certain parameter values the equation has two periodic wave solutions, each representing a train of impulses, at each frequency less than a maximum frequency ωmax. The slower one is found to be unstable and the faster one to be stable, while that at ωmax is found to be neutrally stable. These spatial stability results complement the previous results of Rinzel and Keller (1973, Biophys. J. 13: 1313) on temporal stability, which are applicable to the solutions of initial value problems.

Original language | English (US) |
---|---|

Pages (from-to) | 975-988 |

Number of pages | 14 |

Journal | Biophysical Journal |

Volume | 15 |

Issue number | 10 |

State | Published - 1975 |

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### ASJC Scopus subject areas

- Biophysics

### Cite this

*Biophysical Journal*,

*15*(10), 975-988.

**Spatial stability of traveling wave solutions of a nerve conduction equation.** / Rinzel, J.

Research output: Contribution to journal › Article

*Biophysical Journal*, vol. 15, no. 10, pp. 975-988.

}

TY - JOUR

T1 - Spatial stability of traveling wave solutions of a nerve conduction equation

AU - Rinzel, J.

PY - 1975

Y1 - 1975

N2 - A simplified FitzHugh Nagumo nerve conduction equation with known traveling wave solutions is considered. The spatial stability of these solutions is analyzed to determine which solutions should occur in signal transmission along such a nerve model. It is found that the slower of the two pulse solutions is unstable while the faster one is stable, so the faster one should occur. This agrees with conjectures which have been made about the solutions of other nerve conduction equations. Furthermore for certain parameter values the equation has two periodic wave solutions, each representing a train of impulses, at each frequency less than a maximum frequency ωmax. The slower one is found to be unstable and the faster one to be stable, while that at ωmax is found to be neutrally stable. These spatial stability results complement the previous results of Rinzel and Keller (1973, Biophys. J. 13: 1313) on temporal stability, which are applicable to the solutions of initial value problems.

AB - A simplified FitzHugh Nagumo nerve conduction equation with known traveling wave solutions is considered. The spatial stability of these solutions is analyzed to determine which solutions should occur in signal transmission along such a nerve model. It is found that the slower of the two pulse solutions is unstable while the faster one is stable, so the faster one should occur. This agrees with conjectures which have been made about the solutions of other nerve conduction equations. Furthermore for certain parameter values the equation has two periodic wave solutions, each representing a train of impulses, at each frequency less than a maximum frequency ωmax. The slower one is found to be unstable and the faster one to be stable, while that at ωmax is found to be neutrally stable. These spatial stability results complement the previous results of Rinzel and Keller (1973, Biophys. J. 13: 1313) on temporal stability, which are applicable to the solutions of initial value problems.

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M3 - Article

VL - 15

SP - 975

EP - 988

JO - Biophysical Journal

JF - Biophysical Journal

SN - 0006-3495

IS - 10

ER -