### Abstract

A pair of differential equations is considered whose solutions exhibit the qualitative properties of nerve conduction, yet which are simple enough to be solved exactly and explicitly. The equations are of the FitzHugh Nagumo type, with a piecewise linear nonlinearity, and they contain two parameters. All the pulse and periodic solutions, and their propagation speeds, are found for these equations, and the stability of the solutions is analyzed. For certain parameter values, there are two different pulse shaped waves with different propagation speeds. The slower pulse is shown to be unstable and the faster one to be stable, confirming conjectures which have been made before for other nerve conduction equations. Two periodic waves, representing trains of propagated impulses, are also found for each period greater than some minimum which depends on the parameters. The slower train is unstable and the faster one is usually stable, although in some cases both are unstable.

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

Pages (from-to) | 1313-1337 |

Number of pages | 25 |

Journal | Biophysical Journal |

Volume | 13 |

Issue number | 12 |

State | Published - 1973 |

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### ASJC Scopus subject areas

- Biophysics

### Cite this

*Biophysical Journal*,

*13*(12), 1313-1337.

**Traveling wave solutions of a nerve conduction equation.** / Rinzel, J.; Keller, J. B.

Research output: Contribution to journal › Article

*Biophysical Journal*, vol. 13, no. 12, pp. 1313-1337.

}

TY - JOUR

T1 - Traveling wave solutions of a nerve conduction equation

AU - Rinzel, J.

AU - Keller, J. B.

PY - 1973

Y1 - 1973

N2 - A pair of differential equations is considered whose solutions exhibit the qualitative properties of nerve conduction, yet which are simple enough to be solved exactly and explicitly. The equations are of the FitzHugh Nagumo type, with a piecewise linear nonlinearity, and they contain two parameters. All the pulse and periodic solutions, and their propagation speeds, are found for these equations, and the stability of the solutions is analyzed. For certain parameter values, there are two different pulse shaped waves with different propagation speeds. The slower pulse is shown to be unstable and the faster one to be stable, confirming conjectures which have been made before for other nerve conduction equations. Two periodic waves, representing trains of propagated impulses, are also found for each period greater than some minimum which depends on the parameters. The slower train is unstable and the faster one is usually stable, although in some cases both are unstable.

AB - A pair of differential equations is considered whose solutions exhibit the qualitative properties of nerve conduction, yet which are simple enough to be solved exactly and explicitly. The equations are of the FitzHugh Nagumo type, with a piecewise linear nonlinearity, and they contain two parameters. All the pulse and periodic solutions, and their propagation speeds, are found for these equations, and the stability of the solutions is analyzed. For certain parameter values, there are two different pulse shaped waves with different propagation speeds. The slower pulse is shown to be unstable and the faster one to be stable, confirming conjectures which have been made before for other nerve conduction equations. Two periodic waves, representing trains of propagated impulses, are also found for each period greater than some minimum which depends on the parameters. The slower train is unstable and the faster one is usually stable, although in some cases both are unstable.

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M3 - Article

C2 - 4761578

AN - SCOPUS:0015719581

VL - 13

SP - 1313

EP - 1337

JO - Biophysical Journal

JF - Biophysical Journal

SN - 0006-3495

IS - 12

ER -