### Abstract

Propagation speed of an impulse is influenced by previous activity. A pulse following its predecessor too closely may travel more slowly than a solitary pulse. In contrast, for some range of interspike intervals, a pulse may travel faster than normal because of a possible superexcitable phase of its predecessor's wake. Thus, in general, pulse speeds and interspike intervals will not remain constant during propagation. We consider these issues for the Hodgkin-Huxley cable equations. First, the relation between speed and frequency or interspike interval, the dispersion relation, is computed for particular solutions, steadily propagating periodic wave trains. For each frequency, ω, below some maximum frequency, ω(max), we find two such solutions, one fast and one slow. The latter are likely unstable as a computational example illustrates. The solitary pulse is obtained in the limit as ω tends to zero. At high frequency, speed drops significantly below the solitary pulse speed; for 6.3°C, the drop at ω(max) is >60%. For an intermediate range of frequencies, supernormal speeds are found and these are correlated with oscillatory swings in sub- and superexcitability in the return to rest of an impulse. Qualitative consequences of the dispersion relation are illustrated with several different computed pulse train responses of the full cable equations for repetitively applied current pulses. Moreover, changes in pulse speed and interspike interval during propagation are predicted quantitatively by a simple kinematic approximation which applies the dispersion relation, instantaneously, to individual pulses. One example shows how interspike time intervals can be distorted during propagation from a ratio of 2:1 at input to 6:5 at a distance of 6.5 cm.

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

Pages (from-to) | 227-259 |

Number of pages | 33 |

Journal | Biophysical Journal |

Volume | 34 |

Issue number | 2 |

State | Published - 1981 |

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### ASJC Scopus subject areas

- Biophysics

### Cite this

*Biophysical Journal*,

*34*(2), 227-259.

**The dependence of impulse propagation speed on firing frequency, dispersion, for the Hodgkin-Huxley model.** / Miller, R. N.; Rinzel, J.

Research output: Contribution to journal › Article

*Biophysical Journal*, vol. 34, no. 2, pp. 227-259.

}

TY - JOUR

T1 - The dependence of impulse propagation speed on firing frequency, dispersion, for the Hodgkin-Huxley model

AU - Miller, R. N.

AU - Rinzel, J.

PY - 1981

Y1 - 1981

N2 - Propagation speed of an impulse is influenced by previous activity. A pulse following its predecessor too closely may travel more slowly than a solitary pulse. In contrast, for some range of interspike intervals, a pulse may travel faster than normal because of a possible superexcitable phase of its predecessor's wake. Thus, in general, pulse speeds and interspike intervals will not remain constant during propagation. We consider these issues for the Hodgkin-Huxley cable equations. First, the relation between speed and frequency or interspike interval, the dispersion relation, is computed for particular solutions, steadily propagating periodic wave trains. For each frequency, ω, below some maximum frequency, ω(max), we find two such solutions, one fast and one slow. The latter are likely unstable as a computational example illustrates. The solitary pulse is obtained in the limit as ω tends to zero. At high frequency, speed drops significantly below the solitary pulse speed; for 6.3°C, the drop at ω(max) is >60%. For an intermediate range of frequencies, supernormal speeds are found and these are correlated with oscillatory swings in sub- and superexcitability in the return to rest of an impulse. Qualitative consequences of the dispersion relation are illustrated with several different computed pulse train responses of the full cable equations for repetitively applied current pulses. Moreover, changes in pulse speed and interspike interval during propagation are predicted quantitatively by a simple kinematic approximation which applies the dispersion relation, instantaneously, to individual pulses. One example shows how interspike time intervals can be distorted during propagation from a ratio of 2:1 at input to 6:5 at a distance of 6.5 cm.

AB - Propagation speed of an impulse is influenced by previous activity. A pulse following its predecessor too closely may travel more slowly than a solitary pulse. In contrast, for some range of interspike intervals, a pulse may travel faster than normal because of a possible superexcitable phase of its predecessor's wake. Thus, in general, pulse speeds and interspike intervals will not remain constant during propagation. We consider these issues for the Hodgkin-Huxley cable equations. First, the relation between speed and frequency or interspike interval, the dispersion relation, is computed for particular solutions, steadily propagating periodic wave trains. For each frequency, ω, below some maximum frequency, ω(max), we find two such solutions, one fast and one slow. The latter are likely unstable as a computational example illustrates. The solitary pulse is obtained in the limit as ω tends to zero. At high frequency, speed drops significantly below the solitary pulse speed; for 6.3°C, the drop at ω(max) is >60%. For an intermediate range of frequencies, supernormal speeds are found and these are correlated with oscillatory swings in sub- and superexcitability in the return to rest of an impulse. Qualitative consequences of the dispersion relation are illustrated with several different computed pulse train responses of the full cable equations for repetitively applied current pulses. Moreover, changes in pulse speed and interspike interval during propagation are predicted quantitatively by a simple kinematic approximation which applies the dispersion relation, instantaneously, to individual pulses. One example shows how interspike time intervals can be distorted during propagation from a ratio of 2:1 at input to 6:5 at a distance of 6.5 cm.

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M3 - Article

VL - 34

SP - 227

EP - 259

JO - Biophysical Journal

JF - Biophysical Journal

SN - 0006-3495

IS - 2

ER -