## Reduction of the Hodgkin-Huxley Equations to a Single-Variable Threshold Model (1997)

Venue: | NEURAL COMPUTATION |

Citations: | 67 - 22 self |

### BibTeX

@ARTICLE{Kistler97reductionof,

author = {Werner Kistler and Wulfram Gerstner and J. Leo van Hemmen},

title = {Reduction of the Hodgkin-Huxley Equations to a Single-Variable Threshold Model},

journal = {NEURAL COMPUTATION},

year = {1997},

volume = {9},

number = {5},

pages = {1015--1045}

}

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### Abstract

It is generally believed that a neuron is a threshold element which fires when some variable u reaches a threshold. Here we pursue the question of whether this picture can be justified and study the four-dimensional neuron model of Hodgkin and Huxley as a concrete example. The model is approximated by a response kernel expansion in terms of a single variable, the membrane voltage. The first-order term is linear in the input and has the typical form of an elementary postsynaptic potential. Higher-order kernels take care of nonlinear interactions between input spikes. In contrast to the standard Volterra expansion the kernels depend on the firing time of the most recent output spike. In particular, a zero-order kernel which describes the shape of the spike and the typical afterpotential is included. Our model neuron fires, if the membrane voltage, given by the truncated response kernel expansion crosses a threshold. The threshold model is tested on a spike train generated by t...