Reduction of the Hodgkin-Huxley Equations to a Single-Variable Threshold Model

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Abstract
It is generally believed that a neuron is a threshold element that 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 its kernel 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 that describes the shape of the spike and the typical after-potential 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 the Hodgkin-Huxley model with a stochastic input current. We find that the threshold model predicts 90 percent of the spikes correctly. Our results show that, to good approximation, the description of a neuron as a threshold element can indeed be justified.