REDUCTION OF THE HODGKIN-HUXLEY EQUATIONS TO A SINGLE-VARIABLE THRESHOLD-MODEL

It is generally believed that a neuron is a threshold element that fir es when some variable u reaches a threshold. Here we pursue the questi on of whether this picture can be justified and study the four-dimensi onal neuron model of Hodgkin and Huxley as a concrete example. The mod el 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 postsynapt ic 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 s pike and the typical afterpotential is included. Our model neuron fire s if the membrane voltage, given by the truncated response kernel expa nsion, crosses a threshold. The threshold model is tested on a spike t rain generated by the Hodgkin-Huxley model with a stochastic input cur rent. We find that the threshold model predicts 90 percent of the spik es correctly. Our results show that, to good approximation, the descri ption of a neuron as a threshold element can indeed be justified.