Model Neurons: from Hodgkin-huxley to Hopfield

A new technique is used to reduce the classic Hodgkin-Huxley model to a two-dimensional neuronal model which retains the essential features of the full description. The resulting equations are similar in spirit to those of the FitzHugh-Nagumo model, but they exhibit some essential diierences which provide a more accurate representation of the full dynamics. By further simplifying the two-dimensional model we derive linear and nonlinear integrate and re models and an interesting binary model with time-dependent threshold and hysteresis. All the parameters of these simpliied models are obtained through the reduction procedure from the underlying Hodgkin-Huxley equations. Thus, our methods establish direct connections between highly simpliied models of neuronal dynamics and more realistic descriptions in terms of time-and voltage-dependent conductances.