Higher order effects on rate reduction for networks of hodgkin-huxley neurons

We propose a systematic method of rate reduction for a Hodgkin–Huxley type neural network model. In this context, Shriki et al. assumed that the threshold of the f – I curve for the reduced rate model depends linearly on the leak conductance of the Hodgkin–Huxley equation, while its gain remains constant. First, we show that the threshold and gain have second order dependence on the leak conductance. Second, we show that the Hodgkin–Huxley type network with second order interaction can be naturally reduced to an analog type neural network model with higher order interaction based on this finding. Finally, we construct statistical mechanics for the Hodgkin–Huxley type network with the Mexican-hat interaction through our rate reduction technique.