Integrate-and-fire and Hodgkin–Huxley models with current inputs

Abstract For the integrate-and-fire model and the Hodgkin–Huxley model, we consider how current inputs including α-wave and square-wave affect their outputs. Firstly the usual approximation is employed to approximate the models with current inputswhich quantitatively reveals the difference between instantaneous and noninstantaneous (current) inputs. When the rising time of α-wave inputs is long or the ratio between the inhibitory and excitatory inputs is close to one, the usual approximation fails to approximate the α-wave inputs in the integrate-and-fire model. For the Hodgkin–Huxley model, the usual approximation in general gives an unsatisfying approximation. A novel approach based upon a superposition of ‘coloured’ and ‘white’ noise is then proposed to replace the usual approximation. Numerical results show that the novel approach substantially improves the approximation within widely physiologically reasonable regions of the rising time of α-wave inputs.