Firing patterns of the modified Hodgkin–Huxley models subject to Taylor ’s formula

It is well known that the classical Hodgkin–Huxley (HH) equations exhibit complex nonlinear dynamic properties. This paper examines the ion conductance equations of 28 modified HH models, calculating the ion channel conductance and simulating neuronal firing patterns. It is interesting to discover and confirm that the ion conductance is exactly the coefficient of the fourth-order term at the origin of Taylor’s formula. It is demonstrated that the equations for these models are closely related to the two-variable Taylor’s formula of the conductance around the origin in terms of the channel parameters. We consider one specific model in great detail as an example. We find that the conductance curves are generally of the same form, but that the conductance peaks differ. Several firing patterns are observed in the modified HH models, including bursting and mixed-mode oscillations. The emergence of mixed-mode oscillations in particular may be of interest for future work.