Generalisation of neuronal excitability allows for the identification of an excitability change parameter that links to an experimentally measurable value

Neuronal excitability is the phenomena that describes action potential generation due to a stimulus input. Commonly, neuronal excitability is divided into two classes: Type I and Type II, both having different properties that affect information processing, such as thresholding and gain scaling. These properties can be mathematically studied using generalised phenomenological models, such as the Fitzhugh-Nagumo model and the mirrored FHN. The FHN model shows that each excitability type corresponds to one specific type of bifurcation in the phase plane: Type I underlies a saddle-node on invariant cycle bifurcation, and Type II a Hopf bifurcation. The difficulty of modelling Type I excitability is that it is not only represented by its underlying bifurcation, but also should be able to generate frequency while maintaining a small depolarising current. Using the mFHN model, we show that this situation is possible without modifying the phase portrait, due to the incorporation of a slow regenerative variable. We show that in the singular limit of the mFHN model, the time-scale separation can be chosen such that there is a configuration of a classical phase portrait that allows for SNIC bifurcation, zero-frequency onset and a depolarising current, such as observed in Type I excitability. Using the definition of slow conductance, g_s, we show that these mathematical findings for excitability change are translatable to reduced conductance based models and also relates to an experimentally measurable quantity. This not only allows for a measure of excitability change, but also relates the mathematical parameters that indicate a physiological Type I excitability to parameters that can be tuned during experiments.