  We introduce an efficient discretization of a novel fractional-order adaptive
exponential (FrAdEx) integrate-and-fire model, which is used to study the
fractional-order dynamics of neuronal activities. The discretization is based
on extension of L1-type methods that can accurately handle the exponential
growth and the spiking mechanism of the model. This new method is implicit and
uses adaptive time stepping to robustly handle the stiff system that arises due
to the exponential term. The implicit nonlinear system can be solved exactly,
without the need for iterative methods, making the scheme efficient while
maintaining accuracy. We present a complete error model for the numerical
scheme that can be extended to other integrate-and-fire models with minor
changes. To show the feasibility of our approach, the numerical method has been
rigorously validated and used to investigate several different spiking
oscillations of the model. We observed that the fractional-order model is
capable of predicting biophysical activities, which are interpreted through
phase diagrams describing the transition from one firing type to another. This
simple model shows significant promise, as it has sufficient expressive
dynamics to reproduce several features qualitatively from a biophysical
dynamical perspective.
