Simulating neuronal dynamics in fractional adaptive exponential integrate-and-fire models

TL;DR

This paper introduces an efficient numerical discretization method for a novel fractional-order neuronal model, enabling accurate simulation of complex neuronal dynamics with potential applications in biophysical activity prediction.

Contribution

The paper presents a new implicit discretization scheme for fractional adaptive exponential integrate-and-fire models that is both accurate and computationally efficient, handling exponential growth and spiking mechanisms.

Findings

The numerical method accurately simulates various neuronal spiking oscillations.

The fractional model can predict biophysical neuronal activities.

Phase diagrams illustrate transitions between firing types.

Abstract

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.