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

Alexandru Fikl, Aman Jhinga, Eva Kaslik, Argha Mondal

Fractional Calculus and Applied Analysis (FCAA)

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Abstract

We introduce an efficient discretisation 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 discretisation is based on an extension of L1-type methods that can accurately handle 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 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 the diverse 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.

BibTeX

@article{FiklJhingaKaslikMondal2025,
  title         = {
    Simulating neuronal dynamics in fractional adaptive exponential
    integrate-and-fire models
  },
  author        = {Alexandru Fikl and Aman Jhinga and Eva Kaslik and Argha Mondal},
  year          = 2025,
  month         = mar,
  day           = 24,
  journal       = {Fractional Calculus and Applied Analysis},
  doi           = {10.1007/s13540-025-00392-7},
  issn          = {1314-2224},
  url           = {https://doi.org/10.1007/s13540-025-00392-7},
  language      = {en},
  abstract      = {
    We introduce an efficient discretisation 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
    discretisation is based on an extension of L1-type methods that can
    accurately handle 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 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 the diverse 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.
  },
}