Exploring Complex Neuronal Dynamics Using Atangana–Baleanu Fractal-Fractional Order Morris–Lecar Model

Authors

  • M. J. Abedin, M. J. Islam, M. G. Hafez, Yu-Ming Chu

Keywords:

Atangana–Baleanu Fractal-fractional operator; Morris–Lecar Model; Numerical approximation; Neuronal Dynamics.

Abstract

Fractal-fractional calculus offers a powerful framework for modeling physical and biological systems that exhibit memory-dependent behavior. Unlike traditional calculus, it employs differential and integral operators of non-integer order, allowing it to capture dynamics associated with systems possessing fractional or fractal properties. Given the capacity of fractal-fractional derivatives to effectively represent long-term memory effects in neural responses, we extend the classical Morris–Lecar model into the fractal-fractional order domain to better describe the dynamics of neuronal activity. This generalized Atangana–Baleanu fractal-fractional order Morris–Lecar (FFML) model is constructed to explore the complex spiking behavior inherent to neuron systems, and its performance is compared with that of the original integer-order model. Since exact analytical solutions of the FFML model are generally not obtainable, we rely on numerical approximation techniques to study its behavior. Our simulations reveal that, depending on the fractional derivative order  with fractal time dimension  and input current levels, the model is capable of exhibiting a wide range of dynamics, including quiescent states, regular spiking, and bursting patterns—similar to the classical model but under different parameter regimes. Furthermore, we identify several bifurcation phenomena in the fractional-order, fractal-dimensional, or both fractal- fractional system by varying the input current and the order of the derivative and time dimension. By tuning the fractional order and fractal-dimension, we are able to classify different dynamical regimes of the model. This enhanced flexibility provides a richer understanding of the complex behaviors exhibited by single neuron systems, making the fractal-fractional approach a valuable tool in computational neuroscience.

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Published

2025-06-10

How to Cite

M. J. Abedin, M. J. Islam, M. G. Hafez, Yu-Ming Chu. (2025). Exploring Complex Neuronal Dynamics Using Atangana–Baleanu Fractal-Fractional Order Morris–Lecar Model. Journal of Computational Analysis and Applications (JoCAAA), 34(6), 35–55. Retrieved from https://eudoxuspress.com/index.php/pub/article/view/2940

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Articles