article · Chaos An Interdisciplinary Journal of Nonlinear Science
This research investigates the complex behaviour of nerve cells by examining chaotic poles of attraction in the Hindmarsh-Rose neuron model under an external current input. By integrating fractional differentiation into the existing model, an additional parameter representing the non-integer order of the derivative is introduced. The resulting system is solved numerically using Haar wavelets to assess how membrane potential dynamics change under different conditions. The findings show that in the standard scenario where the derivative order equals one, the nerve cell exhibits irregular behaviour characterised by a pole of attraction that generates a limit cycle. As the value of this fractional parameter is reduced, the irregularity of the membrane potential intensifies, causing the pole of attraction to shift into a chaotic state.
Understanding how neurons process electrical signals is fundamental to mapping nervous system function. By demonstrating that fractional calculus can describe irregular and chaotic membrane dynamics, this study provides mathematical tools to better capture the complex, diverse electrical responses that standard models may not fully resolve.
The abstract does not indicate a direct application pathway, as it represents early-stage theoretical and numerical modelling of mathematical neuroscience without defined end users or market-ready outputs.
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Since the neurologists Hindmarsh and Rose improved the Hodgkin-Huxley model to provide a better understanding on the diversity of neural response, features like pole of attraction unfolding complex bifurcation for the membrane potential was still a mystery. This work explores the possible existence of chaotic poles of attraction in the dynamics of Hindmarsh-Rose neurons with an external current input. Combining with fractional differentiation, the model is generalized with the introduction of an additional parameter, the non-integer order of the derivative σ, and solved numerically thanks to the Haar Wavelets. Numerical simulations of the membrane potential dynamics show that in the standard case where the control parameter σ=1, the nerve cell's behavior seems irregular with a pole of attraction generating a limit cycle. This irregularity accentuates as σ decreases (σ=0.9 and σ=0.85) with the pole of attraction becoming chaotic.
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DOI: 10.1063/1.5083180
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