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Nonlinear dynamics of a fractional-order neuron model for nerve impulse propagation in cell membranes: Chaos, multistability, control, and soliton phenomena
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DOI：10.1016/j.jppr.2026.04.003.png)
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Abstract
En 中文
This study explores the nonlinear dynamics of a nerve impulse neuron model governed by a partial differential equation (PDE) with beta-fractional derivatives, allowing the inclusion of nonlocal temporal dependence in the model formulation. The model exhibits rich dynamical behavior, including chaos, multistability, and chaos control, alongside the emergence of solitary and periodic wave solutions within the cell membrane. Through a suitable transformation, the PDE is transformed into an ordinary differential equation (ODE), which is further reduced to a planar dynamical system via the Galilean transformation. The stability of the system’s equilibrium points is assessed through eigenvalue analysis of the associated Jacobian matrix. Numerical solutions of the reduced integer-order ODE system are performed using a Runge-Kutta method, and chaotic dynamics are induced via a time-dependent periodic forcing term. Chaos is consistently identified using standard diagnostics, including phase portraits, time-series analysis, Lyapunov exponents, Poincaré maps, bifurcation diagrams, power spectra, return maps, and recurrence plots. The Pyragas time-delayed feedback control method is then applied to stabilize the system, resulting in stabilized periodic states. To explore wave solutions, variational and Hamiltonian approaches are employed to derive analytical expressions for bright, bright-dark, kinky-bright, and periodic wave solutions within the fractional-order framework. The influence of the fractional order on temporal dynamics and spatial wave propagation is illustrated through two- and three-dimensional graphical representations. These wave structures may be interpreted as mathematical analogs of excitation, inhibition, and rhythmic firing mechanisms in neuronal dynamics. The results provide new insights into the nonlinear features of the fractional neuron model, linking memory effects, nonlinear excitability to mathematically meaningful interpretations of neuronal firing and signal transmission.
Keywords:
Nonlinear neuron model
Beta derivative
Chaos
Multistability analysis
Chaos control
Solitary wave solutions
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