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Memory-Induced Synchronization in a Time-Fractional Partly Diffusive Coupled Hindmarsh–Rose Network with Nonlinear Diffusion
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DOI：10.3390/fractalfract10080548.png)
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Abstract
En 中文
We present and analyze a time-fractional, partly diffusive model of two electrically coupled Hindmarsh–Rose neurons in which only the membrane potentials undergo spatial transport, through a nonlinear Neumann m-Laplacian, while the temporal evolution is governed by a Caputo derivative of order ρ∈(0,1]. The fractional operator incorporates hereditary relaxation, whereas the m-Laplacian represents gradient-dependent degenerate transport and recovers ordinary diffusion when m=2. In a Gelfand triple adapted to the no-flux boundary condition, we derive a fractional energy inequality, a uniform dissipative estimate, and the existence of a global weak solution by a Faedo–Galerkin approximation, fractional compactness, and Minty’s method. Uniqueness and continuous dependence are obtained in the stated bounded solution class. We prove global Mittag–Leffler synchronization above an explicit coupling threshold and establish a practical synchronization bound under parameter mismatch. A fully implicit L1 finite-volume method is then constructed; every time step is solvable, uniqueness follows under an explicit monotonicity condition, and the scheme is unconditionally energy dissipative and convergent. Manufactured-solution tests recover the expected 2−ρ temporal and second-order spatial rates. In the neuronal simulations, reducing ρ from 1 to 0.90 lengthens the mean bursting period from about 379 to 565 time units, an increase of roughly one half, and raises the number of spikes per burst from about 31.7 to 38.8. Over 2≤m≤4 the temporal rhythm is essentially unchanged, the burst period staying near 362 time units, while the diffusion exponent reshapes the peak amplitude and the spatial gradient profiles of the traveling fronts. The empirical synchronization threshold for the canonical parameter set is approximately 11.0, far below the global sufficient bound 2.3917×104, which quantifies the conservatism of the analytical certificates.
Keywords:
Caputo fractional derivative
coupled Hindmarsh–Rose neurons
nonlinear diffusion
m-Laplacian
partly diffusive system
global weak solution
Mittag–Leffler synchronization
L1 finite-volume scheme.
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