Memory-Controlled Pattern Formation in Fractional Neural Field Models with Struve-Type Synaptic Connectivity
Keywords:
fractional neural field, Struve function, synaptic connectivity, pattern formation, Turing instability, nonlocal modelsAbstract
We study a fractional-in-time neural field equation with Struve-type synaptic connectivity and examine how temporal memory influences spatial instability. Working in a Hilbert space framework, we establish existence and uniqueness of mild solutions. In the one-dimensional setting, the Fourier transform of the Struve kernel yields an explicit dispersion relation, allowing a direct spectral characterization of unstable modes. We show that the linear instability threshold depends solely on the spectral parameter and remains independent of the fractional order. However, for a fixed observation horizon $T>1$, the amplification factor $E_\alpha(\lambda T^\alpha)$ increases strictly with $\alpha$ when $\lambda>0$, demonstrating that memory alters the time scale of pattern growth without shifting the onset of instability. Numerical experiments illustrate the theoretical findings.
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