Fractional Memory Meets Neural Oscillations: Analytical Foundations for the Delayed FitzHugh-Nagumo Model

This paper develops a rigorous and unified analytical framework for the study of implicit fractional-order dynamical systems incorporating distributed memory and discrete time delays. The core contribution is the derivation of novel \emph{multivariate Grönwall--Wendroff-type integral inequalities} tailored to weakly singular fractional kernels and implicit nonlinear structures. These inequalities yield explicit \emph{a priori} bounds for vector-valued solutions and constitute a fundamental analytical tool for controlling nonlocal dynamics induced by fractional memory and delayed feedback. Building on these estimates, we establish general existence, uniqueness, and continuous dependence results for a broad class of implicit fractional differential equations by means of fixed-point arguments in suitably weighted Banach spaces. Furthermore, we prove \emph{Ulam--Hyers stability}, providing quantitative robustness estimates with respect to modeling uncertainties and perturbation errors, and extending classical stability notions to implicit systems with memory effects. The applicability of the proposed framework is demonstrated through a detailed analysis of the delayed fractional FitzHugh--Nagumo (FHN-$\alpha$-$\tau$) model, a canonical system in mathematical neuroscience. We derive explicit and verifiable conditions ensuring the existence of nontrivial, orbitally stable limit cycles corresponding to sustained neuronal spiking. In particular, we show that the interplay between subdiffusive fractional memory ($\alpha<1$) and synaptic delay ($\tau$) can induce and regulate self-sustained oscillations, yielding precise parameter thresholds for excitability and rhythmic activity. Overall, this work bridges a gap between abstract fractional calculus and applied dynamical systems theory, providing a versatile analytical toolkit for systems governed by long-range temporal correlations and delayed feedback, with potential applications in neurodynamics, complex biological networks, materials science, and control theory.