Neuronal Dynamics of an Intrinsically Bursting Neuron Through the Caputo–Fabrizio Fractal–Fractional Hodgkin–Huxley Model

This study introduces a novel fractal–fractional extension of the Hodgkin–Huxley model to capture complex neuronal dynamics, with particular focus on intrinsically bursting patterns. The key innovation lies in the simultaneous incorporation of Caputo–Fabrizio operators with fractional order α for memory effects and fractal dimension τ for temporal scaling, enabling the representation of nonlocal interactions and multiscale dynamics that extend beyond the capabilities of classical models. Our numerical simulations demonstrate that the synergistic combination of α,τ parameters uniquely modulates burst duration, interburst intervals, and spike-frequency adaptation, producing dynamical regimes inaccessible to both integer-order models and single-parameter fractional approaches. Lyapunov stability analysis confirms that the framework maintains biological plausibility while enabling substantially richer temporal patterns. This work establishes a comprehensive mathematical foundation for understanding multiscale neuronal behavior, with significant implications for analyzing pathological rhythms and advancing neuromodulation studies.