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Abstract:
Our paper investigates the dynamic relationship between homoclinic bifurcations and mixed-mode oscillations (MMOs) in a reduced three-dimensional singularly perturbed Hodgkin–Huxley (HH) liked model, developed to characterize complex neurodynamic oscillations in neural systems. Through invariant manifold tracking, we establish the coexistence of elusive homoclinic bifurcations and MMOs. Our manifold analysis reveals critical conditions for the emergence of Shilnikov homoclinic bifurcation and oscillatory stability parameter thresholds. Employing Fenichel’s theorem and Bogdanov–Takens (BT) bifurcation theory within a nonlinear multiscale framework, we construct a locally topologically equivalent system for singular dynamics. This system exhibits three canonical bifurcation curves that elucidate the mechanistic origins of MMOs induced by singular BT bifurcations. The proof strategy for Shilnikov homoclinic bifurcations relies on a systematic examination of invariant manifold intersections in multiscale systems. Application to neuronal models demonstrates rich dynamic phenomena including pseudo-plateau bursting, MMOs, and chaotic-MMOs. Numerical simulations validate theoretical predictions with remarkable consistency, particularly in the bifurcation parameter regions.
