Hopf bifurcation and dynamical transitions in a fractional-order FitzHugh-Rinzel model with multiple time delays

This paper studies the Hopf bifurcation and transitions of firing activities in a fractional-order FitzHugh-Rinzel system with multiple time delays. We first explicitly derive the stability condition of the system without delays and the fractional-order-induced Hopf bifurcation distinguishes between resting and firing. When unstable, there exist complex oscillations, including spiking, mixed-mode oscillations, and diverse types of mixed-mode bursting oscillations. Bifurcation diagrams and Poincaré sections uncover the generation mechanisms of bursters in FFHR and provide a detailed description of the transient patterns. The critical values of delays when FDFHR undergoes Hopf bifurcation under different existence of two delays are derived, presented by Hopf bifurcation curves. The study concludes that the fractional-order derivative and delays, capturing memory and historical influences in neural processing significantly enrich the dynamics compared to its integer-order counterpart, fostering a diverse range of activities. • Fractional-order-dependent HB of FFHR is derived by linearization and stability theory. • Transition mechanisms between MMOs and MMBOs are identified by Poincaŕe section. • HBs of FDFHR with different distribution of τ 1 , τ 2 are identified. • Conditions of HB in FHR, FFHR, and FDFHR interact with each other. • A fractional-order system with multiple delays is suitable in depicting the neural activities.