Tristability and memory effects in a fractional-order 2D-Tagne Nkounga-Yamapi neuronal model

We propose a fractional-order extension of the 2D- Tagne Nkounga–Yamapi (TN–Y) neuronal model, which exhibits the coexistence of a resting state and two oscillatory states. Fractional derivatives introduce long-term memory effects and additional modeling flexibility, enabling richer dynamical behaviors. An equivalent integer-order representation of the fractional equations allows deterministic bifurcation analysis using MATCONT, revealing Hopf and limit-cycle bifurcations, with the fractional orders effectively modulating the location and extent of the multistability domains. Analytical approximations obtained via the residue harmonic balance method yield amplitude–frequency relationships, which are validated against predictor–corrector simulations. The stability of oscillatory states is assessed using the Floquet theory combined with an energy-variation criterion. To account for ion-channel fluctuations, the Gaussian white noise is introduced, and noise-induced bifurcation phenomena are investigated. Specifically, the stochastic dynamics are analyzed through effective pseudo-potential landscapes and stationary probability densities derived from the Fokker–Planck equation. The results show that fractional-order memory and noise intensity strongly modulate tristability and oscillatory behavior, providing insights into neuronal adaptability under deterministic and stochastic influences. • The analysis of effects of fractional order on the deterministic dynamics and tri-rhythmic properties has been investigated. • It is observed that the fractional order shifts the domain of tristability from left to right, leading to interesting neuronal dynamics. • It is revealed that Hopf and limit-cycle bifurcations, with the fractional orders extent of the multi-stability domains. • It is proved that the fractional-order memory and noise intensity strongly modulate tri-stability and oscillatory behavior. • It appeared the combined effects of noise and fractional order are investigated, showing that the fractional order can act as an effective damping parameter.