Exact soliton solutions and bifurcation analysis of the beta time fractional FitzHugh–Nagumo neuronal model in the beta derivative framework

The FitzHugh–Nagumo (FN) equation is an important nonlinear reaction–diffusion equation with significant applications in nerve impulse propagation, circuit theory, and population genetics. Nonlinear evolution equations with fractional-order derivatives have become a promising field in the study of nonlinear dynamics in science and engineering. In this study, we investigate solitary wave solutions of the time-fractional FN model using the generalized Kudryashov method with the beta fractional-order derivative and analyze their dynamical behavior. We derive several novel rational solutions expressed in hyperbolic and exponential functions, which give kink, one-sided kink, anti-kink, and singular solitary waves exhibiting phase-shifting behavior directed by the fractional parameter, highlighting the role of non-local temporal effects. We identify two transcritical bifurcations by varying the key parameters along with a strong restoring force toward the equilibrium state. The sensitivity analysis reveals how the system responds to varying initial perturbations across different dynamic regimes. The results provide new insights into signal propagation in neurons and other excitable media within the fractional-order derivative framework.