Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis

The intricate propagation of bioelectrical impulses in neural tissue can be effectively modeled using fractional reaction–diffusion frameworks that capture memory-dependent effects in neuronal signal transmission. In this study, a time-fractional form of the generalized Huxley equation is examined to obtain localized wave and series solutions. By employing the Riemann–Liouville fractional operator within an analytical reduction framework, the governing equation is transformed into a fractional ordinary differential equation characterized by Erdélyi–Kober-type derivatives. The reduced equation admits an explicit power series solution, whose convergence is rigorously analyzed to ensure analytical validity. Furthermore, exact traveling-wave solutions of the model are constructed through a fractional hyperbolic function approach, resulting in diverse localized wave profiles, including single and multiple singular peak structures. Graphical representations of the localized wave and series solutions reveal rich dynamical patterns, emphasizing the influence of the fractional order on the spatial profiles and responsiveness of the model.