Analytical exploration of fractional neural wave structures and their stability characteristics

The nonlinear fractional soliton neuron model is used in many complicated domains, including fluid mechanics, applied science, neuroscience, nonlinear dynamics, mathematical physics, engineering, biosciences, plasma physics, and geology. It demonstrates how nonlinear waves propagate. This article explores the the fractional nonlinear soliton neuron model, which can play a crucial role in understanding complicated phenomena in neuroscience. This model describes how axons generate and transmit action potentials using a thermodynamic theory of nerve transmission. Signals flowing through the cell membrane (CM) are believed to be single sound pulses, also known as solitons. In this paper we applied the modified Sardar sub-equation method to investigate the exact traveling wave solutions to fractional nonlinear soliton neuron model. In order to discuss the wave patterns of the model, the principal nonlinear equation is transformed to an ordinary differential equation by the usual traveling-wave transformation. The given method provides precise solutions in the form of hyperbolic, trigonometric, and exponential, which results in a well-defined system of solitary wave structures. Consequently, we obtain distinct types of solutions, containing bright solitons, dark solitons, bright-peakon mixtures, peakon and anti-peakon shapes, cuspon and compacton, compacton-kink, kink and anti-kink, dark-bright, bell-shaped solitons, singular periodic, and isolated periodic. The obtained solutions are described in terms of their physical behavior with the help of several visual tools, such as 2D and 3D surface plots, density plots, and contour plots, which have been used to display how the fractional terms reshape the wave profiles, alter their strength, and influence their overall stability. The findings provide a better understanding of the nonlinear wave propagation in neuron-inspired systems and suggest that it may be relevant in any optical media, plasma environment, and computational modeling of wave-based dynamics. Besides these analytical solutions, there is also a modulation instability study, which brings out the fact that minimal perturbation can cause instability of the underlying wave forms. This additional understanding can be used to explain the sensitivity of the model and the circumstances within which wave patterns are stable or start to increase.