Stochastic Nonlinear Dynamics of Neural Pulses: Stability Loss and Chaotic Signal Transport

This work explores the nonlinear dynamics of neural signal propagation in the brain using the prism of a Stochastic Generalized Derivative Nonlinear Schrödinger Equation (SGDNLSE). The model encapsulates important biophysical aspects of neural communication through the inclusion of stochastic perturbations (synaptic noise), nonlinear wave interactions, and dispersion. In deriving exact soliton solutions, we utilize [Formula: see text] method. These solution forms capture localized signal structures pertinent in neural wave phenomena like spike propagation and burst firing. The analysis offers a rich parameter study showing the effects of noise amplitude [Formula: see text], nonlinearity coefficient [Formula: see text], and other wave features on the soliton shape, stability, and dynamics. Numerical solutions and 3D visualizations are presented to show how higher levels of stochasticity modulate the coherence and morphology of neural pulses, providing insight into signal fidelity breakdown in disordered or noisy neural media. In addition, a full bifurcation analysis is performed to categorize the system’s critical points and identify conditions for transition from stable signal conduction to complex or chaotic dynamics. Chaos and sensitivity analyses also provide insight into the system’s reaction to parameter changes and initial condition variations, highlighting the possibility of irregular firing and oscillatory instability in neural circuits. This structure not only presents novel exact analytical solutions of a class of stochastic nonlinear evolution equations but also presents a biologically meaningful mathematical formulation for understanding neurophysiological signal behaviors under uncertainty, with impact on modeling epilepsy, signal degradation, and other cognitive diseases depending on nonlinear and stochastic processes.