Stochastic nonlinear dynamics of the FitzHugh–Nagumo model: traveling waves, bifurcations and chaotic neuronal insight

Purpose This research investigates the nonlinear stochastic FitzHugh–Nagumo equation, with a focus on traveling wave solutions, bifurcation theory, chaotic behavior and emphasizing how noise, such as ion channel oscillations and conductivity changes, affects neuronal pulse behavior. Design/methodology/approach By utilizing the extended modified tanh expansion method, diverse analytical solutions, including hyperbolic, trigonometric and periodic forms, are achieved. The impacts of parameter changes on these solutions are visualized in 2D and 3D, and the effects of stochastic excitation are investigated for real-world neuroscience relevance. Findings The study explores the influence of noise on chaotic behavior and wave patterns, with visualizations revealing complex dynamics between parameters and their effects on neuronal activity. Research limitations/implications While the model provides valuable insights, it doesn't fully account for spatial heterogeneity or higher-dimensional systems, which could be explored in future research. Originality/value This work presents novel insights into the role of stochastic noise in neuronal dynamics, providing analytical solutions and deepening our understanding of nerve impulse communication in computational neuroscience.