Mathematical Neuroscience of Neuronal Excitability Spectral Stability, Critical Slowing, and Nonlinear Transitions in the Hodgkin–Huxley Model

This preprint presents a computational and mathematical neuroscience analysis of neuronal excitability using the classical Hodgkin–Huxley model. The study examines how nonlinear ion-channel dynamics give rise to transitions between stable resting behavior and sustained action-potential firing through spectral stability analysis, Jacobian eigenvalue calculations, bifurcation structure, and indicators of critical slowing. Numerical analysis is used to characterize stability boundaries in the Hodgkin–Huxley system and to examine changes in voltage dynamics, variance, autocorrelation, and recovery behavior as the model approaches critical transitions. The manuscript also develops a broader framework for interpreting neuronal resilience using concepts from nonlinear dynamical systems, biophysics, computational neuroscience, and mathematical modeling. The work is intended as an exploratory mathematical neuroscience study and does not present clinical data or make direct diagnostic claims. Potential applications to neurological phenomena such as altered neuronal excitability, ion-channel dysfunction, seizure-related dynamics, and channelopathies are discussed as areas for future investigation. Keywords: Hodgkin–Huxley model, mathematical neuroscience, computational neuroscience, neuronal excitability, nonlinear dynamics, bifurcation theory, spectral stability, critical slowing, ion channels, neurophysiology.