Linear Approximation of the Nonlinear Hodgkin–Huxley Model Using First-Order Taylor Expansion for Local Neuronal Behavior Analysis

Abstract

The Hodgkin–Huxley model is one of the most important mathematical models used to describe neuronal membrane dynamics and action potential generation. However, the nonlinear structure of the model makes analytical investigation difficult, particularly near equilibrium conditions. This study presents a mathematical linearization of the Hodgkin–Huxley model around the resting membrane potential using first-order Taylor series expansion. The equilibrium point of the system was determined, and perturbation variables representing small deviations from equilibrium were introduced. The sodium, potassium, and leakage currents were then linearized by evaluating partial derivatives at the equilibrium point. In addition, the gating-variable equations were linearized to obtain a complete system of coupled linear differential equations. The resulting model preserves the essential local behavior of the original nonlinear system while providing a mathematically tractable framework for studying neuronal dynamics. The derived linearized system enables the application of analytical methods such as stability analysis and local dynamic response analysis. Overall, the study demonstrates the importance of linearization in simplifying complex neuronal models and facilitating mathematical investigation in computational neuroscience and biomedical engineering.