Fitzhugh-Nagumo Model

The Fitzhugh-Nagumo model is a simplified two-variable dynamical system that captures the essential behavior of neuronal excitability — depolarization, threshold, firing, and recovery — without the four-dimensional complexity of the full Hodgkin-Huxley equations. It was developed independently by Richard FitzHugh in 1961 and Jinichi Nagumo in 1962, and it remains the standard textbook example of a system that exhibits both excitable and oscillatory regimes depending on parameter values.

The model separates the dynamics into a fast voltage-like variable and a slow recovery variable, producing the characteristic cubic nullcline geometry that generates relaxation oscillations and threshold behavior. For parameters below a critical value, the system is excitable: a small perturbation decays, but a perturbation above threshold triggers a single pulse before returning to rest. For parameters above the critical value, the system oscillates spontaneously. The transition between these two regimes is a Hopf bifurcation, and the explosive growth of the limit cycle as the parameter crosses threshold is a canard explosion.

The FitzHugh-Nagumo model is not merely a simplification; it is a dimensional reduction that reveals the geometric skeleton of neuronal dynamics. The full Hodgkin-Huxley equations, despite their biophysical detail, exhibit the same slow-fast structure when the fast sodium and potassium variables are treated as the fast subsystem and the slower recovery processes as the slow subsystem. In this sense, the FitzHugh-Nagumo model is the minimal model that captures the dynamical essence of excitability: a cubic nonlinearity, a threshold, and a recovery mechanism. It has been extended to two spatial dimensions to model propagating action potentials and to coupled networks to study synchronization and pattern formation in neural tissue.