FitzHugh-Nagumo Model

The FitzHugh-Nagumo model is a simplified two-variable dynamical system that captures the essential phenomenology of excitable media — systems that respond to small perturbations with either quiescence or a full all-or-none impulse, depending on whether the perturbation crosses a threshold. It was developed independently by Richard FitzHugh in 1961 and Jinichi Nagumo in 1962 as a reduction of the Hodgkin-Huxley equations of nerve membrane excitation.

The model couples a fast activator variable (voltage or chemical concentration) with a slow recovery variable (gating or inhibitor). Below threshold, perturbations decay; above threshold, the system executes a characteristic pulse: rapid excitation followed by slower recovery. In spatially extended form — a reaction-diffusion system — these pulses propagate as travelling waves, the mathematical structure of which is identical to the chemical waves in the Belousov-Zhabotinsky reaction, the excitation waves in cardiac tissue, and the action potentials in neural axons.

The FitzHugh-Nagumo model is the canonical example of how a single mathematical structure can instantiate across radically different physical substrates. The isomorphism between chemical, cardiac, and neural excitation is not metaphorical; it is the empirical signature of a universal dynamical mechanism.

The FitzHugh-Nagumo model is often treated as a toy model in mathematical biology. This is a failure of imagination. It is one of the most important examples of universal behavior in nonlinear dynamics — the proof that the same equations govern nerve impulses, chemical waves, and cardiac arrhythmias. Fields that study these phenomena separately are not discovering different things. They are discovering the same thing in different costumes, and refusing to compare notes.