the Hodgkin-Huxley model
/ HOJ-kin HUKS-lee /
When you touch something hot, a signal races up your nerve to your brain — a sharp electrical spike that travels without fading and without distorting, a 'nerve impulse' or action potential. How does a thin, leaky biological cable carry a clean pulse over a long distance? The Hodgkin-Huxley model is the mathematical answer, a triumph of biophysics that earned its authors a Nobel Prize. It is one of the most influential reaction-diffusion systems ever written.
The model is a PDE for the voltage V(x, t) along a nerve fibre, coupled to ordinary differential equations for the state of ion channels. The voltage obeys a cable equation of the form V_t = D V_xx + (ionic currents)/C, where D V_xx is diffusion of voltage along the fibre (the cable acting like a leaky wire) and the ionic-current term is a nonlinear reaction depending on gating variables that open and close sodium and potassium channels. So it has the reaction-diffusion shape u_t = D u_xx + f(u): diffusion spreads the disturbance, while the nonlinear reaction f regenerates and sharpens it. The interplay produces a travelling-wave solution — a pulse of fixed shape moving at constant speed — which is exactly the action potential.
Why this matters beyond neuroscience: the Hodgkin-Huxley system is the founding example of an excitable medium, where a small stimulus either dies out or triggers a large, self-sustaining, regenerative pulse — an all-or-nothing threshold. The simpler FitzHugh-Nagumo equations capture the same qualitative behaviour with two variables and are the standard playground for studying nerve pulses, heart-tissue waves, and spiral waves in cardiac arrhythmia. The same reaction-diffusion mathematics underlies chemical waves, flame fronts, and the spread of populations.
Subthreshold versus suprathreshold: poke a nerve gently and the voltage bump just diffuses away and dies (D V_xx wins). Poke it past a threshold and the nonlinear sodium current f(V) fires, regenerating the pulse as it goes, so it travels the whole length of the axon at constant amplitude — the all-or-nothing law of the nerve impulse.
Reaction-diffusion u_t = D u_xx + f(u); diffusion spreads, reaction regenerates a travelling pulse.
Full Hodgkin-Huxley is a coupled PDE-plus-ODE system with four interacting variables, not a single tidy equation; the famous tractable mathematics usually comes from its reduction, the two-variable FitzHugh-Nagumo model. The biology is a model fitted to the squid giant axon, accurate but species- and condition-specific.