HodgkinHuxley

Hodgkin-Huxley is a mathematical model of the electrical properties of excitable cells, named after Alan L. Hodgkin and Andrew Huxley. Introduced in 1952, it describes how action potentials in the squid giant axon arise from voltage-dependent sodium and potassium conductances plus a leak current.

The model represents the cell membrane as a capacitor connected to three conductances: g_Na for sodium, g_K for potassium, and g_L for a passive leak. It uses three gating variables, m, h, and n, which regulate the opening of Na+ and K+ channels. The core equation is C_m dV/dt = I_ext − g_Na m^3 h (V − E_Na) − g_K n^4 (V − E_K) − g_L (V − E_L). The time evolution of m, h, and n follows first-order kinetics with voltage-dependent rate constants α(V) and β(V).

The gating variables capture how ion channel conductances change with membrane potential. The model was derived from voltage-clamp experiments on the squid giant axon and successfully reproduced the generation and propagation of action potentials, including their threshold and refractory behaviors.

In 1963 Hodgkin and Huxley (jointly with John Eccles) received the Nobel Prize in Physiology or Medicine for discoveries concerning ionic mechanisms involved in excitation and inhibition in the nervous system. The Hodgkin-Huxley framework is foundational in neuroscience, enabling detailed simulations of neuronal behavior and influencing later ion-channel models and reduced qualitative descriptions such as FitzHugh–Nagumo. It remains a cornerstone of computational neuroscience and neurophysiology, while acknowledging its assumptions, such as a single isopotential compartment and temperature considerations.