The Hodgkin-Huxley (HH) model, published in 1952 by Alan Hodgkin and Andrew Huxley based on voltage-clamp experiments on the squid giant axon, is one of the most important mathematical models in all of biology. It describes how an action potential is generated through the interplay of voltage-dependent sodium (Na⁺) and potassium (K⁺) ion channels, whose conductances are governed by dynamical gating variables. For this work, Hodgkin and Huxley received the Nobel Prize in Physiology or Medicine in 1963.
The Model Equations
Gating variable dynamics:
dm/dt = α_m(V) · (1 − m) − β_m(V) · m
dh/dt = α_h(V) · (1 − h) − β_h(V) · h
dn/dt = α_n(V) · (1 − n) − β_n(V) · n
m = Na⁺ activation (fast); h = Na⁺ inactivation (slow)
n = K⁺ activation (slow); α, β = voltage-dependent rate functions
The model represents the membrane as an electrical circuit with four parallel branches: a capacitor (the lipid bilayer), a sodium conductance, a potassium conductance, and a leak conductance. Each ionic conductance is the product of a maximal conductance and gating variables raised to integer powers. The gating variables m, h, and n each obey first-order kinetics with voltage-dependent opening (α) and closing (β) rates. The sodium conductance depends on three activation gates (m³) and one inactivation gate (h), while the potassium conductance depends on four activation gates (n⁴).
The Action Potential Mechanism
The action potential emerges from the interplay of these gating dynamics. When the membrane is depolarized above threshold, the fast sodium activation gates (m) open rapidly, allowing Na⁺ influx that further depolarizes the membrane — a positive feedback loop. The slower sodium inactivation gates (h) then close, reducing Na⁺ conductance, while the slow potassium activation gates (n) open, allowing K⁺ efflux that repolarizes the membrane. The result is the characteristic spike waveform: a rapid depolarization followed by repolarization and a brief hyperpolarization (undershoot). The refractory period arises because the Na⁺ inactivation gates require time to recover.
The Hodgkin-Huxley framework established the paradigm of conductance-based modeling that remains the gold standard for biophysically detailed neural simulation. Every subsequent ion channel model — including channels for calcium, HCN, and various modulatory currents — follows the same formalism of voltage- and time-dependent gating variables. The Human Brain Project and the Allen Institute's Brain Observatory use Hodgkin-Huxley-type models with dozens of ion channel types to simulate individual neurons with high fidelity. In mathematical psychology, the HH model is important as the biophysical foundation that grounds more abstract models (like integrate-and-fire and rate models) in membrane biophysics.
The mathematical richness of the HH model has made it a subject of extensive nonlinear dynamics analysis. Depending on parameter values, the model exhibits a variety of dynamical behaviors — quiescence, tonic firing, bursting, and bistability — that correspond to different neural response modes. FitzHugh (1961) and Nagumo, Arimoto, and Yoshizawa (1962) simplified the four-dimensional HH system to two dimensions (the FitzHugh-Nagumo model), preserving the essential qualitative dynamics while enabling phase-plane analysis and providing a more intuitive understanding of excitability and oscillation.