The Hodgkin–Huxley
Neuron Model

Alan L. Hodgkin
1914 – 1998

Andrew F. Huxley
1917 – 2012

Working at the Marine Biological Association laboratory in Plymouth, England between 1947 and 1952, Alan Lloyd Hodgkin and Andrew Fielding Huxley performed a series of elegant voltage-clamp experiments on the Loligo forbesi giant squid axon. This particular axon — 0.5–0.8 mm in diameter — was large enough to insert a wire electrode directly into its interior, making it the ideal preparation for measuring membrane currents with then-unprecedented precision. Their findings, published across five landmark papers in the Journal of Physiology in 1952, constituted the first complete quantitative description of the ionic mechanisms underlying the action potential.^[1]

Loligo forbesi — the longfin inshore squid. Its giant axon (≈0.5 mm diameter) was the key preparation used by H&H.
© Hans Hillewaert / CC BY-SA 4.0

The core insight of Hodgkin and Huxley was that the action potential — the brief, stereotyped electrical impulse by which neurons communicate — arises from the coordinated, voltage-dependent opening and closing of ion-selective channels in the cell membrane. By holding the membrane potential at fixed values (voltage clamp) and measuring the resulting ionic currents, they could isolate and characterize separate sodium (Na⁺) and potassium (K⁺) conductances as functions of both voltage and time.

Their mathematical model, derived entirely from experimental data, was able to predict action potential shape, conduction velocity, refractory period, and threshold — phenomena that had been observed but never explained from first principles. For this work, Hodgkin and Huxley (together with Sir John Eccles for his work on synaptic transmission) received the Nobel Prize in Physiology or Medicine in 1963.^[2]

The Hodgkin–Huxley model represents a small patch of excitable membrane as an electrical circuit. The lipid bilayer acts as a capacitor (C_m), storing charge across its ~7 nm thickness. Embedded ion channels are modeled as variable conductances (g_Na, g_K) in series with batteries representing the Nernst equilibrium potentials (E_Na, E_K). A leak conductance (g_L) captures the passive background permeability — primarily Cl⁻ and small K⁺ leakage — which stabilizes the resting potential.

Applying Kirchhoff's current law to the equivalent circuit gives the membrane voltage ODE. The total membrane current must equal the external applied current:

The gating variables m, h, and n each obey a first-order kinetic equation. They represent the probability that individual gating particles (subunits) are in the permissive (open) state. Each evolves toward its voltage-dependent steady state x_∞(V) with time constant τ_x(V):

The voltage-dependent rate functions (transition rates between closed and open states, in ms⁻¹) were determined empirically by Hodgkin and Huxley from fits to their voltage-clamp data. They use the original H&H convention where V = 0 at rest (= −65 mV absolute); threshold ≈ V+15 mV, AP peak ≈ V+100 mV.

The choice of m³h for Na⁺ and n⁴ for K⁺ was motivated by the shapes of the conductance time courses observed experimentally: the sigmoidal activation delay of g_Na required at least three independent activation gates, while the delayed rectifier K⁺ conductance required four. This was later validated by single-channel recordings showing that Na⁺ channels have three independent activation subunits (S4 voltage sensors) and one inactivation gate, while K⁺ channels are homotetrameric.^[5]

Adjust the parameters and stimulus below to explore action potential generation. Try injecting a sub-threshold current (below ~6 µA/cm²) to see the membrane return to rest without firing, or a supra-threshold step to trigger an action potential. Block Na⁺ (simulates tetrodotoxin, TTX) or K⁺ (simulates tetraethylammonium, TEA) channels to see how each contributes to AP shape.

These curves — derived analytically from the rate functions — show how the steady-state activation/inactivation and time constants of each gate depend on membrane voltage. The vertical dashed line tracks the current simulated membrane potential. Notice how m_∞ (Na⁺ activation) shifts right of h_∞ (Na⁺ inactivation): depolarization quickly opens m before h shuts, creating the brief window of Na⁺ permeability that drives the action potential upstroke.

The phase plane projects the high-dimensional state of the neuron onto two dimensions. Plotting the membrane potential V against the K⁺ activation variable n reveals the limit cycle — the closed trajectory traced by a periodically firing neuron. During rest, the system sits at a stable fixed point (lower left). A sufficiently large perturbation kicks the system onto the unstable spiral, which falls onto the limit cycle. The system then returns to the fixed point after one AP (if sub-threshold) or continues to orbit (if tonically firing).^[3]