The Hodgkin-Huxley Model

The Hodgkin-Huxley (HH) model describes the squid giant axon membrane as an electrical circuit with a capacitor (the lipid bilayer) in parallel with variable resistors (ion channels), each resistor having its own battery (equilibrium potential) set by a Nernst equilibrium potential for that ion. Total membrane current is the sum of capacitive current and three ionic currents: a sodium current, a potassium current, and a small "leak" current carried mostly by chloride and other ions. The critical insight, established through voltage clamp experiments, is that the sodium and potassium conductances are not fixed resistances — they change with both membrane voltage and time, and they do so in characteristic, reproducible ways. Each conductance is written as a maximum conductance multiplied by dimensionless gating variables between 0 and 1, each representing the fraction of channels in a permissive state. Sodium conductance uses two gates: an activation gate m that opens quickly with depolarization, and an inactivation gate h that closes more slowly with depolarization. Potassium conductance uses a single activation gate, n, that opens slowly with depolarization and has no inactivation on this timescale. Each gating variable relaxes toward a voltage-dependent steady state with its own voltage-dependent time constant, governed by first-order kinetics with rate constants α(V) and β(V) that Hodgkin and Huxley fit empirically to voltage-clamp data — they did not derive these rate functions from any deeper molecular theory, since the physical identity of channels and their gating charges was unknown until the 1980s–1990s. The model's power is that these gating parameters, fit only to voltage-clamp step responses, are sufficient — with no additional free parameters — to reconstruct the freely propagating action potential when plugged into the cable equation for membrane current. That the same four numbers (three gates plus voltage) that explained clamped, non-propagating responses also predicted an unclamped, propagating spike of the correct amplitude, shape, threshold, and velocity is the central evidentiary triumph described below.

Step 1 — the membrane as a circuit. Treat a patch of axon membrane as a capacitor of specific capacitance Cm ≈ 1 μF/cm² in parallel with ionic conductances. Total membrane current density is: Im = Cm(dV/dt) + INa + IK + IL where each ionic current follows an Ohmic driving-force form: INa = ḡNa·m³h·(V − ENa) IK = ḡK·n⁴·(V − EK) IL = ḡL·(V − EL) Here ḡNa and ḡK are the maximal conductances (fully-open channel population), and ENa, EK, EL are the Nernst/reversal potentials for each pathway, fixed by ion concentration gradients across the membrane. For squid axon at roughly 6.3 °C, Hodgkin and Huxley used:

Step 2 — why m³h and n⁴. The exponents were not assumed; they were fit to the shape of the current's time course. Sodium conductance turns on with an S-shaped delay (best fit by raising an exponential rise to the third power, m³) and then decays due to inactivation (h). Potassium conductance turns on with a similar delayed, sigmoidal rise, best fit by the fourth power of a single rising exponential (n⁴). Hodgkin and Huxley interpreted m, h, and n as the probabilities that individual "particles" controlling the channel gate are in a permissive position, with three independent activating particles for Na⁺ and four for K⁺ — a phenomenological device, since actual channel structure (four subunits each contributing a voltage sensor, in the real Na⁺ and K⁺ channels) was unknown at the time but turned out to rhyme remarkably well with the fitted exponents.

Step 3 — gating kinetics. Each gating variable x ∈ {m, h, n} obeys a first-order relaxation: dx/dt = αx(V)·(1 − x) − βx(V)·x equivalently written as dx/dt = (x∞(V) − x) / τx(V) where x∞(V) = αx/(αx+βx) is the voltage-dependent steady-state value and τx(V) = 1/(αx+βx) is the voltage-dependent time constant. Hodgkin and Huxley fit αx(V) and βx(V) as explicit exponential/sigmoidal functions of V directly from voltage-clamp records — six independent functions in total (α and β for each of m, h, n), giving the model its characteristic look of "many equations, no unexplained free parameters left over."

Step 4 — the emergent dynamics of a spike. 1. A depolarizing stimulus raises V above threshold (roughly −55 mV from a resting potential near −65 mV). 2. Because τm is very fast (fractions of a millisecond) at depolarized voltages, m rises almost instantly, opening Na⁺ channels and driving INa inward. 3. Inward Na⁺ current further depolarizes V, which further raises m∞ — a positive feedback loop (regenerative depolarization) that produces the fast upstroke. 4. V approaches ENa (+55 mV) as the peak, but two brakes engage on a slower timescale: h falls toward 0 (Na⁺ inactivation) and n rises toward 1 (delayed K⁺ activation). 5. Falling h shuts off INa; rising n opens K⁺ channels, driving outward IK that repolarizes V. 6. V undershoots below resting potential (afterhyperpolarization) because n decays back to baseline more slowly than it rose, keeping K⁺ conductance elevated briefly after V has returned near EK. 7. During this period h is still low and n is still elevated, producing the absolute and relative refractory periods — the biophysical basis of one-way action potential propagation and a maximum firing rate. Because ionic currents also flow axially along the axon core (governed by the cable equation, using core resistance per unit length), a spike at one point in the membrane depolarizes the adjacent patch above threshold, and the entire cycle repeats down the axon — this is what makes the model, originally fit from a space-clamped (non-propagating) preparation, capable of predicting a propagating wave with a specific conduction velocity.

The voltage clamp (Cole, Marmont, Hodgkin, Huxley; Hodgkin, Huxley & Katz, 1952). The central technical obstacle to studying ionic currents was that membrane voltage and current are entangled: current through channels changes voltage, and changing voltage changes channel opening, in a runaway loop that makes the underlying conductance impossible to see directly. Kenneth Cole had introduced the concept of clamping membrane potential; Hodgkin and Huxley, working with Bernard Katz at the Marine Biological Laboratory in Plymouth after the war, built a feedback amplifier system that injected exactly the current needed to hold V at a chosen, fixed level via an internal longitudinal wire electrode threaded down the giant axon (space-clamping the whole segment so V was uniform along its length). With voltage fixed, any measured current had to be ionic current, not capacitive current (since dV/dt = 0 at the clamped steady voltage, aside from the instant of the step) — this decoupling was what let them isolate INa and IK cleanly for the first time.

Ionic substitution to separate Na⁺ and K⁺ currents. In seawater, stepping V from rest to a depolarized level produced a biphasic current: a brief inward phase followed by a sustained outward phase. To determine which ion carried which phase, Hodgkin and Huxley replaced external Na⁺ with the impermeant cation choline in a series of experiments (1949, with Katz), and found the early inward current vanished and reversed as expected for a Na⁺-selective current shifting with ENa, while the late outward current was unaffected — proof that two pharmacologically and temporally separable conductances underlie the response, one Na⁺-selective and one (later shown) K⁺-selective.

Reconstructing the gating kinetics. By stepping to many different clamp voltages and measuring the resulting current time courses, Hodgkin and Huxley extracted gNa(t) and gK(t) at each voltage. They found gNa rose then fell (activation then inactivation) while gK rose and stayed elevated (activation only, on this timescale) — and that both curves could be fit by their m³h and n⁴ forms with voltage-dependent rate constants. This curve-fitting exercise, done by hand and with a mechanical calculator (Huxley later used a hand-cranked Brunsviga machine for numerical integration, since electronic computers were not yet accessible to them for this purpose), is what generated the explicit α(V), β(V) functions.

The decisive test: reconstruction, not just fitting. The critical evidence that the model was correct, rather than merely descriptive, came from using parameters fit only from clamped (non-propagating) step responses to predict the fully dynamic, propagating action potential — a completely different experimental regime never used in the fitting. Hodgkin and Huxley numerically integrated their four coupled equations (by hand, iteratively) and found:

• Predicted action potential peak: about +40 to +50 mV overshoot above 0 mV — matching directly recorded spikes to within roughly 1 mV.
• Predicted threshold behavior, refractory period, and after-hyperpolarization shape, matching recordings qualitatively and quantitatively.
• Predicted conduction velocity for the propagating spike: about 18.8 m/s at 18.5 °C, compared with an experimentally measured value of about 21.2 m/s in the same paper — agreement within roughly 10%, remarkable given the model's parameters came from an entirely separate, non-propagating clamp experiment.
• Predicted "all-or-none" threshold and an ability to reproduce a subthreshold, decaying local response versus a full regenerative spike above threshold, matching the known excitability properties of real axons.

This four-paper series appeared in the Journal of Physiology in 1952 ("A quantitative description of membrane current and its application to conduction and excitation in nerve" being the capstone paper). The agreement between a model built from clamped step responses and an entirely independent propagating-wave measurement is what elevated the HH equations from curve-fitting to a genuine, falsifiable biological theory — and it is why the model survived essentially unmodified in its ionic-current logic for over 70 years, even after the molecular identity of Na⁺ and K⁺ channels (voltage-gated ion channel proteins with S4 voltage-sensor helices) was established decades later by cloning and structural biology, confirming the physical basis for the m³h/n⁴ phenomenology Hodgkin and Huxley had inferred purely from electrical measurements.

Example 1 — computing driving force and current sign at two points on the spike. Take resting values ENa = +55 mV, EK = −72 mV, and consider two moments: At rest, V = −65 mV. Driving force for Na⁺: V − ENa = −65 − 55 = −120 mV (large inward driving force, but gNa is tiny at rest since m∞ is small, so INa ≈ 0). Driving force for K⁺: V − EK = −65 − (−72) = +7 mV (small outward driving force; gK is also small at rest, giving a small resting IK that helps set resting potential together with leak). At the peak of the spike, V ≈ +40 mV. Driving force for Na⁺: V − ENa = 40 − 55 = −15 mV. This is still inward, but small — showing why the spike peak sits close to, but below, ENa: as V rises the net Na⁺ driving force itself shrinks even as m keeps rising, and h has already started falling, which together with the shrinking driving force explains why the upstroke decelerates and turns over near ENa rather than continuing indefinitely. Driving force for K⁺: V − EK = 40 − (−72) = +112 mV, very large and outward — and because n has been rising throughout the upstroke (with a lag, since τn is slower than τm), IK becomes substantial exactly during the falling phase, driving repolarization. This shows quantitatively why the action potential is self-limiting: not because "the channels get tired," but because rising V simultaneously shrinks the Na⁺ driving force, triggers h to fall (Na⁺ inactivation), and — with a delay set by τn — turns on a large outward K⁺ current.

Example 2 — reasoning through a clinical/pharmacological perturbation: tetrodotoxin (TTX) and a demyelinating scenario. Suppose an experimenter applies TTX, a puffer-fish toxin that selectively and essentially completely blocks voltage-gated Na⁺ channels, to a segment of axon, effectively setting ḡNa → 0 in the HH equations while leaving ḡK, ḡL, and Cm unchanged. Predict the outcome using the model: with ḡNa = 0, the INa term vanishes from the membrane current equation entirely, regardless of m or h. Any depolarizing stimulus can no longer trigger the regenerative m-driven inward current that produces the upstroke. The membrane can still passively depolarize in response to injected current (behaving as a simple RC circuit with leak conductance), but there is no threshold, no all-or-none upstroke, and no propagating spike — the axon becomes electrically inexcitable, consistent with TTX's known clinical effect as a paralytic poison (it blocks action potentials in motor and sensory axons alike). Now contrast with a demyelination scenario (relevant to conditions such as multiple sclerosis), which the base HH model does not include Na⁺/K⁺ conductance changes for, but which enters through the cable properties: demyelination increases membrane capacitance per unit length and decreases effective transverse resistance, both of which slow the effective time constant of the cable and reduce the safety factor for regenerative propagation from node to node (in myelinated fibers) or reduce conduction velocity along the cable (in the HH framework applied more generally). The model predicts, correctly, that partial demyelination can still support local subthreshold responses and even a spike at a single patch (INa is intact — m, h, n kinetics are unaffected by loss of myelin), but that conduction can fail at a point down the axon if the local depolarization arriving from the previous patch is insufficient to bring the next patch above threshold before it decays — explaining why demyelinating disease produces conduction block and slowing rather than an inability to generate a spike at all.

Alan Lloyd Hodgkin and Andrew Fielding Huxley worked together at the Physiological Laboratory in Cambridge and at the Marine Biological Association laboratory in Plymouth, using the Atlantic squid's uniquely large axon as a natural experimental gift — an axon wide enough to cannulate with a fine silver wire for internal voltage recording and current injection, something impossible in ordinary vertebrate axons at the time. Their collaboration was interrupted by the Second World War, during which both worked on radar and aviation medicine; they resumed the squid axon work in 1947–1949 and completed the definitive voltage-clamp and modeling papers by 1952. Huxley in particular carried out the laborious hand-integration of the four coupled nonlinear differential equations using a mechanical calculating machine, since digital computers were not practically available to them for this purpose — a striking contrast with how trivially the same integration is done today in a few lines of code.

The HH formalism is now the universal template for modeling any excitable cell, not just neurons: cardiac myocyte models (used to study arrhythmia and to test antiarrhythmic drugs in silico before clinical trials), pancreatic β-cell models of insulin-secreting bursting activity, and smooth-muscle models all extend the same architecture of maximal conductance × gating variables × driving force, simply substituting different channel populations (L-type Ca²⁺ channels, HCN "funny" channels, KATP channels) for the squid's Na⁺ and K⁺ conductances. Local anesthetics such as lidocaine and antiepileptic drugs such as phenytoin and carbamazepine work, mechanistically, by binding voltage-gated Na⁺ channels and shifting or slowing their m/h kinetics — a fact directly interpretable in HH terms as reducing effective ḡNa or shifting h∞(V), predicting reduced excitability and raised threshold exactly as the equations describe. In neurology, conduction studies for peripheral neuropathies and demyelinating disease (multiple sclerosis, Guillain-Barré syndrome) are interpreted using cable-theory extensions of the HH framework, since conduction velocity and its failure under demyelination follow directly from the coupling between local HH kinetics and axial cable properties. In biotechnology, HH-style conductance-based models are the computational core of large-scale simulations of neural circuits (e.g., detailed cortical microcircuit models), letting researchers test hypotheses about disease mutations in specific channel genes (channelopathies, such as SCN1A mutations in Dravet syndrome) by altering the corresponding gating parameters and predicting resulting excitability changes before any wet-lab experiment. Pharmaceutical safety testing for cardiac toxicity (the hERG channel assay pipeline) is built on the same conductance-based logic Hodgkin and Huxley introduced: measure a channel's voltage-dependent gating in isolation, then predict its consequences for the whole-cell electrical event.

Ask the tutor: Can you show me how changing ḡK in the equations changes the shape and duration of the predicted action potential?