The Hodgkin–Huxley model

This is the entry that explains why this site has two directions rather than one. The Hodgkin–Huxley model is the clearest case of an idea crossing the border, being transformed, and crossing back.

A membrane that broke the theory

By the late 1930s the membrane was already being measured as a circuit. At Woods Hole, Kenneth Cole and Howard Curtis put a squid giant axon into an alternating-current Wheatstone bridge and reported in 1939 that during the impulse the membrane resistance fell from about 1,000 Ω·cm² to an average of 25, while the capacitance changed by only about 2 per cent. Excitation was a fortyfold rise in a conductance in parallel with a nearly fixed capacitor: the impulse measured as an impedance before anyone knew which ions carried it.

At first sight that fitted the classical account. Bernstein’s membrane theory, dating from 1902, held that the resting membrane is selectively permeable to potassium and that excitation is a momentary breakdown of that selectivity — and a breakdown can take the membrane potential to zero but cannot reverse it. In 1939 Hodgkin and Huxley at Plymouth, and Curtis and Cole at Woods Hole, pushed a fine capillary electrode down the inside of the axon, a fibre about half a millimetre across. Once the measurements were refined, the action potential came out 40–50 mV larger than the resting potential: the inside swung positive.

War work intervened: more than five years of airborne radar for Hodgkin, gunnery for Huxley. The explanation, worked out by Hodgkin, Bernard Katz and Huxley and tested by Hodgkin and Katz in 1947, was the sodium hypothesis: during the impulse the membrane becomes selectively permeable to sodium, so the potential heads towards sodium’s Nernst potential, positive inside. Swap the sodium in the seawater for choline and the action potential shrinks; the resting potential does not.

Separating the currents

The sodium hypothesis named the ion but not how the permeabilities moved, and a free action potential is the worst place to find out, because conductance and voltage drive each other. The voltage clamp, which Cole and George Marmont had used at Woods Hole in 1947, cuts that loop: a feedback amplifier holds the membrane at a commanded voltage, and the current it supplies is the measurement. At Plymouth in the summer of 1948 Hodgkin, Huxley and Katz used two internal electrodes where Cole had used one: silver wires 20 µm in diameter, pushed 20–30 mm along the axis of the axon, one reading the voltage and the other carrying the current. A conductor along the axis also holds a whole stretch of membrane at one potential, so current does not spread as it would along a cable.

Under the clamp, a depolarising step drew an early inward current and a late outward one. In choline seawater the early inward current disappeared, replaced by a hump of outward current, while the late outward current was only slightly altered. Comparing records at the same voltage in normal and low-sodium solutions split the ionic current into sodium and potassium components, each with its own time course, and the voltage at which the early current reversed moved with external sodium as the potential of a sodium electrode would.

The records, taken at Plymouth in the summer of 1949, were analysed in Cambridge over the next two years and published in 1952 as five papers in The Journal of Physiology — the first with Katz, the other four by Hodgkin and Huxley alone.

Borrowed: the axon as a circuit

Hodgkin and Huxley did not set out to invent a branch of biophysics. They had voltage-clamp data from a squid giant axon and needed a language to describe it, so they reached for the one already available in electrical engineering. The membrane became a capacitor. Each ion species became a conductance in series with a battery, the battery being its Nernst potential. Current conservation gave them

$$C_m \frac{dV}{dt} = -\bar{g}_{\mathrm{Na}} m^3 h (V - E_{\mathrm{Na}}) - \bar{g}_{\mathrm{K}} n^4 (V - E_{\mathrm{K}}) - g_L (V - E_L) + I_{\text{ext}}$$

The move that made it work was letting the conductances depend on voltage, through gating variables $m$, $h$, and $n$ that each relax toward a voltage-dependent steady state with a voltage-dependent time constant. Sodium conductance rises with depolarisation, which depolarises further — positive feedback, and the regenerative upstroke of the action potential. Potassium conductance rises more slowly and pulls the voltage back down. Negative feedback, delayed. An excitable system.

The delay had shown up on an impedance bridge before any clamp record existed. In 1941 Cole and Baker found that the squid membrane behaved as though it contained an inductance, and Cole pointed to thermistors and carbon-filament lamps, which do the same: apply a constant voltage and the current keeps rising as the resistance falls, as it would through an inductor in series with a resistor. A potassium conductance that switches on with a lag is that kind of element, and Huxley later named it as one reason he and Hodgkin built a delay into potassium in their 1947 trial calculations, before any clamp could show one.

Integrated by hand

Fitting clamp records guaranteed nothing about an axon left to itself, where the ionic currents charge the membrane instead of being drawn off by the feedback amplifier. Hodgkin and Huxley had the equations and constants by March 1951 and meant to solve them on EDSAC, the Cambridge University computer, which was then taken out of service for about six months. Huxley solved them by hand, on a Brunsviga mechanical calculator. A membrane action potential, with the whole axon firing at once, took days; the propagated action potential took about three weeks.

In the propagated case, membrane current must also feed the current flowing along the axon, which adds the cable term and turns the equations into a partial differential equation, impracticable by hand. Assuming a wave of fixed shape moving at constant velocity collapses it back to an ordinary differential equation, with the velocity as an unknown. Guess the velocity, integrate forward from rest, and the solution runs away — to one infinity if the guess was too slow, to the other if too fast. Only the true value brings the voltage back to rest after the spike. It is a shooting method, run by hand in time steps of 0.01–0.02 ms through the upstroke, with the voltage kept to the microvolt.

The search settled on $K$ = 10.47 ms⁻¹, a constant that bundles the velocity with the cable properties and depends only on the membrane. Unpacking it gives the conduction velocity, $\theta = \sqrt{Ka/(2R_2C_m)}$, with the fibre’s radius $a$, the axoplasm’s resistivity $R_2$ and the membrane capacitance per unit area $C_m$ all measured on the axon being matched — 238 µm and 35.4 Ω·cm for the first two. That gives 18.8 m/s. The same fibre, at the same 18.5 °C, conducted at 21.2 m/s, and no constant had been adjusted to fit propagation. The formula also says that, for a given membrane, velocity grows only as the square root of the fibre’s radius.

Huxley said the hand computations taught him “the complete inadequacy of one’s intuition in trying to deal with a system of this degree of complexity.”

Why it mattered on the biology side

The fit was extraordinarily good, and it was predictive rather than descriptive: the model implied conduction velocities and threshold behaviour the authors had not fitted to.

It also implied machinery. The exponents on $m^3h$ and $n^4$ are not curve-fitting decoration — they suggest independent gating particles that must all be in place for conduction to occur. That inference pointed straight at discrete molecular gates, twenty-four years before the patch clamp recorded a single channel in a cell membrane. A model built out of borrowed circuit theory told biologists what to go looking for.

Finding the particles

By the time Hodgkin and Huxley shared the 1963 Nobel Prize in Physiology or Medicine with John Eccles, nobody had seen a gating particle, and the final 1952 paper had been careful not to claim one: the equations’ success, it said, was no evidence for the permeability mechanism the authors had pictured while writing them.

It was nonetheless specific about what to look for. Near rest, the sodium conductance rose e-fold for every 4 mV of depolarisation — steep enough, they argued, to imply that some component of the membrane moves as though it carried a large charge or dipole moment. A moving charge is a current, so opening should come with a small current of its own. They saw none, and bounded it at a few per cent of the peak sodium current. They also concluded that the moving components must be sparse, with many sodium ions crossing at each active patch — ion channels in all but name.

Measuring that current was an instrumentation problem, and it took twenty-one years. Clay Armstrong and Francisco Bezanilla blocked the ionic currents with impermeant ions and tetrodotoxin, then paired each test pulse with an equal pulse of opposite polarity that moved no gating charge; summing the two cancelled the linear leak and capacitive transients. What remained was too small and fast for photographed oscilloscope traces, so they used a borrowed waveform eductor — a hundred capacitors switched in sequence, each accumulating one time sample over repeated sweeps: in effect, an analog signal averager. In 1973 they reported a current with the voltage and time dependence expected of sodium gating; their fuller account a year later put its peak at roughly a fiftieth of the sodium current in normal seawater, inside the 1952 bound. Richard Keynes and Eduardo Rojas reported the same current from squid in 1973.

Then the molecules. In 1984 Masaharu Noda, Shosaku Numa and colleagues cloned the electric eel’s sodium channel: one protein of 1,820 amino acids with four repeated domains, each carrying a segment of clustered positive charges that the authors suggested belonged to the gating structure. Mutagenesis later pointed to those charges as the voltage sensor — the charged particle, as a stretch of protein. In 1998 Declan Doyle, Roderick MacKinnon and colleagues published the atomic structure of a bacterial potassium channel: four identical subunits around one pore, with a selectivity filter only 12 Å long, held open at the right size for potassium ions but not the smaller sodium ions. Four, as in $n^4$; MacKinnon had already shown in 1991 that a voltage-gated potassium channel, Shaker, is built from four subunits.

Handed back: silicon neurons

Then the traffic reversed. If a neuron is a capacitor with voltage-dependent conductances, and a subthreshold MOS transistor has an exponential current–voltage relationship of roughly the right shape, you can build the equations rather than integrate them. The gating dynamics become transistor physics; the membrane capacitor becomes a capacitor.

This is the foundation of analog neuromorphic design, and the appeal is not that it is a tidy analogy. It is that the silicon runs the dynamics in real time at a tiny fraction of the energy a numerical solver would need for the same equations. The model that biology borrowed from circuits came back as a circuit.

Electronic nerves were older than that idea. In 1961 Richard FitzHugh captured the excitability of the four-variable Hodgkin–Huxley system in a two-variable generalisation of van der Pol’s relaxation oscillator, and in 1962 Jin-ichi Nagumo and colleagues built it as a transmission line of tunnel diodes, whose current–voltage curve has the cubic shape the equations need. They pitched it as a self-restoring pulse line that strips out signals below threshold, suited to information processing. It reproduced excitability, deliberately, not the conductances.

Circuits built on the conductances themselves followed. In 1968 Edwin Lewis described electronic models of nerve membrane that combined the Hodgkin–Huxley conductances with Eccles’s synaptic ones, and wired a few of them into small neural nets. In 1991 Misha Mahowald, who had built the silicon retina with Carver Mead, and Rodney Douglas published A silicon neuron: an analog chip whose ionic conductances were five-transistor transconductance amplifiers running below threshold, justified on the grounds that conduction in silicon devices and in biological membranes rests on similar physics. In both, a charge crosses a voltage-controlled energy barrier, so a transistor circuit can stand in for a population of gating particles rather than merely imitate their curves. The chip ran in real time on little power, and many such neurons could be built on one die. The price is precision: circuits like this approximate the equations qualitatively rather than solve them exactly.

What the model leaves out

Worth being clear about, since the model’s success has made it something of a default:

• It is a uniform membrane: one patch, or at most a uniform cylinder of it. Real dendrites have geometry, and geometry does computational work that a uniform cable cannot represent.
• The gating variables are phenomenological. They fit the data beautifully without describing any actual channel protein’s conformational states.
• Squid axon is not cortex. Channel complements differ enormously across cell types, and the parameter set does not transfer.
• The independence is wrong in detail. If closing needed only one of three independent particles to return, the gating current at the end of a pulse would decay three times more slowly than the sodium current; it decays at most 1.2 times more slowly. Nor is inactivation a separate particle multiplied in: an inactivated sodium channel holds part of its gating charge in place until it recovers.
• Squid spikes are expensive. Hodgkin noted that the sodium entering per impulse, 3–4 × 10⁻¹² mol/cm², was about three times the charge needed to swing a 1 µF/cm² membrane through 120 mV, because the sodium and potassium currents overlap. Brain energy budgets had assumed a factor of about four; in 2009 Henrik Alle, Arnd Roth and Jörg Geiger measured rat hippocampal mossy-fibre axons at 1.3 times the minimum, with fast sodium decay and late potassium onset keeping the overlap small.

None of which detracts from the point. A description good enough to predict unobserved molecular machinery, and later to be fabricated, has earned its place twice over.