Hodgkin-Huxley model

The Hodgkin-Huxley model is a set of four coupled differential equations, published by Alan Hodgkin and Andrew Huxley in 1952, that describes how the voltage across the membrane of a nerve fiber changes during a nerve impulse. It does so by tracking two ionic currents, sodium into the fiber and potassium out of it, each switched by voltage. The equations were fitted to measurements on the giant axon of the squid and reproduce the shape and speed of its action potential. Hodgkin and Huxley shared the 1963 Nobel Prize in Physiology or Medicine with John Eccles "for their discoveries concerning the ionic mechanisms involved in excitation and inhibition in the peripheral and central portions of the nerve cell membrane".^[1]

The model is a description of currents. It says how much sodium and potassium conductance the membrane has at each voltage and time, and it does not say what the conducting parts are. Those were later identified as ion channels, whose single openings are recorded with the patch clamp as steps of a few picoamperes, the quantity a nanopore instrument also measures.

By 1939 the accepted account was Julius Bernstein's: at rest a nerve membrane is permeable to potassium alone, and during an impulse it becomes permeable to everything, so the voltage across it falls to zero. Kenneth Cole and Howard Curtis had confirmed that the electrical impedance of the membrane drops during an impulse.^[2][3] The squid supplied the preparation. Its giant nerve fibers, which J. Z. Young had found, run along the body wall and drive the muscles that expel water from the mantle cavity, and some are as much as a millimeter across, wide enough to thread an electrode down the inside.^[4]

In the summer of 1939 Hodgkin and Huxley at the Plymouth marine laboratory, and Curtis and Cole at Woods Hole, made the first recordings from inside a giant axon. At rest the inside of the fiber was about 45 millivolts negative. At the peak of the impulse it was about 40 millivolts positive, which was not what Bernstein's account allowed. Hodgkin and Huxley left Plymouth on August 30 because war was obviously imminent, and published the result as a short letter to Nature with no explanation of the overshoot.^[2][5]

Cole used a feedback circuit in 1947 to hold the voltage across a squid membrane at a chosen value, because the impulse is explosive and its currents cannot otherwise be measured. Hodgkin, Huxley and Bernard Katz did their first experiments with the method in the summer of 1948.^[2] With the voltage fixed, the current the amplifier must supply is the current through the membrane. Replacing the sodium of the bathing solution with choline separated the sodium current from the potassium current.^[6][7] The series of papers that resulted appeared in the Journal of Physiology in 1952, and the last of them gave the equations.^[8]

The model treats a patch of membrane as a capacitor in parallel with three conductances:

Three variables between 0 and 1, m, h and n, each change at rates α and β that depend on the voltage: dx/dt = α_x(1 − x) − β_xx. Sodium conductance depends on an activation variable m, which rises fast when the membrane is depolarized, and an inactivation variable h, which falls more slowly, so the sodium current starts quickly and then shuts itself off. Potassium conductance follows n, which rises more slowly still. The exponents 3 and 4 were chosen to fit the data.

In the form found in textbooks, for an axon at 6.3 °C, the membrane capacitance is 1 µF/cm² and the maximum conductances are 120 mS/cm² for sodium, 36 for potassium and 0.3 for leak, with reversal potentials of +50, −77 and −54.4 mV and a resting potential near −65 mV.^[a] A simulation of this form with a 1 ms pulse of 10 µA/cm² produces a spike that peaks near +39 mV about 2.5 ms after the start of the pulse, with a width at half height of about 1.4 ms, then undershoots to about −76 mV before the membrane settles at −65 mV. The peak sodium current is about 0.8 mA/cm² inward.

Huxley computed the solutions by hand. For a propagating impulse he assumed a constant velocity, which turns the partial differential equation into an ordinary one, and he adjusted the velocity until the solution stopped running off to infinity. Hodgkin recalled that the propagated action potential took about three weeks to compute on a hand-operated Brunsviga calculating machine.^[2][9] The computed speed was 18.8 meters per second against a measured 21.2 at 18.5 °C.^[8]

Hodgkin and Huxley did not decide how the ions cross. In the words of the first paper of the series, "Details of the mechanism will probably not be settled for some time", and their equations are consistent with carriers as well as pores.^[10] Huxley wrote in 2002 that they interpreted the results as ions crossing through channels opened and closed by the voltage, a view that has been amply confirmed, though he did not think they had considered it in their speculations of 1946 to 1948.^[2]

Hodgkin and Keynes inferred in 1955 from the flow of radioactive potassium that ions move through the potassium path in single file. Neher and Sakmann recorded the currents of single channels in 1976, and the sodium channel was cloned in 1984.^[10][11][12] In 1998 Roderick MacKinnon's group solved the structure of a bacterial potassium channel, four identical subunits around a pore, with a selectivity filter 12 ångströms long. The model's potassium conductance is proportional to the fourth power of one variable.^[13]

The model's currents are densities per square centimeter of membrane, about 0.8 mA/cm² at the sodium peak in the simulation above. A single open channel passes a current of the order of a picoampere. The amplifier that records it is the same kind as the one that records the current through a nanopore, and the Axopatch 200B, designed for patch clamping, appears in the methods of many nanopore papers.^[11] A 2021 simulation of ions in two slits one molecule thick found that their history-dependent conduction was enough to reproduce the voltage spikes of the Hodgkin-Huxley model, without proteins or gates.^[14]

• Ion channel
• Patch clamp
• Ionic current
• Erwin Neher and Bert Sakmann
• Animal electricity
• Brain lab