In Chapter 6 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe the Hodgkin-Huxley model of membrane current in a nerve axon.

“Considerable work was done on nerve conduction in the late 1940s, culminating in a model that relates the propagation of the action potential to the changes in membrane permeability that accompany a change in voltage. The model [Hodgkin and Huxley (1952)] does not explain why the membrane permeability changes; it relates the shape and conduction speed of the impulse to the observed changes in membrane permeability. Nor does it explain all the changes in current…Nonetheless, the work was a triumph that led to the Nobel Prize for Alan Hodgkin and Andrew Huxley.”The paper we cite (A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve, Journal of Physiology, Volume 117, Pages 500-544) is one of my favorites. Whenever I teach biological physics, I assign this paper to my students as an example of mathematical modeling in biology at its best. In 1981 Hodgkin and Huxley wrote a “citation classic” article about their paper, which has now been cited over 9300 times. They concluded

"Another reason why our paper has been widely read may be that it shows how a wide range of well-known, complicated, and variable phenomena in many excitable tissues can be explained quantitatively by a few fairly simple relations between membrane potential and changes of ion permeability—processes that are several steps away from the phenomena that are usually observed, so that the connections between them are too complex to be appreciated, intuitively. There now seems little doubt that the main outlines of our explanation are correct, but we have always felt that our equations should be regarded only as a first approximation that needs to be refined and extended in many ways in the search for the actual mechanism of the permeability change's on the molecular scale."As one who does mathematical modeling of bioelectric phenomena for a living, I can think of no better way to honor Huxley than to show you his equations:

This set of four nonlinear ordinary differential equations, plus six expressions relating how the ion channel rate constants depend on voltage, not only describes the membrane of the squid giant nerve axon, but is the starting point for models of all electrically active tissue. Russ and I consider this model to be so important that we dedicate six pages to exploring it, and present in our Fig. 6.38 a computer program to solve the equations. For anyone interested in electrophysiology, becoming familiar with the Hodgkin-Huxley model is job one, just as analyzing the Bohr model for hydrogen is the starting point for someone interested in atomic structure. Remarkably, 60 years ago Huxley solved these differential equations numerically using only a hand-crank adding machine.

How can your learn more about this great man? First, the Nobel Prize website contains his biography, a transcript of his Nobel lecture, and a video of an interview. Another recent, more detailed interview is available on youtube in two parts, part1 and part 2. Huxley wrote a fascinating description of the many false leads during their nerve studies in a commemorative article celebrating the 50th anniversary of his famous paper. Finally, the Guardian published an obituary of Huxley yesterday.

I will conclude by quoting the summary at the end of Hodgkin and Huxley’s 1952 paper, which was the last of a series of five articles describing their voltage clamp experiments on a squid axon.

SUMMARY

1. The voltage clamp data obtained previously are used to find equations which describe the changes in sodium and potassium conductance associated with an alteration of membrane potential. The parameters in these equations were determined by fitting solutions to the experimental curves relating sodium or potassium conductance to time at various membrane potentials.

2. The equations, given on pp. 518-19, were used to predict the quantitative behaviour of a model nerve under a variety of conditions which corresponded to those in actual experiments. Good agreement was obtained in the cases:

(a) The form, amplitude and threshold of an action potential under zero membrane current at two temperatures.

(b) The form, amplitude and velocity of a propagated action potential.

(c) The form and amplitude of the impedance changes associated with an action potential.

(d) The total inward movement of sodium ions and the total outward movement of potassium ions associated with an impulse.

(e) The threshold and response during the refractory period.

(f) The existence and form of subthreshold responses.

(g) The existence and form of an anode break response.

(h) The properties of the subthreshold oscillations seen in cephalopod axons.

3. The theory also predicts that a direct current will not excite if it rises sufficiently slowly.

4. Of the minor defects the only one for which there is no fairly simple explanation is that the calculated exchange of potassium ions is higher than that found inSepiaaxons.

5. It is concluded that the responses of an isolated giant axon ofLoligoto electrical stimuli are due to reversible alterations in sodium and potassium permeability arising from changes in membrane potential.

If in 1981 they state their equations should only be regarded as a first approximation, what does the year-2012-approximation set of equations look like? Also, do we know what the maximum rate of change of V(t) can be before the conductances no longer track according to the above equations?

ReplyDeleteExcellent/Important questions! Straight answer: this remains unclear.

Delete