Kenneth ("Kacy") Cole, in
the 1930s and 1940s, employed both experimental and analytic techniques
to measure and describe cell membranes in terms of : their conductance,
currents,and the voltage across across them.
Development of the Voltage Clamp: Marmont, (working with Cole) made a technological leap by developing the "space clamp", a guarded region of membrane surrounding an axial electrode in a squid giant axon where the voltage throughout this region was uniform. Cole made an enormous conceptual break-through. He realized that he could "break" the normal feedback between current and voltage in generating an action potential by controlling the membrane potential with an electronic feedback circuit (instead of the current control employed by Marmont). Furthermore, a step change in voltage allowed him to separate the ionic current from the capacitive current flowing in the membrane (Fig 3:16. in Cole's book, Membranes, Ions, and Impulses).
Meanwhile, Alan (later Sir Alan) Hodgkin and Andrew (later Sir Andrew) Huxley (HH or H&H) were also carrying out experiments and thinking carefully about how action potentials might be generated.
Numerical integration: It was Huxley who initiated the numerical integration techniques applied to excitable membranes. Prior to the second World War, Alan and Andrew had considered a carrier mechanism model for the generation of impulses. Alan's ideas were recounted in "Chance and design in electrophysiology" (J. Physiol. 1976. 263: 1-22). Huxley had carried out extensive calculations and made predictions about how action potential would change with variations in the concentration of the extracellular sodium. The following summer he went on a honeymoon and Alan collaborated with Bernard Katz at Plymouth to carry out the experiments on varying the sodium concentration in the medium bathing squid giant axons - which bore out the predictions of Huxley's calculations.
After World War II, Hodgkin visited Cole and was shown the current records with delayed onset upon depolarization and fast exponential decay upon repolarization. Alan and Andrew had theorized previously that "the sodium conductance was a continuous function of membrane potential multiplied by an inactivation variable that fell with first-order kinetics towards another function of membrane potential". This formulation predicted that a voltage step would cause a jump in the sodium conductance followed by an exponential decay. Alan, fresh from his radar and electronics experience during World War II, questioned Kacy sharply about the delayed onset, arguing that the electronics may have been too slow to follow a fast change - which had worked so well in Huxley action potential calculations. Hodgkin also was aware of errors introduced by the resistance of the axial electrode and developed several ways to improve on Kacy's voltage control of the axon membrane. The fact that the delay in onset was very obvious in the records which he made later with Huxley and Katz (using their much improved experimental setup) may have played an important role in revising his thinking about the details of the ionic channel kinetics and conductances.
Because (as noted above) Alan and Andrew had thought extensively about the kind of system which might generate an action potential, they had a clear view of what data they would need to describe the underlying mechanisms. Perhaps this is why they were able to develop such a monumental model from the extraordinarily small number of experiments (41 is the largest number referenced in their figures or tables) carried out on a few squid giant axons. Many of these were in deteriorated condition (at least by the end of the experiment), yet they were able to scale and so fully analyze this data and encapsulate them in an extraordinary set of equations which are, some 45 years later, still the defacto standard expression for simulations. Furthermore they also serve as a framework by which equations for other excitable membranes are described. The remarkable form of the equations, raising a first order reaction to a power came about for at least two reasons:
Hodgkin and Huxley felt that they could not be sure that they understood the ionic basis of the action potential until they could reproduce its shape with these equations. Kacy told me that Alan and/or Andrew had expressed to him their reservations about whether or not they would achieve this goal. After having chosen the form of the equations to be used, they consolidated all of their data to find the appropriate voltage sensitive rate constants. Then, because the Cambridge Univ. computer was "off the air for 6 months or so, while it underwent major modifications", in the spring of 1951, Huxley began the slow work of using a Brunsviga 20 manually cranked calculator with numbers entered by a set of adjusting levers (projecting from the wheels that were rotated by the hand crank). The output was a line of digits on the wheels to be read and transcribed to paper. First, he found that the time and voltage-sensitivities of the ionic conductances could be reproduced. Then the long process of numerical integration of the action potential began. Tabular records of the rate and state variables were entered into the the levers and transcribed from the dials for small increments of time. Huxley used a tedious iterative, error-correcting, numerical integration method to estimate and correct for numerical integration errors. The fact that the whole process for calculation of a 5mS interval, showing the initiation of and recovery following an action potential, could be accomplished in 8 hours is astonishing. The calculated action potentials were - with the exception of a small "gratuitous bump" late in the falling phase - excellent reproductions of the experimental observations under a variety of conditions.
This monumental accomplishment was not only tremendously gratifying to H & H and won for them a Nobel prize, but has stood as an outstanding example of how to tackle such problems. It is also a remarkable achievement that the HH equations have not been replaced in the intervening four & one-half decades but remain the default standard equations used for simulations of the squid axon and also the usual form in which equations for other cell membranes are cast.
Even more astonishing than the calculation of a membrane action potential
for a uniform patch was that Andrew was able to use these equations to
extract the velocity of impulse propagation in an axon of uniform diameter.
Although it was essentially impossible to solve the partial differential
equation describing propagation in time and space, he was able to cast
the problem into an ordinary differential equation for a traveling wave
as shown in Cole's Fig 3.49 (in Membranes,
Ions, and Impulses).
The velocity of 18.8 m/sec he found was very close to that observed experimentally (21.2m/sec) and serves to this day as a solid known reference standard which one can use to test any program used to simulate propagation of HH impulses.
Alan and Andrew sent galley proofs of their spectacular papers to Kacy, then Scientific Director of the Naval Medical Research Institute (NMRI). As the sole professional in his lab at the time, I made additional calculations from their equations and plotted them out by hand. Thus I was one of the first to experience the awe aroused in readers of these papers.