Hodgkin and Huxley's data analysis and fitting

The ultimate goal of Hodgkin and Huxley's experiments was to elucidate the mechanism of the action potential. In the minds of these Nobel Laureates, they did not achieve their goal. Hodgkin writes in his memoirs:

"...although we had obtained much new information the overall conclusion was basically a disappointment. We had started off to test a carrier hypothesis and believed that even if that hypothesis was not correct, we should nevertheless be able to deduce a mechanism from the massive amount of electrical data that we had collected. These hopes faded as the analysis progressed."

They had to "settle," then for the lesser goal (in their minds) of reconstructing an action potential from analytical expressions of the experimental observations. They hoped to integrate numerically the equations for the kinetics and amplitudes of both the Na and K currents.

Having separated the Na and K components of the currents observed under voltage clamp, Hodgkin and Huxley knew that they needed to find analytic expressions to fit their data. They realized that the key would be equations describing the currents in terms of the underlying ionic conductances and driving forces. Thus, they converted all of their current records to conductances.

Hodgkin writes:

" We soon realized that the carrier model could not be made to fit certain results, for example the nearly linear instantaneous current voltage relationship, and that it had to be replaced by some kind of voltage-dependent gate. As soon as we began to think about molecular mechanisms it became clear that the electrical data would by themselves yield only very general information about the class of system likely to be involved. So we settled for the more pedestrian aim of finding a simple set of mathematical equations which might plausibly represent the movement of electrically charged gating particles. But even that was not easy, as the kinetics of the conductance changes were unlike anything we had come across before, particularly the S-shaped 'on' and exponential 'off' of the conductance curves. I think we both appreciated the need to involve several particles, but it was Andrew Huxley who eventually came up with the ideas which led to the m^3 * h and n^4 formulation."

The kinetics of the ionic conductances were unexpected.

First-order kinetics could not account for the lag in the onset of either conductance.

The time courses of the conductances were unusual.
The patterns of both the Na and the K conductances exhibited a delayed, S-shaped onset in response to a step depolarization. The conductances turned off differently from one another, however. The Na conductance showed an exponential decay with time in spite of continued depolarization. The K conductance, on the other hand, persisted during the voltage step, decaying quickly and exponentially upon return of the voltage to its resting level. The problem became one of describing both of these perplexing conductance curves with equations.

The kinetics of a first-order system were inadequate.
Biological processes and chemical reactions in general may be described by a simple, ordinary differential equation, such as that describing the curve illustrated here, or a sequence of such equations. No such simple equation could describe the time course of either the Na or the K conductance. The time course of the K conductance seemed simpler than that of the Na conductance yet a simple equation could not describe both its delayed onset and its fast exponential decay.

The conception of charged particle movement was the key.
From Hodgkin's memoirs, quoted above, we know that Hodgkin and Huxley attempted all along to conceive of a molecular mechanism to guide their thinking about a mathematical formulation. They reasoned that the voltage-sensitive increase in either conductance might be due to a change in the position of a charged particle in the membrane. The particle's position would move from one location to another upon a change in the electric field across the membrane. In order to express this idea mathematically, the probability of the particle's location at a particular place would vary between zero and one. An electric field change would elicit a probability change that would follow a first-order time course, as in the figure above.

This starting notion was adequate to fit the exponential decay of the K conductance following a depolarizing pulse. However, it was incompatible with the lag in conductance onset.

Second-order kinetics provided a lag but still could not account for the asymmetry in onset and offset kinetics.

In a second-order differential equation (typical of a system with a mass, a spring, and friction) the response lags at both onset and reset. Although the K conductance lagged at onset, there was no lag at offset. The problem of envisioning a molecular mechanism (with a mass, spring, and friction) aside, this type of equation also would not describe the conductances.

The mathematical description of the ionic conductance patterns was unique and brilliant.

Huxley expressed conductances as probabilities raised to a power.
In a stroke of genius, an epiphany, Huxley realized that expressing the conductances as probabilities that a particle would be in the correct position for a "gate" to be open coupled with raising a first-order process to a power would solve the problem. Such a formulation would describe not only the kinetics of the conductances but also the S-shaped dependence of their amplitudes on voltage.

Their original model envisioned an ion carrier in which a "single particle" was moved by a change in the electrical field across the membrane, thus translocating the ion. As they changed their notion from a carrier to a gate, they also made the mental leap that perhaps more than one "particle" needed to move to open the gate. The kinetics of the opening of the gate would then be described by a power function. They chose the variables n and m to represent the probability of a single particle moving between two positions to open the K and Na gates, respectively. If movement of two particles (or charges or conformational changes) were required to open the K gate, for example, n would be squared.

Power functions can describe the kinetics.
A first-order reaction raised to a power not only provided for both the delayed onset and exponential decay of the conductances but also offered a simple way to incorporate their voltage sensitivities. A power function, first of all, would describe the lag in the onset and also the fast return of the K conductance. Moreover, by expressing the conductances as probabilities, the power function provided (1) an S-shaped lag at the onset of depolarization, (2) a nonlinear relation between the steady amplitude and level of depolarization, and (3) a fast exponential decay upon repolarization.

The kinetic form of the K conductance involves a power function.

Hodgkin and Huxley used n to represent the probability of a single particle being on the side of the membrane promoting the opening of a K "gate." If the opening of the gate required that more than a single particle be simultaneously on that side, the conductance (now expressed as the probability of a gate being open) would be proportional to n raised to a power.

They found that their K conductance data could be fit reasonably well if n were raised to the fourth power, implying that four "particles" had be on the proper side of the membrane to open the K gate. Interestingly, a better match to the long delay at the onset required a higher power, but this small improvement was judged not to be worth the additional calculations required.

This figure shows how the power function changes the kinetics and amplitude of the K conductance curve from that of a first-order expression. The black curves show first-order kinetic changes of probability for a weak and strong depolarization. The blue curves plot the values of the black curves raised to the 4th power. The blue curves then represent the K conductance or, as we know nowadays, the fraction of K channels that are open at any given time. Note that the power function attenuates the smaller conductance change far more than it does the larger conductance change. This attenuation, when plotted for a family of conductance changes, is an S-shaped relationship that describes the data.

Armed with the n^4 form for the analytic expression for the K conductance, Hodgkin and Huxley then took on the tedious process of finding the rate constants to fit this form. They first had to find and tabulate the rate constants describing n as a function of membrane potential. This achieved, they then had to find analytical expressions for the rate constants as a function of voltage.

The kinetic form of the Na conductance requires two power functions.

The Na conductance is a product of activation and inactivation processes.

Hodgkin and Huxley applied a similar approach to fitting the kinetics of the Na conductance. Because the onset of the conductance also showed a lag upon depolarization, they again employed a first-order kinetics process, choosing m to represent the gating particle and raising it to a power, in this case 3. Unlike the K conductance, the Na conductance turned off during the sustained depolarizing voltage steps, and this process had to be described by yet another variable, h. It turned out that a first-order kinetic process could represent the decline in Na conductance. The product of these two processes, "m-cubed" and h, provided the appropriate fit to the transient conductance patterns for Na. We now conceive of the h process as the ball on the end of the chain that swings into the mouth of the Na channel to block (inactivate) it.

Again the implication was that four particles had to be in their proper places to meet the conditions required to open the Na channel, three m particles and one h particle.

The probability of the Na conductance being activated or inactivated is a function of time.

The two accompanying figures illustrate the kinetics of the variables affecting the Na conductance (the population of Na channels). Each figure plots, as a function of time after a depolarizing step, the probability value for an m or an h particle to be in the correct location for a channel to be open. The depolarizing step that elicited these changes in probability was delivered from rest. Thus h begins at a value of 0.6, indicating that the position of the h particle at t = 0 permits only 60% of the channels to be activated (40% are already inactivated).

(In order for the h particle to be in the position required to remove all inactivation (h = 1), the depolarization would have to be preceded by a strong hyperpolarization and for a time sufficient to allow the particle movement to be accomplished.)

In this upper graphs, the black curve shows m following a first-order kinetic process. The turn-on of the Na conductance does not follow first-order kinetics, however, but is better described by the red curve, where m is cubed. Neither curve describes the process of turning off the conductance. Another process, inactivation, must be added.

This lower graph represents the m^3 kinetics of the turn-on process as shown above (red), the kinetics of the inactivation process , h (blue), and the product of these two processes, m^3 * h (green).

Although at the onset of the step there is no inactivation for 60% of the channels, with time during the step the probability that the ball is hanging free falls with first-order kinetics. That is, the depolarizing step moves the inactivating particle into the mouth of the channel so that inactivation proceeds. This notion of h is perhaps one of the more confusing concepts to grasp in understanding the Hodgkin-Huxley equations. h is not a particle, not the inactivating ball on the chain. It is a probability, the probability that the ball is hanging free.

It may be helpful to realize that the conductance will be zero -- the channels will not open -- if either m equals zero (the probability is zero that a gating particle moves to open the channel) or h equals zero (the probability is zero that the ball is hanging free, so all of the channels are blocked).

Note that repolarization quickly returns m to zero but h returns slowly to its high values. In terms of the population of channels, with time the inactivation particle detaches from the pore of each channel and as time goes on the fraction of channels that are unblocked increases.

Having arrived at the " m^3" * h form for the analytic expression for the Na conductance, Hodgkin and Huxley once again (as they did for n for the K conductance) took on the tedious process of finding the values of the rate constants for both m and h for each depolarization to fit this form. Because the values of these rate constants depend on the membrane potential, they had to find analytical expressions to fit the dependence of the rate constants on the voltage.

The Hodgkin-Huxley equations have proved extraordinary useful.

The Hodgkin-Huxley formulations have proved to be so successful in describing the conductances of squid axon membranes that they have had a phenomenal lifetime of at least one-half century and have also served as a framework by which equations for other excitable membranes are described.