1. Hodgkin and Huxley's Equations Describe a Concrete Model

As Aristotle might say, model is said in many ways. For Schaffner a model is an ideal, prototypical mechanism whose scientific importance derives from analogical relations to other systems. This seems to fit Hodgkin and Huxley's (HH's) characterization of some of their work as “devoted to calculations of the electrical behavior of a model nerve whose properties are defined by the equations [like their total current equation] which were fitted to the voltage clamp records” (Hodgkin and Huxley Reference Hodgkin and Huxley1952c, 518). Models in this sense are neither true nor false, neither explanatory nor descriptive. Those notions apply to what Craver and Weber call the HH model—a theory about the propagation of the action potential.

In fact, the ideal system notion of a model is relatively unimportant with regard to the research results reported in the 1952 presentation of the HH model. The passage I just quoted refers to calculations of some important electrical quantities from equations they devised to describe the action potential. We needn’t think about ideal systems in order to understand the calculations or the use to which HH put them. Saying that they compared the electrical behavior of the ideal nerve to that of real nerves is a picturesque way of saying that they tested their equations by comparing their calculations to values of quantities they and others calculated from experimental evidence (Hodgkin and Huxley Reference Hodgkin and Huxley1952c, 518–540). Although neuroscience texts tend to talk about the HH model as an ideal system or its description, I think we can learn more about what HH were doing and its contribution to subsequent research by considering another kind of model.

Most of HH's data came from experimental manipulations on giant axons removed from squids, immersed in salt solutions, and subjected to electrical stimulation and recording. The squid axon preparations are models in the sense of the real-world systems that investigators study with hope of learning about other real-world systems. Scale models in wind tunnels experimented on by aeronautical engineers to predict the behavior of airplane wings in flight are models in this sense.Footnote ¹ So are the animal models (worms, flies, rodents, and their body parts) biologists use to investigate biological mechanisms in other species. Such models are not theoretical or hypothetical constructs. Because they are neither abstractions nor idealizations I’ll call them concrete models. Unlike the models philosophers of science usually talk about, concrete models are starting points rather than end products of research. HH put their preparations together to use in research they undertook for the purpose of developing and evaluating the theory that is customarily called the HH model of the action potential.

Concrete though they are, the squid axon preparations resemble Schaffner's prototypes in some important respects. They are multilevel causal systems. They are simplified in the sense that they are deliberately prepared so as to lack features that complicate (and in some cases rule out) obtaining and interpreting data from naturally occurring action potentials. Their components can be modified to make them something like physical analogs or counterparts of idealizations. Their behavior can be described by generalizations of different degrees of abstraction and breadth of application. And like Schaffner's models, they owe their importance to analogical relations in which they stand to systems of interest. Their usefulness depends on how well and with respect to what factors their electrochemical behavior resembles that of normal neurons in normal organisms (including humans) in normal environments. To draw general conclusions about naturally occurring neurons, HH had to exploit known (or assumed) similarities and devise ways to work around or correct for errors arising from differences. I believe the HH equations derive their importance from the resemblances between the concrete model (squid axon preparation) and other systems in vitro and in vivo.

As HH acknowledge and Craver argues, the current equations do not explain the action potential. Instead, they describe the electrical behavior of giant squid axon preparations in a mathematically convenient form. They belong to what Schaffner calls a heuristically valuable ‘emergent simplification’ that ignores enough fine-grained peculiarities of individual and kinds of neurons to serve as ‘unifiers’. By ‘unification’ the standard philosophical literature usually means establishing reductive relations that bring descriptions and explanatory principles together. Although what I have to say is compatible with Schaffner's version of this notion of unification, I emphasize the role the HH equations played in allowing investigators to use findings from the concrete model to raise questions about action potential mechanisms and to look for answers by devising research strategies and designing, executing, and interpreting a variety of in vitro and in vivo experiments on a variety of different systems.

2. The Function of the HH Equations Is Not to Secure the Application of Physics Generalizations to an Explanation of the Action Potential

Weber supposes that the HH current equations contribute to our understanding of neuronal signaling by securing the application of Coulomb's law, along with Nernst's and other highly general equations, to the explanation of cross-membrane ion flows that constitute the action potential. Some of these equations (e.g., Coulomb's law) are fundamental physical laws. Some (e.g., Nernst's equation) derive from fundamental physical laws. In what follows I will sometimes refer to these along with lesser generalizations like Ohm's “law” as fundamental equations. In the original version of his contribution to this symposium and some earlier writings, Weber held that to explain an event or series of events, one must show that it is necessitated by the laws of physics, where necessitation is to be understood in terms of non backtracking counterfactuals supported by fundamental laws. This is the view I discuss here.Footnote ²

Weber held that the HH current equations do not meet this condition for explanation. Instead, they and many other biological principles further the explanatory enterprise by helping investigators bring fundamental physical laws to bear on explananda. Taken together with the rest of the HH model, the current equations help us understand the action potential by singling out a class of systems to which fundamental laws can be applied. We need Coulomb's law, Nernst's equation, and the rest because, without them, “it would be a brute fact that selective ion permeabilities can generate a membrane potential and that there is an ion-motive force that tends to restore the rest potential” (Weber, pers. comm.).

But assumptions of brute fact abound in Weber's own account of explanation. To give a Weberian explanation of the relationship between membrane potentials and ion permeabilities, one must appeal to maximally widespread brute facts and counterfactuals about attraction, repulsion, diffusion, and so on, purportedly described by fundamental physical laws.

HH looked for explanations of another kind. Like Machamer, Darden, and Craver (Reference Machamer, Darden and Craver2000), they thought an explanation of the action potential should describe a physical mechanism, enumerate its parts, and show what they do to initiate and control cross-membrane ion flows in response to changes in membrane potential. This sort of explanation assumes brute facts about what things do and what results from the activities they engage in, for example, that oppositely charged particles attract and similarly charged particles repel one another, and that particles respond to attraction by moving toward the attractor and to repulsion by moving away. Trying to explain the action potential by showing that such things are necessitated by Coulomb's and other fundamental laws is no way to avoid appeals to brute facts like these.

The probabilistic and irregular nature of the processes through which action potentials are propagated presents Weber with another difficulty. It is hard to see how the results of stochastic processes can be explained if to explain something shows that the laws of physics necessitate it.Footnote ³

The fundamental equations Weber mentions do not deliver explanations that meet his own conditions of adequacy. For example, in order to describe the effect of changing sodium concentrations in their preparations on the sodium current without tampering with Coulomb's law, HH assumed for simplicity that each ion moves independently of the positions of the others. But this assumption flatly contradicts Coulomb's law except in application to highly unrealistic imaginary arrangements of charged particles. To avoid the independence assumption, they could have followed Ussing's example and supplemented Coulomb's law with crotchets to correct for frictional and other effects of changes in the relative positions of ions flowing through the membrane (Hodgkin and Huxley Reference Hodgkin and Huxley1952a, 467ff.). But Ussing's alternative equations are quantitatively faithful only to highly unrealistic, arbitrarily contrived imaginary systems (Hille Reference Hille2001, 357ff.). Thus, ions do not flow across the real membranes HH studied as Coulomb's law or Ussing's equation requires them to. For Weber, it should follow (what I take to be the case) that neither principle explains their motion.

Finally, the neuronal signal is an emergent phenomenon in the sense of Wimsatt's nonaggregativity (Reference Wimsatt2000, 275ff.). Its propagation results from a complex, tightly orchestrated sequence of changes in membrane permeability to sodium, potassium, and leakage ions in response to an equally elaborate sequence of changes in membrane potential. Although fundamental equations certainly apply (albeit roughly) to ion currents, membrane potentials, and so on, they do not pretend to tell us how these factors conspire to produce the action potential. No equations of any considerable generality deliver Weberian explanations of how coordinated patterns of local polarization, depolarization, and inward and outward flows of charged ions emerge from the specific behaviors the fundamental equations describe. I think the probabilistic, nonlinear emergence of action potentials from electrochemical interactions among ions and membrane components fits Nancy Cartwright's idea that “natural objects are much like people in societies. Their behavior is constrained by some specific laws and by a handful of general principles, but it is not determined in detail, even statistically. What happens on most occasions is dictated by no law at all” (Cartwright Reference Cartwright1983, 49). In particular, I think there is no law to dictate the orchestrations from which action potentials emerge.

The moral of these objections to Weber is that what the HH equations do to help us understand the action potential neither consists of nor depends upon enabling us to explain the action potential or its constituent cross-membrane currents as necessitated by physical laws.

3. How HH Used Ohm's Law, Nernst's Equation, and Other Fundamental Generalizations

Craver says HH derived their equations from fundamental equations including Ohm's as well as Coulomb's law. I agree if derivation is construed so broadly that what is derived needn’t be a mathematical or other deductive consequence of what it is said to derive from. Laura Ruetsche says physicists sometimes say B is derived from A when there is an argument “in which A features as a consideration and for which B serves as a conclusion—not in the sense of deductively following from anything else, but in the sense of being where the argument ends” (pers. comm.). HH did use fundamental equations in derivations loosely construed. Here are a few examples.

The final installment of Hodgkin and Huxley's 1952 papers includes a diagram said to represent the electrical behavior of a small bit of prepared squid giant axon membrane. The membrane is represented as if its electrical activity accorded perfectly and without exception to Ohm's law (Hodgkin and Huxley Reference Hodgkin and Huxley1952c, 500–501). In fact, axon membranes are not Ohmic conductors. HH found that electrical activity in a squid axon membrane accords, at best, reasonably well to Ohm's law and only for an instant immediately after a sudden change in membrane potential (Hodgkin and Huxley Reference Hodgkin and Huxley1952b, 477) and only in solutions that don’t include much less sodium than sea water (Hodgkin and Huxley Reference Hodgkin and Huxley1952b, 480–482). HH did not consider the violations of Ohm's law they found to be a particularly bad result. That, they say, is because all they wanted from Ohm's law was help in developing a general description of ion currents “under conditions which allow a normal action potential to be propagated” (Hodgkin and Huxley Reference Hodgkin and Huxley1952b, 482). I suppose HH could tolerate the relatively small observable inaccuracies of Ohm's law even for axons in sea water as long as they didn’t frustrate the usefulness of the general description of membrane currents Ohm's law helped them develop.

The Ohmic circuit the diagram portrays is equivalent to the membrane circuit in the sense that its variables and their values correspond roughly to electrical quantities in an axon membrane. But HH's notation reflects their appreciation of significant differences. In the diagram, I signifies an ideal current. Its magnitude varies with the sum of a capacitance current fixed for simplicity at a constant value and three component currents symbolized by I _Na, I _K, and I _l. The latter are related in strict accordance with Ohm's law to E (the difference between voltage at different points along an ideal Ohmic conductor) and conductances g _Na, g _K, and g _l (the reciprocals of equivalent circuit resistances R _Na, R _K, and R _l). HH use different symbols to describe currents in their concrete model. With regard to prepared axons, I symbolizes a current that varies with a capacitance current, voltage quantities $V-V_{\mathrm{K}\,}$, $V-V_{\mathrm{Na}\,}$, and $V-V_{l}$, and permeability quantities symbolized by $\overline{g}^{\mathrm{n}\,4}_{\mathrm{K}\,}$, $\overline{g}^{\mathrm{m}\,3\mathrm{h}\,}_{\mathrm{Na}\,}$, and $\overline{g}_{l}$. These last take the places of g _K, g _Na, and g _l in the diagram. V is the difference between the membrane's resting potential and its actual membrane potential at an instant. V _K is the difference between actual membrane potential and the potassium equilibrium potential—the difference in voltage across the membrane such that there is no net cross-membrane potassium current flow and similarly for V _Na and V _K. The terms $\overline{g}_{\mathrm{K}\,}$, $\overline{g}_{\mathrm{Na}\,}$, and $\overline{g}_{l}$ symbolize maximal potassium, sodium, and leakage (i.e., chlorine and other ion current) conductances, while n4 and m3h are weighting crotchets with no physical interpretation (Hodgkin and Huxley Reference Hodgkin and Huxley1952c, 507, 541, 518); $\overline{g}^{\mathrm{n}\,4}_{\mathrm{K}\,}$ and the rest are ‘permeability coefficient[s]’, each of which has ‘the dimensions of a conductance’ (Hodgkin and Huxley Reference Hodgkin and Huxley1952c, 500).

HH needed Ohm's law because experiments on axon preparations could yield data on current, voltage, and some other quantities but not on permeability. Magnitudes of the latter had to be estimated from the former. To this end, HH applied Ohm's law to real-world quantities as if they were denizens of the equivalent circuit. In particular, they treated each ion permeability as if it varied with membrane potential and current according to Ohm's law (Hodgkin and Huxley Reference Hodgkin and Huxley1952b, 477ff.). Ohm's law played a derivational role in the sense that these calculations were necessary for the development of the HH current equations. Thus, Ohm's law was used as a calculation tool rather than an explanatory principle, or the premise of a derivation strictly construed.Footnote ⁴

Here is a second example of the use of a fundamental equation. HH used Nernst's equation to calculate values with which to test the following hypothesis: the inward cross-membrane currents elicited by lowering membrane potentials are carried by sodium ions whose flow depends upon “a driving force which is the resultant of the effects of the concentration difference and the electrical potential difference across the membrane” (Hodgkin and Huxley Reference Hodgkin and Huxley1952a, 450–451). The driving force for sodium is the difference between V, the difference between actual and resting membrane potentials, and V _Na, the difference between actual membrane potential and the sodium equilibrium potential. To test the hypothesis, HH had to calculate driving-force magnitudes. To this end, they had to estimate the magnitude of V _Na for membranes immersed in solutions containing different amounts of sodium. They produced data for this purpose by measuring electrical responses of prepared squid giant axons to electrical stimuli in a variety of salt solutions. To interpret the data, they assumed with Nernst that ions flow down their electrical and concentration gradients in accordance with Coulomb's and thermodynamic laws and that each ion species has a mathematically well-behaved equilibrium potential. Armed with these assumptions, they used Nernst's equation to calculate the sodium equilibrium potential; that is, “the critical value … of membrane potential such that the [inward and outward ion] fluxes are equal and the net sodium current is therefore zero. … Its value should be given by the Nernst equation $E_{\mathrm{Na}\,}=RT/ F$ $\mathrm{\log}_{e}[ \mathrm{Na}\,] _{i}/ [ \mathrm{Na}\,] _{o}$” (Hodgkin and Huxley Reference Hodgkin and Huxley1952a, 451).

$[ \mathrm{Na}\,] _{i}/ [ \mathrm{Na}\,] _{o}$ is the ratio of inward to outward Na⁺ concentrations (Hodgkin and Huxley Reference Hodgkin and Huxley1952a, 451).Footnote ⁵ Having calculated magnitudes of V _Na, they could go on to estimate driving forces as required to test the hypothesis against experimentally determined temperatures, sodium concentrations, currents, and voltages (Hodgkin and Huxley Reference Hodgkin and Huxley1952a, 454–455).

Let us consider a third example. As discussed by Schaffner, HH used Laplacian cable equations to calculate total and individual ionic conductances during the propagation of an action potential for a fiber of specified radius from their total current equation (Hodgkin and Huxley Reference Hodgkin and Huxley1952c, 530). The results were compared to experimental evidence to find out how well the HH equations describe the electrochemical behavior of the axon preparations.

To summarize, HH used fundamental equations to calculate ion permeabilities, conductances, resting potentials, driving forces, and so on, from experimental data. Having constructed equations to describe relations among the these quantities and other factors, they used fundamental equations to generate predictions to be tested against new magnitudes calculated from new experimental evidence. The purpose of the tests was to find out how well the equations described the electrochemical behavior of squid axon preparations. In these cases, HH used fundamental equations not as explanations but as tools for making calculations with which to develop and test current equations and hypotheses (such as the one given above) about electrical quantities of interest.

These were not the only uses to which fundamental equations are put. They also supply and constrain expectations about the operation of mechanisms responsible for action potential propagation. Coulomb's law tells us to expect similarly charged particles to repel rather than attract each other. Ohm's law tells us to expect current flows to vary with voltage and conductance. In providing and constraining such expectations, fundamental equations helped investigators develop what Craver calls “how possibly explanations.” For example, although neither Coulomb's nor Ohm's law explains or delivers quantitatively accurate descriptions of cross-membrane ion flows, they provide good reason to suppose that whatever influences membrane permeability and concentrations of ions on either side of the membrane can help explain how action potentials are propagated and damped.

Furthermore, to the extent that they apply (to some satisfactory degree of approximation) to in vitro and in vivo electrical behaviors of other neurons, the fundamental equations helped justify inductive arguments from descriptions and explanations of electrical activity in the concrete model to conclusions about the electrical behavior of other neurons in other settings (Bogen Reference Bogen2005). In all of these uses, fundamental equations contributed to the development and application of explanations of action potential phenomena, but none deliver explanations that fit Weber's or such received accounts of explanation as covering law, probabilistic, statistical relevance, interventionist, conserved quantity transmission, or unification.

4. Nonexplanatory and Inaccurate as They Are, What's So Good about the HH Equations?

The HH total current equation,

equates I, the total current crossing the neuronal membrane, with the sum of a capacitance current [$C_{M}\left(dV/dt\right)$] and three ion currents: [$\overline{g}^{\mathrm{n}\,4}_{K}( V-V_{K}) $], the potassium current; [$\overline{g}^{\mathrm{m}\,3\mathrm{h}\,}_{\mathrm{Na}\,}( V-V_{\mathrm{Na}\,}) $], the sodium current; and [$\overline{g}_{l}( V-V_{l}\,\,) $], the “leakage current” (Hodgkin and Huxley Reference Hodgkin and Huxley1952b, 477, 487). Recall that m3h and n4 are uninterpreted weighting constants. As Craver explains, the most HH have to say in favor of these weighting constants is that they yield the simplest equation they can devise that is acceptably faithful to relations among experimentally determined electrical magnitudes.Footnote ⁶ HH insist that the total current equation and the equations for K⁺, Na⁺, and leakage currents are descriptive rather than explanatory. Craver's discussion convincingly urges us to take HH at their word when they say they have no good reason to think their “equations are anything more than an empirical description of the time courses of in permeability to sodium and potassium” (Hodgkin and Huxley Reference Hodgkin and Huxley1952c, 541).Footnote ⁷

If the equations don't explain the action potential, what about their descriptive and predictive accuracy? HH conclude their 1952 papers by pointing out quantitative inaccuracies and limitations. First, “they apply … only to the isolated squid giant axon” (Hodgkin and Huxley Reference Hodgkin and Huxley1952a, 541), and, although there is reason to believe the action potential mechanisms are basically the same in different organisms and environmental settings, “the great differences in the shape of action potentials show that even if equations of the same form as ours are applicable in other cases, some at least of the parameters must have very different values” (Hodgkin and Huxley Reference Hodgkin and Huxley1952a, 543). For example, ionic gradients in living animals must be maintained by some “additional process” that is not present in squid axon preparations. And even with regard to the preparations they used, the current equations “cover only the short term responses of the [axon] membrane [to electrical stimulation]” (Hodgkin and Huxley Reference Hodgkin and Huxley1952a, 541). It should be noted that the artificial stimulation to which the prepared axons are subjected is typically significantly different from the stimuli that give rise to in vivo action potentials in natural settings. Allowing for that, there are still observable disparities between observed electrical behavior in squid giant axon preparations and the behavior the HH equations predicted (Hodgkin and Huxley Reference Hodgkin and Huxley1952a, 541).

So what is it about these nonexplanatory and somewhat inaccurate equations that makes people consider them foundational to our understanding of neuronal signaling? My answer is a variation on Schaffner's theme of exploiting analogies to apply what is known about one system to the investigation of other systems.

Like fundamental equations, HH equations are important because for all their faults they support inductive inferences from findings about the concrete model to conclusions about other axons in vitro and in vivo. Barring empirical or theoretical arguments to the contrary, one should expect action potential mechanisms in squid axon preparations to resemble their counterparts in other systems whose electrical behavior conforms qualitatively or to an acceptably good quantitative approximation to the HH equations. If so, or if one can account for significant disparities, suitably modified explanations and descriptions of the behavior of the concrete model can be applied to in vivo and in vitro action potentials in other species.

The HH equations were just as, if not more, important for their contributions to neuroscientific research practices. I said earlier that in order to use one physical system as a model for the study of another, investigators must exploit similarities and finesse differences between them. This applies as much to devising and applying experimental techniques and interpreting experimental results as it does to justifying generalizing claims about one system for application to another. Here are some of the ways investigators used HH equations to direct and conduct their research.

Craver observes that in the earlier stages of a research program, investigators may not be able to develop anything more than explanation sketches in which crucial details are left “unspecified or hidden behind filler terms” and that the HH equations are a case in point (see Craver Reference Craver2007, 113ff.). HH were satisfied enough with the ways their predictions described relations among experimentally established electrical quantities to think that the way to find an explanation for the action potential would be to look for physical counterparts to m, n, and h. As it turned out, no straightforward physical counterparts for these weighting constants turned up. But neuroscientists discovered a great deal of what is now known about the existence, physical makeup, and operation of voltage-gated ion channels by pursuing progressively more sophisticated versions of this research program and developing new lines of inquiry as they discovered and learned from new failures of predictions based on the HH equations (Hille Reference Hille2001, 37–60).

In some cases, ideas about action potential mechanisms were modified to bring them into better accord with the HH equations. Thus, some proposals about voltage gating were rejected and others accepted according to how well their predictions agreed with time courses predicted by the HH equations. But some intractable disparities were taken to result from defects in the equations themselves. When this happened, investigators looked for ways to replace or correct for mistaken assumptions in their construction. For example, we saw that HH simplified some of their calculations by assuming that the positions of other ions make no difference to the motion of any given ion as it passes through the membrane. Their successors found they could improve their explanations of the action potential by developing more realistic stories about ion flow (Hille Reference Hille2001, 356ff.).

Naundorf, Wolf, and Volgushev's (Reference Naundorf, Wolf and Volgushev2006) dramatic recent proposal that the rate at which an ion channel opens depends in part upon what is going on in neighboring channels is another example of how violations of the current equations could lead to new ideas. Naundorf et al. developed their idea to account for changes in electrical activity at the beginnings of action potentials whose abruptness and magnitudes violate HH predictions (Gutkin and Ermentrout Reference Gutkin and Ermentrout2006, 999–1000; Naundorf et al. Reference Naundorf, Wolf and Volgushev2006, 1060). It is hoped that an explanation of how activity in a sodium channel is influenced by activity in its neighbors may help explain how neurons process information from rapid inputs. Here, as with the independence assumption, discrepancies between experimental results and HH predictions were used to find more realistic assumptions about the mechanisms under investigation.

These examples feature the use of computer technology that did not exist in 1952. Its development made it possible for investigators to run computer simulations of neuronal signaling based on HH current equations supplemented with a variety of background specifications and additional hypotheses. Following HH's example, they devised and tested predictions from their simulations against experimental results. This is an analogy-based extension of research strategies described in the 1952 papers to the development, execution, and interpretation of results from experiments employing new technologies.

The fact that the Hodgkin-Huxley equations are qualitatively faithful to neurons in many species suggests which electrical quantities should be measured in new neurons, what kinds of experimental interventions should be undertaken with regard to them, and what instruments and experimental techniques one might use to obtain useful data.

Finally, the HH equations suggest what kinds of factors one might look to in order to explain the action potential. For example, the fact that if ion permeability changes with membrane potential made it natural to look for voltage-gated membrane channels that open and close to regulate ion flows. Sodium, potassium, and leakage currents have observably different time courses. This motivated investigators to look for more than one kind of channel or gating process.

Surprisingly, some investigators following this line of thought were able to shed light on neuronal signaling by studying cell membranes in organisms that have no neurons. A striking example is MacKinnon's and his associates’ Nobel Prize–winning crystallographic studies of the structure and possible operation of the KvAP K⁺ channel in a bacterium that flourishes without the benefit of a nervous system in conditions that are entirely unsuitable for habitation by creatures with nervous systems. The relevance of their research to the understanding of action potentials derives from analogies between the KvAP channels and the K⁺ channels responsible for features of squid giant axon action potentials (Jiang et al. Reference Jiiang, Lee, Chen, Ruta, Cadene, Chait and MacKinnon2003a, Reference Jiiang, Ruta, Chen, Lee and MacKinnon2003b).Footnote ⁸

These are examples of what might be called investigative, research, or experimental unification. It exploits analogies between the concrete model as different from it as the human axon and MacKinnon's bacterial potassium channels. It brings them together in service to the same umbrella research program. Investigative unification also connects diverse experimental designs, techniques, technologies, and strategies for data analysis and interpretation by bringing them to bear on the study of the action potential and related phenomena. Thus, I submit that the HH equations are foundational because, and to the extent that, they are indicative of crucial points of analogy between the concrete model and neurons of other species in other in vivo and in vitro environments.