Chapter 2 (2) : HH Conductance Model

The previous post summarized, in somewhat qualitative terms, the HH model of the action potential. Since then, I've written up a supplement to Hille's description of their quantitative model. It is available online as HH_Conduct.pdf. It goes into a little more detail than Hille about their model. For instance, it motivates and describes the equation they actually used to fit their g_k data, thus explicitly connecting their speculative lower-level model (in terms of charged 'gating particles') to observed macroscopic conductances. The first paragraph follows.

Hodgkin and Huxley (HH) were able to reliably measure the voltage- and time-dependent changes in sodium and potassium conductance (g_na and g_k) during the action potential (Figure 2.11 [Hille]). The story that emerged from such experiments is now part of the core knowledge of all neurophysiologists. After a suprathreshold voltage step, g_na quickly rises and falls while g_k slowly increases to a steady state (Figure 2.11 [Hille]). In a cell that is not voltage-clamped, this increase in g_na causes V_m to rapidly approach E_na. However, the subsequent g_k increase, and g_na decrease, causes V_m to quickly fall back down to the resting potential near E_k. HH quantified these conductance changes in a model with explanatory and predictive power that only continues to grow. In what follows, I will describe in more detail their model of g_k and g_na.

Each time I revisit the HH corpus, I am impressed by their profound creativity, technical skills (both quantitative and experimental), and luck.

Chapter 2 (1): Classical Biophysics of the Action Potential

Last updated 9/01/05

In Chapter 2, Hille provides an excellent summary of the classical period in biophysics that culminated in the Hodgkin-Huxley papers of 1952. This work on the squid giant axon laid the foundation for thinking about the electrical properties of neurons that has persisted to this day.

Bernstein's membrane hypothesis
Bernstein was the first physiologist to hypothesize that action potentials result from transient changes in membrane permeability to ions. Based partly on the work of Nernst, he hypothesized that at rest the neuronal membrane was selectively permeable to K ions, and that during an action potential there was a reversible 'breakdown' of the cell membrane so that it became nonselectively permeable to all ions. Such a breakdown would cause Vm to shoot up to zero.

Right before WWII, when Cole and Hodgkin's groups were finally able to accurately measure membrane voltage with intracellular electrodes, they discovered that something had to be wrong with Bernstein's hypothesis: Vm shot up past zero during an action potential (Figure 2.1). After the war, Hodgkin's group was finally able to experimentally attack the cause of the overshoot. They surmised that, instead of becoming permeable to all ions, the membrane became selectively permeable to sodium ions, which they knew from Nernst had a resting potential well over zero mV.

Hodgkin and Huxley crack the action potential
Hodgkin and Katz, in 1949, confirmed their sodium hypothesis by decreasing the Na-concentration in the extracellular fluid and observing a reduction in the amplitude of the action potential. They also showed that, when the extracellular K concentration was reduced, the resting potential of the cell became less hyperpolarized.

The development of the technology to clamp the voltage across the membrane of the squid giant axon was the driving force behind their solution to the action potential problem. It allowed them to hold Vm at a particular voltage (V_clamp) while measuring the current across the membrane (I_m) (Figure 2.6). By combining voltage clamp with ionic substitution experiments they were able to more precisely characterize the ionic basis of the action potential. For instance, they predicted and confirmed that when V_clamp was above E_na, the current would switch from negative (inward) to positive (outward) (Figure 2.7). They hypothesized that the action potential could be generated solely by voltage-dependent Na and K currents (i.e., I_m=I_na+I_k). When they replaced extracellular Na with an impermeant cation, clamping the cell at any voltage above E_k generated only outward currents, which they took to be I_k. By subtracting I_k from the membrane current I_m observed in normal conditions, they were able to infer I_na during an action potential (Figure 2.8). They applied this method at multiple values of V_clamp, generating Figure 2.9, which plots steady-state I_k and peak I_na as a function of V_clamp.

Since from Ohm's law we know that for ion species i, g_i=I_i/(V_m-E_i), they were able to track the voltage and time-dependent changes in g_na and g_k during the action potential (Figure 2.11). The story that emerged from such experiments is now well-known to all neurophysiologists. After a suprathreshold voltage step, g_na quickly increases(and then decreases), while g_k increases to a steady state (Figure 2.11). This causes V_m to quickly approach E_na, but then when g_k increases (and g_na decreases), the cell quickly comes back down to the resting potential near E_k.

What causes these changes in g_k and g_na? HH hypothesized that the g_na change is actually due to two types of processes in the channel: activation (m) and inactivation (h). These processes are referred to as 'gates' or 'gating variables'or 'particles'. Each of these gates can be in a permissive (activated/de-inactivated) or nonpermissive (de-activated/inactivated) state: a channel is open only when all of the gates are in the permissive state. Unlike the sodium channel, the voltage-gated K channels have only one type of gate, which is either activated or deactivated (n). At E_rest, most Na channels are closed: while h tends to be in the permissive (deinactivated) state, the m gate tends to be in the nonpermissive, deactivated state. When the voltage rises, m tends to become activated very quickly. At this time, when m and h are both in the permissive state, the Na channel opens and sodium flows into the cell. This action current generates a brief and fast rise in V_m: the action potential. This rapid depolarization causes the Na channel to inactivate (i.e., h tends toward the nonpermissive state), shutting off the current flow. Compounding this insult to Na, the depolarization also causes m, the gating element in the potassium channel, to enter the permissive state (i.e., the K channel is activated), which causes V_m to drop back down toward E_rest. This hyperpolarization resets the whole process. Back down at baseline, the K channels inactivate and the Na channels become de-inactivated (while remaining deactivated), so the cell is poised to spike once again. The above menagerie of permissive and nonpermissive gate states can be confusing, but it's the biological reality evolution stuck us with, and is so important in neuroscience it is worth taking the time to grok.

In the next post, I will summarize Hille's more quantitative description of the Hodgkin-Huxley model and add more details to the above qualitative summary. Then, I will describe gating currents, about which two Nature papers were published just last month, which settles an open question at the time of publication of ICEM.

Differential equations primer

Since first-order differential equations come up fairly frequently in Hille (and in any quantitative treatment of anything in science, for that matter), I have spent the past week writing up a primer on first-order differential equations for neuroscientists. You can download it here: Diff_Eq.pdf.

Here is the introductory section of the manuscript:

Ordinary first-order differential equations come up repeatedly in neuroscience. They are used to model many fundamental processes such as passive membrane dynamics and gating kinetics in individual ion channels. When the equations come up, most electrophysiology texts provide the solution, but do not provide any explanation. This manuscript tries to fill the gap, providing an introduction to some of the mathematical facets of first-order differential equations. While the main goal of this manuscript is to examine the equations neurophysiology texts present without justification, I pushed the proofs to the end of the manuscript so that those who wish to skip them don't have to waste time working out which pages to skip. The manuscript, including the proofs, presupposes a little knowledge of first-year calculus, much of which is reviewed when needed.

The manuscript has three sections. Section One provides a brief statement of the general problem and its solution. Section Two examines the solution for a special case that comes up quite frequently in practice, and also examines a concrete example, the equivalent circuit model of a patch of neuronal membrane. Section Three contains a simple derivation of the general solution given in Section One.

There is a Matlab function that accompanies it which you can use to simulate the equivalent circuit model I go over in the document. You can get the code here: equiv_circuit_plot.m. Feel free to suggest improvements to the code.

I welcome any criticisms of content, clarity, or formatting. I hope it can serve as a useful reference to explain all those taus and infinities in Hille's treatment of the Hodgkin-Huxley equations in Chapter 2, which I'll post about next week.

Update 8/27/05: Version 0.2 of the document is up. I have retooled the document so that the connection with Hodgkin-Huxley is more transparent, and so those who want to avoid proofs can do so while still learning some differential equations (NB: skipping proofs causes warts).

Summaries update

I am presently writing a summary of Chapter 2 of ICEM. I'll be on vacation the next few days, but it should be done sometime next week. While I am happy to continue providing summaries, I'd be even happier if anybody out there wants to contribute one! If interested, please email me to let me know what chapter and we'll discuss the details.

Useful software

David Touretzky and others at CMU have written a great program called HHsim: Graphical Hodgkin-Huxley Simulator. If you want to actually tinker with the variables discussed in Chapters 1 and 2 of ICEM, it provides a great platform. It is a stand-alone implementation of the HH model of the squid giant axon (Chapter 2 of ICEM), and lets you play around with current clamp, voltage clamp, and tweak all the parameters in the HH model while seeing the results in real time. As they describe in a set of possible virtual experiments, you can also use HHsim to explore the Nernst equation as well as resting potential, so it is a useful supplement to Chapter 1 of ICEM as well. Also, installation is painless (just download the executable and double click: it will install and then you can click on the desktop icon to run it).

HHsim is educational software good for exploring some key biophysical concepts. If you actually want to construct your own models using the channels discussed in ICEM, then you will need to go to a modeling platform, most likely Genesis (their book is now available free online here), Neuron (they are finally putting out a book this year), or XPP-Aut, an underappreciated analysis tool maintained by Bard Ermentrout (book available here and helpful tutorial available here). If I have missed any software, please let me know and I'll put it in the sidebar.

Note: I am following along ICEM using Genesis, and may post some Genesis-specific posts separate from the other posts. If you are interested in doing the same, and are on a Windows machine, you should probably go through the installation hell now, as it will likely take you a few hours of labor to get it working. Once working, it is worth the initial effort.

Chapter 1: Introduction

In this introductory chapter Hille provides general background that should serve as a useful reference for the rest of the book. Even in this chapter, the density of the book, and its sometimes frustrating tendency to quickly make arcane technical points as if they were part of the average neuroscientist's working knowledge, is exhibited in full force (see pages 14-15 for his derivation of the Nernst equation for an example). This is why I thought it would be a good book for discussion.

Chapter 1: Introduction
There are two main types of transport processes that give excitable cells their electrical phenotypes: ion channels and carrier molecules. The carrier molecules establish the overall ion gradients across neuronal membranes, while ion channels are responsible for fast changes in the electrical properties of cells. Ion channels, such as voltage-gated sodium channels, are water-filled pores through which ions travel. Carrier molecules, such as the sodium-potassium pump, are large proteins embedded in the cell membrane whose binding sites are thought to be alternatively exposed on the extracellular and cytoplasmic surfaces. Carrier molecule function is still not very well understood.

Neurons are usefully represented by equivalent circuit models which recreate their I-V relations. Cells are often represented by a capacitor in parallel with resistors, with the capacitor standing in for the cell membrane and the resistors standing in for ion channels. Recall from physics that the prototypical capacitor consists of two conducting plates separated by an insulating material. The cell membrane acts as a capacitor, with the cytoplasmic and extracellular spaces acting as conductors while the membrane is relatively insulating. Resistors in circuit theory are conductors of current, specifically conductors through which energy (electrical potential) is lost as heat. Hence, the overall picture we have of a neuron is a capacitor (cell membrane) in parallel with a resistor (the ion channels) [Figure 1.2].

In neurons, current flow is much more complicated than in electrical circuits because the current flow is caused by ion species traveling through aqueous pores, and these same ions set up the voltage gradient across the membrane. That is, voltage differences across the membrane is not caused by energy lost due to heat, but because the ion flow itself creates differences in the net charge across the membrane, and this induces an electrical field and an associated potential field. The second half of the book goes into great detail about the physics of these processes, but some important preliminary details are given in this chapter.

The Nernst Equation (Equation 1.10) describes how ion flow works in the simplest possible case: a membrane selectively permeable to only one ion species (e.g., K+), initially set up with different concentrations of that ion on the two sides of the membrane (Figure 1.4). There are two forces acting in such cases: the concentration difference effectively sets up diffusional forces that push the ion from the side of higher concentration to that of lower concentration. Passage of ions down this diffusion gradient then sets up a voltage gradient, whose force acts in the direction opposite to that of the diffusional gradient. Mathematically, you can solve for the case in which the forces due to diffusion and electrical potential cancel out, and the solution is the Nernst equation:

E_eqm = E_in-E_out = (RT/F) ln ([ion]_out/[ion]_in)

In words, the potential difference across the membrane will exactly counteract the diffusion pressure when enough ions have passed through the membrane to set up E_eqm. When the voltage across the membrane is equal to E_eqm, the net ion flow through the membrane is zero. Note that in physiological systems it takes very few ions to set up the required voltage difference, so the ion concentrations remain relatively unchanged. For instance, the concentration of K+ ions remains very high on the inside of a cell even when it is at E_k.

At rest, the neural membrane is not permeable to only one ion species, so to calculate the potential at which the net current flow is zero across the membrane, we cannot simply rely on the Nernst equation. In such cases, the GHK (Goldman-Hodgkin-Katz) equations describe how to calculate the voltage-difference E_r (the resting potential) at which the net current flow through the membrane is zero.

Because of the ionic basis of membrane currents, we can't simply apply Ohms law (I= gV) to cell membranes. Such an analysis would predict that zero current would be flowing through an ion channel when the voltage difference across the membrane is zero. We typically have zero current flowing when V=E_eqm, calculated using the GHK equation, so we need to replace Ohm's law from electric circuits with I=g(V-E_eqm) to accurately recreate neuronal behavior. When a channel is selectively permeable to only one ion, then E_eqm for that channel can be calculated using the Nernst equation.

Finally, as Hille shows by working through in a few examples in Figure 1.6, the conductance in the equation I=g(V-E_eqm)is rarely fixed in time. It is often a function of the voltage across the membrane, which gives neurons some of their most interesting behavior. Before moving on to Part I of ICEM, I would recommend thorougly working through the examples in Figure 1.6, as it is a very useful intuition pump in which Hille first considers the behavior of simple Ohmic channels and builds up to more realistic voltage-dependent channels with nonzero equilibrium potentials. It is probably the most important figure of the chapter.

Kandel and Schwartz (Chapter 6) and this site have good discussions of the Nernst equation. Johnston and Wu derive it from basic principles.