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:
Eeqm = Ein-Eout = (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 Eeqm. When the voltage across the membrane is equal to Eeqm, 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 Ek.
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 Er (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=Eeqm, calculated using the GHK equation, so we need to replace Ohm's law from electric circuits with I=g(V-Eeqm) to accurately recreate neuronal behavior. When a channel is selectively permeable to only one ion, then Eeqm 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-Eeqm)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.