Changes

==Resistance or Conductance versus Permeability, and the Electrical Equivalent Circuit for an Ion Channel==

Figure 1 shows a simple schematic view of a nerve cell membrane, greatly magnified. It is important to focus on two features of the membrane:

* Most of the membrane consists of a lipid bilayer.

** Another way to think about it is this: if one applied a fixed current, then a measure of the resistance would be how much the voltage changed for that fixed current. If the resistance is low, there would be a small voltage change; if the resistance is high, there would be a larger voltage change.

Another way to describe the ease with which an ion can move through a channel is to take the '''reciprocal''' of the resistance, which is known as the '''conductance'''. The conductance, usually represented by the symbol $$g$$ and measured in units of siemens, is thus equal to $$1/R$$.

* Qualitatively, the relationship between resistance and conductance is clear: a high conductance implies a low resistance, and a low conductance implies a high resistance.

* To use conductances with Ohm's law, divide both sides of the equation by $$R$$ to obtain $$V/R = I$$; since $$1/R = g$$, Ohm's law becomes $$g V = I$$.

Channels that are permeable to a specific ion will be associated with a specific Nernst or equilibrium potential. That is, the current flow through a channel will not only depend on the voltage across the membrane, but will also depend on how different the voltage is from the Nernst or equilibrium potential.

* If the membrane voltage is equal to the Nernst or equilibrium potential, then there should be no net flux of those particular kinds of ions across the membrane. For example, suppose you have a neuron with a high internal concentration of potassium, and the membrane potential is at the equilibrium potential for potassium. Imagine that the flux of potassium ions caused by the concentration gradient (an ''outward flux'') became a bit larger due to a random fluctuation in the movement of the ions. This would cause the inside of the membrane to become more negative and the outside more positive, leading to a hyperpolarization of the cell. This would increase the strength of the electrical gradient; in turn, the flux of potassium ions caused by the electrical gradient (an ''inward flux'') would strengthen, and the random fluctuation would be undone. Thus, there would be a balance between the flux of potassium ions out of the nerve cell (flux due to the concentration gradient) and the flux of potassium ions into the nerve cell (flux due to the electrical gradient), as illustrated in Figure 2.

C \frac{dV}{dt} = I. \quad\text{(Equation 4)}$$

* In circuit diagrams, capacitors are represented by a symbol that shows two plates of equal length with a gap between; charges can be stored on the plates, but cannot directly jump across them. However, as positive charges are added or removed to one plate, positive charges will be removed or added to the other plate (respectively), so as '''Equation 4''' states, changing amounts of charge, or current, will lead to changes in voltage.

* We can now represent all of the ungated potassium channels in the membrane, and how they will respond to charges injected into the nerve cell, with an electrical equivalent circuit that shows the capacitor in parallel with the ion channels. The capacitor is in parallel because ions can either flow through ion channels, or affect charges on the other side of the membrane if they land on the membrane, and thus there are two parallel paths for the movement of charge. In Figure 4, the capacitance of the membrane is labeled $$C_m$$.

* An analogy, which can be shown to be mathematically equivalent, is to think about what happens to a hose with flexible sides when water pressure is applied to one end. The water pressure is equivalent to voltage, the center of the hose is equivalent to the resistor, and the flexible sides are equivalent to the capacitor.

==The Electrical Equivalent Circuit for the Resting Membrane==

* We developed the entire electrical equivalent circuit by focusing solely on the potassium ions for simplicity. Since this is the ion to which the nerve membrane is primarily permeable at rest, this is a reasonable first approximation. However, as we saw in the previous unit, real nerve cells are permeable not only to potassium, but also to sodium and chloride ions.

* We can represent the other conductances in the same way that we did for the potassium ion channels: as a conductance for those channels in series with a battery, representing the equilibrium or Nernst potential for each ion. The result is shown in Figure 5.

* The total current flowing across the membrane is the sum of the currents flowing through each of the branches of the equivalent circuit. In symbols, we can state this as <math display="block">

I_m = I_{\text{K}^+} + I_{\text{Na}^+} + I_{\text{Cl}^-}. \quad\text{(Equation 5)}$$

* At the resting potential, the net flux of currents across the membrane is zero, even though the net flux of each of the ions is not zero. Figure 6 shows a schematic diagram for two of the three ions, for simplicity.

* If the fluxes were not in balance, the membrane would depolarize or hyperpolarize, and the change in potential would increase the fluxes of ions further from their Nernst potential, and reduce the fluxes of ions closer to their Nernst potential; the net effect would be to change the membrane potential back to the resting potential.