Previously in this lecture, we described in detail how the membrane potential is established and the factors that govern the value of the membrane potential. Briefly, if there is only one type of ion channel in the plasma membrane and those channels are open, the resting membrane potential will be at the equilibrium potential for that ion. For example, if there are only K⁺ channels in a cell plasma membrane, then V_m in this cell will be the same as the K⁺ equilibrium potential (V_K), which can be obtained from the Nernst potential for K⁺ (see Nernst potential calculator). Therefore, in this cell, V_m = V_rest = V_K, which is what you expect because K⁺ distributes itself across the membrane according to the Nernst equation. As you recall, the Nernst equation describes the situation at equilibrium for one ion (see Nernst Equilibrium Potential). Of course, in this example, the K⁺ channels have to be open to allow K⁺ to redistribute itself across the membrane according to the chemical and electrical potential gradients described previously. At V_K, there is no net movement of K⁺ across the membrane. Since in solutions, electrical current is carried by ions, in this cell at V_K, the net current carried by K⁺ across the membrane is zero (i.e., I_K = 0). Thus, at equilibrium, the net current across the membrane is zero. Remember from the previous section that Ohm's law can be used to obtain the current carried by an ion across the plasma membrane according to Equation 1:

Similarly, if there are only Na⁺ channels in the cell (and are open), at equilibrium, V_m = V_rest = V_Na. And at equilibrium, I_Na = 0. In the same fashion, if there are only Cl⁻ channels in the cell (and are open), at equilibrium, V_m = V_rest = V_Cl. And at equilibrium, I_Cl = 0. These principles were described fully previously in this lecture (see Nernst Equilibrium Potential).

If there are more than one type of ion channels in the cell plasma membrane and the channels are open, all participating ions contribute to the establishment of the membrane potential. In this case, the value of the membrane potential can be determined by using the Goldman-Hodgkin-Katz equation (GHK equation). Now, the resting potential can no longer be described as an equilibrium potential because no ion is at equilibrium (V_m = V_rest ≠ V_Eq.). Rather, each ion moves down its own electrochemical potential gradient. Because the membrane potential is not at the equilibrium potential for any ion, there is a driving force that acts on each ion according to DF = V_m − V_Eq., causing the ion to move into or out of the cell depending on the direction (i.e., arithmetic sign) of the driving force. The system as a whole, however, is said to be at steady-state because the resting membrane potential remains stable (so long as no perturbation is applied to the system).

At steady-state, the resting membrane potential is stable (i.e., not changing), however, each of the ions contributing to the membrane potential moves down its own electrochemical potential gradient according to Equation 1. Assuming that a cell plasma membrane has channels only for K⁺, Na⁺, and Cl⁻ and these channels are open (i.e., G_K ≠ 0, G_Na ≠ 0, and G_Cl ≠ 0), the following current equations can be written for K⁺, Na⁺, and Cl⁻:

Based on the driving force and the nature of ionic species (cation or anion), each ion may move into or out of the cell. As each ion moves down its own electrochemical gradient across the plasma membrane, it leads to the generation of a transmembrane current that may be described by Equations 2 through 4 shown above. As noted above, the arithmetic sign of the current is determined by the arithmetic sign of the driving force. Therefore, in order to determine the total current (or net current) moving across the plasma membrane (I_m), we must add the individual currents resulting from the transmembrane movement of each ion. Thus, the total membrane current (I_m) may be expressed as the algebraic sum of all the individual ion currents (Equation 5).

Since at steady-state, the membrane potential is not changing, it follows that at steady-state, the net movement of ions across the plasma membrane is zero. By extension, at steady-state the net current across the membrane is zero (I_m = 0). Therefore, at the steady-state, the following will be true (Equation 6):

The chord conductance equation is useful in many ways. The membrane potential is described as a weighted average of the equilibrium potentials, and the magnitude of the conductance for all the ions contributing to the membrane potential. The magnitude of the conductance for each ion is related to total number of open channels for that ion, and determines the contribution of that ion to V_m. The larger the conductance of a given ion, the larger the contribution of that ion will be in setting the membrane potential. For example, it can be seen that if G_K is very large (compared with G_Na and G_Cl), V_m will be closer to V_K than it will be to V_Na or V_Cl. Notice that if the channels for a given ion are all closed (i.e., the conductance for that ion is zero), the above expression is reduced and simplified to include only the terms for the other two ions. Remember that the current carried by each ion will tend to move the membrane potential towards the equilibrium potential for that ion. Thus, if G_K is much larger than G_Na and G_Cl, then V_m will be closer to V_K, although it will never be exactly at V_K unless both G_Na = 0 and G_Cl = 0. Of course, if both G_Na = 0 and G_Cl = 0, then it can be seen that Equation 11 will be reduced to simply V_m = V_K. In this case, K⁺ is the only ion that contributes to the establishment of the membrane potential and, hence, V_m will be exactly equal to V_K as predicted by the Nernst equation.

A close examination of the chord conductance equation also reveals that this equation shares a similarity with the Goldman-Hodgkin-Katz (GHK) equation. Both equations rely on the total number of open channels for the ions in the system to determine the resting membrane potential. In the chord conductance equation, the conductance for each ion is used as a major determinant of that ion's contribution to the membrane potential. In the Goldman-Hodgkin-Katz equation, the relative permeability for each ion is used as a major determinant of that ion's contribution to the membrane potential. Both conductance and permeability are related to the total number of open channels in the plasma membrane. In fact, conductance is the electrical equivalent of membrane permeability for an ion.