Our online Goldman-Hodgkin-Katz (GHK) calculator can be used to calculate the membrane potential (V_m) under a variety of conditions. The Goldman-Hodgkin-Katz equation can be derived based on simple thermodynamic principles in the same manner that the Nernst equation can be derived based on simple thermodynamic principles (see Derivation of the Nernst Equation). Notice that whereas the extracellular concentrations of K⁺ and Na⁺ appear in the numerator, the intracellular concentration of Cl⁻ is in the numerator. Moreover, whereas the intracellular concentrations of K⁺ and Na⁺ are in the denominator, the extracellular concentration of Cl⁻ is in the denominator.

In the equation above, p_K, p_Na, and p_Cl are the relative membrane permeabilities for K⁺, Na⁺, and Cl⁻, respectively. Permeability refers to the ease with which ions cross the membrane, and is directly proportional to the total number of open channels for a given ion in the membrane. Therefore, if many K⁺ channels are open, p_K will be high. If only a few K⁺ channels are open, p_K will be small. If all K⁺ channels are closed, p_K will be zero.

As can be seen from the equation above, two major factors contribute to the membrane potential: concentration gradients and the relative permeabilities of the ions. Under most physiological conditions, the intracellular and extracellular concentrations of ions remain the same. Similarly, temperature remains the same in most cases. Therefore, the relative permeability values become the most important factors that contribute to the value of the membrane potential. The magnitude of the permeability (i.e., how many open channels) for each ion determines the relative contribution of that ion to V_m. The larger the permeability of a certain ion, the larger the contribution of that ion will be in setting the membrane potential. Notice that if the channels for a certain ion are all closed (i.e., the permeability for that ion is zero), the above expression is reduced and simplified to include only the terms involving the other two ions. It is important to see that the movement of each ion down its own electrochemical gradient will tend to move the membrane potential towards the equilibrium potential for that ion.

Normally, permeabilities are reported as relative permeabilities with p_K having the reference value of one (because in most cells at rest p_K is larger than p_Na and p_Cl). For a typical neuron at rest, p_K : p_Na : p_Cl = 1 : 0.05 : 0.45. Given these values and typical mammalian intracellular and extracellular concentrations (see here), you should be able to calculate the membrane potential for a neuron (or any other cell) at rest. Using the GHK equation, and the ionic concentrations and relative permeabilities given above, it can be calculated that V_m ≈ −68 mV in a typical neuron. In most cells, the membrane potential (V_m) is relatively stable with little or no significant deviation from the resting value. Therefore, in most cells, V_m = V_rest. This is because in these cells, the relative ionic permeability values do not change appreciably over time. In excitable cells (such as neurons, muscle cells and some endocrine cells), on the other hand, there are large transient changes in the relative permeability values for ions and, therefore, the membrane potential transiently deviates from the "normal" resting membrane potential.

Figure 1 demonstrates how changes in the relative permeability values for K⁺, Na⁺, and Cl⁻ influence the value of the membrane potential. Changes in the relative permeability of ions can come about as a result of channel opening or closing caused by physiological stimuli (e.g., opening of voltage-gated channels) or by the application of agonistic or antagonistic drugs. If the values of p_Na and p_Cl are set to zero (i.e., if all Na⁺ and Cl⁻ channels are closed), it can be see that K⁺ will be the only contributor to the membrane potential. In this case, the value of the membrane potential will be exactly the same as the K⁺ equilibrium potential (V_m = V_K) (2 in Fig. 1). On the other hand, if the values of p_K and p_Cl are set to zero (i.e., if all K⁺ and Cl⁻ channels are closed), it can be see that Na⁺ will be the only contributor to the membrane potential. In this case, the value of the membrane potential will be exactly the same as the Na⁺ equilibrium potential (V_m = V_Na) (3 in Fig. 1). Similarly, if all K⁺ and Na⁺ channels are closed (i.e., p_K = 0 and p_Na = 0), then V_m = V_Cl (4 in Fig. 1).

We should also consider more realistic situation where p_K, p_Na, and p_Cl have non-zero values. For example, it can also be seen that if p_K is much larger than p_Na and p_Cl, then V_m will be closer to V_K than it will be to V_Na or V_Cl because K⁺ is now the major contributor to the membrane potential (5 in Fig. 1). Thus, if p_K is much larger than p_Na and p_Cl, then V_m will be closer to V_K, although it will never be exactly at V_K (unless p_Na = 0 and p_Cl = 0). Similarly, if p_Na is much larger than p_K and p_Cl, then V_m will be closer to V_Na (6 in Fig. 1). Finally, if p_Cl is much larger than p_K and p_Na, then V_m will be closer to V_Cl (7 in Fig. 1).

In summary, based on the information presented here and the equilibrium potential values for common ions found in physiological solutions (see here), it can be seen that under physiological ionic concentrations, the equilibrium potential for K⁺ (V_K) sets the negative limit for the membrane potential (∼ &minus90 mV), whereas the equilibrium potential for Ca²⁺ (V_Ca) sets the positive limit for the membrane potential (∼ +137 mV). As discussed above, for these extreme values to be reached, K⁺ or Ca²⁺ has to be the only ion contributing to the membrane potential. Thus, under the physiological conditions of mammalian cells, membrane potentials outside of this range are not possible. Of course, experimentally, the membrane potential may be changed to almost any value (within limits of about ±200 mV where the membrane begins to break down) by changing the ionic concentrations or by injecting positive or negative current into the cell (see lecture on the Neuronal Action Potential).