Applying The Goldmanhodgkinkatz Equation To The Action Potential

There are two important positively charged ions (K+ and Na+), and the membrane potential appears to be governed by the relative permeabilities of these two ions.

c ra

VRest -60

VRest -60

Time (msec)

FIGURE 8 Sketch of a nerve action potential.

Time (msec)

FIGURE 8 Sketch of a nerve action potential.

As a result, the Goldman-Hodgkin-Katz equation can be used:

Figure 8 is a sketch of an action potential. One important observation is that the action potential traverses a region that is bounded by ENa on one extreme and Ek on the other. Because the action potential traverses this bounded region, it is possible to use the Goldman-Hodgkin-Katz equation to roughly predict any value of the action potential simply by adjusting the ratio of the Na+ and K+ permeabilities. For the resting level, we have already seen that the ratio of Na+ and K+ permeabilities is 0.01. Thus, we can substitute these values into the Goldman-Hodgkin-Katz equation and calculate a value of approximately —60 mV.

Assume that the Na+ permeability is very high. Then a is a very large number, and the Na+ terms dominate the Goldman-Hodgkin-Katz equation. In the limit, the Goldman-Hodgkin-Katz equation reduces to the Nernst equation for Na+. So, when there is a high Na+ permeability and a low K+ permeability, we can calculate a potential that approximates the peak amplitude of the action potential. During the repolarization phase of the action potential, we can simply assume that the ratio of Na+ and K+ permeabilities returns back to normal, substitute this value into the Goldman-Hodgkin-Katz equation, and calculate a membrane potential of —60 mV. The hyperpolarizing afterpotential could be accounted for by a slight decrease in Na+ permeability to less than its resting level or by a K+ permeability greater than its resting level. The important point is that by adjusting the ratio of Na+ and K+ permeabilities, it is possible to predict the entire trajectory of the action potential.

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