Functional role of voltage-gated potassium channels
The meaning of reversal potential

Ion channels do not themselves have reversal potentials except for via a weighted average of the ions that they conduct. The reversal potential for a given ion is given by the Nernst equation and besides physical constants is a function only of the relative concentrations of ions inside and outside the cell.

The equilibrium potential of the whole cell is given by the Goldman equation, which is similar to the Nernst equation but covers any number of ions. Note that you can also calculate a "reversal potential" for a channel using the Goldman equation by considering all the ions that a given channel will conduct.

The reason that a cell's equilibrium potential at rest is not equal to the potassium reversal potential is because cells are not permeable only to potassium at rest; although potassium dominates, there is some conductance to other ions, which depolarizes the cell slightly.

There is no way for a potassium-selective channel to have a reversal potential any different from the potassium reversal potential, unless it were to be permeable to other ions as well.

What would happen if there were no potassium channels?

If there were no potassium channels at all, cells would rest near 0 mV, give or take a few mV. The "leak" channels that allow cells to rest at a negative potential are primarily potassium channels. However, it seems like your question is mostly directed towards voltage-gated potassium channels.

What would happen if there were no voltage-gated potassium channels?

As you suggest, cells would repolarize much more slowly. However, your question still has inaccuracies.

1) This repolarization would not be from sodium diffusing away, it would be from potassium leaving the cell through leak channels.

2) The sodium-potassium ATPase is not directly responsible for the membrane potential or for repolarization in an action potential. Ion concentrations change very little during an action potential. The thing that changes is the permeability to different ions. The sodium-potassium ATPase is important, however, for maintaining the relative inside concentrations of sodium and potassium ions over the long term.

In addition, as you also suggest, there would be no afterhyperpolarization.

Is there a purpose to the afterhyperpolarization? Is it just unavoidable?

There is somewhat of a purpose. Although the afterhyperpolarization isn't strictly necessary to get action potentials to fire, there are some hyperpolarization-activated channels in some cells, and in all cells the hyperpolarization facilitates the removal of the inactivation block of sodium channels. In general words, it serves as a bit of a 'reset signal' beyond mere repolarization.

However, the afterhyperpolarization is also pretty much unavoidable without a very fine-tuned engineering of the repolarization process (you would need to rapidly close potassium channels as they approached rest, which would also slow down the return to rest).



Overall, this material would be well-covered in an introductory neuroscience textbook. My personal recommendation is Purves:

Purves, D., Augustine, G. J., Fitzpatrick, D., Hall, W. C., LaMantia, A. S., McNamara, J. O., &amp; White, L. E. (2014). Neuroscience, 2008. De Boeck, Sinauer, Sunderland, Mass.

(any other edition is fine, too, and there a lot of other good alternatives as well)