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My professor said that inward rectifying channels help move the membrane potential back to the resting potential during the undershoot phase of the action potential. The membrane potential would have to be lower than the equilibrium potential of potassium for this to occur (inward potassium driving force), but the membrane potential never goes below the equilibrium potential of potassium during the action potential. So, when are potassium rectifying channels actively passing ions?

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I hope I can maybe shed some light on this concept for you. Rather than complicate anything more than necessary, let's start small and build. To directly answer your question, these channels can function as leak channels (so ions are always flowing through them at some rate) or can be open at select voltages (ions will only flow upon voltage-selective opening of the channel).

Here's a little more detail. These channels are involved in a number of physiological processes (not only in neurons, but smooth muscle and cardiac cells as well). One such process is, as your professor pointed out, to drive the membrane potential back to rest after you hyperpolarize the cell following an action potential. I completely understand your confusion with this concept because in order to properly answer your question, I've gone through a number of resources to make sure I give the most complete answer possible. Here goes:

Maybe the most important part!!! These inward rectifiers were originally called “anomalous” rectifiers because, let's face it, they're goofy. They don't do what you expect them to do (as you have pointed out). Part of the reason is that there are molecular mechanisms that cause these channels to be blocked and preferentially permit an inward conductance, despite what you'd expect. There's a decent review article on these channels that might help your understanding (the introduction is the most important part). You can find it here http://physrev.physiology.org/content/90/1/291#ref-233.

Now I'll try to get into a different explanation that kind of oversimplifies things, but I believe might help.

Remember that the gradient is not solely electrical in nature, it is an electrochemical gradient. Now you're probably thinking to yourself "I know that, dude, don't patronize me!" I only bring this up because it appears your biggest concern is how the electrical properties of the gradient regulate K+ flow across the membrane (heavily considering the electrical driving force by focusing on membrane potential minus the Nernst potential for K+). However, flip that thinking for a second. Specifically, think about what happens during an action potential solely in terms of concentration. Your rising phase is sodium influx to the cell (you go from lots of sodium outside the cell to lots of sodium inside the cell). On your falling phase, sodium channels close and potassium channels open. Thanks to the good ol' Na+/K+ pump, you have a gradient of little K+ outside the cell and a ton inside. So what happens? All that K+ rushes out of the cell (as you would also predict from the electrical component of your gradient now!). That means you had a lot of K+ inside and a little outside, but you now have a lot outside and a little inside. This goes on too long and you get an undershoot/hyperpolarization, but you have to get back to your resting membrane potential. Enter the inward rectifiers. These K+ channels open and (again, thinking solely about concentration) the K+ is going to flow down the gradient back into the cell. If you throw in the electrical component, remember this: Neurons exist in a multi-ion system which complicates things like reversal potential for a single ion. This is a complicated subject that I won't get into, but it boils down to reversal potential being a different property from equilibrium potential, which becomes important in a multi-ion system.

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