Are you accidently hitting the mode button at any time? The one just below the little green light. My friend does that all the time when playing games on my comp. If it's off, the joystick will stop responding because it's used when you hook more than one controller together. The light should be on at all times. If you are hitting it, just turn it back on and the joystick functionality will return in a second or two.
I've ended up taping a piece of cardboard over my button so my friend didn't hit it all the time. It's never been a problem for me, but he's more of a button masher than I am
