DOH! I finally figured out this big clue...
You always know when rats are going to appear/attack you by looking for the evil rat chalk drawings on the walls or floor of the asylum. How did I not see this before? DUH.
Also, Shegore has a bad habit of peeking out at you in the same areas (but, not all areas that she peeks out at you will have a rat attack). I think the game programmers did this on purpose since she has a high pitched squeak that sounds similar to the rats. By the time you realize those squeaking noises aren't coming from her, the rats are upon you.
Rats will attack in two waves. On the lower portions of the asylum tower, this means two rats will attack first and then there will be one more rat coming in a second wave.
As you climb higher in the asylum tower, there are more rats in each wave of attack. This is where the Psi-Chain-Blast comes in handy, killing the rats you don't see. If the second wave of rats doesn't appear (but you still hear them) shut off Invisibility for a second or two to force them to come and attack you. Then turn on Invisibility, get a safe distance away and then blast them.
After going through the game over and over again, I'm finding that Invisibility works great if you hit it the minute you hear them start squeaking. At this point in the game, the only powers I have loaded are Psi Blast, Invisibility and Leviatation. Shield may be good for the first wave of the rat attack but usually the second wave of the rat attack will come when your shield runs out.