Brake Thumps and Pulsates

There a several possibilities that I can think of without actually experiencing the problem. The #1 cause of brake shudder is warped rotors. That's also the easiest thing to fix. But if it's related to the engine thing, it might be related to your power brake assist system. I'm not really familiar with that area, but basically there might be a problem with a vacuum hose or a sensor that's f'ing with your ECU. Or it could be an ABS malfunction, which would be the worst problem. If your "Check Engine" light is coming on, I would take it to a Lexus shop (or anybody else who can do an engine diagnostsic) and find out what the malfunction code is. That would help narrow down your possibilities.