HTC Desire HD :: Weird Happenings With Desk Night Clock?

Both my wife and I have Desire HDs and we are both experiencing this same problem. We have our phones charging overnight on the bedside with the night clock activated, but after a random amount of time the phone starts going what I can only describe as 'wappy'. It's as if there is someone tapping the screen and selecting options, all manner of things are happening on the screen, apps are starting, search boxes opening. No clock though. I've made sure there is nothing touching the screen and then sat and waited for it, it just starts happening for no apparent reason. Any ideas? My logical mind has already ruled out paranormal activity.