One could models the process that makes you and Brittney Spears as two different processes. Or you could view it as one process that moves the two states forward. In this sense you could say that you wake up as both but off course their brainstates don't know about each others inner workings. If you undergo lobotomy or multiple personality disorder you would probably still identify with all the pieces even if they can't (fully) communicate with each other. (Or rather the pieces would acknowledge their common origin and close ties they have with each other)