It's the old question ... just imagine a teleporter which doesn't only assemble the new person elsewhere but leaves the old one in one piece. If you enter it - who of them is you? I'ld say, just the old person which steps out of the teleporter and complains that it didn't work. :)
I'm also wondering whether I've got some hints straight. There are some suggestions that much of the experience is somewhat metaphoric, e.g. once the doctor has blinded herself nobody reads your texts anymore or notices you at all (expect Dr. Gray, that is). On the other hand, a room full of bunnies suggests that things are actually real and are repeating themselves only because they get set up anew, e.g. each slaughtered bunny will be replaced by a new one.