In order that a "self" may exist there must be some continuity of mental experiences and, particularly, continuity bridging gaps of unconsciousness. For example, the continuity of our "self" is resumed after sleep, anaesthesia, and the temporary amnesias of concussion and convulsions.
"Eccles' Model of the Self Controlling Its Brain: The Irrelevance of Dualist-Interactionism" by Donald E. Watson and Bernard O. Williams, 2003.