In that case I think it would heavily rely on the whole "everything critical is duplicated in software" part. It's the same ship of theseus, just one level up - instead of neurons and pathways, it's entire sections of the brain. If my biological short term memory withers away from dementia, boom, the chip already has that capability backed up and running. If my long term memory starts going because of Alzheimers, boom, also have those uploaded to the cloud on demand. Got ALS? Motor functions already rerouted through the chip.

At no point is the entire biological brain destroyed at once and "transferred"; there is a continuity of consciousness throughout the process, a continuity of self and pattern. That's the part most critical to me.