This is the finest essay on this subject I have read anywhere. I will add a couple of things to Matthew's argument.

The first is to expose the idea that science can tell us something about what the brain can and cannot do (or can and cannot be found in the brain) is simply silly. Science tells us nothing about higher functions. What form does data like memories take? Where is it stored? How is it accessed and changed? There are not even working theories for these questions. We do not know how the brain is truly organized in a way that would yield its function. There is no basis to say that there is no place in the brain for this or that. Yes there are heaps of information about these issues, but no explanations.

Second, is the question of self and in particular Buddhism. To say I look for a self and find nothing (until I stop looking) and therefore conclude there is no self is rather simplistic. If I look in consciousness for any part of the brain, I will not find it. So? Is this really science? As for Buddhism, it says that there is no permanent, unchanging, indivisible thing we call a self. And therefore we should not get attached to it. That is all. The Buddha typically did not respond when asked about the self. When he was pushed hard by his disciples he answered, 'There is non-self and there is no non-self". Both extremes were untrue. Western Buddhism is fairly obsessed with this but it is a deep and difficult teaching that few understand. There are translation issues (non-self is different from not self) and commentary over thousands of teachings. Better to just not get too attached to "I am a .........." (butcher, baker, doctor, lawyer) as all that could change. There is always a deeper you made up of values and an even deeper you made up of simple awareness that can watch even your consciousness.

Be prepared for loss of what seems like yourself today, celebrate what you are now, share what you can, and cope when you must. That is how to handle the self.