There is a fantastic thread in this forum called "do you talk to brain dead patients"? It has lots and lots of links to brain death and the criteria for same.
Off the top of my head though, I think the best way of looking at this was the one outlined in "The Intensive Care Manual" by T.E.Oh (can't remember exactly which edition though
) where he stated that we do not make a distinction between brainstem death and brain death. (think about it - the tests for the reflexes are ALL brainstem reflexes).
Conversely can you then have survival of the brainstem with death of the cerebrum? YES!!! Essentially this is what occurs in PSV. Persistent Vegatative State.
A persistent vegetative state (commonly, but incorrectly, referred to as "brain-death") sometimes follows a coma. Individuals in such a state have lost their thinking abilities and awareness of their surroundings, but retain non-cognitive function and normal sleep patterns. Even though those in a persistent vegetative state lose their higher brain functions, other key functions such as breathing and circulation remain relatively intact. Spontaneous movements may occur, and the eyes may open in response to external stimuli. They may even occasionally grimace, cry, or laugh. Although individuals in a persistent vegetative state may appear somewhat normal, they do not speak and they are unable to respond to commands.