Your question: I noticed that even after my mother was considered brain dead she kept on breathing over the rate set on the ventilator. Why do people keep breathing even after they are considered brain dead?
My humble answer: One of the neat things about the human body is it has a lot of fail safe mechanisms that work kind of like the checks and balance system of the U.S. Government. The heart can keep beating long after the brain is dead because it creates its own electrical impulses. Likewise, the respiratory center is at the base of the brain, a region such that it is often unaffected by damage to the rest of the brain. In this way, even while there is brain damage you continue to breathe. Although if the body is unfed organ failure will ultimately occur.