Death is defined by irreversible cessation of breathing, of heartbeat and irreversible cessation of functions of the brain. We can quibble over what that means, but "signs of brain activity" i.e. electrical activity does not equal a functioning brain, especially after the heart has irreversibly stopped beating - and so is no longer pumping blood to the brain - and breathing has irreversibly stopped. 
"According to the Uniform Determination of Death Act, an individual is dead when he or she "has sustained either (1) irreversible cessation of circulatory and respiratory functions, or (2) irreversible cessation of all functions of the entire brain, including the brain stem."  What that essentially means is that the brain, an extremely complex organ, no longer helps the patient function."
Presumably the same criteria apply to all animals equipped with a heart, lungs and a brain.