The brain stem was the part of a humanoid brain that controlled all vital functions.
The psi-wave device used by the Prytt was connected directly to the brain stem. (TNG: "Attached")
In late 2366, the Zalkonian known as "John Doe" damaged his brain stem when his ship crash landed on a planet. (TNG: "Transfigurations")
In 2369, while performing an autopsy on Hon-Tihl, Dr. Julian Bashir noted that Hon-Tihl's membrane permeability in the columnar cells in his brain stem were off. (DS9: "Dramatis Personae")
In 2373, The Doctor's 'daughter' Belle's brain stem was compromised after she severely hit her head. (VOY: "Real Life")