Their ability to contract or dilate in response to impulses from the brain absolutely. Sever the nerves of the sympathetic nervous system and you'll never be able to raise your heart rate again. Sever the parasympathetic nerves and you'll never be able to lower your heart rate. Everything has to do with your brain.
Also to add on, it's possible to still have voluntary or involuntary control of a body part, but receive no sensory input from it. You could sever those afferent nerves, making it so you could get a boner, but not feel anything from it, which would lead to a lot of complications in itself. In most forms of flaccid paralysis, though, you neither have feeling or control. Either way, getting paralyzed in one way another means your sex life is over.