Sometimes it doesn’t, but a lot of autonomic nervous function is controlled by the [vagus nerve](https://en.m.wikipedia.org/wiki/Vagus_nerve) which runs independently from the nerves of the spinal column.
Fun fact: finger/anal stimulation of the vagus nerve in the anus has shown to cure [hiccups](https://www.newscientist.com/article/dn10207-ig-nobel-prizes-hail-digital-rectal-massage/).
The exact nature of impairment depends on the cause of the paralysis, the extent of the paralysis, and where the paralysis begins. [Here is a diagram of the spinal column](https://s3.amazonaws.com/reeve-assets-production/BrainSpinalCordlabld.jpg) with the vertebrae labeled with the functions of the spinal nerves that emerge from them. Someone with an injury at or above the C4 vertebrae is going to have breathing problems and will probably require some form of mechanical ventilation, as the C4 vertebra enervates the diaphragm. While the brain does regulate the heart to some degree, the heart beats on its own. You know this already because people with transplanted hearts are still alive! The heart will beat on a sort of “default” setting, although a person will probably have some amount of cardiac dysfunction.
The beating of the heart is controlled by nerves in the heart itself, originating from the sinoatrial node. This is why a heart transplant isn’t a problem. Indeed a heart can continue beating outside the body for hours, as long as it’s provided with a supply of oxygenated blood.
Breathing reflex is controlled by the brain stem via the spinal cord. Respiratory paralysis is a concern in a serious head or neck injury.
People who are quadriplegic generally need an artificial ventilator.
The heart can be *influenced* by the vagus nerve,(parasympathetic nervous system) and the cardiac nerve which originates from the sympathetic ganglia. Both of these are largely autonomous and aren’t necessarily affected by a neck injury.
The vagus nerve works to slow the heartbeat in certain situations by producing acetylcholine. The cardiac nerve works to increase heart rate and increases force of contraction. But again these don’t produce the heartbeat itself.
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