Perhaps the most obvious way is that neither the brain nor any other organ can survive without a supply of oxygen and removal of its carbon dioxide. It’s the lungs that provide these services.
A less obvious way has to do with the rhythm of breathing. Our heartbeat and respiration are the two most conspicuous and widely known rhythms of the human body. The heart has its own pacemaker and will beat rhythmically even when removed from the body, but the lungs do not. Their pacemaker system is in the brainstem—particularly the medulla oblongata and pons. Every breath has to begin with signals from the brain that travel down the spinal cord and out through nerves leading to the muscles of the diaphragm and between the ribs. Thus, severance of the spinal cord in the upper neck causes respiratory arrest; signals from the brain can’t get to the respiratory muscles any more. Conversely, respiratory arrest for any other reason quickly leads to irreversible brain damage.
Furthermore, there are many voluntary aspects of respiration that are controlled by the brain, with signals originating in the cerebral cortex—decisions to modify our respiration in speaking, shouting, whispering, singing, playing wind instruments, coughing, breath-holding, and so on.
