Final answer:
The Hering-Breuer reflex, involving the vagus nerve, prevents lung overinflation by sending inhibitory signals to the inspiratory center in the brainstem.
Step-by-step explanation:
The neural pathway that sends continual inhibitory impulses to the inspiratory center in the brainstem to prevent overinflation of the lungs is known as the Hering-Breuer reflex. This reflex involves the vagus nerve, which provides sensory input to the brain when the lungs are inflated to a certain volume. The reflex then triggers the pneumotaxic center located in the pons. This center sends inhibitory signals to the dorsal respiratory group (DRG) of the medulla oblongata, effectively controlling the rate of breathing and preventing overinflation of the lungs during quiet breathing.