The answer is the first one (A): by stimulating neurons in the dorsal respiratory group in the medulla.
Explanation for each option:
(A) by stimulating neurons in the dorsal respiratory group in the medulla - This is correct. The apneustic center in the pons promotes deep, prolonged inhalation by stimulating the neurons in the dorsal respiratory group (DRG) in the medulla, which are responsible for initiating inspiration.
(B) by inhibiting the activity of neurons in the dorsal respiratory group in the medulla - This is incorrect. Inhibition of the DRG would reduce the drive for inspiration, which is contrary to the function of the apneustic center.
(C) by stimulating the pneumotaxic center in the pons - This is incorrect. The pneumotaxic center actually works to inhibit the apneustic center and limit the duration of inspiration, thus promoting a normal breathing rhythm rather than deep breathing.
(D) by suppressing neurons in the ventral respiratory group in the medulla - This is incorrect. The ventral respiratory group (VRG) is primarily involved in forced expiration and active breathing, not the deep, prolonged inspiration stimulated by the apneustic center.
Summary:
The apneustic center in the pons stimulates deep breathing by stimulating neurons in the dorsal respiratory group in the medulla.