**Core Concept**
The depth of inspiration is a critical component of the respiratory cycle, controlled by the integration of various neural and mechanical signals. The primary mechanism of depth control is mediated by the Hering-Breuer inflation reflex, which senses changes in lung volume and sends feedback to the brainstem.

**Why the Correct Answer is Right**
The correct answer is controlled by the **Hering-Breuer inflation reflex**, which is mediated by the activation of **stretch receptors** in the bronchi and bronchioles. These stretch receptors are sensitive to changes in lung volume and send afferent signals to the **dorsal respiratory group (DRG)** in the medulla oblongata. The DRG then integrates this information with other signals, such as the rate and depth of breathing, to regulate the depth of inspiration. The Hering-Breuer inflation reflex helps to prevent over-inflation of the lungs and ensures that inspiration is not too deep.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because it is not directly related to the control of inspiration depth. While it is true that the diaphragm is a primary muscle involved in inspiration, it does not control the depth of inspiration.

**Option B:** This option is incorrect because the vagus nerve is primarily involved in the parasympathetic control of the heart rate and the regulation of digestive functions, rather than the control of inspiration depth.

**Option C:** This option is incorrect because the pons is involved in the control of respiration, but it is not the primary structure responsible for controlling the depth of inspiration.

**Clinical Pearl / High-Yield Fact**
The Hering-Breuer inflation reflex is an important mechanism for preventing over-inflation of the lungs and is often disrupted in patients with chronic obstructive pulmonary disease (COPD).

**Correct Answer: C. Pons.**