**Core Concept**
Lateral medullary syndrome, also known as Wallenberg's syndrome, is a clinical entity resulting from the occlusion of a specific posterior circulation artery in the brainstem. This syndrome is characterized by a constellation of neurological deficits, including ataxia, dysphagia, and hoarseness.

**Why the Correct Answer is Right**
The correct answer is the vertebral artery or its posterior inferior cerebellar artery (PICA) branch, which supplies the lateral medulla. The occlusion of this vessel leads to ischemia of the lateral medulla, resulting in the characteristic clinical features of Wallenberg's syndrome. The lateral medulla contains critical structures such as the nucleus ambiguus, the spinal trigeminal nucleus, and the inferior cerebellar peduncle, which are responsible for the clinical manifestations of this syndrome.

**Why Each Wrong Option is Incorrect**
**Option A:** The anterior inferior cerebellar artery (AICA) primarily supplies the anterior inferior cerebellum and not the lateral medulla.
**Option B:** The middle cerebral artery (MCA) is a major artery supplying the lateral surface of the brain's hemispheres and is not related to the lateral medulla.
**Option C:** The posterior cerebral artery (PCA) primarily supplies the occipital lobe and the medial and posterior aspects of the temporal lobe, but not the lateral medulla.

**Clinical Pearl / High-Yield Fact**
The clinical features of lateral medullary syndrome can be remembered using the acronym "C-2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12", which stands for the cranial nerves that are affected in this syndrome, including the vagus nerve (C10), the glossopharyngeal nerve (C9), and the hypoglossal nerve (C12).

**Correct Answer: C. Posterior inferior cerebellar artery (PICA) branch of the vertebral artery.**