"Old-fashioned telephone systems relied on operators to take incoming calls and connect them to the appropriate persons they were intended for. Which structure of the brain functions in an analogous manner?"

The thalamus of the brain functions in this analogous manner.

Explanation:

The thalamus (dorsal and ventral thalamus) is a structure located in the forebrain, at the base of the cerebrum. It acts as a relay center with nerve fibers project out to the cerebral cortex in all directions.

The thalamus is mainly made up of gray matter and contains a small amount of white matter. The grey matter, which is divided into the anterior part, medial and dorsal part contains several nuclei that interconnect brain activity between different areas of the brain. The anterior part contains anterior thalamic nuclei, medial part contains dorsomedial nuclei and the lateral part contains dorsal tier and ventral tier nuclei. Interlaminar nuclei , reticular nuclei, medial geniculate body , lateral geniculate body, etc are other nuclei present in the thalamus.

The thalamus receives sensory information from the various receptors in the body. It relays this sensory information to the cerebral cortex, where it is interpreted as pain, touch, temperature, etc. The various functions of thalamus include the integration of sensory information, emotional control, motor control, integration of sensations with emotions, regulation of consciousness, sleep, and alertness.

this would be a math