The is involved in coordinating eye and body movement and works with the pons to coordinate sleep and arousal

Mid-brain.

Explanation:

According to my research on studies conducted by neurologists, I can say that based on the information provided within the question the location of the brain being described is called the Mid-brain. This location is also known as the mesencephalon and is the biggest part of the adult brain, which is responsible for all that is mentioned in the question.

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the solute can move "downhill," from regions of higher to lower concentration, relying on the specificity of the protein carrier to pass through the membrane. this process is called passive transport or facilitated diffusion, and does not require energy.