Thanks for asking. Obviously, the behavioral manifestation will depend largely upon two issues: (1) the actual deterioration taking place and (2) the brain structure affected. Given that your question is general, I will broaden the reply.
Alzheimer's tends to be the poster child for deterioration, but other degenerative disorders can hit the brain (Parkinson's of Huntington's, for example). Typically, the most obvious manifestation is if the process hits either the hippocampus (affecting memory), or the cerebellum (affecting fluid movement and balance). Alzheimer's hits the memory centers and Parkinson's & Huntington's hit the movement centers most profoundly - Parkinson's creates small motor tremors and Huntington's creates "chorea," or large muscle tremors/spasms. Of course, they will all eventually kill the person (think Ronald Reagan with Alzheimer's or Mohammad Ali for Parkinson's) because as more of the brain is affected, deeper areas begin to feel the effects of the degeneration.
BTW - chronic long-term alcoholism will also "degenerate" the brain and create a condition known as "Wernicke-Korsakoff Syndrome." Although it is a bit different, think of it as a self-imposed Parkinson's. In the vernacular, most folks call it "wet brain."
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