The neurons you're born with stay with you for the most part. Most of their complex organization is formed through a series of one time events early in development that can't really be replicated and then stays with you for the rest of your life. You get shingles when you're older because the same neurons were with you that got infected by chicken pox when you were younger are still there. There's a few limited areas in the nervous system where new neurons might be formed, but in general neurons are life long cells so be nice to your nervous system. Most other cell types in your body are turning over as you said, including glia and other types of cells in your brain.