I'm writing a story that features modified humans that have a very long lifespan/ practically immortality. Immortal characters in fiction aren't exactly rare, but I want to write something more realistic (which might be ironic considering the premise). 

If we suppose a person wouldn't physically die given extremely long age, what would happen to the brain of the person? Could a person that has lived for thousands of years act as a part of society or would he have trouble with recalling memories and other functions, etc?

Edit: I'm assuming that diseases due to immortal nature are not a factor. Though I'm no expert in any of this, so nothing's to say there aren't some diseases that would affect the brain on the long run despite immense recovery capabilities I'd link with immortality. The immortals have stunning recovery capabilities, and are for example capable of regenerating bodily damage almost in an instant, though I'm not exactly sure what this would mean to the brain.