Why do neurons not regenerate?

Well, the answer is that the brain's neurones have an architecture that's what's called post-mitotic: there are only a few restricted areas in the brain and central nervous system where there are new nerve cells being born in an adult brain.

This means that, for the most part, you must rely on the complement of nerve cells that you are born with - and which continue to divide for a very short window after you were born - meaning that what you're born with is what you have to make last a lifetime.

There's a reason for this, because if brain cells were dividing all over the place - and remember that brain cells have long connections that they make from one cell to the other, and those connections are crucial to you being able to do the right thing, say the right thing, have memories and for your brain to be able to work properly - if those cells were dividing all over the place and making aberrant connections, then it will be very, very difficult to preserve that architecture. So there's kind of method in the madness.

The problem is that, as that is a fixed structure, it's very hard to repair it by getting the cells to re-divide because basically, if you have an injury that's bad enough to destroy a part of your brain or your nervous system, evolutionarily speaking the chances are you'd be dead anyway. So, we haven't really evolved the ability to repair the brain and spinal cord.

In some animals though, that can happen and things like gold fish, lampreys, and also even salamanders can restore whole limbs, and bits of their nervous systems. If you take the eye out of a frog, turn it around and put it back in again, it will rewire itself back into the brain, only, because the eyes now are upside down, the animals see upside down and it does the wrong thing. If you hold a fly in front of it, instead of jumping forward at the fly, it jumps backwards and takes a bite out of the deck.

That won a Nobel Prize for Roger Sperry a few years ago and proves that some animals can regenerate their nervous system, but certainly, not us unfortunately.