Under what circumstances will a human neuron divide?
Mature, differentiated neurons do not divide (undergo mitosis), but apparently there is a small population of self-renewing neural stem cells in adults that can produce new neurons. Neurogenesis predominantly occurs in the subventricular and subgranular zones of the brain.

Peripheral nerves can regenerate along its axon as long as the endoneurial tube and the Schwann cells are intact. Here's a picture of a neuron regenerating.