High saturated fat and sugar diet causes reduced BDNF in the hippocampus. This is an animal model. The effect was a reduction in spacial learning. Dietary choice can effect the neuroplasticity and learning capacity of the hippocampus.
The absolute best way to promote neurogenesis is through exercise. This is through an increase in BDNF. There are ways to augment this: curcumin, n-3, nutrition... The concept that the brain doesn't heal is out the window.