Differentiated neural cells (brain cells) are characterized by their lack of cell division (mitosis) as you described in your question. Cell division is a highly regulated process with different factors in the cell that tell the cell to start dividing. These factors include cyclins, which are proteins that are involved in cell cycle progression. These cyclins are regulated by other proteins that turn cyclin activity on or off. Interestingly, the molecular signals that initiate mitosis are actually produced in brain cells but may not necessarily be active or be located in the wrong area of the cell. Initiation of cell division requires these factors to enter the nucleus where the DNA is. However, if the factors remain in the cytoplasm they will not cause mitosis to occur.

Researchers believe that diseases such as Alzheimers may actually be caused neurons that accidentally start dividing again. The cells start dividing but are not prepared to divide, resulting in incomplete division and cell death. Thus, inducing mitosis may actually cause more harm than good. Instead, increasing the amount of neural stem cells (undifferentiated cells that have the potential to become neurons) or inducing these neural stem cells to become differentiated neurons may help to replace damaged neurons.