How can neurons divide without centrioles?
You talk about memory loss. The hippocampus is involved in the formation and storage of memories and indeed one of the few places in the brain where new neurons are formed. The formation of new neurons here is indeed linked to the formation of memories (Coras et al., 2010). 

Hippocampal neurons do not lack centrioles (Dotti &amp; Banker, 1991; Poppov &amp; Tsygunova, 1996). In fact, centrioles may not be necessary for cell division to occur (source: BSCB), although the lack of centrioles in cell models may result in aberrant cell division (Rodrigues-Martins et al., 2008).

In the hippocampus, however, it is not the neurons themselves that regenerate, but rather the resident neural stem cells and progenitor cells (Kirby et al., 2014).   

References
- Coras et al., Brain (2010); 133(11): 3359-72
- Dotti &amp; Banker, J Cell Sci (1991); S15: 75-84
- Kirby et al., PNAS (2014); 112(13): 4128–33
- Poppov &amp; Tsygunova, Neuroscience Letters; 203(2): 135–8
- Rodrigues-Martins et al., Cell Cycle (2008); 7(1): 11-6