In this BBC news article a study shows that during sleep brain cells shrink to open up the gaps between neurons and allow fluid to wash the brain clean. But do the cells shrink and undergo the whole procedure during REM sleep and lucid dreaming?
For those who have access, here's the full text.
Having read the paper, I can't give you a good answer. This study was performed in mice, not humans, and "sleep" was achieved in a few ways. Mice normally sleep most of the day, so it was easy to get mice sleeping normally. They also induced "sleep" through anesthesia (incidentally, they saw these same results in anesthetized mice, which I think is quite interesting). The state of "sleep" was confirmed mainly through the measurement of brain waves, in particular the delta wave. Delta waves correlate to deep sleep, and that's what they measured. Panels D and G from figure 1 (below) basically confirm just that - the mice were asleep - and that's all we know so far. More exciting research to be done!