Study shows brain cleaning through sleep:
http://www.urmc.rochester.edu/news/story/index.cfm?id=3956

Quote
The new findings hinge on the discovery last year by Nedergaard and her colleagues of a previously unknown system of waste removal that is unique to the brain. The system responsible for disposing cellular waste in the rest of the body, the lymphatic system, does not extend to the brain. This is because the brain maintains its own closed "ecosystem" and is protected by a complex system molecular gateways - called the blood-brain barrier - that tightly control what enters and exits the brain.

My take
Sleep was first a chance to end the eating/digestion period, regroup needed water and hormones for next day digestion, and prepare waste out -bowel movement.

This evolved to include a brain resting period to clean out the brain (See study and video above) But more exactly a cleaning out of all BBB blood brain barrier places (brain, testes, eyes, ears, joints, spine)

Now why the NREM and REM?
My first guess is NREM is 'house maintenance' that includes getting digestive tract ready, flush all BBB sites, and deliver some hormones.
REM is waste out or a type of flush out of waste after the NREM period.

Still looking at the BBB, blood brain barrier and sleep. Start with the above study about the glymphatic system cleansing the brain at night .

Maybe it's part of a bigger cleansing - that might include other parts of the blood barrier system (testis, joints, spine, etc). Maybe this is why we sleep.

Found this - see info below: What was interesting to me and perhaps a connection, is that
the Pineal Gland is outside the BBB, AND secretes melatonin.

So what is the connection

Pineal gland is awake during sleep, secretes the sleep melatonin, is outside the BBB, is awake while glymphatic cleansing of brain in sleep.
So does that mean that sleep is for glymphatic cleaning of all blood barriers, or is it all body of which this is part or....

INFO CONNECTED"

The blood-brain barrier prevents toxic substances, large molecules, and neurotransmitters released in the blood from entering the brain.

Four areas of the brain are not protected by the blood-brain barrier. These areas include the posterior pituitary gland, pineal gland, the median eminence of the hypothalamus and the area postrema.

Posterior Pituitary Gland
The pituitary gland releases the hormones oxytocin, which controls milk release, and vasopressin, which regulates water balance. The pituitary gland is not covered by the blood-brain barrier because the hormones it secretes need to go into circulation and pass through the rest of the body.

Pineal Gland
The pineal gland secretes the hormone melatonin, which controls circadian and seasonal rhythms, or sleep/wake cycles. The pineal gland is not covered by the blood brain-barrier because it secretes the hormone into the bloodstream.

Median Eminence of the Hypothalamus
The median eminence of the hypothalamus connects the hypothalamus to the pituitary gland. The median eminence is not covered by the blood brain-barrier because hormones secreted by the pituitary gland collect in this region before being secreted into the bloodstream.

The area postrema is not covered by the blood brain-barrier because it senses toxins in the blood that the other parts of the brain are protected from. The area postrema triggers nausea and vomiting to prevent further ingestion of toxins.
http://www.livestrong.com/article/162348-4-parts-of-the-brain-not-protected-by-the-blood-brain-barrier/