Can parts of a human brain be asleep independently of each other, or vary in the times required for them to fall asleep?
Have you ever tried to read a complicated book after several hours of hard work that required a high concentration? Imagine that you have forced a particular area of your brain. After a hard work, the areas of the brain that we used to finish a work seem don't respond to any stimulation and make us feeling tired and as we can't understand what we are doing anymore. This happens because we have forced that particular area of the brain and it needs to reinforce the connections it made and to do this, it has to sleep, such as we do during the conscious (you know you are going to sleep) sleeping. Eventually we can say that some particular areas of our brain must sleep if forced to a hard work and they do it when you feel like you can't get something which you know you can. A recent study has demonstrated this by understanding physiological functions of sleeping.

Nature article of Vyazovsky V. et al. Vol. 472, pages 443-447, April 2011 (I am not copying the link anyway, have to search it by yourself). Or to Science of Bushey D. et al. Vol. 332, pages 1576-1581, June 2011. And on Nature Review Neuroscience of Diekelmann S. And Born J. Vol 11 number 2, pages 114-126, February 2010.