Your brain does create waste as well as your body. However it's not quite as effective as your body as getting rid of the waste, especially whilst we're awake. A recent study by the University of Rochester Medical Centre looked at how the brain deals with waste. They found that activity to get rid of waste products in our brains was relatively low during waking hours but when we slept activity increased 10 fold. That's a massive increase!
What does this mean?
Effectively it means that without adequate sleep our brains can really struggle to get rid of the waste products. You could also say with a fair bit of confidence that one of the main reasons we sleep is to clear out toxins from our brains. The way in which our brains clear out the waste takes quite a bit of energy. It would appear that we've evolved to be either awake and aware, or asleep and cleaning up.
What should you do?
My advice stays the same. Get adequate amounts of sleep on a daily basis to promote the health of your brain. Getting adequate sleep will mean that you're giving your body and brain enough chance to do what it wants to in terms of waste disposal. With regards to getting enough sleep it does vary from person to most. Most will need around 8 hours per night of quality sleep - without lots of interruptions. Judge for yourself though; if you feel tired when you wake up then maybe you didn't get enough. Listen to your body!