@
Bertie Well, no, they can't go to sleep completely.
The brain is overseeing keeping the rest of our body operating, but is also carrying out its own housekeeping and maintenance.
It closes our main three senses - hearing, touch and sight - down to watch-keeping alarm levels, so we ignore normal stimuli but a loud noise or unexpected contact will still wake us. Our sight is still "on" - we close our eyes but our eye-lids are just translucent enough for a switched-on lamp or strong daylight to wake us. I don't know if our sense of smell will still register a strong odour, such as smoke, though we might notice that by throat irritation.
The sense of touch has an important sleeping role. It detects when we have been lying still for too long and triggers our turning over, so we are not applying pressure to any area of the body or keeping muscles static for too long at a time. A time-lapse video of someone sleeping normally is startling in showing just how many times we move about and roll over.
You see similar behaviour in a pet cat or dog - and a sleeping cat especially noticeably moves its ears about, making me think the animal is still monitoring its surroundings. (We can't waggle our outer ears.)
The brain is also monitoring things like bladder capacity via the organ's nerves - waking us with the right warning sensations when necessary!
We know the imagination and memory bits get a bit wild and busy at times - creating dreams - but what is curious is that sometimes they seem to solve real problems or have little bursts of creativity like composing fragments of music. Perhaps these are a special sort of dream inspired by us having been concentrating on the subject.