This actually works to reduce the perceived impact of missed sleep to a degree for me (you can’t just stop sleeping and do this instead, and it’s definitely not healthy or doing the same thing as sleep for your brain but it works for the occasional lost night or lost hours of sleep), but it’s annoying.

I lie down in a comfortable position, keep my eyes shut all night, and just daydream/meditate/think (not ruminating or worrying or anything like that, just calm thoughts- I play rain sounds and think about being a forest frog, dozing under a leaf and aware that my predators are all avoiding the rain and inactive. It works for me, but I suspect others have different calming scenarios they like to think through) for eight hours. It can be boring and difficult to make yourself do, but I’m more alert, don’t get a migraine, have a normal appetite, and am not struggling to keep from falling asleep the next day