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So I have a planet with multiple species on it. My planet is tidally locked. I know it is generally thought a tidally locked planet couldn't harbor life because of the extreme temperatures, but my argument is that it is theoretically possible if the planet has a thick enough atmosphere, which helps with cooling, and if there is steady airflow from light side to dark side. This does cause very extreme weather, especially around the area where it goes from light to dark, but still, it keeps the temperatures survivable most places on the planet, except the spots closest to and furthest away from the star.

But here's my question. I am assuming on a planet like this, circadian rhythms would not have evolved since there is no day night cycle, and therefore species would not sleep, at least not in the way we do. The planet is at a tilt, and therefore some latitudes do have daylight during summer and night during winter, so in those places I can see hibernation being a thing. But otherwise, I don't think they would sleep.

So my question is what could replace sleep as a restorative process for their bodies? Entropy would still cause their cells/neurons to degenerate, and so I imagine they would still need some kind of regular, restorative process to keep them functioning. What could this be? How exactly does sleep help restore our own brains? If I understood how that works, I might be able to come up with a reasonable process to replace it.

Die, and be reborn.

In general, if you have stuff, you can maintain what you have. Or you can throw it out when it is worn and get new. Our bodies do both. Epithelial cells (like skin) fulfills its task and then dies and is sloughed. Blood cells live their lives and then die and are recycled.

Neurons are not recycled. They are long lived, and so are maintained. Presumably sleep is part of this maintenance. But there is no particular reason for them not to die and be replaced like other cells in the body. The dying neuron would hand off its information to its replacement before it died. Your organisms could do this periodically, like a snake shedding its skin. The in-between period offers narrative potential.

Re the bigger questions of sleep - interesting for a fiction but these big, unresolved questions. There is a question of memory - how it is stored and where. Is sleep necessary for such things? Why do we sleep? Why does every animal sleep? Would it not be evolutionarily advantageous to not have to sleep? These are open questions and the subject of ongoing research. You could riff off of these for your world - your Sleepless Ones might have advantages and disadvantages compared to sleepy Earth creatures.

Half sleep.

Dolphins sleep one half of the brain at a time. The other half tends to basic functions, like going up to get air,

https://www.livescience.com/44822-how-do-dolphins-sleep.html

Instead, these animals undergo an unusual form of sleep called
"unihemispheric slow-wave sleep." Also known as deep sleep, slow-wave
sleep is a type of sleep thought to help the brain consolidate new
memories and recover from its daily activities.

When it's time to rest, a dolphin will shut down only one hemisphere
of its brain, and close the opposite eye (the left eye will be closed
when the right half of the brain sleeps, and vice versa). During this
time, the other half of the brain monitors what's going in the
environment and controls breathing functions.

In your world of perpetual day, if you want to have some kind of sleep you could adapt something like this. It would make for an interesting story to think about a human that slept like this - maybe your aliens are of human stock originally and have evolved to sleep like dolphins. A creature in half sleep would be somewhat less than it was when completely awake. Its manner might depend on which half was asleep and which half was awake.

My cat sleeps all the time...

I'm going to advocate that sleep need not be lost in your circumstances. The Circadian rhythm is convenient, but I'm not convinced my cat knows what it is. Or, perhaps, our Latin brothers and sisters, who enjoy siesta.

Rest can come at any time, whether through sleep or meditation.

However, as I think about it, it's not necessarily true that the Circadian rhythm doesn't exist on your planet, either. It's an assumption that it exists because of the day/night cycle. But it's just as likely that life accommodated the day/night cycle. In other words, it's easily believed that a periodic cycle exists with your creatures despite no day/night cycle.

Even on a tidally-locked planet, there are cycles

Life on a tidally-locked planet must have cycles. Things are born, grow old, and die. Energy is put into flowering, then fruit, then restoring. You run from the dragon until you get tired and are forced to stop and rest. Well... and then you die a gruesome death... but that's actually important.

Darkness cometh from palm fronds

I can easily imagine that part of the development of Earth's Circadian rythym came from the need for protection. You get tired, and not wanting to be et by the dragon, you find someplace well protected... out of sight... dark, where the dragon can't find you. (That you need protection regularly at night simply reinforces the rhythm.) A lot of the "rhythm" of your planet (and perhaps any planet) would come from the predator-prey fight/hide-for-your-life behavior.

Which brings us to palm fronds. Dragons are tall and, looking down, they can't see you sleeping beneath the fronds where you found a comfy place to sleep of the chase.

Conclusion

• Rhythms are going to happen for many reasons, the lack of a day/night cycle wouldn't stop that.




• The need for protection would naturally create an artificial "night" due to the need to hide from predators. Burrows in the ground and palm fronds leading to buildings via the dawn of intellect.




• Like my cat, one can rest at any time. Eat a big turkey meal and see how easy it is to fall asleep.




• Therefore, I don't think you'd need to replace sleep at all.

Circadian rhythms regulate the timing of sleep. They do not create the need for it.

A body in an area without day-night cycles would still require sleep. People and animals who lived in Arctic regions before artificial light still slept in the height of summer or winter, as well as year-round, despite drastic changes in the amount of sun each day.

People who are completely blind (with zero awareness of light or who simply don't have eyes) still sleep and need to sleep (they're not just doing it for cultural reasons). But they often don't have 24-hour rhythms and may have disorders that stem from that.

One of the most important things we need sleep for is the generation of Growth Hormone. GH is released primarily during sleep. It, and its downstream products such as IGF-1, regulate insulin use in the body, strength bones, produce breastmilk in lactating women, help with detoxification pathways, and dozens of other things.

GH also rebuilds muscle. If you're familiar with the rule for weightlifting that you only work out a set of muscles every other day or less, this is why. Exercise breaks down your muscles and restorative sleep releases GH that builds your muscles back up (bigger and stronger in many cases, but also what you need to maintain strength).

The myth is that only growing children need GH. The reality is that everyone needs it (not too much of it, but just the right amounts). Adults with GH deficiency (which can be a holdover from childhood or it can be adult-onset) are often tired and weak because their muscles aren't getting built back up after everyday activity like they should.

While other activities can release GH from the pituitary, deep sleep is really what you need to do it right. All humans need sleep and this is a large reason why.

Other reasons for sleep include: a full relaxation of the voluntary muscles (very hard to do without being asleep, though not impossible) and dreaming. The brain differences in various stages of sleep and dreaming are important. They affect both our physical health and our mental health.

To replace sleep, you would still need a cyclical restorative process.