Time to break out the ichthyology book. :D (The fact that this excites me says a lot about my personality.) Fish don't sleep like mammals sleep. The book defines fish sleep as when "a fish assumes a typical resting position for a prolonged period, uses some form of shelter, and is relatively insensitive to disturbances." Most species of fish sleep, however scientists believe that some don't (like Croakers). So my guess would be that when bettas feel the need to fish sleep they will try to do it close to surface so as not to move much when going for air (I think this is the reasoning behind the ZooMed betta hammock). If they can't find a place close the surface, they will just have to wake up and go get air. I have witnessed my bettas doing this on numerous occasions. Especially when I have them in quarantine bowls.
A little more on fish sleep: the science community is still debating the significance of fish sleep. The only reason they can surmise for sleep is energy conservation (fish isn't moving and therefor using less energy). Another fun fish sleeping fact is that some fish like wrasse and some parrotfishes will secrete a cocoon of mucus around their body while they sleep to mask their smell from predators.
I always assumed that my bettas would sleep inbetween breaths. I've never seen them sleep on any of the plants on my tank. Since they like to hide in their Urn or cuddle up next to it, I figured they would swim up when they had to. I guess the frantic swim up to the surface doesn't bug my fish.
most of my bettas sleep on the bottom of the tank. Sniper, Lulu, and Scout are the only ones who sleep near the top(Lulu lays in her plant, really cute. gonna hafta get a pic one day, Sniper wedges between the tank wall and a leaf near his nest, and Scout floats up top in his grass-plant).... i think, during that time, they're resting, so they don't NEED to breath..... they slow down so they don't have to breath for hours.... *shrugs* just my thoughts.
Sleep in fishes still needs much study, but some sharks can go into a resting phase and actively pump water over their gills, nurse sharks being the example that comes to mind. Normally sharks need to swim to keep water flowing over their gills. So I would guess the ones that can't pump water go into a "resting on the go" phase where they are not actively hunting. They could also never go into a true sleep phase but just be in a not-hunting phase. I imagine this would be most true for open-ocean sharks.
But you also have to remember there will be behavioral difference between animals in captivity (which is where most studies take place) and animals in the wild. Animals in captivity get regular feedings and won't be hunting as voraciously as wild animals. I saw an interesting example of this yesterday. I have 2 cats. Two older and one younger. The older cats had been allowed to go outside when they were younger, but we since stopped that because my neighbor asked us to because of her birdfeeders. The youngest has never been outside. Yesterday one of the older cats found a mouse inside and quickly set about killing it. The youngest jumped in and wanted a piece of the action; as I observed her techniques I saw they weren't as efficient at killing as the older cat. I surmised this was because the older cat had been able to develop his hunter instincts when he was outside; the younger cat hadn't which was why she appeared more to be playing with it than attempting to kill it.
To bring this back to fish, studying sleep in fish that know they will be getting a meal probably isn't going to be as true to life as studying it in the wild who are always looking for the next meal. And sharks are some of the hardest fish to study due to the fact that they often cross international boundaries.
As far as fish having brainwave changes like mammals do when they sleep (If you know what REM sleep is, poke your cat or dog when they are in it; so much fun!), I don't think it has been studied yet. I can imagine hooking those sensors up to fish would be rather hard. You would be amazed how little is known about fish by the scientific world. So many species have undocumented breeding habits.
Sorry for the long post. Observational science is one of my hobbies.
Wow, koimaiden, I am so impressed! Awesome, I just learned so much from you. It's amazing to think that there may be a creature that never has a true sleep cycle. Wow. Because I'm not a science person, I admire people who are. :)
In regards to your cat example, I definitely understand that. When I got one of my cats, he was a tiny kitten who I believe was probably separated from his mother quite young. I literally had to teach him how to play. He didn't know how to even bat at a feather toy.
And my cat squeaks when he's in REM. Twitches like crazy and squeaks.
I'm not sure if dolphins do the same thing as bettas do, but when dolphins sleep, one half of their brain is still 'awake' so it can still breathe and such, while the other half of their brain is at rest. I'm not sure if anyone else said something remotely like this but, its what I learned in science class haha