I'm not sure. It's pretty much proven when animals and humans go to REM sleep, they become essentially paralized, because if they weren't, they'd act out what they dream. I know humans and some animals dream scary things which apparently prepare them for living in the real world during rem sleep (remember when you used to have scary dreams of being chased by monsters as a little kid?)... hence why nature has created a way for your body to paralize itself during REM sleep.
Apparently some scientists did an experiment with rats, cats, dogs, etc, where they disabled the paralasis, and the cats, rats, and dogs would walk/run around while dreaming, much like they would awake (they were sleep walking). I'm thinking betta fish might, after all, there are times when you walk in on your betta, and he's sleeping, not moving, and doesn't react to you for a bit. What if that's their form of paralasis during REM sleep? Or they might be in that restful trance Aluyasha said.