It isn't a complete paralysis - people will still twitch and mutter, and some may even kick around and cause their bed partners a great deal of consternation. But it works well enough for what it needs.
This paralysis is associated with REM sleep, and is called "REM atonia." It is regulated by a hormone, which usually wears off after REM sleep passes. However, sometimes this switch gets stuck for a little while after you wake up. If you don't know what's happening, this can be a terrifying experience.
People who have experienced a bout of sleep paralysis report that they can see and hear, but cannot move. They may feel as if they are suffocating, and panic frequently sets in. (Understandably so!) Most episodes last only a few seconds, but it can seem like it takes a lot longer.
To compound the terror, sleep paralysis is often accompanied by lingering nightmare visions which can seem very real indeed.
Sleep paralysis is known in every human culture, and every culture has its own explanation for it. In Europe and in Cajun culture it was called "the Old Hag." She was thought to be an invisible spirit who sat on you during the night, thus explaining the feeling of chest compression.
Other explanations have included demons, devils, waking nightmares, spirit possession, black magic, and more. Most legends of sleep paralysis involve an invisible entity sitting on the chest, or holding the body down.
It's difficult to get accurate statistics, but many studies indicate that sleep paralysis is surprisingly common. You may have noticed that there aren't really any mainstream American mythical explanations for sleep paralysis. Considering how commonplace it is, this is surprising.
Many people theorize that stories of alien abduction, ghosts, spirit possession, and childhood satanic ritual abuse may in fact be manifestations of sleep paralysis. People view events through the lens of their culture, which means that in a culture that believes in demons as a literal fact, someone who goes through an episode of sleep paralysis will reach for demons as an excuse.
In our culture, a feeling of floating, suffocation, being watched, and possibly of seeing shadowy figures flitting around the edges of vision could easily be framed as an alien abduction. Consider how many stories of alien abduction begin with "I had gone to sleep." The same is true of ghost stories.
Both ghosts and aliens seem only to come out at night, which has always struck me as interesting. I know of no alien abduction stories which take place in the middle of the day. And spirit possession always seems to happen in the middle of the night. (I have heard of daytime ghostly apparitions, noises, and other subtle manifestations, though.)
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