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The first REM period may begin only a half hour after falling asleep.

In the sentence, I'm confused with the use of the phrase after falling asleep.
Please explain it to me.

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Rapid Eye Movement is a process the brain uses to associate events of the day into memory and reassociate long and short-term memory.

When one falls asleep one loses some consciousness of the outside world, eyes are closed, breathing becomes less rapid, and one immediately dives into a deep sleep which to an observer is similar to being unconscious

It is during this deep sleep that REM can occur and is observable by noticing rapid twitches in the sleeper's eyes.

Falling asleep is an idiom used to describe the act of becoming non-wakeful.
After is the preposition to describe after which time something will occur.
After falling asleep does not designate a duration of time, only the beginning of a time interval.

After falling asleep, he had a bad dream.
After falling asleep, he woke up again.

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SUPPLEMENTAL to Peter's Answer (not for upvoting):

There is no participle phrase here. The -ing verbform falling is not employed as a participle, which is a verbform acting as an adjective, but as a gerund, which is the -ing verbform acting as a noun.

 PARTICIPLE: The running man ...  
             The man running down the street ...   
     GERUND: Running can make you healthier.  
             I am too old to enjoy running.  

In your sentence, falling asleep is a gerund clause.

  • Internally falling is a non-finite verb, meaning that it has no tense or specific time reference. It takes the complement asleep, and its subject is omitted because what is meant is any situation where anybody falls asleep. Asleep is a predicative complement, an adjective designating a state or condition employed here as the figurative destination or final location which verbs of motion like fall ordinarily take (etymologically, in fact, asleep is a preposition phrase, on sleep).
  • Externally falling asleep is a nominal (noun-like) phrase acting as the object of the preposition after. In that context it designates a particular point in time, the moment when somebody falls asleep.

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