The Invasion

After the space shuttle Patriot crashes on Earth, a fungus-like alien lifeform is discovered on the remaining parts scattered over US territory. Once people come into contact with the organism, they are controlled by it when they enter REM sleep. One of the first people infected is Tucker Kaufman, a CDC director investigating the crash.
Tucker's ex-wife, psychiatrist Carol Bennell, begins to feel something is amiss when people seem to have "changed". Her patient Wendy Lenk describes how her husband "is not her husband", and one of her son's friends acts detached and emotionless. At a neighborhood Halloween party, Carol's son Oliver finds a "skin" on a partier which is initially believed to be costume make-up. Carol speculates that the skin may be an organism connected to reports of a fast-spreading flu. Carol takes the organism to her doctor friend Ben Driscoll to have it checked. Carol attends a friendly dinner meeting between Russian diplomat Yorish and Czechoslovakian diplomat Belicec (along with female companion Luddie) and discusses her post-modern feminism views over caviar and champagne. Meanwhile, Tucker uses the CDC to spread the disease further, disguising the spores as flu vaccine.
Ben and Dr. Stephen Galeano, a biologist, discover how the spore takes over the brain during REM sleep. During this investigation Luddie calls Ben, worried about Yorish's behavior. Driscoll and Galeano also find that people who had brain affecting illnesses, such as encephalitis or ADEM, are immune to the spore because their previous illnesses prevents the spore from "latching on" to the brain matter. Carol's son, Oliver, is immune to the spore because of the ADEM he had as a young child. Carol decides to get her son, who might show a way to a cure, back from Tucker. Before she drives to Tucker's house, she joins Ben's team who is called to the house of the Belicecs, the Czech ambassador and his wife, in a case of emergency. There they witness the transformation of Yorish, the Russian ambassador and the Belicecs' friend.
When Carol arrives at Tucker's house, he and several colleagues close in on her. He explains that the changed humans, devoid of irrational emotions, are offering a better world, and asks her to join them. When Carol resists, he holds her to the ground and infects her by spurting his saliva on her. She escapes and returns to Ben at the Belicecs' house. They flee when Belicec returns with more transformed people intent on infecting anyone in the house. Galaneo and one of his assistants head to a base outside Baltimore where they and other scientists attempt to find a cure for the alien virus. Carol and Ben separate to find Oliver, who texts his location, the apartment of Tucker's mother, to Carol.
Finally Ben arrives, but Carol realizes that he too has become one of the infected. He successfully seduces her to give in to the new society, but also frankly states that there is no room for people like Oliver who are immune. Carol shoots Ben in the leg and flees with Oliver. She is picked up by helicopter and flies to the center preparing the inoculations. In time, the alien virus is reversed and society reverts to its violent ways.

Question: When does the spore take over the brain?
Answer: During REM sleep