Having taken SSRI for anxiety, it feels more like a second order effect. The brain is anxious, the serotonin goes in, the serotonin is not exactly "anti-anxiety" signal but "feel-good" signal.
Repeat constantly every day for months and the brain thinks, "ok we constantly have this euphoria going on, time to turn the anxiety off no need for it anymore"
This would also be why the serotonin increase is instant when starting SSRIs, but anti-anxiety effects take months and are gradual
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