That's a point of debate, actually. SSRIs in the long term appear to actually work by reducing serotonin in the brain tissues.
Serotonin is really more of a torpor hormone that depresses the whole body than a "happy" neurotransmitter. It's associated with hibernation, for example. The "turkey causes sleepiness" meme is about tryptophan in the meat boosting serotonin levels.
