I confess, I don't really understand Twitter*.
So, I wondered, what else you could do with a simple, popular and highly mobile communications network - Why, build a brain, of course! - but it might take some organization, and I don't think it'll win any prizes for speed.
In effect, Twitter* becomes
the synapse. Each Twit plays the part a Neuron. When enough of them exist and interact we might create a form of Artificial Intelligence.
When the first Twit detects a description which tickles its twitter-buds it posts a response - Other Twits Follow, and when they detect a pattern they like, they propagate their own factoid, and so on. Hey presto, a brain! (OK, perhaps this is rather simplistic)
The following is a vision example, but other Twits could be commenting on sounds, tastes, observations, smells, concepts etc.
Each "Twit" describes their own interest. e.g. "I like a particular shade of blue-green-grey in the top-left quadrant of an image".
The Twit then assigns itself to Follow the musings of another Twit which is describing everything it sees.
Other Twits, interested when a Pale-Cyan is in view, Follow.
Sometime later, somewhere, someone with a penchant for the Mona Lisa receives the final piece of their Jigsaw Puzzle and makes their first Tweet!
*If there's anything else out there on the web that behaves similarly, then please take the name "Twitter" as a generic mass-communications protocol.