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Guess - Isn't it as fast as electricity can pass in the nervous system, so about 250 mph?
Tennis players at Wimbledon are certainly not consciously reacting to the ball coming over the net - it all happens too quickly. Instead their visual system is sending the information about the trajectory of the ball through a paralel processing pathway involving the brain's cerebellum. This is the "mini-brain" that sits at the back of the head and controls movement. It's involved in the smooth execution of learned movements. This pathway also involves very few cell to cell links (synapses), so it's very swift.
One interesting piece of work that may interest Paul was done by Molly Potter at MIT many, many years ago. She found that if you show somebody a picture of a scene for only about 100ms, they are able to remember and describe a great deal of the information presented in that time. This is interesting primarily because 100ms is a very short time period for neurons. A signal can only travel down a few neurons in that time. This comes back to the issue of trying to tie the speed of neuronal signals to the speed of thought. (If you don't think scene recognition is thought, consider that a great deal of interpretation must be done to go from the visual imprint of some line drawings to the representation of Don Quixote on a hill.)