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This is a complicated question, so others might be able to contribute more to the discussion.
Let's start with the speed of signals in the brain. This page might be helpful:
"When the Prussian genius Hermann von Helmholtz in 1849 measured for the first time the speed of a nerve impulse, he recorded in his lab notes and ultimately published, in 1850, a figure of 27 meters per second, or about 60 mph."
"Von Helmholtz happened to measure a slow nerve – fast ones can conduct an impulse at 265 mph. But this is not, obviously, the speed of light."
So it's acknowledged that the speed of a nerve impulse is slow, but that somehow nerves may be able to communicate a lot of information in each spike, even if it travels slowly.
Now we can consider the computing speed of processors. See this page:
"Today we are witnessing the development of computer chips with clock speeds of over 1 GHz. This is 109 cycles per second, or 1 cycle per nanosecond. In a nanosecond, light can only travel 30 cm (about 1 foot). So if you want to, for example, get a response from a random-access memory within 1 cycle, none of that memory can be more than 15 cm or 6 in away from the central processor."
"If you use electrical signals rather than light for communication, it's even worse, because in practice electrical signals really only propagate at about half the speed of light (depending on the materials), so really none of your memory can be more than 3 in away."
So if you're simply comparing pure speeds, and ignoring the amount of information contained in each "packet" that is sent, you are really just comparing 55 mph to about half the speed of light.
The speed of light is 299,792,458 meters per second, and you can use this page to convert it to miles per hour:
Now if you take half the speed of light, you get about 335,207,752 mph.
And that's where the claim about the speed of a computer signal being millions of times faster than the brain signal propagation might come from.