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Jack Gallant perches on the edge of a swivel chair in his lab at the University of California, Berkeley, fixated on the screen of a computer that is trying to decode someone's thoughts.

On the left-hand side of the screen is a reel of film clips that Gallant showed to a study participant during a brain scan. And on the right side of the screen, the computer program uses only the details of that scan to guess what the participant was watching at the time.

Anne Hathaway's face appears in a clip from the film Bride Wars, engaged in heated conversation with Kate Hudson. The algorithm confidently labels them with the words 'woman' and 'talk', in large type. Another clip appears — an underwater scene from a wildlife documentary. The program struggles, and eventually offers 'whale' and 'swim' in a small, tentative font.

“This is a manatee, but it doesn't know what that is,” says Gallant, talking about the program as one might a recalcitrant student. They had trained the program, he explains, by showing it patterns of brain...