The brain would need to leverage quantum effects to achieve hyper-computation. Otherwise it follows the same rules as other turing machines. Here's a quote from Scott Arronson:
"Indeed, I see no evidence, from neuroscience or any other field, that the cognitive information processing done by the brain is anything but classical." -- http://www.scottaaronson.com/blog/?p=1951
Notably this is from a post discussing quantum effects as a potential explanation for consciousness. There's still no evidence for that either. Also, if true, it would only sit atop a classical cognitive process which acted as a turing machine. That or a horde of tiny turing machines.
