how to translate 大脑迅速转动起来?
Figuring things out
There is a very similar metaphor in English, but usage is much more restricted:

When I told John the money was gone, I could see the wheels turning (in his head).

Very often the use is:

A person learns some new or surprising information.
This information causes the person to think, and they slowly start figuring out the implications of it.
They look like they're staring into space, eyes sort of glazed over.

An expression with almost the exact same use as &quot;see the wheels turning (in X's head)&quot; is &quot;(X started to) put the pieces together&quot;. To continue with the tale of deception and treachery above:

By the time he had put the pieces together, it was too late--Bill had already fled the country with all of the cash.

Concentrating deeply
I don't think there are any rotation-oriented metaphors for the scenario where you are actively trying to think about something (rather than being spurred into thought by the information). A common enough, if somewhat silly idiom is &quot;Put on your thinking cap&quot;:

Hey class, let's put on our thinking caps and try to solve this problem!

I'm open for alternative suggestions (that are less suited to elementary school).