"across his mind" is a correct and commonly used phrase in written English.
It can be used to describe a thought or idea that briefly passes through someone's mind. Example: As he sat by the window, staring out at the rain, memories of his childhood suddenly flashed across his mind.
Memories are waiting all around, beckoning him, and when he tosses a hazelnut into the water, the ripples carry across his mind as he remembers "everything".
Hence Ginsberg's compositional method, the moment-by-moment transcription of thoughts and images as they passed across his mind.
As he headed for his favorite spot in Manhattan, Mr. Light said, an unwelcome thought flashed across his mind: "It's an Afghani restaurant".
He packed the place on weekend nights, and roamed it like a party host — holding up the show until he finished telling you about the movie you were about to see, and whatever else flitted across his mind.
The third time — the fixture on the landing — the thought zigzagged across his mind that these little explosions were a sign, like the two dogs he had come across in the autumn, greyhound and bulldog, locked together on the grassy slope of the local park.
I said: 'Mr Allen, you will have the last word in the film.' He said: 'Sorry, did I say anything wrong?' He had no idea he'd said anything special – that he'd asked the ultimate question: 'Why are there so many of them...?'" He still looks delighted as the scene plays back across his mind's eye.
A few years ago, Ms. Valentine read a book by Mel Levine, an expert on learning disabilities, about schoolchildren who have trouble focusing, and came across his term "mind trips" to describe such moments of distraction.
He walks across Los Angeles, losing his mind as he devolves from a white-collar professional to a man who destroys a convenience store in protest of the inflation of soft drink prices (he wants to pay 50 rather than 85 cents for a can of soda) to a vigilante murderer of Latino street gang members who harass him to, very nearly, a cop-killer.
The concept of a man without a price has never, until now, passed across the desert of his mind.
He wears his polymath intelligence lightly as his mind roams across geology, archaeology, fauna, flora, architecture, art, literature and urban design, retrieving small surprises everywhere he walks.
Soering explains that, as his mind raced across what he knew from books and movies, it alighted on Sydney Carton, of "A Tale of Two Cities".
