If there were studies, they might be conducted using an MRI and a non-magnetic keyboard like the one in this TED video.
Very likely, they would find certain parts of the brain light up during programming, and there might be some differences between different programming languages and certainly between natural and programming languages.
WRT mathematics driving people to madness (or music, or many other pursuits), I suspect many of those occurrences are coincidental. We all know people who have eccentricities, but I suspect that might be more a factor of working in isolation either individually or as a team that doesn't participate in (or enforce) broader social norms.
Also, I would check some books on neuroscience about speaking Italian changing your feelings vs. changing your capabilities or mode. It is commendable to speak multiple languages, but I think sometimes what our brain tells us about our brain might not be fully reliable. A lot of early psychology was considered less reliable after science provided better tools for measurement. I think David Eagleman writes about the brain in a way that is accessible to lay people and discusses similar issues.
Also, while the movie Pi was innovative and mind bending, and I remember enjoying it and wanting more, I would not bring its paranoia and obsession to bear on problems. Pi is just a number. Writing code, is just writing code.