I had a question very similar to this. I plan on doing my neurosurgical residency at the University of Michigan. At most hospitals including U of M, you can pursue a PHD during your residency but it will add at minimum two years. I was told by someone that one of his friends when he was doing his residency pursued a PHD from a University in Toronto when he was a Umich resident. It is not uncommon for a residency program to award you an MPH during your residency. But trust me, PHDs, from what I have learned thus far, are worthless. You would be 37 at minimum by the time you would be able to pursue a career in Neurosurgery. Plus, you get paid the big bucks from doing surgeries, not research.
PM me if you have any questions.
PS Take a look at this
http://www.mayo.edu/msgme/residenci...urgery/neurologic-surgery-residency-minnesota
Take it from me, you would be better off doing an MD PHD program that a PHD during residency. It might help with you getting into a very good residency.