Did your stutter begin after an accident or a fall? Are you having other symptoms, like difficulty swallowing, muscle weakness, or memory loss? If no to all of those, then neither a neurologist nor a brain scan are called for. Find an slp who has some experience treating stuttering and is willing to see you all the way through your treatment, including carry-over and maintenance. If you don't know any slp's, you can search at asha.org. Good luck!