As far as I know, and I know about 20 stutterers, the vast majority of stutterers can speak perfectly fluent when they are alone. This is why I will say that for most stutterers it is not a physiological issue - or at least not solely a physiological/neurological issue.
And just FYI there is no sound research that stuttering is caused by a miscommunication between the brain and body. This is one of the theories, but experts still don't know exactly what causes stuttering.
Have you looked into Rational Emotive Behavior Therapy? I've found this book very helpful on my journey. http://www.amazon.com/Stuttering-Fluency-Manage-Emotions-Fully/dp/1494302586/ref=sr_1_1?ie=UTF8&qid=1443657095&sr=8-1&keywords=from+stuttering+to+fluency