What we have is a false parallel. The argument is designed to making you choose between the prospect that the mind and the brain are so tied together that they are the same thing or that they are so separate that the what happens to the brain doesn't affect the mind. The thing is that there is no reason to conclude that if the mind is affected by what happens to the brain that the mind/soul do not live on after the physical body (brain) dies. There is no evidence that the relationship should be so. I have yet to read or hear anyone successfully argue why we should make that assumption. It seems that there is no reason why they both can't be true - that the mind is affected by what happens to the brain and that the mind/soul survives physical death - brain death in other words.
A lot of people admit that if near-death experiences (NDE) are proof that the mind/soul survives death, then it's proof that the mind and the brain are not the same thing. This is why many atheists argue very hard against NDE's. They recognize that if accept the possibility that our minds survive the death of our bodies it puts their atheism in a precarious condition because we can't really know anything about what happens after we die using our own natural ways of observing the material world. Worse than that. It means that all truth cannot be determined using the natural methods many of them have pinned their faith on. It means you have to truly consider if the Bible is true. You have to admit that it matters if the Bible is true or not.
Dr. Gary Habermas gives a talk about data reported in peer-reviewed medical journals involving near-death experiences.
Listen to the talk at the following link: Near-Death Experiences: Evidence for an Afterlife? - Gary Habermas MP3 Audio
Also he has a FAQ section on NDEs on his website. Read it at http://www.garyhabermas.com/qa/qa_index.htm#nea