If the mind is not the brain, then we need to say that there is an additional thing out there, which is the mind. It will be a very mysterious thing.
That thing needs to interact with the brain somehow. We know, from experimental psychology, that many psychological functions are functions of the brain — the brain is implicated in memory, learning, sensory and motor functions, executive functions, language, etc.
So we need to say that the mind and the brain bear some sort of relationship to each other, and both contribute in their own ways to psychological functioning.
It makes sense to me to say that the brain serves functions which are more “mechanical,” lower-level, than the functions served by the mind. The mind, then, would serve higher-level functions. Under this view, we would say things such as these:
* My brain is the thing that computes sums, but my mind is the thing that enjoys doing math.
* When I compose music in my head, my temporal lobes contain a representation of the music, but my mind is the thing that is performing the creative act of composition.
* When I feel pain, there is a chain of signaling proceeding from my peripheral neurons, and eventually into my brain, but that which feels the pain qualia is my mind.
These examples give a rough idea of the sort of division of functions I am hypothesizing. I can make the idea more concrete with a metaphor.
Imagine that my body is a giant robot, like the robots in Gundam Wing, and my mind is the pilot of this robot. The brain is the set of instruments, in the cockpit, with which the pilot interacts with the world.
* The occipital lobe is a video display.
* The temporal lobe is a sound system.
* The motor cortex is a control panel, with joysticks, buttons, etc.
* The frontal lobe is a sort of digital assistant, which automatically performs various helpful tasks. Among other things, it has a pocket calculator, a calendar, and software that gives me important reminders from time to time.
* The brain also has a filesystem which stores all of my memories.
This crude analogy is intended to illustrate the sort of relationship that the mind and the brain might bear to each other. A finished theory would give precise descriptions of the respective functions of the mind and the brain; but I do not have that information.