Is there anything which rules out any possibility of finding out how consciousness and brain activity are related? (for any coherent definition of consciousness where brain activity is relevant)
Many people here agree with some form of what Steve has written a lot about, subjective reality. That perspective sees consciousness as all that there is (in other words consciousness is defined as "everything"), yet within that perspective there is the seemingly individual subjective experience of self-awareness where self is a subset of consciousness and centred on each individual human.
That self-awareness is part of the common interpretation of consciousness, which, again in part, is our subjective experience of existence. An 'objective reality' perspective sees this definition of consciousness as a result of brain activity (though the mechanisms are not fully understood).
In either (or any other) perspective, if we want a complete understanding we need to explain how that subjective experience of existence results in observable brain activity. That's where the question comes in. Does the SR perspective of consciousness rule out observable neural activity? I don't think so (since even if it's not 'real', it still happens. At least something happens, which is interpreted as brain activity and subjective experience). And if not, or if we take the OR perspective, can the link between neural activity and consciousness be found? If not, why not?
(Incidentally this deals with the
'hard' question of consciousness. I'm not expecting an answer to that question in full though, so don't fry any brain cells trying to figure it out

)