Interesting, thought-provoking question!
And the answer is....!
(Drumroll please)
"It depends on who you ask!"
This is a very philosophical question, as science says consciousness can be explained through the firing of brain neurons, as you've stated.
Those who are spiritual may take it a step further by claiming it has nothing to do with physical matter at all.
I currently believe it is a bit of both: the mind being a separate entity from the brain, yet relying on the brain to perform so long as we are attached with the flesh.
For example, when we go "unconscious" through either anesthesia, a head trauma, or any other means.. our very state of awareness and sense of self is affected (state of being unconscious).
However when we die and separate from the flesh our conscious (or spirit/state of being) still exists, whether it chooses to occupy a new body, hang out in Heaven, or gets stuck in purgatory for a while.
When someone says: "You seem like an old soul.." I can't help but feel this directly relates to a spirit that has chosen to occupy several bodies in it's time.
You got me thinking about this a lot. I found a very interesting article with multiple scientific theories explained (Did you come across this one as well?):
iep.utm.edu
Scroll down to
"4. Specific Theories of Consciousness"
(It's about halfway down the page)