Broadly speaking, neural correlates of a percept, e.g. seeing a face, are what happens with neurons when we see a face. For example, a bunch of neurons would fire when we see Jennifer Aniston. What do neural correlates teach us about perception, or more generally about the mind-body problem?
The interest in neural correlates of perception is largely subtended by the implicit belief that there must be a mapping between perception and brain state seen as a physical system. That is, the percept of seeing Jennifer Aniston's face corresponds to a particular brain state; the percept of a sound being at a particular spatial position corresponds to another one. There are considerable conceptual difficulties with this belief. Consider these two thoughts experiments.
1) Imagine we can instantaneously freeze the brain so that its components, ions, tissues, etc are preserved in the same state (it's a thought experiment!). Does the brain still experience seeing Jennifer Aniston's face?
2) Imagine we record the precise spatiotemporal activity of all neurons in response to Jennifer Aniston's face. Then we inactivate all synapses, and we replay the activity pattern optogenetically. Would the brain still experience seeing Jennifer Aniston's face?
Intuitively, our answer to both questions is negative. If you answered the second question positively, then consider this one:
2b) Imagine we record the precise spatiotemporal activity of all neurons in response to Jennifer Aniston's face. Then we replay the activity pattern on a set of light diodes. Would the diodes experience seeing Jennifer Aniston's face?
If our intuition is correct, then brain states, even understood more broadly as “firing patterns” are not constitutive of percepts. It appears that whatever a percept is, it must involve not just the state or even the activity of neurons, but the interaction between neurons. Therefore when we describe neural correlates of perception in terms of neural “activity”, we appear to be missing a crucial ingredient, which has to do with interactional properties of neurons. To be honest, I must admit here that “interactional properties of neurons” is a loosely defined concept, but apparently there seems to be a need for a concept that goes beyond the concept of activity “pattern”, a concept to be clarified (see afterthought below).
Underlying the problematic concept of neural correlates of perception is the representational view of perception; the idea that whatever we perceive must somehow be “represented” in the brain, like neural paintings of the world. I have pointed out the deep problems with the representational view on this blog (for example here and there) – and obviously I am not the first one to do so (see e.g. Gibson, Brooks, Merleau-Ponty, O'Regan etc). Let us simply reflect on the following one. When we look at Jennifer Anniston's face, we experience the percept of seeing her face at different moments. It seems as if at any instant, we are experiencing the same percept (along with others, of course). Possibly this is an illusion and experience is actually discrete in time, but in any case the perceptual “grain of time” is no more than a few tens of ms. Therefore when looking for neural correlates of the percept, then we cannot be happy to rely on average activity, over time or over trials. We do not experience percepts “on average”, but at any instant (this is related to my points on the rate vs. spike debate). What we should be looking for is something in the interactional properties of neurons that is invariant through the entire time during which the percept is experienced. The concept is quite different from the more traditional “neural paintings” concept.
So, in the current state of research, what neural correlates of perception tell us about perception, more specifically about the mind-body problem, is disappointingly: not so much.
Afterthought: an interesting analogy is the concept of temperature in physics. Since temperature corresponds to the movement of particles, you cannot really define the temperature of a physical object at any given time. Temperature corresponds to the activity, not the position or nature of the particles. What's more, the concept of temperature makes no sense except when considering the interaction between agitated particles. Temperature is perhaps an example of “interactional property” of a set of particles.