It is fair to say that we have little idea about how neural activity gives rise to consciousness, and about the relationship between neural activity and conscious states (i.e., what you are experiencing). This is the mind-body problem. In my opinion, there has been relatively little fundamental progress on this question because it has been addressed mainly within the computationalist framework (ie in terms of information processing), which is very inappropriate for this question (this is partly Chalmers' criticism). So below I am listing a number of unanswered questions on this matter, which I believe requires a very different kind of approach. First of all, let me remark that because being conscious is always being conscious of something, understanding consciousness is largely about understanding perception at the phenomenal level (perception in the broadest sense, e.g., perceiving your thoughts).
1) How can perception be stable?
Why is it that a pure tone feels like a stable percept when 1) the acoustic wave is time-varying, 2) the activity of neurons everywhere in the brain is dynamic? The same can be said of all senses; in vision, the eyes move at high frequency even when fixating an object, and there is no visual percept if they are forced to be still. More generally: if there is a mapping between states of the brain and percepts, then why is it that percepts are not changing all the time?
A thought experiment. Imagine the state of the brain is held fixed. Someone scratches her nose and time is stopped. Would you still experience something? Any conscious experience seems to require a change, not just a state. This suggests that the relevant mapping is actually not from brain states to percepts, but from brain activity to percepts. This immediately raises a problem, because a conscious state can be defined at any point in time, but it is not immediate that brain activity can (as this would reduce activity to state). This is not a fatal problem, though, for there is a precedent in physics: a gas is composed of individual particles, but the pressure of a gas at a given instant cannot be defined as a function of the state of the particles at that moment, because pressure corresponds to the force exerted by the particles impacting a surface. It might be that the relation between neural activity and conscious states is of a similar kind as the relation between mechanics and thermodynamics.
Two more thoughts experiments. 1) Record the firing of all neurons in the brain, then play them on a set of unconnected light diodes, does that set feel the same experience? 2) (adapted from Chalmers) Replace randomly every other neuron in the brain by an artificial neuron that interacts with other neurons in exactly same way as the neuron it replaces, would there be a conscious experience? My personal answers would be: (1) no and (2) yes, and this suggests to me that the right substrate to look at is not neural activity as a state (e.g. firing rates of all neurons) but neural activity as an interaction between neurons.
2) What is time for a conscious brain?
A fundamental property of consciousness is its unity: a single conscious entity sees, hears and thinks. If visual and auditory areas where independent and, say, control speech, then one conscious entity would report visual experience and another conscious entity would report auditory experience. It could not be a single conscious entity since the two relevant parts are physically disconnected. Thus the unity of consciousness requires an interdependence between all the elements that compose it. This is, as I understand it, the issue that is addressed by a number of biological theories of consciousness, for example Edelman's “reentrant loops” or Tononi's integrated information theory.
However, as far as I know, there is another crucial aspect to this problem, which is the unity of consciousness, or lack of it, in time. There is no general unity of consciousness across time: two things that happen at, say, 1 minute of interval produce distinct percepts, not a single one. Clearly, consciousness is dynamic. But the big question is: how can there be a unique conscious state at any given moment in time when all the elements of the conscious network interact with some delay (since they are physical elements), typically of a few milliseconds? And what is time for such a network? Imagine there is a (physical) visual event at time t1 and an auditory event at time t2. At what time do they occur for the network, as they are sensed at different times by all its elements?Why is it that electricity changes on a millisecond timescale in the brain but conscious states seem to change at a much slower rate?
3) How can there be an intrinsic relation between neural activity and percepts?
Why is it that a particular pattern of neural activity produces the experience of redness? Most biological explanations are of this kind: I experience redness because when some red object is presented, neurons fire in that specific way. This is the coding perspective. The problem in the coding perspective is of course: who decodes the code? Ultimately, this kind of explanation is strongly dualist: it is implicitly assumed that, at some point, neural activity is transformed into the redness experience by some undetermined process that must be of a very different nature.
I would like to point out that proposals in which perception lies in the interaction between the organism and the environment (e.g. the sensorimotor theory) do not solve this problem either. I can close my eyes and imagine something red. It could be that redness corresponds to a particular way in which visual inputs change when I move my eyes or the surface, which I am anticipating or imagining, but this does not explain what is intrinsically red about the pattern of neural activity now. If we cannot explain it without referring to what happened before, then we are denying that the pattern of neural activity itself determines experience, and again this is a strong dualist view.
An experiment of thought. Consider two salamanders, and each of them has only one neuron, which is both a sensory neuron and motor neuron; say, its firing produces a particular movement. The salamanders are very similar, but their visual receptors are tuned to different wavelengths. In the first salamander, the neuron reacts to red stimuli; in the second salamander, the neuron reacts to blue stimuli. What might happen in terms of visual experience when the neuron fires? Does the first salamander see red and the other see blue? If we think that neural activity alone determines experience, then in fact the two salamanders should experience exactly the same thing – and this is also independent of the sensorimotor contingencies in this case.
4) What is the relationship between the structure of experience and the structure of neural activity?
Subjective experience is highly structured. There might be some dispute about how rich it actually is, but it is at least as rich as what you can describe with words. A striking fact about language is that the meaning of sentences is not only implied by the words but also by the relations between them, i.e., the syntax. For example, a visual scene is composed of objects with spatial relations between them, and with attributes (a red car in front of a small house). In fact, there must be more to it than syntax, there must also be semantics: if neural activity completely determines subjective experience, it must not only specify that there is a car, but also what a car is. A useful notion in psychology of perception is the concept of “affordance” introduced by James Gibson: the affordance of an object is what it allows you to do (e.g. a car affords driving). Affordances are potentialities of interaction, and they gives some meaning (rather than labels) to perceptual objects. This brings an inferential structure to experience (if I did that, this would happen).
This stands in sharp contrast with the central perceptual concept in neuroscience, the notion that “cell assemblies” represent particular percepts. A cell assembly is simply a set of neurons, and their co-activation represents a particular percept (say, a particular face). Let us say that one neuron represents “red”, another represents “car”, then the assembly of the two neurons represents the red car. The problem with this concept is that it is very poorly structured. It cannot represent relations between objects, for example. This type of representation is known as the “bag-of-words” model in language processing: a text is represented by its set of words, without any syntactic relationship; clearly, the meaning of the text is quite degraded. The concept of cell assembly is simply too unstructured to represent experience.
If we are looking for a mapping between neural activity and percepts, then 1) we must find a way to define some structure on neural activity, and 2) the mapping must preserve that structure (in mathematical terms, we are looking for a morphism, not a simple mapping).
I can summarize this discussion by pointing out that to make progress on the mind-body problem, there are two crucial steps: 1) to understand the articulation between physical time and the time of consciousness, 2) to understand the articulation between the structure of neural activity and the structure of phenomenal experience.