For neuroscientists, treating "The Hard Problem of Consciousness" outside of bar-room speculation is a risky career move. This is why we have true doctors of philosophy, and why the philosophy paper "Neural Plasticity and Consciousness" by Susan Hurley and Alva Noë is a good thing. Hurley and Noë's thesis relates to some recent activity on WeAlone [1,2, maybe 3] , so I will attempt to summarize the article in a language that makes most sense to me.
First Hurley and Noë note that the "hard problem of consciousness" is equivalent to what they call an "absolute gap", i.e. "why should we assume that neural activity is solely responsible for conscious perception at all ?". My interpretation is that Hurley and Noë say "we can't, this is a leap of faith", and for the purposes of the paper accept as an axiom that neural activity corresponds to perception. The meat of the paper then, discusses why some neural activity should take on a particular quality, like seeing, and other neural activity should take on a distinct quality, like hearing.
Lately, I've been throwing around the term "neural topology" and "manifold structure" in an embarrassingly non-rigorous manner. I'd like to say "the topology of qualia acquires the topology of stimuli via learning of the intrinsic statistical structure of the stimuli, and in a sense, the stimulus stimulus model constitutes the nature of qualia", but this is vague. Hurley and Noë express, I believe, a similar sentiment clearly and without abusing terms from mathematics :
It is argued that the different characteristics of input activity from specific sources (visual vs. auditory) generate not just representational structure specific to that source but also source-specific sensory and perceptual qualities.
That is to say, when the brain learns the topology of stimuli ( possibly in union with the topology of motor outputs as they modify stimuli ), the brain acquires the qualia corresponding to said stimuli.
Earlier we talked about the possibility of defining an algebraic structure representing the shape of information coded in the brain. The take-away point was that it might be possible to rigorously say "these two areas have effectively the same abstract structure, since you can relate them by some structure preserving relationship". The Hurley and Noë paper provides anecdotes which suggest that, when two physically distinct neural circuits have the same abstract structure (topology), then the subjective experience (qualia) are also the same. Specifically, they discuss experiments in which blind patients were able to acquire visual qualia through a tactile stimulation device that translates camera images into stimulation of the skin.
After a period of adaptation (as short as a few minutes), subjects report perceptual experiences that are distinctively non-tactile and quasi-visual. … However, Bach-y-Rita emphasizes that the transition to quasi-visual perception depends on the subject’s exercising active control of the camera. … Perceivers can acquire and use practical knowledge of the common laws of sensorimotor contingency that vision and TVSS-perception share. For example, as you move around an object, hidden portions of its surface come into tactile-visual view, just as they would if you were seeing them."
This experiment suggests that giving a system a new topology induces qualia of that topology, and that learning the new topology does not necessarily require expensive and lengthly re-wiring. That camera control was necessary for inducing visual qualia from tactile stimulation suggests that the structure of visual stimuli and the experience of seeing must necessarily incorporate how our actions : movement of the eyes and head, and translation in space, alter the content of visual stimuli. Thus, when we talk about the "topology" of a stimulus, we must also incorporate how our actions change the stimulus (how our motor operators transform the stimulus space).
Hurley and Noë cover a number of other interesting anecdotes, including what happens when the brain fails to adapt its structure to reality, and pointing out that, in a left-right reversal of vision, reversing the interpretation of visual data is topologically equivalent to reversing the coordinates of motor output and proprioception, such that many different possible explanations of neural adaptation may be topologically equivalent.
So, I really do feel like, if we can make this notion of "neural topology*" more rigorous, we will have a satisfying answer to the portion of "the hard problem" that is amenable to scientific and mathematical investigations.
*neural topology : the idea that, in high dimensional sensory spaces, the distribution of probable stimuli occupy a reduced subset of said high dimensional space, and that one can move about this subset in a differentiable manner to transition smoothly between probable stimuli. This is a vague notion. It is related to "statistical structure" and "manifold", although I should note that we don't have enough information to say that the space of probably sensory-motor states is actually a manifold.