Sorry this is late, but check out Dr. Giulio Tononi's mathematical apparatus called Integrated Information Theory. In a nutshell, it considers a system and all the causal relations within it, measures some quantity related to the number of states that system can take, and then considers the graph cut that minimizes this quantity in the subsystems. The difference between the uncut and cut graphs gives a degree of "integrated information", which can be argued is a useful analogue or even the very definition of consciousness.
