The most interesting definition I have seen of consciousness is [1] "the mind thinking about the mind thinking", i.e. some sort of meta-level "diagnostic process".
What I find interesting is that this definition supports awareness and an the existence of an 'I', but doesn't imply any sort of control or direct influence on the unconscious mind for which there seem to be experimental evidence against. Instead the result of this (literal) 'reflection' would be fed back to the unconscious mind as a sensorial input as any other.
Such a reflective process could possibly arise evolutionarily as it is a way to improve the efficiency of the thought processes.
I know absolutely nothing about neural network, but I wonder what would happen if the time series of the internal signals of a NN was fed to another NN and its output back into the first NN. Probably nothing stable at all and training would be 'interesting'.
Another interesting thing is that consciousness would be 'just' a process and not necessary for intelligence to arise. Peter Watts in 'Blindsight' [2] explores this last point in depth.
[1] Unfortunately I can't remember were I read this reference nor I have been able to track it.
[2] An extremely good first contact hard scifi novel, can be read for free from the author website.