I only read the article so my understanding of Jayne is limited to that, but he doesn't seen to be saying that a mental model of the world is what gives us consciousness. I think many animals have a mental model of the world, with some of the most intelligent animals having a very sophisticated model. My understanding (of Jayne) is that humans have consciousness because they are able to use language to further contemplate and analyze a mental model of the world using metaphors and analogies.
One of the things that actually struck me about this theory was that it really boosts the idea that we should respect the minds of non-human animal mind because they may not experience the world differently than we do.