Christof Koch has a Scientific American article on the ideas of Giulio Tononi (here). Tononi is putting forward a panpsychic idea of consciousness: everything from a hydrogen atom to a human has some degree of consciousness. Well, I suppose that it would depend on your definition of consciousness! He defines consciousness in terms of an elaboration of information theory, integrated information theory (IIT) of consciousness,
IIT is based on two axiomatic pillars.
First, conscious states are highly differentiated; they are informationally very rich. You can be conscious of an uncountable number of things…Think of all the frames from all the movies you have ever seen or that have ever been filmed or that will be filmed! Each frame, each view, is a specific conscious percept.
Second, this information is highly integrated. No matter how hard you try, you cannot force yourself to see the world in black-and-white, nor can you see only the left half of your field of view and not the right. When you’re looking at your friend’s face, you can’t fail to also notice if she is crying. Whatever information you are conscious of is wholly and completely presented to your mind; it cannot be subdivided. Underlying this unity of consciousness is a multitude of causal interactions among the relevant parts of your brain. If areas of the brain start to disconnect or become fragmented and balkanized, as occurs in deep sleep or in anesthesia, consciousness fades and might cease altogether. …
To be conscious, then, you need to be a single, integrated entity with a large repertoire of highly differentiated states. … And the more interconnected, the more meaningful they become. Indeed, Tononi’s IIT postulates that the amount of integrated information that an entity possesses corresponds to its level of consciousness.
These ideas can be precisely expressed in the language of mathematics using notions from information theory such as entropy. Given a particular brain, with its neurons and axons, dendrites and synapses, one can, in principle, accurately compute the extent to which this brain is integrated. From this calculation, the theory derives a single number, fi . Measured in bits, fi denotes the size of the conscious repertoire associated with any network of causally interacting parts. Think of fi as the synergy of the system. The more integrated the system is, the more synergy it has, the more conscious it is. If individual brain regions are too isolated from one another or are interconnected at random, fi will be low. If the organism has many neurons and is richly endowed with specific connections, fi will be high—capturing the quantity of consciousness but not the quality of any one conscious experience. (That value is generated by the informational geometry that is associated with fi but won’t be discussed here.)…
The cortex and its gateway, the thalamus…on the other hand, are essential for consciousness, providing it with its elaborate content. Its circuitry conjoins functional specialization with functional integration thanks to extensive reciprocal connections linking distinct cortical regions and the cortex with the thalamus. This corticothalamic complex is well suited to behave as a single dynamic entity endowed with a large number of discriminable states.
Somehow there is a ‘with-one-almighty-leap’ between the mathematics and the actual awareness and ‘fi’ is almost impossible to calculate for a system of any size. But these are the sort of problems that beset other theories of consciousness too.