Consciousness is supported by near-critical slow cortical electrodynamics

Pretty cool. I am surprised that low frequency dynamics are so strongly associated with reports of aware conciousnes states given that the high frequency 40 Hz activity of cortico-thalamo-cortical loops are necessary (if not sufficient) for consciousness as well. I'd put my money on the low frequency dynamics being a downstream effect from the cessation of the ~40 Hz cortico-thalamo-cortical loops under anesthesia. I'd also like it if they had included a gas anesthetic along with the receptor mediated anesthetics.

Can anyone ELI5 this?

Eh, not really.

It's hard to describe features of chaos / nonlinear dynamics simply.

There are features of chaotic systems that undergo phase changes from one type of behavior to another. Think a signal on one side of the transition to look like a pure sine wave and on the other side of the transition to look like purely random white noise.

What these guys are reporting is that they found a parameter, a signal they can measure in brains that seems to correlate with consciousness. When the signal is right up near the edge of this kind of chaotic phase transition, the subject is conscious, when nowhere near the phase transition, you aren't. So they have a way to sort of measure and quantify consciousness and perhaps even control it a little by measuring and doing things to manipulate this signal.

It might be sort of like an OPEN sign in a store, just a signal that happens to reflect consciousness, or this brain wave behavior might be a fundamental part of consciousness and higher thinking. It all seems quite interesting.

Projecting guesses out for this is that chaotic dynamics are a fundamental part of how our brains, especially at the highest levels work. One would guess given this evidence and other things that one of the things that makes consciousness possible is a complex system kept balanced at a knife edge of a certain kind of chaotic phase transition. It's also the kind of thing that can open a backdoor for free will in a deterministic universe, your brain in this chaotic state might behave deterministically but the actual state would never be knowable with enough precision to predict the outcome of inputs for any amount of time (butterfly effect).

What does "a critical point or phase transition [of the] electrodynamics of the cortex" mean? Electricity doesn't have "phases" AFAIK (well there's two/three-phase power but I don't think that's what they're talking about).

They're talking about the dynamics of the system's behavior not something physical about electrons.