the neurons to fluctuating external signals. Testing this idea has a difficulty that has nothing to do with neurons. Going back to equilibrium statistical mechanics, many models in which spins (or other degrees of freedom) interact with one another are mathematically equivalent to a collection of spins responding independently to a fluctuating field (Appendix \[SNM\]). Thus, correlations are almost always interpretable as independent responses to unmeasured fluctuations, and for neurons there are many possibilities, including internal states of neurons, that would not fall into the usual classification of network effects vs common input.

We can exclude the visual stimulus alone as the explanation of what we see, since similar behavior—e.g., the growth of a peak in specific heat with increasing $N$—appears in response to random checkerboard movies or even full field flicker (Appendix \[cond\_ind\], Fig \[compare\_ens\]). Our analysis focused on the total distribution over activity states, not distinguishing between the correlations due to a common stimulus input and those due to intrinsic circuit interactions. While a network of conditionally independent neurons might give rise to signatures of criticality as observed here, such models are incompatible with our data. We can construct conditionally independent neurons by shuffling the repeated presentations of 