specific heat when we learn models for $N=100$ neurons from even one–tenth of our data, and the variance across fractions of the data is only a few percent (Appendix \[limited\_data\], Fig \[Cv\_finitesize\]).

While the inference of maximum entropy models might be accurate, some less interesting models might mimic the signatures of criticality [@1overF]. In particular, it has been suggested that independent neurons with a broad distribution of spike rates could generate a distribution of $N$–neuron activity patterns $\{\sigma_{\rm i}\}$ that mimics some aspects of critical behavior [@bas]. But in an independent model built from the actual spike rates of the neurons, the probability of seeing the same state twice would be less than one part in a billion, dramatically inconsistent with the measured $P_c \sim 0.04$. Such independent models also cannot account for the faster than linear growth of the heat capacity with $N$ (Fig \[Cv\]), which is an essential feature of the data and its support for criticality.

In maximum entropy models, the probability distribution over patterns of neural activity is described in terms of “interactions” between neurons, such as the terms $J_{\rm ij}$ in Eq (\[HKpair\]); an alternative view is that the correlations result from the response of 