here, we can’t impose this scaling ourselves—either it emerges from the data or it doesn’t. We can make the emergence of the thermodynamic limit more precise by noting that, at a fixed value of $S/N$, the value of $E/N$ extrapolates to a well defined limit in a plot vs. $1/N$, as in the inset to Fig \[fig3\]A. The results of this extrapolation from the real data are strikingly simple: the entropy is equal to the energy, within (small) error bars. .

Interpreting $S(E)$
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What thermodynamic behavior is predicted by Fig \[fig3\]? If the plot of entropy vs energy is a straight line with unit slope, then Eq (\[fixT\]) is solved not by one single value of $E$ but by a whole range. Not only do we have $d^2 S/dE^2 = 0$, as at an ordinary critical point, but all higher order derivatives also are zero. Thus, the results in Fig \[fig3\] suggest that the joint distribution of activity across neurons in this network is poised at a very unusual critical point.

Figure \[fig3\] is telling us that the tradeoff between the probability and numerosity of states in the network takes a very special form. We expect that states of 