states (Fig \[fig1\]) and the maximum entropy models (Fig \[models\]) give us two complementary ways of estimating the function $n(E)$ and hence the entropy vs energy in the same data set. Results are shown in Fig \[fig3\], with some technical issues discussed in Appendix \[app:DoS1\].

As emphasized above and in Appendix \[thermoP\], the plot of entropy vs energy contains all of the thermodynamic behavior of a system, and this has a meaning for any probability distribution, even if we are not considering a system at thermal equilibrium. Thus, Fig \[fig3\] is as close as we can get to constructing the thermodynamics of this network. With the direct counting of states we see less and less of the plot at larger $N$, but the part we can see is approaching a limit as $N \rightarrow \infty$, and this is confirmed by the results from the maximum entropy models. This by itself is a significant result. If we write down a model like Eq (\[HKpair\]), then in a purely theoretical discussion we can scale the couplings between neurons $J_{\rm ij}$ with $N$ to guarantee the existence of a thermodynamic limit [@amit+al_87], but with $J_{\rm ij}$ constructed from real data, as we do 