the solution to the equation $${{dS(E)}\over{dE}} = 1.
\label{fixT}$$ For most systems, the energy $E({\mathbf\sigma})$ has only small fluctuations around $E^*$, $\langle (\delta E)^2 \rangle/(E^*)^2 \sim 1/N$, and in this sense most of the states that we see have the [*same*]{} value of log probability per degree of freedom. But hidden in the function $S(E)$ are all the parameters describing the interactions among the $N$ degrees of freedom in the system. At special values of these parameters, $[d^2 S(E)/dE^2]_{E=E^*} \rightarrow 0$, and the variance of $E$ diverges as $N$ becomes large. This is a critical point, and is mathematically equivalent to the divergence of the specific heat in an equilibrium system [@schnabel+al_11].

These observations focus our attention on the “density of states” $n(E)$. Rather than asking how often we will see specific combinations of neurons spiking while the others remain silent, we ask how many states there are with a particular probability. If we can estimate $n(E)$ for patterns of neural activity, then we can construct a thermodynamics for the network.

![[**Counting states in the response of retinal ganglion cells.**]{} [**(A)**]{} A single frame from the naturalistic movie, red ellipse indicates the approximate extent of a receptive field center for 