of models in which the strength of correlations changes but spike rates are fixed. Such a family is given by $$\begin{aligned}
P(\{\sigma_{\rm i}\}; \alpha) &=& {1\over {Z(\alpha)}} \exp\left[ -E_\alpha (\{\sigma_{\rm i}\})\right]\label{PHalpha} \\
E_\alpha (\{\sigma_{\rm i}\})&=& - \sum_{{\rm i}=1}^N h_{\rm i}' (\alpha )\sigma_{\rm i} 
\nonumber\\
&&\,\,\,\,\,\,\,\,\,- \alpha\left[ {1\over 2} \sum_{{\rm i}, {\rm j} =1}^N J_{\rm ij} \sigma_{\rm i} \sigma_{\rm j}
+ V \left(K \right) \right],  \label{HKpairalpha}\end{aligned}$$ where changing $\alpha$ changes the strength of correlations, and we adjust all the $h_{\rm i}' (\alpha )$ to hold mean spike rates fixed at their observed values. The behavior of this family of models is explored in Fig \[alpha\_fig\].

![[**Changing correlations at fixed spike rates.**]{} [**(A)**]{} Three maximum entropy models for a 120-neuron network, where correlations have been eliminated (left, $\alpha=0$), left at the strength found in data (middle, $\alpha=1$), or scaled up (right, $\alpha=2$). The first row shows 10000 most frequent patterns (black=spike, white=silence) in each model. The second row shows the distribution of pairwise correlation coefficients. [**(B)**]{} Entropy vs. energy for the networks in A. [**(C)**]{} Entropy per neuron as a function of $\alpha$, for different subnetwork sizes $N$. [**(D)**]{} Heat capacity per neuron exhibits a peak close to $\alpha=1$. Error bars are 