that reproduces these data within experimental errors. \[models\]](fig2.pdf){width="\linewidth"}

![image](fig3){width="\linewidth"}

The maximum entropy approach to describing activity in networks of neurons has been explored, in several different systems, for nearly a decade [@schneidman+al_06; @tkacik+al_06; @shlens+al_06; @tang+al_08; @tkacik+al_09; @shlens+al_09; @ohiorhenuan+al_10; @ganmor+al_11; @simplest; @sdme], and there have been parallel efforts to use this approach in other biological systems [@lezon+al_06; @tkacik_07; @bialek+ranganathan_07; @seno+al_08; @weigt+al_09; @stephens+bialek_10; @mora+al_10; @marks+al_11; @sulkowska+al_12; @bialek+al_13; @mukherjee+al_13; @bialek+al_14; @santolini+al_14]. Recently we have used the maximum entropy method to build models for the activity of up to $N=120$ neurons in the experiments described above [@tkacik+al_14a]. As summarized in Fig \[models\], we take from experiment the mean probability of each neuron generating a spike ($\langle \sigma_{\rm i}\rangle$), the correlations between spiking in pairs of neurons ($\langle \sigma_{\rm i}\sigma_{\rm j}\rangle$), and the probability that $K$ out of the $N$ neurons spike in the same small window of time \[$P(K)$\]. Mathematically, the maximum entropy models consistent with these data have the form $$\begin{aligned}
P(\{\sigma_{\rm i}\}) &=& {1\over Z} \exp\left[ - E(\{\sigma_{\rm i}\})\right]\label{PH} \\
E (\{\sigma_{\rm i}\})&=& - \sum_{{\rm i}=1}^N h_{\rm i} \sigma_{\rm i} - {1\over 2} \sum_{{\rm i}, {\rm j} =1}^N J_{\rm ij} \sigma_{\rm i} \sigma_{\rm j}
- V \left(K \right) ,  \label{HKpair}\end{aligned}$$ where $K 