  Recent work emphasizes that the maximum entropy principle provides a bridge
between statistical mechanics models for collective behavior in neural networks
and experiments on networks of real neurons. Most of this work has focused on
capturing the measured correlations among pairs of neurons. Here we suggest an
alternative, constructing models that are consistent with the distribution of
global network activity, i.e. the probability that K out of N cells in the
network generate action potentials in the same small time bin. The inverse
problem that we need to solve in constructing the model is analytically
tractable, and provides a natural "thermodynamics" for the network in the limit
of large N. We analyze the responses of neurons in a small patch of the retina
to naturalistic stimuli, and find that the implied thermodynamics is very close
to an unusual critical point, in which the entropy (in proper units) is exactly
equal to the energy.
