  Ising models with pairwise interactions are the least structured, or
maximum-entropy, probability distributions that exactly reproduce measured
pairwise correlations between spins. Here we use this equivalence to construct
Ising models that describe the correlated spiking activity of populations of 40
neurons in the salamander retina responding to natural movies. We show that
pairwise interactions between neurons account for observed higher-order
correlations, and that for groups of 10 or more neurons pairwise interactions
can no longer be regarded as small perturbations in an independent system. We
then construct network ensembles that generalize the network instances observed
in the experiment, and study their thermodynamic behavior and coding capacity.
Based on this construction, we can also create synthetic networks of 120
neurons, and find that with increasing size the networks operate closer to a
critical point and start exhibiting collective behaviors reminiscent of spin
glasses. We examine closely two such behaviors that could be relevant for
neural code: tuning of the network to the critical point to maximize the
ability to encode diverse stimuli, and using the metastable states of the Ising
Hamiltonian as neural code words.
