Ising models for neural activity inferred via selective cluster expansion: structural and coding properties

We describe the selective cluster expansion (SCE) of the entropy, a method for inferring an Ising model which describes the correlated activity of populations of neurons. We re-analyze data obtained from multielectrode recordings performed in vitro on the retina and in vivo on the prefrontal cortex. Recorded population sizes N range from N = 37 to 117 neurons. We compare the SCE method with the simplest mean field methods (corresponding to a Gaussian model) and with regularizations which favor sparse networks (L1 norm) or penalize large couplings (L2 norm). The network of the strongest interactions inferred via mean field methods generally agree with those obtained from SCE. Reconstruction of the sampled moments of the distributions, corresponding to neuron spiking frequencies and pairwise correlations, and the prediction of higher moments including three-cell correlations and multi-neuron firing frequencies, is more difficult than determining the large-scale structure of the interaction network, and, apart from a cortical recording in which the measured correlation indices are small, these goals are achieved with the SCE but not with mean field approaches. We also find differences in the inferred structure of retinal and cortical networks: inferred interactions tend to be more irregular and sparse for cortical data than for retinal data. This result may reflect the structure of the recording. As a consequence, the SCE is more effective for retinal data when expanding the entropy with respect to a mean field reference S − SMF, while expansions of the entropy S alone perform better for cortical data.