The quality and complexity of pairwise maximum entropy models for large cortical populations

We investigate the ability of the pairwise maximum entropy model to describe the spiking activity of large populations of neurons recorded from the visual, auditory, motor, and somatosensory cortices. To quantify this performance, we use (1) Kullback-Leibler (KL) divergences, (2) the extent to which the pairwise model predicts third-order correlations, and (3) its ability to predict the probability of multiple simultaneously active neurons. We compare these with the performance of a model with independent neurons and study the relationship between the different performance measures, while varying the population size, mean firing rate of the chosen population, and the bin size used for binarizing the data. We find excellent performance for small populations (N<20), and that smaller mean firing rates and bin sizes generally improve performance. But this is not the case for larger populations. For large populations, pairwise models may be good in terms of predicting third-order correlations and the probability of multiple neurons being active, but they do significantly worse in terms KL-divergences. We show that these results are independent of the cortical area and of whether approximate methods or Boltzmann learning are used for inferring the pairwise couplings. Finally, we compared the scaling of the inferred couplings with N and find it to be well explained by the Sherrington-Kirkpatrick model, whose strong coupling regime shows a complex phase with many metastable states. We find that, up to the maximum population size studied here, pairwise correlations can be fitted by a pairwise model outside its complex phase.