Neurons within a population are strongly correlated, but how to simply capture these correlations is still a matter of debate. Recent studies have shown that the activity of each cell is influenced by the population rate, defined as the summed activity of all neurons in the population. However, an explicit, tractable model for these interactions is still lacking. Here we build a probabilistic model of population activity that reproduces the firing rate of each cell, the distribution of the population rate, and the linear coupling between them. This model is tractable, meaning that its parameters can be learned in a few seconds on a standard computer even for large population recordings. We inferred our model for a population of 160 neurons in the salamander retina. In this population, single-cell firing rates depended in unexpected ways on the population rate. In particular, some cells had a preferred population rate at which they were most likely to fire. These complex dependencies could not be explained by a linear coupling between the cell and the population rate. We designed a more general, still tractable model that could fully account for these non-linear dependencies. We thus provide a simple and computationally tractable way to learn models that reproduce the dependence of each neuron on the population rate.
Significance Statement: The description of the correlated activity of large populations of neurons is essential to understand how the brain performs computations and encodes sensory information. These correlations can manifest themselves in the coupling of single cells to the total firing rate of the surrounding population, as was recently demonstrated in the visual cortex, but how to build this dependence into an explicit model of the population activity is an open question. Here we introduce a general and tractable model based on the principle of maximum entropy to describe this population coupling. By applying our approach to multi-electrode recordings of retinal ganglion cells, we find complex forms of coupling, with the unexpected tuning of many neurons to a preferred population rate.