  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.
