Simpler models of neuronal activity via compression of interactions *

New experimental methods make it possible to record the simultaneous activity of thousands of neurons in the brain. Theoretical models that capture the behavior of these large populations, though, continue to be a challenge, often leading to an explosion of complexity that needs simplification. In condensed matter physics, simple models often work, quantitatively, but simplifications rest in part in the pairwise and local microscopic interactions. Here we explore the idea that the influence of the whole system on one degree of freedom can be compressed, in the sense of information theory. We study populations of neurons in the salamander retina and the mouse hippocampus, and focus on interactions between a single neuron and groups of eight other cells, a scale for which data allow complete sampling. We find that we can capture the influence of the group on one cell with only ten states, significantly less than the 256 possible states. Inspired by the renormalization group, we iterate this strategy and find, surprisingly, that the number of states needed to describe the interactions grows linearly with the number of neurons included. Compared with the expected exponential growth, compression of interactions seems to provide a path to build simpler models of neuronal activity.