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Neurons in living things work cooperatively and efficiently to processincoming sensory information, often exhibiting sparse and widespread populationactivity involving structured higher-order interactions. While there arestatistical models based on continuous probability distributions for neurons'sparse firing rates, how the spiking activities of a large number ofinteracting neurons result in the sparse and widespread population activityremains unknown. Here, for homogeneous (0,1) binary neurons, we providesufficient conditions under which their spike-count population distributionconverges to a sparse widespread distribution of the population spike rate inan infinitely large population of neurons. Following the conditions, we proposenew models belonging to an exponential family distribution in which the signand magnitude of neurons' higher-order interactions alternate and shrink as theorder increases. The distributions exhibit parameter-dependent sparsity on abounded support for the population firing rate. The theory serves as a buildingblock for developing prior distributions and neurons' non-linearity forspike-based sparse coding.

Alternating Shrinking Higher-order Interactions for Sparse Neural Population Activity