Phase Transitions in <i>in vivo</i> or <i>in vitro</i> Populations of Spiking Neurons Belong to Different Universality Classes

The “critical brain hypothesis” posits that neural circuitry may be tuned close to a “critical point” or “phase transition”—a boundary between different operating regimes of the circuit. The renormalization group method was developed to explain the emergence of scale invariance in the statistics of systems tuned to a critical point. In the brain this scale invariance has been hypothesized to have several computational benefits, including increased collective sensitivity to changes in input and robust propagation of information across a circuit. However, our theoretical understanding of critical phenomena in neural circuitry is limited because standard renormalization group methods apply to systems with either highly organized or completely random connections. Connections between neurons lie between these extremes and may be either excitatory (positive) or inhibitory (negative) but not both. In this work we present a scaling theory for the population statistics of critical neural activity on networks with some realistic biological constraints on the connectivity. This scaling theory derives from a renormalization group method we develop for models of spiking neural populations—the spike response model with escape noise, also known as the nonlinear Hawkes process. We show that the scaling theories differ for models of versus circuits—they belong to different “universality classes”—and that both may exhibit “anomalous” scaling at a critical balance of inhibition and excitation. We verify our theoretical results on simulations of neural activity data and discuss how our scaling theory can be further extended and applied to real neural data, as well as other biophysical phenomena.