Phase transitions in

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 and theory of critical phenomena explain how systems tuned to a critical point display scale invariance due to fluctuations in activity spanning a wide range of time or spatial scales. 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 develop a renormalization group method that applies to models of spiking neural populations with some realistic biological constraints on connectivity, and derive a scaling theory for the statistics of neural activity when the population is tuned to a critical point. We show that the scaling theories differ for models of