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The "critical brain hypothesis" posits that neural circuitry may be tunedclose to a "critical point" or "phase transition" -- a boundary betweendifferent operating regimes of the circuit. The renormalization group andtheory of critical phenomena explain how systems tuned to a critical pointdisplay scale invariance due to fluctuations in activity spanning a wide rangeof time or spatial scales. In the brain this scale invariance has beenhypothesized to have several computational benefits, including increasedcollective sensitivity to changes in input and robust propagation ofinformation across a circuit. However, our theoretical understanding ofcritical phenomena in neural circuitry is limited because standardrenormalization group methods apply to systems with either highly organized orcompletely random connections. Connections between neurons lie between theseextremes, and may be either excitatory (positive) or inhibitory (negative), butnot both. In this work we develop a renormalization group method that appliesto models of spiking neural populations with some realistic biologicalconstraints on connectivity, and derive a scaling theory for the statistics ofneural activity when the population is tuned to a critical point. We show thatthe scaling theories differ for models of in vitro versus in vivo circuits --they belong to different "universality classes" -- and that both may exhibit"anomalous" scaling at a critical balance of inhibition and excitation. Weverify our theoretical results on simulations of neural activity data, anddiscuss how our scaling theory can be further extended and applied to realneural data.

Phase transitions in in vivo or in vitro populations of spiking neurons belong to different universality classes