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Phase Transitions in in vivo or in vitro Populations of Spiking Neurons Belong to Different Universality Classes

Braden A. W. Brinkman

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PRX Life 3, 043003 – Published 9 October, 2025

DOI: https://doi.org/10.1103/kjx5-1hwy

Abstract

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 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. 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.

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  145. Had we not been careful to define z in terms of the difference between the membrane potential and its running set point θs then our stability analysis would have found a WF fixed point that was unstable toward a complex-valued “spinodal” critical point that may describe critical metastable transitions.
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  148. For the in vivo networks, if we worked to higher order in z̃, then we would obtain nonzero contributions to the anomalous scaling, giving β*=ν*2(d−2−4δη*X) and Δ*=ν*2(d+2−4δη*X), where δη*X=η*X−(d+2)/4. The anomalous terms appear identically in the exponents, which means that in one of the exponent relations it has an incorrect sign. Since we find δη*X>0, the sign is incorrect in β*. The resolution to this issue is to go beyond the local potential approximation.

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PRX Life 3, 043003– Published 9 October, 2025
  • Received 30 January 2023
  • Accepted 8 September 2025
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