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

TL;DR

The authors address how phase transitions in neural populations can fall into different universality classes. They develop a non-perturbative renormalization group framework for leaky integrate-and-fire spiking networks, deriving Widom scaling forms and identifying two universality classes: directed percolation for in vitro-like absorbing-state transitions and Ising for in vivo-like asynchronous-to-bistable transitions, with the spectral dimension of the synaptic-weight spectrum governing the effective dimension $d$. Mean-field scaling emerges when an outlier eigenvalue dominates (infinite $d$), while edge spectra can yield anomalous scaling, a prediction borne out by simulations on excitatory lattices and excitatory-inhibitory networks. The results provide a concrete foundation for interpreting neural avalanche data, connect critical exponents to network topology, and suggest experimental paradigms to probe universal scaling in real neural circuits. Overall, the work establishes that spiking neural populations near criticality can be categorized by well-known statistical physics universality classes, with a tractable RG approach bridging biology and theory.

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 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 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.

Figures (9)

• Figure 1: Phase transitions in in vitro versus in vivo neural populations. Neural activity may differ between recorded from tissue maintained or grown in a pitri dish ("in vitro"; top row) or recorded directly from neurons in a living organism ("in vivo"; bottom row). These differences partly reflect external input: in vitro tissue may require experimenter-provided stimulation to maintain the activity of neurons, while in vivo neurons are constantly bombarded with input from other brain areas or body systems, leading to spontaneous activity. In both cases qualitative changes in population-level activity may be observed as properties of the network, such as the overall strength of synaptic connections, are modulated. A. Phase diagram of an in vitro network: if the equilibrium resting potential of the neurons is perturbed (e.g., due to external tonic current input), the network's firing can be suppressed ($\mathcal{E} < 0$) or promoted ($\mathcal{E} > 0$). At the equilibrium potential $\mathcal{E} = 0$ (normalized units) the network activity will decay away if the strength of synaptic connections is less than a critical value $J_c$. For synaptic strengths $J > J_c$ the network activity is self-sustaining. At the critical value $J_c$ the activity decays to quiescence, but very slowly. B. Example raster plot of spiking activity in a network with subcritical $J = J_c/2$ (circle), along the line $\mathcal{E} = 0$, showing a fast decay of activity. C. Spiking activity in a network at the approximate critical point $J = J_c$ (star), showing slow decay of activity. D. Spiking activity in a supercritical network with $J = 2J_c$ (square), showing sustained activity. E. Phase diagram of an in vivo network: perturbing the equilibrium resting potential will increase or decrease neural firing. Along a critical line $\mathcal{E} = \mathcal{E}_c$ (dashed diagonal line) the network will fire asynchronously. For synaptic strengths $J > J_c$ there exist states of low or high firing, which the network can transition spontaneously between in finite networks. F. Example raster plot of spiking activity in a network with subcritical $J = J_c/2$ (circle), along the approximate critical line $\mathcal{E} = \mathcal{E}_c$, showing asynchronous activity. G. Spiking activity in a network at the approximate critical point $J = J_c$ (star), showing intermittent high and low spiking activity activity. H. Spiking activity in a supercritical network with $J = 2J_c$ (square), showing apparent transient metastable transitions between high and low firing rate states that are possible in finite-sized network simulations. Excitatory neurons are colored red, inhibitory neurons are colored blue.
• Figure 2: Mean-field behavior of the spiking network for in vitro networks (top row) and in vivo networks (bottom row). A,D) A typical nonlinearity $\nu = \phi(\psi)$ for each of the two network types. In the in vitro networks the nonlinearity is rectified, such that the firing rate is zero when a neuron's membrane potential is negative. In in vivo networks the firing rate is never zero---there is always a non-zero, though possibly small, probability of firing. B,E) The decay of $\nu(t)$, the population- and trial-averaged membrane potential, starting from an initial value of $\nu(0) \approx 1$ in in vitro networks and $\nu_+(0) \approx 1$ and $\nu_-(0) \approx 0$ in in vivo networks. C,F) Widom scaling collapses using Eq. (\ref{['eqn:ASwidomscaling']}) for in vitro networks and Eq. (\ref{['eqn:doublescalingform']}) for in vivo networks and the mean-field exponents given in Table \ref{['tab:latticeexponents']}.
• Figure 3: Widom scaling collapses for simulated activity on excitatory lattices. Top row (A-D): $d=2$. Bottom row (E-H): $d=3$. A, E. Population-and-trial-averaged spike trains $\nu(t)$ versus time in in vitro networks as the synaptic strength $J$ is tuned from subcritical ($J < J_c$, blue curves) to supercritical ($J > J_c$, green-gold curves). B, F. Widom scaling collapse of the data using Eq. (\ref{['eqn:ASwidomscaling']}). Data below and above $J_c$ collapse onto different curves, with tails given by Eq. (\ref{['eqn:FAStails']}). C, G. Population-and-trial-averaged spike trains versus time in in vivo networks, starting from a high firing rate initial condition $\nu_+(0) \approx 1$ and a low firing rate initial condition $\nu_-(0) \approx 0$. Curves correspond to equilibrium potentials $\mathcal{E} > \mathcal{E}_c$ (green-gold curves) to $\mathcal{E} < \mathcal{E}_c$ (blue curves). D, H. Widom scaling collapse of $\nu_+(t)-\nu_-(t)$ according to Eq. (\ref{['eqn:doublescalingform']}), with tails given by Eq. (\ref{['eqn:Fbartails']}). The critical exponents used to collapse the data, inset in each collapse, are the known values of the critical exponents for the directed percolation (DP) and Ising model (IM) universality classes, given in Table \ref{['tab:latticeexponents']}.
• Figure 4: Widom scaling collapses for simulated activity on networks with random regular excitatory-excitatory connections. All excitatory neurons make $k=3$ excitatory connections to other neurons. Top row: Simulation results for purely excitatory networks. A. Decay of the population-averaged spiking activity in an absorbing state network for several values of coupling strength $J$, and B. its corresponding data collapse using mean-field predictions for the critical exponents. C. Decay of the population-averaged spiking activity in a spontaneously active network for several values of the input current $\mathcal{E}$, and D. its corresponding data collapse using mean-field predictions for the critical exponents. Middle row: Simulation results for an excitatory population with effective inhibitory connections between neurons. E-F and G-H are the same as A-B and C-D, but using anomalous values of the critical exponents. Bottom: Simulation results for a model of separate excitatory and inhibitory populations that reduces to the effective model (Appendix \ref{['sec:EIreduction']}). I-J and K-L are the same as E-F and G-H, using the same values of the anomalous exponents. In the absorbing state collapses (second column), the analytically estimated asymptotic Widom scaling forms Eq. (\ref{['eqn:FAStails']}) are plotted in red, scaled by non-universal factors to match the data. Similarly for the spontaneous network collapses (fourth column) using Eqs. (\ref{['eqn:doublescalingform']}) and (\ref{['eqn:Fbartails']}); see also Appendix \ref{['sec:scalingformderivation']}.
• Figure 5: Mean-field versus anomalous scaling in Excitatory-Inhibitory networks of varying degree.A-B. Simulated activity in an effective EI network with degree $4$ random regular excitatory-excitatory connections, and B. its corresponding scaling collapse using mean-field exponents. The phase transition occurs at a value of $J_c \approx 1.38$ that is less than the mean-field prediction $J_c = 4/(2\sqrt{3}-1) \approx 1.62$, whereas in our other cases our RG analysis predicts that $J_c$ is larger than the mean-field prediction. C-D. Effective EI network with $20\%$ degree $4$ and $80\%$ degree $3$ excitatory-excitatory connections. The phase transition occurs at $J_c \approx 1.98$, less than the mean-field prediction $J_c \approx 2.26$. The data can be collapsed using mean-field exponents. E-F. Effective EI network with $10\%$ degree $4$ and $90\%$ degree $3$ excitatory-excitatory connections. The estimated critical coupling is $J_c \approx 2.2$, comparable to the mean-field prediction of $J_c \approx 2.23$. The data, however, collapses using the same anomalous exponents as the pure degree $3$ effective EI network.