Latent Dynamical Variables Produce Signatures of Spatiotemporal Criticality in Large Biological Systems

Understanding the activity of large populations of neurons is difficult due to the combinatorial complexity of possible cell-cell interactions. To reduce the complexity, coarse graining had been previously applied to experimental neural recordings, which showed over two decades of apparent scaling in free energy, activity variance, eigenvalue spectra, and correlation time, hinting that the mouse hippocampus operates in a critical regime. We model such data by simulating conditionally independent binary neurons coupled to a small number of long-timescale stochastic fields and then replicating the coarse-graining procedure and analysis. This reproduces the experimentally observed scalings, suggesting that they do not require fine-tuning of internal parameters, but will arise in any system, biological or not, where activity variables are coupled to latent dynamic stimuli. Parameter sweeps for our model suggest that emergence of scaling requires most of the cells in a population to couple to the latent stimuli, predicting that even the celebrated place cells must also respond to nonplace stimuli.

Figure 1

(a) Activity variance of coarse-grained variables at each coarse-graining iteration, fit to $\propto K^{\alpha }$, $\alpha =1.37\pm 0.01$. This is within error of the experimental observation $\alpha =1.4\pm 0.06$ [14], shown in blue. (b) Average free energy, Eq. (5), at each coarse-graining iteration, fit to $\propto K^{\tilde{\beta }}$, $\tilde{\beta }=0.84\pm 0.01$, again close to experimentally found $\tilde{\beta }=0.88\pm 0.01$ [14]. (c) Eigenvalue spectrum of cluster covariance for cluster sizes $K=32$, 64, 128 against the scaled rank, averaged over clusters. We observe scaling as in Eq. (6) for $\sim 1.5$ decades with $\mu =0.65\pm 0.01$, within error of the experimental $\mu =0.71\pm 0.06$ [14]. For all panels, error bars are standard deviations over randomly selected contiguous quarters of the simulation.

Figure 2

(a) Average autocorrelation function for cluster sizes $K=2,4,\dots ,256$ as a function of time. (b) Same data, but with time rescaled by the appropriate ${\tau }_c$ for each cluster size. (c) Time constants ${\tau }_c$ extracted from each curve in (a) obey ${\tau }_c\propto K^{\tilde{z}}$, $\tilde{z}=0.27\pm 0.01$, for roughly one decade. Experimentally found $\tilde{z}$ is shown in blue [14]. Error bars are standard deviations over randomly selected contiguous quarters of the simulation.

Figure 3

Distribution of coarse-grained variables for $k=N/16,N/32,N/64,N/128$ modes retained under momentum-space coarse graining, with a Gaussian distribution (gray dashed line) shown for comparison. Note that the momentum-space coarse-grained variables may take negative values. The distribution of coarse-grained variables approaches a non-Gaussian limit as $k$ decreases. Error bars are standard deviations over randomly selected contiguous quarters of the simulation.