Scaling and criticality in a phenomenological renormalization group

We present a systematic study to test a recently introduced phenomenological renormalization group, proposed to coarse-grain data of neural activity from their correlation matrix. The approach allows, at least in principle, to establish whether the collective behavior of the network of spiking neurons is described by a non-Gaussian critical fixed point. We test this renormalization procedure in a variety of models focusing in particular on the contact process, which displays an absorbing phase transition at $\lambda ={\lambda }_c$ between a silent and an active state. We find that the results of the coarse graining do not depend on the presence of long-range interactions and, overall, the method proves to be able to distinguish the critical regime from the supercritical one. However, some scaling features persist in the supercritical regime, at least for a finite system, as we see in a contact process above ${\lambda }_c$. Our results provide both a systematic test of the method and insights on the possible subtleties that one needs to consider when applying such phenomenological approaches directly to data to infer signatures of criticality.

Figure 1

The correlation matrix for the 2D contact process at $\lambda ={\lambda }_c$. (a) Correlation matrix of the raw variables. (b) Correlation matrix of clusters of 16 variables. Notice how the coarse graining seems to preserve and unravel the nontrivial correlation structure.

Figure 2

(a) Scaling of the variance, Eq. (1). (b) Scaling of the free energy, Eq. (3). Both are fitted with the corresponding power laws shown in Sec. 2. Notice that both the critical and the supercritical regime show power-law behavior but with different exponents: in particular, the silence probability is smaller and decays much faster in the supercritical case due to the proliferation of the activity. However, we do not find full compatibility between the supercritical contact process and the independent case.

Figure 3

(a) Scaling of the eigenvalues of the covariance matrix inside clusters of size $K=32,64,128$. (b) Scaling of the autocorrelation times during coarse graining. The distinction among the two phases is rather clear. For the spectrum, in the critical case we find an exponent $\mu =0.63\pm 0.02$ compatible with the expected value, whereas the eigenvalues show less variability at $\lambda >{\lambda }_c$. The same holds for the autocorrelation times, which follow a power-law behavior at criticality and become negligible above it.

Figure 4

(a) Evolution of the probability distribution of the nonzero normalized activity during the coarse graining via maximally correlated pairs, as in Eq. (2). (b) Evolution of the probability distribution of the coarse-grained variables in momentum space, as in Eq. (8). We show the results for $K=32,\cdots ,256$ and for $N/8,\cdots ,N/128$ modes (from brighter to darker color). The distribution in direct space is very different due to the critical contact process being typically close to the absorbing state. In momentum space, the supercritical contact process converges to a Gaussian fixed point in agreement with to the central limit theorem [22], whereas at criticality we see the presence of non-Gaussian tails.

Figure 5

(a) Scaling of the variance (1) during the coarse graining for $\lambda ={\lambda }_{\text{nc}}$. (b) Scaling of the free energy (3) at $\lambda ={\lambda }_{\text{nc}}$. Even if this is a supercritical contact process, the exponents are far from being comparable with the independent case $\tilde{\alpha }=1=\tilde{\beta }$. Overall, the scaling is more similar to the real critical case, see Fig. 2. Notice in particular how the silence probability is nonvanishing even for large clusters.

Figure 6

(a) Scaling of the eigenvalues of the covariance matrix for $K=32,64,128$ (from brighter to darker color). (b) Scaling of the autocorrelation times during the coarse graining for $\lambda ⪆{\lambda }_c$. The spectrum of the covariance matrix does not show a power-law decay, and the scaling of the autocorrelation times is not as convincing as the critical case shown in Fig. 3.

Figure 7

Evolution of the joint probability during the coarse graining for $\lambda ⪆{\lambda }_c$. We show the results (a) for $K=32,\cdots ,256$ and (b) for $N/8,\cdots ,N/128$ modes. Both of them are comparable with the supercritical case, and in particular in momentum space we do not see the non-Gaussian tails typical of criticality.

Figure 8

(a) Failure of the scaling of the variance in the binomial model. (b) The non-Gaussian distribution resulting from the procedure in momentum space.

Figure 9

(a) Scaling of the eigenvalue at the critical temperature of the Ising model. Notice that a dominant eigenvalue is present, because the model starts to order, but it scales in the same way as the rest of the spectrum. (b) Convergence to a non-Gaussian fixed form of the distribution of coarse-grained variables in momentum space, Eq. (8).

Figure 10

Results for the contact process in a small network, simulated with a synchronous update algorithm. (a) Scaling of the variance; (b) scaling of the free energy; (c) scaling of the eigenvalues; (d) scaling of the autocorrelation times; (e) joint probability distribution of normalized activity; and (f) joint probability distribution in momentum space.

Figure 11

The scaling of the autocorrelation times in the critical contact process is robust with respect to the definition of the autocorrelation time itself.

Figure 12

If we rescale the times by $t\to t/\tau$, with $\tau$ being the autocorrelation time, the time-autocorrelation functions at different $K$ (a) collapse into a single curve (b).