Abstract
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, but some scaling features persist in the super-critical system up to a distance of $10\%$ from $\lambda_c$. Our results provide insights on the possible subtleties that one needs to consider when applying such phenomenological approaches directly to data to infer signatures of criticality.
Highlights
The possibility that living systems may be poised at criticality is a fascinating hypothesis [1,2,3], and in recent years it has been explored in a vast variety of areas [4,5,6,7].Tools from statistical mechanics, such as the renormalization group [8,9,10], teach us that at criticality the macroscopic, collective behavior of the system is described by a few relevant attributes, such as the embedding dimension of the system and its symmetries, while most the microscopic details of the system become irrelevant
We do not see the non-Gaussian tails that we previously found at the critical point, which is expected since away from criticality the variables are much less correlated with one another and they are eventually dominated by the central limit theorem
We have found that the super- and subcritical regimes can be recognized, even though the nature of the phase transition is qualitatively different from the one of the Ising model
Summary
The possibility that living systems may be poised at criticality is a fascinating hypothesis [1,2,3], and in recent years it has been explored in a vast variety of areas [4,5,6,7].Tools from statistical mechanics, such as the renormalization group [8,9,10], teach us that at criticality the macroscopic, collective behavior of the system is described by a few relevant attributes, such as the embedding dimension of the system and its symmetries, while most the microscopic details of the system become irrelevant. At the critical point the physical properties are determined by a nontrivial fixed point in the space of the possible models. In the broad landscape of natural systems one often has to deal directly with data without an explicit model, and the systems are typically finite, so that most of the time it is hard to come up with a definitive answer about whether they are poised near a critical point [3]. Data from single-neuron recordings, from the hippocampus of a mouse running along a virtual track, were directly analyzed by the authors in order to understand if this coarse-graining procedure (which we recall in Sec. II) drives the system towards a nontrivial fixed point in the renormalization group sense, if the neural dynamics is critical and details
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