Signatures of criticality arise from random subsampling in simple population models

The rise of large-scale recordings of neuronal activity has fueled the hope to gain new insights into the collective activity of neural ensembles. How can one link the statistics of neural population activity to underlying principles and theories? One attempt to interpret such data builds upon analogies to the behaviour of collective systems in statistical physics. Divergence of the specific heat—a measure of population statistics derived from thermodynamics—has been used to suggest that neural populations are optimized to operate at a “critical point”. However, these findings have been challenged by theoretical studies which have shown that common inputs can lead to diverging specific heat. Here, we connect “signatures of criticality”, and in particular the divergence of specific heat, back to statistics of neural population activity commonly studied in neural coding: firing rates and pairwise correlations. We show that the specific heat diverges whenever the average correlation strength does not depend on population size. This is necessarily true when data with correlations is randomly subsampled during the analysis process, irrespective of the detailed structure or origin of correlations. We also show how the characteristic shape of specific heat capacity curves depends on firing rates and correlations, using both analytically tractable models and numerical simulations of a canonical feed-forward population model. To analyze these simulations, we develop efficient methods for characterizing large-scale neural population activity with maximum entropy models. We find that, consistent with experimental findings, increases in firing rates and correlation directly lead to more pronounced signatures. Thus, previous reports of thermodynamical criticality in neural populations based on the analysis of specific heat can be explained by average firing rates and correlations, and are not indicative of an optimized coding strategy. We conclude that a reliable interpretation of statistical tests for theories of neural coding is possible only in reference to relevant ground-truth models.
Author summary Understanding how populations of neurons collectively encode sensory information is one of the central goals of computational neuroscience. In physics, systems are often characterized by identifying and describing critical points (e.g. the transition between two states of matter). The success of this approach has inspired a series of studies to search for analogous phenomena in nervous systems, and has lead to the hypothesis that these might be optimized to be poised at ‘thermodynamic critical points’. However, translating concepts from thermodynamics to neural data analysis has been a challenging endeavour. We here study the data analysis approaches that have been used to provide evidence for criticality in the brain. We find that observing signatures of criticality is closely linked to observing activity correlations between neurons– a ubiquitous phenomenon in neural data. Our study questions the experimental evidence that neural systems are optimised to exhibit thermodynamic critical behaviour. Finally, we provide practical, open-source tools for analyzing large-scale measurements of neural population activity using maximum entropy models.