Response functions disambiguate intrinsic versus inherited criticality in spiking neural networks

Criticality has been proposed as a defining feature of neural systems (J.M. Beggs, 2022). Proponents point to the array of time- and length-scales exhibited by critical phenomenon as a mechanism by which brains may compute and propagate signals beyond the innate scales of their constituent neurons (L. Cocchi et al., 2017). Critics of this hypothesis rightly point to the possibility that heavy-tailed input distributions (see e.g. D.L. Ruderman, 1997) could impart heavy-tailedness to neural systems, demonstrating this inherited criticality in simple neural models with shared input (J. Touboul & A. Destexhe, 2017). Measuring this criticality hypothesis against empirical data thus requires a way to distinguish between these competing possibilities. We derive a mean-field theory of spiking neural networks with noisy input where the network and input can be separately tuned to criticality. At the bifurcation in its mean firing rates, the network exhibits heavy-tailedness in its autocovariance and perturbation response functions (averaged over population). Critical input can also drive heavy-tailed network correlations, but does not impart this characteristic on the response functions. This result suggests leveraging empirical response functions to distinguish between intrinsic and inherited criticality.