The recovery of parabolic avalanches in spatially subsampled neuronal networks at criticality

Scaling relationships are key in characterizing complex systems at criticality. In the brain, they are evident in neuronal avalanches-scale-invariant cascades of neuronal activity quantified by power laws. Avalanches manifest at the cellular level as cascades of neuronal groups that fire action potentials simultaneously. Such spatiotemporal synchronization is vital to theories on brain function yet avalanche synchronization is often underestimated when only a fraction of neurons is observed. Here, we investigate biases from fractional sampling within a balanced network of excitatory and inhibitory neurons with all-to-all connectivity and critical branching process dynamics. We focus on how mean avalanche size scales with avalanche duration. For parabolic avalanches, this scaling is quadratic, quantified by the scaling exponent, $\chi =\text{2}$ , reflecting rapid spatial expansion of simultaneous neuronal firing over short durations. However, in networks sampled fractionally, $\chi$ is significantly lower. We demonstrate that applying temporal coarse-graining and increasing a minimum threshold for coincident firing restores $\chi =\text{2}$ , even when as few as 0.1% of neurons are sampled. This correction crucially depends on the network being critical and fails for near sub- and supercritical branching dynamics. Using cellular 2-photon imaging, our approach robustly identifies $\chi =\text{2}$ over a wide parameter regime in ongoing neuronal activity from frontal cortex of awake mice. In contrast, the common 'crackling noise' approach fails to determine $\chi$ under similar sampling conditions at criticality. Our findings overcome scaling bias from fractional sampling and demonstrate rapid, spatiotemporal synchronization of neuronal assemblies consistent with scale-invariant, parabolic avalanches at criticality.