Fig. 3: Temporal coarse-graining recovers avalanche scaling of χ = 2 under subsampling conditions when the network is critical.

a Sketch of the neuronal network with N = 10⁶ neurons (80% excitatory, E; 20% inhibitory, I) and external Poisson drive of rate λ = 20/N per time step. The E/I balance is controlled by the scalar g, which scales inhibitory weight matrices W_II = W_EI as a function of excitatory weight matrices W_EE = W_IE = J. b Change in number of avalanches as a function of threshold Θ normalized by sampling fraction f. c Power law in size (left) and duration (right) distributions for avalanches become shallower with temporal coarse-graining k. Note cut-off regimes for S > ~10³ and duration L > ~50. Inset: Corresponding slopes α(k) and β(k). d Temporal coarse-graining uncovers χ_sh = 2 for short-duration avalanches (L = 1–10), whereas χ_lg remains ~1–1.2 for long-duration avalanches (L > 10). e Summary of change in χ_sh and χ_lg with k for 5 sampling fractions f. A decrease in f requires a higher k to recover χ_sh = 2. Note failure of recovery for very low f. χ_lg does not depend on k. f Temporal coarse-graining recovers avalanches up to the finite-size cut-off of Φ ≅ 400 time steps. Note plateau in maximal scaling range Φ for χ_sh = 2 plotted in simulation time steps as a function of k. g Temporal coarse-graining recovers χ_sh = 2 for critical network dynamics but fails for subcritical dynamics. Note that weak subcritical (g = 3.75) and critical conditions (g = 3.5) can exhibit similar χ_sh at the original temporal resolution yet diverge with temporal coarse-graining. Broken, black lines: visual guide to the eye. Results obtained from T = 10⁸ simulation time steps.