Critical Branching Process

Avalanche size distribution P(s) ~ s^{-3/2} at criticality σ=1

At criticality (σ=1), the branching process generates avalanches with a power-law size distribution P(s)~s^{-3/2} (Zinn-Justin exponent τ=3/2). Subcritical (σ<1) avalanches die out exponentially; supercritical (σ>1) avalanches explode. This critical state is proposed as the operating point of cortical neural networks — "neural avalanches" (Beggs & Plenz 2003) show the same exponent.