The Heavy-Tail Stabilizer

The criticality hypothesis — that biological neural networks operate near a phase transition — has an embarrassing problem. In standard Gaussian mean-field theories, the critical regime is fragile. It requires precise tuning of parameters. Move slightly off the critical point and the network either falls silent or explodes into runaway activity. Biology doesn’t fine-tune.

Mensi et al. show that heavy-tailed synaptic connectivity eliminates the tuning problem entirely.

When connection strengths follow a Cauchy distribution instead of a Gaussian, the macroscopic dynamics collapse to a one-dimensional gradient flow with a global Lyapunov potential. The phase transition is still there — collective activity still grows as the square root of the distance to criticality. But the critical regime is now wide, not sharp. Static susceptibility diverges as the square root rather than linearly, creating a broad zone of high sensitivity.

The mechanism is an emergent automatic gain control. At high activity levels, the heavy-tailed fluctuations generate noise that suppresses effective gain. Near the critical point, the same fluctuations preserve high responsiveness. The network self-regulates: too much activity triggers its own suppression, while low activity allows sensitivity to remain high.

Gaussian theories predicted fragility because they assumed well-behaved fluctuations. Heavy tails — which empirical measurements of synaptic strengths consistently show — provide a microscopic mechanism for the robustness that biology demands. The critical regime isn’t delicately balanced. It’s fat.