Neural networks operate near criticality — the edge between quiescent and active regimes — where sensitivity to inputs is maximal and information processing is most efficient. The problem is that criticality is a phase transition, and phase transitions require precise parameter tuning. A system must sit exactly at the critical point. Move slightly off and it's either too quiet or too active. How do biological neural networks, with their noisy, variable, unreliable components, maintain this precision?
Heavy-tailed synaptic connections solve the problem without solving it. When the distribution of synaptic weights has heavy tails (specifically, Cauchy-distributed), the resulting dynamics naturally produce near-critical behavior over a wide range of parameters, not just at a single point. The mechanism is an emergent automatic gain control: at high activity levels, the activity-dependent noise from heavy-tailed connections suppresses the effective gain, preventing runaway excitation. At low activity, the suppression lifts, restoring high susceptibility. The network regulates itself.
The mathematical reduction is clean. The full network dynamics collapse to a one-dimensional gradient flow with a global Lyapunov potential — meaning the system always descends toward its attractor, the near-critical state. Collective activity grows as the square root of the distance from the critical point, the standard mean-field exponent for a continuous phase transition. The heavy tails do not change the universality class; they change the basin of attraction, making the critical regime accessible from a broad range of initial conditions.
The structural claim is about the source of robustness. The usual story is self-organized criticality — the system tunes itself through feedback. Here, no tuning occurs. The heavy-tailed connectivity is a static structural feature, not a dynamical process. The disorder IS the mechanism. The same variability that makes biological neural networks seem imprecise is what holds them near the regime where they work best.
(arXiv:2603.18478)