The criticality hypothesis --- that biological neural networks operate near a phase transition to maximize dynamic range and computational power --- has a tuning problem. In Gaussian mean-field theory, the critical point is a knife-edge: move slightly away and the system is either quiescent or chaotic. Kojima shows that heavy-tailed synaptic distributions dissolve the knife-edge entirely. For Cauchy-distributed couplings, the dynamical mean-field theory reduces to a one-dimensional gradient flow with a global Lyapunov potential. The phase transition is continuous, with collective activity growing as the square root of the distance from criticality, but the static susceptibility diverges only as the square root rather than linearly. This weaker divergence is the key: it means the system's sensitivity to perturbation grows near the critical point but never becomes catastrophically sharp. An emergent automatic gain control mechanism arises naturally --- activity-dependent noise fluctuations suppress the effective gain at high activity levels while preserving high susceptibility near criticality.
The result generalizes to all symmetric alpha-stable distributions, identifying heavy tails as the microscopic origin of robust near-critical dynamics. The mechanism is not fine-tuning to a critical point but rather the broadening of the critical region itself. Heavy-tailed connectivity creates a regime where the system is “near-critical” over a wide range of parameters, not just at a single value. The Lyapunov potential ensures global stability --- no chaotic trajectories, no runaway activation --- while the square-root scaling ensures the system responds to inputs across a broad range. The mathematical structure is cleaner than the Gaussian case because the one-dimensional reduction is exact, not approximate.
Robustness in complex systems rarely comes from precision. It comes from the distribution's shape. A Gaussian distribution concentrates its weight, creating sharp transitions and narrow operating windows. A heavy-tailed distribution spreads its weight, creating broad transitions and wide operating windows. The criticality tuning problem in neural networks is not a problem about criticality --- it is a problem about Gaussianity. Replace the distribution and the problem vanishes. The general principle: when a desirable operating regime requires fine-tuning under one distributional assumption, the first question is whether the assumption is correct, not whether the fine-tuning is achievable.
(arXiv:2603.18478)