Susceptibility for extremely low external fluctuations and critical behaviour of Greenberg-Hastings neuronal model

We consider the scaling behaviour of the fluctuation susceptibility associated with the average activation in the Greenberg-Hastings neural network model and its relation to microscopic spontaneous activation. We found that, as the spontaneous activation probability tends to zero, a clear finite size scaling behaviour in the susceptibility emerges, characterized by critical exponents which follow already known scaling laws. This shows that the spontaneous activation probability plays the role of an external field conjugated to the order parameter of the dynamical activation transition. The roles of different kinds of activation mechanisms around the different dynamical phase transitions exhibited by the model are characterized numerically and using a mean field approximation.

💡 Research Summary

The paper investigates the finite‑size scaling of the fluctuation susceptibility χ associated with the average activity in the Greenberg‑Hastings (GH) neuronal cellular automaton. The GH model assigns each node three discrete states—quiescent (0), excited (1), and refractory (2)—and updates them synchronously in discrete time steps. The control parameter is a uniform activation threshold T; spontaneous activation occurs with probability r₁, while refractory nodes recover with probability r₂. The authors focus on Watts‑Strogatz small‑world networks (average degree ⟨k⟩ = 12, rewiring probability π = 0.6) and explore system sizes N ranging from 5 × 10³ to 3.2 × 10⁵.

Two regimes are examined. In the absorbing case (r₁ = 0) the system inevitably falls into a quiescent absorbing state, requiring quasi‑stationary methods to extract meaningful statistics. The authors implement two variants: a “reactivation algorithm” that restarts the system with 30 % of nodes excited whenever absorption occurs, and a “fixed‑time algorithm” that discards any run that reaches the absorbing state before a preset maximum time. Both algorithms yield the average activity fₐ and susceptibility χ = N(⟨a²⟩ − ⟨a⟩²) as functions of T. χ exhibits a pronounced, asymmetric double‑sigmoidal peak whose height scales with system size as max χ ∝ N^{γ′/νd}. Power‑law fits give γ′/νd ≈ 0.24–0.28, β/νd ≈ 0.26–0.29, and 1/νd ≈ 0.83–1.21. These exponent values differ from those of mean‑field percolation (γ′/νd = 0) and from the Directed Percolation (DP) universality class, indicating that the GH model belongs to a distinct non‑equilibrium universality class.

The second regime explores the effect of a small but finite spontaneous activation probability r₁. For relatively large r₁ (≈10⁻³) the susceptibility peak is absent, reproducing earlier reports that the external field smears out criticality. As r₁ is reduced toward zero, a clear size‑dependent peak re‑emerges, and the extracted exponents converge to the same values obtained in the r₁ = 0 case. This demonstrates that r₁ acts as an external conjugate field to the order parameter (total activity) and that the critical behavior is recovered only in the limit r₁ → 0.

Beyond scaling, the authors analyze how different activation mechanisms contribute to the order parameter near criticality. In low‑connectivity networks, single‑neuron activation triggered by spontaneous events makes a substantial contribution to fₐ, whereas in higher‑connectivity networks the cooperative activation of multiple neighbors dominates. This shift underscores the importance of network topology and quenched disorder (synaptic weights drawn from an exponential distribution) in shaping the critical dynamics.

Mean‑field approximations predict DP exponents for the GH model, but the numerical simulations on small‑world graphs do not match any known universality class, suggesting that quenched disorder and the effectively infinite dimensionality of Watts‑Strogatz networks invalidate simple mean‑field predictions.

In summary, the study establishes that the spontaneous activation probability r₁ functions as an external field, and that the GH neuronal model exhibits a genuine finite‑size scaling of susceptibility only when this field is removed. The resulting critical exponents define a new universality class distinct from both mean‑field percolation and Directed Percolation. These findings deepen our understanding of how intrinsic noise and network structure influence critical phenomena in models of large‑scale brain activity.