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

TL;DR

The paper investigates how the fluctuation susceptibility of the average activation in the Greenberg–Hastings neural network exhibits finite-size scaling at a dynamical critical point, and how the spontaneous activation probability $r_1$ acts as an external field conjugate to the order parameter. By analyzing the model on Watts–Strogatz networks both with $r_1=0$ and $r_1>0$, the authors demonstrate that removing spontaneous activity reveals clear absorbing-state critical behavior with finite-size scaling for the susceptibility, while nonzero $r_1$ suppresses this scaling. They derive a mean-field description that predicts a continuous transition at high $T$ with a Tc that scales roughly as $T_c \approx \ln(k)/\lambda$ and exponents akin to directed-percolation, but numerical results show exponents that do not match known universality classes, likely due to quenched disorder and rare-region effects. The study also quantifies how activation mechanisms shift from single to cooperative as network connectivity grows, highlighting the limitations of mean-field approaches in intermediate regimes and the potential for rare-region phenomena to influence critical behavior. Overall, the work clarifies the role of external fields in dynamical phase transitions of neural networks and points to ongoing questions about universality in disordered, out-of-equilibrium systems with absorbing states.

Abstract

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.

Figures (8)

• Figure 1: Susceptibility as a function of the threshold $T$ for different system sizes $N$ when $r_1=0$, obtained from different quasi-stationary algorithms. The simulations correspond to the GH model defined on a Watts-Strogatz network of size $N$ with average degree $\langle k \rangle=12$ and rewiring probability $\pi=0.6$. The continuous lines correspond to fittings using appropriate fitting functions (asymmetric double sigmoidal functions). The insets show the maxima of the different curves as a function of $N$ estimated from the fittings, with power law fittings (red lines) $\max\chi \sim N^{\gamma'/\nu d}$. The critical exponents estimations are shown in Table \ref{['tab:exponents']}. (a) Curves obtained using the reactivation algorithm. (b) Curves obtained using the fixed time algorithm.
• Figure 2: (a) Pseudo critical threshold $T_c^*$ estimated from the fittings of Fig.\ref{['fig:algoritmosr1=0']} and (b) activity $f_a$ evaluated at $T_c^*$, as a function of network size $N$ for $r_1 = 0$. Blue circles correspond to data from the reactivation algorithm and red squares to the fixed time algorithm. The continuous lines are fitting curves using $T^*_c \sim T_c - N^{-1/\nu d}$ and $f_a(T=T^*_c) \sim N^{-\beta/\nu d}$. The estimated values of $T_c$, $1/\nu d$ and $\beta/\nu d$ from the fittings are shown in Table \ref{['tab:exponents']}. Inset: $T_c-T_c^*$ in log-log scale. Dashed lines are the corresponding power law fittings.
• Figure 3: Susceptibility as function of the threshold $T$ in Watts-Strogatz networks with $\langle k \rangle=12$ and $\pi=0.6$ for different network sizes $N$ and three typical values of the spontaneous activation probability: (a) $r_1 = 10^{-3}$, (b) $r_1 = 5\times10^{-4}$ and (c) $r_1 = 10^{-5}$. The continuous lines are a guide to the eye.
• Figure 4: Critical exponents as a function of the spontaneous activation probability $r_1$ estimated through finite-size scaling data fittings of (a) susceptibility ($\sim N^{\gamma'/\nu d}$), (b) activity $f_a$ at $T_c^*$ ($\sim N^{-\beta/\nu d}$), and (c) pseudo critical point of the maximum of susceptibility/variance ($\sim T_c - N^{-1/\nu d}$). Examples of data used to extract these exponents are shown in Fig.\ref{['fig:suscep-nonull-r1']}. Black dashed lines correspond to the mean values (for the first three points) (see Table \ref{['tab:exponents']}). Blue dot line and red dashed-dot line correspond to the critical exponent values obtained with the reactivation and fixed time algorithms, respectively (see Table \ref{['tab:exponents']}). Shadows indicate the corresponding uncertainties.
• Figure 5: Schematic drawing of the three mechanisms through which a quiescent neuron can be activated: spontaneous (sp), single activation (s), and cooperative activation (c). Red triangles are excited neurons, and black ones are quiescent.