The phase diagram of random threshold networks

Threshold networks are used as models for neural or gene regulatory networks. They show a rich dynamical behaviour with a transition between a frozen and a chaotic phase. We investigate the phase diagram of randomly connected threshold networks with real-valued thresholds h and a fixed number of inputs per node. The nodes are updated according to the same rules as in a model of the cell-cycle network of Saccharomyces cereviseae [PNAS 101, 4781 (2004)]. Using the annealed approximation, we derive expressions for the time evolution of the proportion of nodes in the “on” and “off” state, and for the sensitivity $\lambda$. The results are compared with simulations of quenched networks. We find that for integer values of h the simulations show marked deviations from the annealed approximation even for large networks. This can be attributed to the particular choice of the updating rule.

💡 Research Summary

This paper investigates the dynamical phase diagram of randomly connected threshold networks (RTNs), a class of binary-state models widely used to represent neural circuits and gene‑regulatory systems. Each node receives a fixed number K of inputs, each weighted by +1 or –1, and updates its state according to a Heaviside rule with a real‑valued threshold h: a node turns “on” if the weighted sum of its inputs exceeds h, otherwise it stays “off”. The authors adopt the same update rule that underlies the celebrated Saccharomyces cerevisiae cell‑cycle model (PNAS 101, 4781, 2004), which includes a special case: when the input sum equals the threshold exactly, the node is forced to the “off” state.

The analytical framework is the annealed approximation, in which the network’s connectivity is randomly reshuffled at every time step, rendering the inputs to each node statistically independent. Under this approximation the authors derive a closed‑form recursion for the fraction ρ(t) of active (“on”) nodes: ρ(t + 1) = F(ρ(t); K, h), where F is obtained by treating the distribution of the weighted sum as a binomial (or, for large K, a Gaussian) and integrating over the region above the threshold. Fixed points ρ* satisfy ρ* = F(ρ*), and their stability determines whether the system settles into a low‑activity or high‑activity regime, or exhibits bistability.

A second key quantity is the sensitivity λ, defined as the average number of nodes whose state changes in response to a single‑node perturbation. Within the annealed picture λ = K·P_flip, where P_flip is the probability that the weighted sum lies within one unit of the threshold, i.e. that a flip of a single input can change the output. The sign of λ‑1 classifies the network: λ < 1 yields a frozen (ordered) phase where perturbations die out, while λ > 1 leads to a chaotic phase where they proliferate.

To test the analytical predictions, the authors perform extensive simulations of quenched networks (fixed connectivity) with sizes up to N ≈ 10⁵. They explore both integer and non‑integer values of h. For non‑integer thresholds the simulation results match the annealed theory remarkably well: the time evolution of ρ(t), the location of fixed points, and the measured λ all converge to the predicted values as N grows, confirming that the independence assumption of the annealed approximation holds when the probability of the input sum equalling h is negligible.

In stark contrast, integer thresholds produce systematic deviations. Because the update rule forces a node to “off” whenever the weighted sum equals h, the probability of exact equality becomes non‑negligible for integer h, especially when K is small. This creates strong correlations between node states that are not captured by the annealed averaging. Consequently, simulated λ values can exceed 1 even when the annealed theory predicts λ < 1, and the observed steady‑state activity ρ* can differ substantially from the theoretical fixed points. The discrepancy diminishes as K increases (the sum distribution becomes more Gaussian) or as the network size grows, but it remains pronounced for realistic biologically relevant parameter ranges.

The authors interpret these findings as evidence that the microscopic details of the update rule—particularly the handling of the equality case—can dramatically reshape the macroscopic phase behavior. They argue that models of real biological networks, which often employ integer thresholds (e.g., a gene is activated only when a certain number of transcription factors are present), must be treated with caution: annealed approximations may over‑simplify the dynamics, and quenched analyses or explicit simulations become necessary.

In conclusion, the paper provides a comprehensive analytical description of RTN dynamics in terms of K and h, delineates the frozen‑chaotic transition via the sensitivity λ, and highlights a critical limitation of the annealed approximation for integer thresholds. These insights are valuable for researchers constructing Boolean‑type models of neural or genetic systems, emphasizing the need to consider both the statistical properties of thresholds and the precise formulation of update rules when predicting system‑level behavior.