Propagation of external regulation and asynchronous dynamics in random Boolean networks

Boolean Networks and their dynamics are of great interest as abstract modeling schemes in various disciplines, ranging from biology to computer science. Whereas parallel update schemes have been studied extensively in past years, the level of understanding of asynchronous updates schemes is still very poor. In this paper we study the propagation of external information given by regulatory input variables into a random Boolean network. We compute both analytically and numerically the time evolution and the asymptotic behavior of this propagation of external regulation (PER). In particular, this allows us to identify variables which are completely determined by this external information. All those variables in the network which are not directly fixed by PER form a core which contains in particular all non-trivial feedback loops. We design a message-passing approach allowing to characterize the statistical properties of these cores in dependence of the Boolean network and the external condition. At the end we establish a link between PER dynamics and the full random asynchronous dynamics of a Boolean network.

💡 Research Summary

The paper tackles the largely unexplored territory of asynchronous dynamics in random Boolean networks (RBNs) by introducing the concept of Propagation of External Regulation (PER). In an RBN each node implements a Boolean function of its inputs; traditionally, studies have focused on synchronous (parallel) updates, but many real‑world systems—gene regulatory networks, distributed computing, neural circuits—operate asynchronously, updating one node at a time. The authors ask how information supplied by a set of externally fixed input variables spreads through the network when updates are asynchronous, and what structural remnants remain after this spread.

First, the authors formalize PER. A fraction α of the network’s variables are designated as external inputs and are held constant throughout the dynamics. The remaining variables are updated according to their Boolean functions, which are randomly drawn from a prescribed distribution (including constant, copy, negation, AND, OR, XOR, etc.) with in‑degree K = 2. PER proceeds iteratively: any node whose all inputs are already fixed becomes fixed itself, and this fixing propagates outward. By writing a mean‑field recursion for the fraction ρ(t) of still‑undetermined nodes at discrete time t, they obtain an analytical expression that depends on α and on the function distribution. The recursion exhibits two regimes. When α exceeds a critical value αc, ρ(t) decays exponentially to zero, meaning that the external information eventually determines the entire network. Below αc, ρ(t) converges to a non‑zero fixed point ρ*; the nodes that remain unfixed constitute a “core” that contains all non‑trivial feedback loops.

To characterize this core, the authors develop a message‑passing (belief‑propagation) scheme. Each variable‑function edge carries a message indicating whether the variable is already fixed or still ambiguous. By iterating these messages to a fixed point they compute the average size of the core, its distribution, and the statistics of loops embedded in it. The analysis reveals a sharp percolation‑like transition: as α is lowered past αc the core suddenly expands, and its internal topology becomes increasingly tangled, leading to a rich set of possible long‑term behaviors (multiple attractors, long cycles). This transition mirrors the frozen‑to‑chaotic transition known from synchronous RBNs, but the asynchronous setting adds an extra layer of complexity because the core’s internal dynamics are not synchronized.

The crucial insight is that PER and the full asynchronous dynamics are intimately linked. In a truly asynchronous update rule, at each microscopic time step a single node is chosen uniformly at random and updated using the current values of its inputs. Nodes that have already been fixed by PER never change again, so the stochastic evolution of the whole network is confined to the core identified by PER. Consequently, statistical properties of the core (size, loop structure) directly determine macroscopic observables of the asynchronous system: the probability of reaching a fixed point, the typical length of limit cycles, and the sensitivity to perturbations. The authors validate this connection with extensive numerical simulations, showing that predictions derived from the PER analysis match the outcomes of full asynchronous simulations across a wide range of α and function ensembles.

Beyond the theoretical contribution, the work has practical implications. In biological contexts, external signals (e.g., nutrients, hormones) correspond to the fixed inputs; PER describes how such signals can deterministically lock large portions of a gene‑regulatory network, leaving only a resilient core that may generate oscillations or bistability. In engineered systems, PER offers a diagnostic tool: by measuring which components become immediately constrained by boundary conditions, designers can anticipate which parts of a circuit will remain flexible under asynchronous operation and thus require additional control or redundancy.

In summary, the paper provides a comprehensive analytical and numerical framework for understanding how external information propagates in random Boolean networks under asynchronous updates. It introduces PER as a tractable intermediate dynamics, uses message‑passing to quantify the residual core, and demonstrates that the core’s statistical features govern the full asynchronous behavior. This bridges a gap between the well‑studied synchronous RBN literature and the more realistic asynchronous setting, opening avenues for future research on control, robustness, and dynamical phase transitions in complex Boolean systems.