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Attractor identification in asynchronous Boolean dynamics with network reduction

Elisa Tonello, Loïc Paulevé

TL;DR

This work addresses the computational challenge of identifying attractors in asynchronous Boolean networks by introducing a reduction-driven pipeline. It systematically eliminates non-autoregulated components to form a smaller network, computes minimal trap spaces, and samples attractors from the reduced system to form candidate states that are lifted back to the original state space via a lifting map $\mathcal{S}^n$. These candidates are screened first through trap-space containment and then via reachability/model-checking to confirm true attractors, yielding significant runtime reductions in experiments on biological models and random benchmarks. The method is implemented by integrating Colomoto tooling (minibnnaldi, trappist) with AEON and mtsNFVS, providing a practical, scalable workflow for attractor analysis in complex networks. The results suggest that reduction rarely introduces many nonminimal or nonunivocal attractors and can markedly speed up analysis, with potential extensions to broader dynamical-system tasks in systems biology and network design.

Abstract

Identification of attractors, that is, stable states and sustained oscillations, is an important step in the analysis of Boolean models and exploration of potential variants. We describe an approach to the search for asynchronous cyclic attractors of Boolean networks that exploits, in a novel way, the established technique of elimination of components. Computation of attractors of simplified networks allows the identification of a limited number of candidate attractor states, which are then screened with techniques of reachability analysis combined with trap space computation. An implementation that brings together recently developed Boolean network analysis tools, tested on biological models and random benchmark networks, shows the potential to significantly reduce running times.

Attractor identification in asynchronous Boolean dynamics with network reduction

TL;DR

This work addresses the computational challenge of identifying attractors in asynchronous Boolean networks by introducing a reduction-driven pipeline. It systematically eliminates non-autoregulated components to form a smaller network, computes minimal trap spaces, and samples attractors from the reduced system to form candidate states that are lifted back to the original state space via a lifting map . These candidates are screened first through trap-space containment and then via reachability/model-checking to confirm true attractors, yielding significant runtime reductions in experiments on biological models and random benchmarks. The method is implemented by integrating Colomoto tooling (minibnnaldi, trappist) with AEON and mtsNFVS, providing a practical, scalable workflow for attractor analysis in complex networks. The results suggest that reduction rarely introduces many nonminimal or nonunivocal attractors and can markedly speed up analysis, with potential extensions to broader dynamical-system tasks in systems biology and network design.

Abstract

Identification of attractors, that is, stable states and sustained oscillations, is an important step in the analysis of Boolean models and exploration of potential variants. We describe an approach to the search for asynchronous cyclic attractors of Boolean networks that exploits, in a novel way, the established technique of elimination of components. Computation of attractors of simplified networks allows the identification of a limited number of candidate attractor states, which are then screened with techniques of reachability analysis combined with trap space computation. An implementation that brings together recently developed Boolean network analysis tools, tested on biological models and random benchmark networks, shows the potential to significantly reduce running times.
Paper Structure (12 sections, 1 theorem, 6 equations, 4 figures, 5 tables)

This paper contains 12 sections, 1 theorem, 6 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Boolean networks
  4. Reduction
  5. Method
  6. Implementation
  7. Software
  8. Elimination order and other considerations
  9. Results
  10. Biological models
  11. Random benchmarks
  12. Discussion

Key Result

Theorem 2.2

If $\mathcal{A}$ is an attractor of $f$, then there exists at least one attractor for $\tilde{f}$ in $\pi(\mathcal{A})$, and for each $x \in \pi(\mathcal{A})$ contained in an attractor of $\tilde{f}$, $\mathcal{S}^n(x) \in \mathcal{A}$.

Figures (4)

  • Figure 1: Asynchronous state transition graphs of the Boolean networks $f$, $g$ and $h$ of \ref{['ex:fgh']}. $\Gamma(f)$ has one minimal univocal attractor, $\Gamma(g)$ has two minimal nonunivocal attractors, and $\Gamma(h)$ has a minimal univocal and a nonminimal attractor.
  • Figure 2: Idea behind the approach: states in attractors of reduced networks $\tilde{f}$ can be used to find candidate states in attractors of $f$.
  • Figure 3: Asynchronous state transition graphs of the Boolean networks $\hat{f}$, $\hat{g}$, $\hat{h}$ of \ref{['ex:attractors-can-split', 'ex:attractors-can-increase', 'ex:not-all-states']}.
  • Figure 4: Main steps of the algorithm.

Theorems & Definitions (7)

  • Example 2.1
  • Theorem 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6