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Bringing memory to Boolean networks: a unifying framework

Maximilien Gadouleau, Loïc Paulevé, Sara Riva

TL;DR

This work develops a unifying framework to incorporate memory into Boolean networks by formalizing memory-based update modes that govern trajectory evolution. It systematically defines and interrelates seven modes—asynchronous $\mathtt{A}$, history-based $\mathtt{H}$, trapping $\mathtt{T}$, most permissive $\mathtt{M}$, subcube-based $\mathtt{S}$, interval $\mathtt{I}$, and cuttable $\mathtt{C}$—and characterizes their reachability, trajectories, and memory structures. A central contribution is linking these dynamics to trapspaces and principal trapspaces, establishing a three-way equivalence between trapping networks, trapspace collections, and principal trapspace collections, and introducing trapping closure $f^{\mathrm{T}}$ and min-trapping closure $f^{\mathrm{M}}$. The results show that trapping dynamics form transitive asynchronous graphs and that most-permissive and trapping updates coincide on minimal trapspaces, while exhibiting exponential differences elsewhere, providing new tools for modeling delayed and memory-driven regulation in complex systems. Overall, the framework advances the theory of BN dynamics with memory, connects to commutative networks, and offers a combinatorial-algebraic lens through which to analyze attractors and trapspaces with potential applications in biology and beyond.

Abstract

Boolean networks are extensively applied as models of complex dynamical systems, aiming at capturing essential features related to causality and synchronicity of the state changes of components along time. Dynamics of Boolean networks result from the application of their Boolean map according to a so-called update mode, specifying the possible transitions between network configurations. In this paper, we explore update modes that possess a memory on past configurations, and provide a generic framework to define them. We show that recently introduced modes such as the most permissive and interval modes can be naturally expressed in this framework. We propose novel update modes, the history-based and trapping modes, and provide a comprehensive comparison between them. Furthermore, we show that trapping dynamics, which further generalize the most permissive mode, correspond to a rich class of networks related to transitive dynamics and encompassing commutative networks. Finally, we provide a thorough characterization of the structure of minimal and principal trapspaces, bringing a combinatorial and algebraic understanding of these objects.

Bringing memory to Boolean networks: a unifying framework

TL;DR

This work develops a unifying framework to incorporate memory into Boolean networks by formalizing memory-based update modes that govern trajectory evolution. It systematically defines and interrelates seven modes—asynchronous , history-based , trapping , most permissive , subcube-based , interval , and cuttable —and characterizes their reachability, trajectories, and memory structures. A central contribution is linking these dynamics to trapspaces and principal trapspaces, establishing a three-way equivalence between trapping networks, trapspace collections, and principal trapspace collections, and introducing trapping closure and min-trapping closure . The results show that trapping dynamics form transitive asynchronous graphs and that most-permissive and trapping updates coincide on minimal trapspaces, while exhibiting exponential differences elsewhere, providing new tools for modeling delayed and memory-driven regulation in complex systems. Overall, the framework advances the theory of BN dynamics with memory, connects to commutative networks, and offers a combinatorial-algebraic lens through which to analyze attractors and trapspaces with potential applications in biology and beyond.

Abstract

Boolean networks are extensively applied as models of complex dynamical systems, aiming at capturing essential features related to causality and synchronicity of the state changes of components along time. Dynamics of Boolean networks result from the application of their Boolean map according to a so-called update mode, specifying the possible transitions between network configurations. In this paper, we explore update modes that possess a memory on past configurations, and provide a generic framework to define them. We show that recently introduced modes such as the most permissive and interval modes can be naturally expressed in this framework. We propose novel update modes, the history-based and trapping modes, and provide a comprehensive comparison between them. Furthermore, we show that trapping dynamics, which further generalize the most permissive mode, correspond to a rich class of networks related to transitive dynamics and encompassing commutative networks. Finally, we provide a thorough characterization of the structure of minimal and principal trapspaces, bringing a combinatorial and algebraic understanding of these objects.
Paper Structure (27 sections, 24 theorems, 55 equations, 3 figures)

This paper contains 27 sections, 24 theorems, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Update modes for dynamics with memory
  3. Asynchronous updates
  4. Asynchronous updates $\mathtt{A}$:
  5. History-based updates
  6. History-based updates $\mathtt{H}$:
  7. Trapping updates
  8. Trapping updates $\mathtt{T}$:
  9. Most permissive updates
  10. Most permissive updates $\mathtt{M}$:
  11. Subcube-based updates
  12. Subcube-based updates $\mathtt{S}$:
  13. Interval updates
  14. Interval updates $\mathtt{I}$:
  15. Cuttable updates
  16. ...and 12 more sections

Key Result

Theorem 3.1

The hierarchy by reachability and trajectory for the update modes listed in Section section:update_modes is given in Figure figure:hierarchy.

Figures (3)

  • Figure 1: Hierarchy of different update modes.
  • Figure 2: Possible trajectories of certain BNs according to different update modes.
  • Figure 3: Example of a history-based and cuttable trajectory that is not interval. The example is also valid concerning reachability.

Theorems & Definitions (61)

  • Example 1.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Theorem 3.1
  • Example 3.2: An interval trajectory that is not trapping
  • ...and 51 more