Creation of fixed points in block-parallel Boolean automata networks

TL;DR

This paper investigates how block-parallel update schedules in Boolean automata networks can create new fixed points, especially in positive cycles, revealing exponential growth in fixed points and challenging previous invariance results.

Contribution

It demonstrates that block-parallel schedules can generate exponentially many fixed points in simple feedback structures, extending understanding of fixed point dynamics.

Findings

Block-parallel schedules can produce exponentially many fixed points.

Positive cycles' fixed point invariance can be broken by certain update schedules.

Numerical quantification of fixed point creation in elementary feedback structures.

Abstract

In the context of discrete dynamical systems and their applications, fixed points often have a clear interpretation. This is indeed a central topic of gene regulatory mechanisms modeled by Boolean automata networks (BANs), where a collection of Boolean entities (the automata) update their state depending on the states of others. Fixed points represent phenotypes such as differentiated cell types. The interaction graph of a BAN captures the architecture of dependencies among its automata. A first seminal result is that cycles of interactions (so called feedbacks) are the engines of dynamical complexity. A second seminal result is that fixed points are invariant under block-sequential update schedules, which update the automata following an ordered partition of the set of automata. In this article we study the ability of block-parallel update schedules (dual to the latter) to break this fixed point invariance property, with a focus on the simplest feedback mechanism: the canonical positive cycle. We quantify numerically the creation of new fixed points, and provide families of block-parallel update schedules generating exponentially many fixed points on this elementary structure of interaction.