Negative circuits and sustained oscillations in asynchronous automata networks

TL;DR

This paper proves René Thomas's conjecture that negative feedback circuits are necessary for sustained oscillations in asynchronous automata networks, a key model for gene network behavior, and establishes a fixed point theorem related to the interaction graph.

Contribution

It formally proves the conjecture for asynchronous automata networks and derives a fixed point theorem linking negative circuits to the existence of fixed points.

Findings

Negative circuits are necessary for sustained oscillations.

If no negative circuit exists, a fixed point is guaranteed.

The results apply to gene network models.

Abstract

The biologist Ren\'e Thomas conjectured, twenty years ago, that the presence of a negative feedback circuit in the interaction graph of a dynamical system is a necessary condition for this system to produce sustained oscillations. In this paper, we state and prove this conjecture for asynchronous automata networks, a class of discrete dynamical systems extensively used to model the behaviors of gene networks. As a corollary, we obtain the following fixed point theorem: given a product $X$ of $n$ finite intervals of integers, and a map $F$ from $X$ to itself, if the interaction graph associated with $F$ has no negative circuit, then $F$ has at least one fixed point.