Negative circuits and sustained oscillations in asynchronous automata networks

2009/07/29 by Adrien Richard, Richard, Adrien
Biochemistry, Genetics and Molecular Biology · Computer Science · #Cellular Automata and Applications #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Gene Regulatory Network Analysis #Single-cell and spatial transcriptomics

paper · pdf · doi:10.48550/arxiv.0907.5096

openalex publication_date 2009/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The biologist René 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.

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