Negative circuits and sustained oscillations in asynchronous automata networks
Discrete Mathematics
2009-07-30 v1
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 of finite intervals of integers, and a map from to itself, if the interaction graph associated with has no negative circuit, then has at least one fixed point.
Keywords
Cite
@article{arxiv.0907.5096,
title = {Negative circuits and sustained oscillations in asynchronous automata networks},
author = {Adrien Richard},
journal= {arXiv preprint arXiv:0907.5096},
year = {2009}
}