Negative circuits and sustained oscillations in asynchronous automata networks

作者：Richard Adrien^*

来源：Advances in Applied Mathematics, 2010, 44(4): 378-392.

DOI：10.1016/j.aam.2009.11.011

• Discrete dynamical system
• Automata network
• Boolean network
• Genetic regulatory network
• Interaction graph
• Discrete Jacobian matrix
• Feedback circuit
• Negative circuit
• Oscillation
• Fixed point

The biologist Rene 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.

• 出版日期2010-5