Negative local feedbacks in Boolean networks

TL;DR

This paper investigates Boolean networks lacking local negative cycles, revealing they can have no fixed points and may possess antipodal attractive cycles, challenging previous assumptions about their dynamical properties.

Contribution

It provides the first negative answers to whether such networks always have fixed points or only fixed point attractors, expanding understanding of their dynamics.

Findings

Networks without local negative cycles can lack fixed points.

Such networks can have antipodal attractive cycles.

These results challenge previous assumptions about network stability.

Abstract

We study the asymptotic dynamical properties of Boolean networks without local negative cycle. While the properties of Boolean networks without local cycle or without local positive cycle are rather well understood, recent literature raises the following two questions about networks without local negative cycle. Do they have at least one fixed point? Should all their attractors be fixed points? The two main results of this paper are negative answers to both questions: we show that and-nets without local negative cycle may have no fixed point, and that Boolean networks without local negative cycle may have antipodal attractive cycles.