Limit cycles in piecewise-affine gene network models with multiple interaction loops

TL;DR

This paper analyzes complex gene network models using piecewise affine differential equations, establishing conditions for the existence of stable periodic solutions or global attraction, extending previous work on simpler feedback loops.

Contribution

It provides an alternative theorem for complex gene networks with multiple loops, generalizing prior results on single negative feedback loops.

Findings

Existence of stable periodic solutions under certain conditions

Trajectories are either periodic or attracted to the origin

Applicable to networks with multiple intricate interaction loops

Abstract

In this paper we consider piecewise affine differential equations modeling gene networks. We work with arbitrary decay rates, and under a local hypothesis expressed as an alignment condition of successive focal points. The interaction graph of the system may be rather complex (multiple intricate loops of any sign, multiple thresholds...). Our main result is an alternative theorem showing that, if a sequence of region is periodically visited by trajectories, then under our hypotheses, there exists either a unique stable periodic solution, or the origin attracts all trajectories in this sequence of regions. This result extends greatly our previous work on a single negative feedback loop. We give several examples and simulations illustrating different cases.