Non-oscillation of asynchronous stochastic logical networks through negative circuits of the joint interaction graph

Oscillation behaviors not only exert a notable impact on the stability and functionality of nonlinear systems, but also play an important role in a wide range of application fields. This paper explores the non-oscillation of asynchronous stochastic logical networks (ASLNs) through negative circuits of the joint interaction graph. First, the joint interaction graph and local interaction graph of ASLNs are established, and some basic properties of them are revealed. Based on these properties, a feasible sufficient condition is presented to verify the non-oscillation of ASLNs via the existence of negative circuits in the joint interaction graph. Second, when the joint interaction graph of ASLNs has no negative circuits, the node removal technique is introduced to find all the fixed points of ASLNs, and a new criterion is built for the global set stability in distribution of ASLNs. Finally, the effectiveness of the main results is demonstrated by using the ASLN model in the processing of Toll and EGF pathways involved in Drosophila development.