Asymptotical Stability and Stabilization of Continuous-Time Probabilistic Logic Networks

Discrete-time probabilistic logic networks (DT-PLNs), of which probabilistic Boolean networks (PBNs) are a special type, are an important qualitative model for gene regulatory networks (GRNs). Although a DT-PLN can predict the long-term behavior of a GRN, using it to describe the transient kinetics at the microtimescale level remains inconvenient. In this article, we investigate the problems associated with the stability and stabilization of continuous-time probabilistic logic networks (CT-PLNs). First, we demonstrate that the concept of finite-time stability for DT-PLNs cannot be extended to CT-PLNs owing to the nonsingularity of transitional probability matrices. Thus, we introduce the concept of asymptotical stability, which is defined as the convergence in distribution of the network state. Second, by developing the theory of invariant subsets for CT-PLNs, a necessary and sufficient condition for asymptotical stability with respect to a subset is proposed, which is expressed in terms of the transition rate matrix of probability. Third, for a CT-PLN with input nodes, termed a continuous-time probabilistic logic control network, we discuss the subsets that are invariant under piece-wise constant input. Based thereupon, we propose a necessary and sufficient condition under which asymptotically stabilizing sampled-data feedback exists. A method for designing sampled-data feedback is proposed. Last, Monte Carlo simulation algorithms are proposed to efficiently simulate a CT-PLN in the time domain. Examples are provided to demonstrate the methods proposed in this article.