Qualitative Study Of Boolean Networks Under Stochastic Function Perturbation

With the rapid development of systematic biology and the improvement of information technology,the research of gene regulatory network and its mathematical model has been continuously promoted.As a kind of discrete-time dynamic logic network,the state of each node in Boolean network can only take two values of 0 or 1,and the update of node state depends on the corresponding logic function.Boolean network can not only effectively analyze and simulate the structure and relationship of genes,but also play an important role in information science,automatic control,game theory and other fields.Because of its simple expression form,abstract description of the actual system,and good simulation of nonlinear dynamics in the process of network control,it has attracted the attention of many researchers.In the process of gene expression,gene mutation is a common phenomenon,which can be described by the concept of function perturbation in mathematics.In this thesis,qualitative problems of Boolean networks under stochastic function perturbation are studied in the framework of semi-tensor product theory of matrices.As is known to all,the qualitative problem includes two parts:stability and synchronization.In this thesis,the two aspects of the content are included in the following analysis and research.In the aspect of stability,the robust stability in distribution of Boolean networks under multi-bits probabilistic and markovian function perturbation is studied.Compared with the deterministic case,the stochastic case considered in this thesis is more general and complex.Firstly,definition of multi-bits stochastic function perturbation is given and an identification matrix is introduced to present each case.Then,Boolean networks under multi-bits stochastic function perturbation can be equivalently converted into stochastic switching systems.After constructing respective transition probability matrices which can unify multi-bits probabilistic and markovian function perturbation in a consolidated framework,robust stability in distribution can be handled.On such basis,necessary and sufficient conditions for robust stability in distribution of Boolean networks under stochastic function perturbation are given respectively.Finally,two numerical examples are presented to verify the validity of theoretical results.In the aspect of synchronization,the cluster synchronization problem of Boolean networks and Boolean control networks under probabilistic function perturbation is studied.By introducing the concept of cluster set,the definition of cluster synchronization is given.Algebraic expression of Boolean networks under probabilistic function perturbation is derived by transforming it into an equivalent stochastic logical system.Then,a necessary and sufficient condition is provided to distinguish whether Boolean networks can achieve cluster synchronization with probability one under probabilistic function perturbation.Furthermore,cluster synchronization of Boolean control networks is discussed by designing corresponding state feedback controllers.Validity of results is demonstrated by subsequent numerical examples.