Synchronization And Controllability Analysis Of Coupled Boolean Networks

Since Boolean networks can be well used to model genetic regulatory networks and biological systems, Boolean model has been a newly developing research topic. Synchronization and control-lability, which have attracted much attention from many researchers, are fundamental concepts in control theory. Investigations on these topics are not only of great theoretical values but also of widely practical significance. This thesis firstly investigates synchronization of coupled Boolean net-works, and then discusses controllability and reachability of Boolean control networks with pinning controllers. The main work of this thesis is listed as follows:In the first chapter, the concepts of synchronization and controllability in Boolean networks are briefed.The second chapter briefly present definition and main properties of semi-tensor product of matrices, matrix presentations of logics, Boolean network and its algebraic representations are briefed.The third chapter presents an analytical study of synchronization in an array of output-coupled Boolean networks with time delay. Two kinds of models are considered. For the first model, the outputs contain time delay, while in the second one the outputs do not contain time delay. One restriction in this model is that the state delay and output delay are restricted to be equal to each other. By referring to the algebraic representations of logical dynamics and semi-tensor product, some necessary and sufficient conditions are derived for synchronization of delay-coupled Boolean networks. Examples including a practical epigenetic example are given for illustration.The forth chapter discusses synchronization problem in an array of output-coupled temporal Boolean networks. A temporal Boolean network is a logical dynamic system developed to model Boolean networks with regulatory delays. Both state delay and output delay are considered, and these two delays are assumed to be different. By referring to the algebraic representations of logical dynamics and semi-tensor product, output-coupled temporal Boolean networks are firstly converted into a discrete-time algebraic evolution system, and then the relationship between states of coupled temporal Boolean networks and initial state sequence is obtained. Then, some necessary and sufficient conditions are derived for synchronization of an array of temporal Boolean networks with an arbitrary given initial state sequence. Two numerical examples including one epigenetic model are finally given to illustrate the obtained results.The fifth chapter investigates synchronization of master-slave Boolean networks with impulsive effects. Necessary and sufficient conditions on synchronization for master-slave Boolean networks with impulsive effects are derived by converting the logical systems into algebraic expression with the method of semi-tensor product. Some convenient conditions are proposed to judge whether a Boolean network with impulsive effects can be synchronized or not.The sixth chapter presents analytical investigations on reachability and controllability of Boolean control networks with pinning controllers. Based on semi-tensor product and matrix expression of logic, Boolean control networks with pinning controllers are converted into a discrete-time algebraic system. Then, a formula is derived to calculate the number of different control sequences steering Boolean control networks between two states in a given step. According to this formula, several necessary and sufficient criteria for reachability and controllability of Boolean control networks with pinning controllers are derived. Moreover, we also make a comparison between three forms of Boolean control networks, which have similar algebraic representations. Finally, we obtain some efficient conditions to judge the dynamic structure of Boolean control networks.Finally, the seventh chapter summarizes the research work of this dissertation. Moreover, the possible improved theories are proposed and the prospect for the work is made.