Reduction and Analysis of Boolean Control Networks by Bisimulation

The paper deals with problems on the stability and stabilization of Boolean networks with respect to a given part of variables characterizing the Boolean network. Using the semi-tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean (control) network can be converted to discrete time linear (bilinear) dynamics called the algebraic form of the Boolean (control) network. The two main results are as follows: (i) From the algebraic form of a Boolean (control) network, two partial stability definitions are introduced. One is partial stability with respect to another variable set, and the other is partial stability with respect to arbitrary initial states. (ii) Based on the algebraic form, necessary and sufficient conditions for the stability of global partial states with respect to the variable set and arbitrary initial states are obtained. Examples are provided to illustrate the effectiveness of our results.

Nominal Boolean network is a kind of Boolean network, which is obtained from a Boolean control network with all controls disconnected. In this paper, the relationship between the transition matrix of the original Boolean control network and the transition matrix of the corresponding nominal Boolean control network is investigated. Due to the fact that transition matrices can be derived from structure matrices, the relationship between structure matrices of the original Boolean control network and structure matrices of the nominal Boolean network is studied. Via constant values corresponding to disconnected controls, structure matrices of the nominal Boolean network can be derived from structure matrices of the original Boolean control network. Finally, a concrete algorithm is presented to derive transition matrices of nominal Boolean networks.

Controllability and observability of Boolean control networks subject to stuck-at fault

Serval existing definitions for observability of Boolean control networks (BCNs) are revisited and a new definition termed observability via output feedback is proposed. This definition restricts the input sequences used to distinguish distinct initial states to be generated by output feedbacks. A technique called parallel extension is proposed for observability analysis, the main idea of which is to interconnect a BCN with its duplication in parallel and transfer the observability problem of the original BCN to the set reachability problem of the interconnected BCN. Novel necessary and sufficient conditions for observability and observability under arbitrary inputs have been given, respectively. In addition, by restricting the output feedback to be static, a sufficient condition for observability via output feedback is proposed. An example is given to illustrate the main results.

On the ensemble controllability of Boolean control networks using STP method

The stability of Boolean networks and the stabilization of Boolean control networks are investigated. Using semi-tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean (control) network can be converted to a discrete time linear (bilinear) dynamics, called the algebraic form of the Boolean (control) network. They provide a framework for this study. Main results consist of two parts: (i) Using logic coordinate transformation, the known sufficient condition based on incidence matrix has been improved. It is also used in stabilizer design. (ii)Based on algebraic form, necessary and sufficient conditions for stability of Boolean networks and stabilization of Boolean control networks respectively are obtained.

Disturbance decoupling of Boolean networks via robust indistinguishability method

We show some new results on the observability of Boolean control networks (BCNs). First, to study the observability, we combine two BCNs with the same transition matrix into a new BCN. Then, we propose the concept of a reachable set that results in a given set of initial states, and we derive four additional necessary and sufficient conditions for the observability of BCNs. In addition, we present an algoriTheoRemark and construct an observability graph to determine the observability of BCNs. Finally, we illustrate the obtained results using three numerical examples.

Minimum energy control and optimal-satisfactory control of Boolean control network

In Fauré etal. (2006), the dynamics of the core network regulating the mammalian cell cycle is formulated as a Boolean control network (BCN) model consisting of nine proteins as state nodes and a tenth protein (protein CycD) as the control input node. In this model, one of the state nodes, protein Cdc20, plays a central role in the separation of sister chromatids. Hence, if any Cdc20 sequence can be obtained, fully controlling the mammalian cell cycle is feasible. Motivated by this fact, we study whether any Cdc20 sequence can be obtained theoretically. We formulate the foregoing problem as the invertibility of BCNs, that is, whether one can obtain any Cdc20 sequence by designing input (i.e., protein CycD) sequences. We give an algorithm to verify the invertibility of any BCN, and find that the BCN model for the core network regulating the mammalian cell cycle is not invertible, that is, one cannot obtain any Cdc20 sequence. We further present another algorithm to test whether a finite Cdc20 sequence can be generated by the BCN model, which leads to a series of periodic infinite Cdc20 sequences with alternately active and inactive Cdc20 segments. States of these sequences are alternated between the two attractors in the proposed model, which reproduces correctly how a cell exits the cell cycle to enter the quiescent state, or the opposite.

This paper mainly investigates the reachability of a kind of Boolean control networks with delays in both state and control. By a system transformation, the reachability problem of the delayed Boolean control network becomes that of a Boolean control network without delays. The relationship between the transformed network and the original network is revealed. A necessary and sufficient condition for the reachability of the delayed Boolean control network is obtained. An example is given to illustrate the effectiveness of the obtained theoretical result.

Input observability of Boolean control networks

In this article, the controllability of asynchronous Boolean control networks (ABCNs) with time-invariant delay is studied. Using semi-tensor product, Boolean control networks under Harvey’s asynchronous update with time delay are converted into linear representation, and a general formula of control-depending network transition matrices is achieved. Firstly, a necessary and sufficient condition is proposed to verify that only control-depending fixed points of ABCNs can be deterministically controlled. Secondly, respectively for three kinds of controls, the controllability of the non-deterministic logical dynamics is investigated, and formulas are given to show: a) the reachable sets at time s under a group of specified controls; b) the reachable sets at time s under arbitrary controls and c) the probability values from a given initial state to different target states. Moreover, we also discuss how to find the particular control sequence to reach a specified target state with the maximum probability. Examples are shown to illustrate the feasibility of the proposed scheme.

Boolean control networks have recently been attracting considerable interests as computational models for genetic regulatory networks. In this paper, we present an approach of impulsive control for attractor transitions in Boolean control networks based on the recent developed matrix semitensor product theory. The reachability of attractors is estimated, and the controller is also obtained. The general derivation proposed here is exemplified with a kind of gene model, which is the protein-nucleic acid interactions network, on numerical simulations.

In this article, the controllability of asynchronous Boolean multiplex control networks (ABMCNs) is studied. First, the model of Boolean multiplex control networks under Harvey' asynchronous update is presented. By means of semi-tensor product approach, the logical dynamics is converted into linear representation, and a generalized formula of control-depending network transition matrices is achieved. Second, a necessary and sufficient condition is proposed to verify that only control-depending fixed points of ABMCNs can be controlled with probability one. Third, using two types of controls, the controllability of system is studied and formulae are given to show: (a) when an initial state is given, the reachable set at time s under a group of specified controls; (b) the reachable set at time s under arbitrary controls; (c) the specific probability values from a given initial state to destination states. Based on the above formulae, an algorithm to calculate overall reachable states from a specified initial state is presented. Moreover, we also discuss an approach to find the particular control sequence which steers the system between two states with maximum probability. Examples are shown to illustrate the feasibility of the proposed scheme.