Attractor-Based Simultaneous Design of the Minimum Set of Control Nodes and Controllers in Boolean Networks

Design of control strategies for gene regulatory networks is a challenging and important topic in systems biology. In this paper, the problem of finding both a minimum set of control nodes (control inputs) and a controller is studied. A control node corresponds to a gene that expression can be controlled. Here, a Boolean network is used as a model of gene regulatory networks, and control specifications on attractors, which represent cell types or states of cells, are imposed. It is important to design a gene regulatory network that has desired attractors and has no undesired attractors. Using a matrix-based representation of BNs, this problem can be rewritten as an integer linear programming problem. Finally, the proposed method is demonstrated by a numerical example on a WNT5A network, which is related to melanoma.

Verification of Real-Time Pricing Systems Based on Probabilistic Boolean Networks

In this paper, verification of real-time pricing systems of electricity is considered using a probabilistic Boolean network (PBN). In real-time pricing systems, electricity conservation is achieved by manipulating the electricity price at each time. A PBN is widely used as a model of complex systems, and is appropriate as a model of real-time pricing systems. Using the PBN-based model, real-time pricing systems can be quantitatively analyzed. In this paper, we propose a verification method of real-time pricing systems using the PBN-based model and the probabilistic model checker PRISM. First, the PBN-based model is derived. Next, the reachability problem, which is one of the typical verification problems, is formulated, and a solution method is derived. Finally, the effectiveness of the proposed method is presented by a numerical example.

Control of random Boolean networks via average sensitivity of Boolean functions

In this paper, we discuss how to transform the disordered phase into an ordered phase in random Boolean networks. To increase the effectiveness, a control scheme is proposed, which periodically freezes a fraction of the network based on the average sensitivity of Boolean functions of the nodes. Theoretical analysis is carried out to estimate the expected critical value of the fraction, and shows that the critical value is reduced using this scheme compared to that of randomly freezing a fraction of the nodes. Finally, the simulation is given for illustrating the effectiveness of the proposed method.

Self-Triggered Set Stabilization of Boolean Control Networks and Its Applications

Set stabilization is one of the essential problems in engineering systems, and self-triggered control(STC) can save the storage space for interactive information, and can be successfully applied in networked control systems with limited communication resources. In this study, the set stabilization problem and STC design of Boolean control networks are investigated via the semi-tensor product technique. On the one hand, the largest control invariant subset is calculated in terms of the strongly connected components of the state transition graph, by which a graph-theoretical condition for set stabilization is derived. On the other hand, a characteristic function is exploited to determine the triggering mechanism and feasible controls. Based on this, the minimum-time and minimum-triggering open-loop, state-feedback and output-feedback STCs for set stabilization are designed,respectively. As classic applications of self-triggered set stabilization, self-triggered synchronization, self-triggered output tracking and self-triggered output regulation are discussed as well. Additionally, several practical examples are given to illustrate the effectiveness of theoretical results.

SOLVABILITY AND CONTROL DESIGN FOR SYNCHRONIZATION OF BOOLEAN NETWORKS

This paper addresses the synchronization problem of Boolean networks.Based on the matrix expression of logic,solvability conditions and design procedures of the synchronization of Boolean networks with outputs are given for both open-loop and feedback control.Necessary and sufficient conditions on open-loop control are proposed first with a constructive design procedure.Then sufficient condition for the feedback control case is obtained,and corresponding design procedure is proposed with the help of algorithms to solve logic matrix equations.Numerical examples are also provided to illustrate the proposed control design.

Set Stability of Probabilistic Time-Delay Boolean Networks with Impulsive Effect

This paper investigates the set stability of probabilistic time-delay Boolean networks(PTDBN)with impulsive effect.Firstly,using the algebraic state space representation,an equivalent stochastic system is established for PTDBN with impulsive effect.Then,based on the probabilistic state transition matrix,a necessary and sufficient condition is presented for the set stability of PTDBN with impulsive effect.Finally,the obtained new result is applied to the networked evolutionary game with memories.

ON THE OBSERVABILITY OF FREE BOOLEAN NETWORKS VIA THE SEMI-TENSOR PRODUCT METHOD

This paper investigates the observabihty of free Boolean networks by using the semi-tensor product method,and presents some new results.First,the concept of observability for free Boolean networks is proposed,based on which and the algebraic form of Boolean networks,a kind of observabihty matrix is constructed.Second,by the observability matrix,a new necessary and sufficient condition is given for the observability of Boolean networks.Third,the concept of observabihty index for observable Boolean networks is defined,and an algorithm is established to calculate the observability index.Finally,a practical example of D.Melanogaster segmentation polarity gene networks is studied to support our new results.The study of the illustrative example shows that the new results obtained in this paper are very effective in investigating the observability of free Boolean networks.

Recent developments in Boolean networks control

The aim of this survey paper is to provide the state of the art of the research on control and optimal control of Boolean control networks,under the assumption that all the state variables are accessible and hence available for feedback.Necessary and sufficient conditions for stabilisability to a limit cycle or to an equilibrium point are given.Additionally,it is shown that when such conditions are satisfied,stabilisation can always be achieved by means of state feedback.Analogous results are obtained for the safe control problem that is investigated for the first time in this survey.Finite and infinite horizon optimal control are subsequently considered,and solution algorithms are provided,based on suitable adaptations of theRiccati difference and algebraic equations.Finally,an appropriate definition of the cost function allows to restate and to solve both stabilisation and safe control as infinite horizon optimal control problems.

Synchronization of switched Boolean networks with impulsive effects

This paper studies the state/output synchronization of switched Boolean networks （SBNs） with impulsive effects via the algebraic state space representation （ASSR） approach. First, an algebraic form is established for SBNs with impulsive effects via ASSR. Second, based on the algebraic form, some necessary and sufficient conditions are presented for the state/output synchronization of SBNs with impulsive effects under arbitrary switching signals. Third, two special kinds of switching signals, that is, free switching signal and feedback switching signal, are considered for the state synchroniza-tion of SBNs with impulsive effects. Finally, two illustrative examples are worked out to show the effectiveness of the obtained results.

Stability of Boolean networks with state-dependent random impulses

We investigate the stability of Boolean networks(BNs)with impulses triggered by both states and random factors.A hybrid index model is used to describe impulsive BNs.First,several necessary and sufficient conditions for forward completeness are obtained.Second,based on the stability criterion of probabilistic BNs and the forward completeness criterion,the necessary and sufficient conditions for the finite-time stability with probability one and the asymptotical stability in distribution are presented.The relationship between these two kinds of stability is discussed.Last,examples and time-domain simulations are provided to illustrate the obtained results.

Robust network structures for conserving total activity in Boolean networks

One of the typical properties of biological systems is the law o f conservation o f mass,that is,the property that the mass must remain constant over time in a closed chemical reaction system.However,it is known that Boolean networks,which are a promising model of biological networks,do not always represent the conservation law.This paper thus addresses a kind of conservation law as a generic property of Boolean networks.In particular,we consider the problem of finding network structures on which,for any Boolean operation on nodes,the number of active nodes,i.ev nodes whose state is one,is constant over time.As a solution to the problem,we focus on the strongly-connected network structures and present a necessary and sufficient condition.

Physical generation of random numbers using an asymmetrical Boolean network

Autonomous Boolean networks(ABNs)have been successfully applied to the generation of random number due to their complex nonlinear dynamics and convenient on-chip integration.Most of the ABNs used for random number generators show a symmetric topology,despite their oscillations dependent on the inconsistency of time delays along links.To address this issue,we suggest an asymmetrical autonomous Boolean network(aABN)and show numerically that it provides large amplitude oscillations by using equal time delays along links and the same logical gates.Experimental results show that the chaotic features of aABN are comparable to those of symmetric ABNs despite their being made of fewer nodes.Finally,we put forward a random number generator based on aABN and show that it generates the random numbers passing the NIST test suite at 100 Mbits/s.The unpredictability of the random numbers is analyzed by restarting the random number generator repeatedly.The aABN may replace symmetrical ABNs in many applications using fewer nodes and,in turn,reducing power consumption.

Observability of Periodically Switched Boolean Control Networks

In this paper,observability is studied for periodically switched Boolean control networks(PSBCNs),which are managed with periodic switching signal and consist of some Boolean control networks.Firstly,via semi-tensor product of matrices,PSBCNs are expressed as algebraic forms.Secondly,a parallel system is constructed by combining two same PSBCNs,based on which,the observability problem of the original PSBCN can be transformed into the set reachability problem of this parallel system.Then,two necessary and sufficient conditions are obtained to detect reachability of parallel systems and observability of PSBCNs.In addition,the proposed conditions are extended to the case of state constraints.Finally,a practical example and a numerical example are provided to illustrate the results.

Asurvey onob vabilit of Boolean control networks

Observability is a fundamental property of a partially observed dynamical system,which means whether one can use an input sequence and the corresponding output sequence to determine the initial state.Observability provides bases for many related problems,such as state estimation,identification,disturbance decoupling,controller synthesis,etc.Until now,fundamental improvement has been obtained in observability of Boolean control networks(BCNs)mainly based on two methods-Edward F.Moore's partition and our observability graph or their equivalent representations found later based on the semitensor product(STP)of matrices(where the STP was proposed by Daizhan Cheng),including necessary and sufficient conditions for different types of observability,extensions to probabilistic Boolean networks(PBNs)and singular BCNs,even to nondeterministic finite-transition systems(NFTSs);and the development(with the help of the STP of matrices)in related topics,such as com-putation of smallest invariant dual subspaces of BNs containing a set of Boolean functions,multiple-experiment observability verification/decomposition in BCNs,disturbance decoupling in BCNs,etc.This paper provides a thorough survey for these topics.The contents of the paper are guided by the above two methods.First,we show that Moore's partition-based method closely relates the following problems:computation of smallest invariant dual subspaces of BNs,multiple-experiment observ-ability verification/decomposition in BCNs,and disturbance decoupling in BCNs.However,this method does not apply to other types of observability or nondeterministic systems.Second,we show that based on our observability graph,four different types of observability have been verified in BCNs,verification results have also been extended to PBNs,singular BCNs,and NFTSs.In addition,Moore's partition also shows similarities between BCNs and linear time-invariant(LTI)control systems,e.g.,smallest invariant dual subspaces of BNs containing a set of Boolean functions in BCNs vs unobservable subspaces of LTI control systems,the forms of quotient systems based on observability decomposition in both types of systems.However,there are essential differences between the two types of systems,e.g.,"all plausible definitions of observability in LTI control systems turn out to be equivalent"(by Walter M.Wonham 1985),but there exist nonequivalent definitions of observability in BCNs;the quotient system based on observability decomposition always exists in an LTI control system,while a quotient system based on multiple-experiment observability decomposition does not always exist in a BCN.

Controllability of Boolean control networks with multiple time delays in both states and controls

In this paper,the problem of controllability of Boolean control networks(BCNs)with multiple time delays in both states and controls is investigated.First,the controllability problem of BCNs with multiple time delays in controls is considered.For this controllability problem,a controllability matrix is constructed by defining a new product of matrices,based on which a necessary and sufficient controllability condition is obtained.Then,the controllability of BCNs with multiple time delays in states is studied by giving a necessary and sufficient condition.Subsequently,based on these results,a controllability matrix for BCNs with multiple time delays in both states and controls is proposed that provides a concise controllability condition.Finally,two examples are given to illustrate the main results.

Observability of Boolean Control Networks with Time-Variant Delays in States

This paper gives an equivalent condition for the observability of Boolean control networks（BCNs） with time-variant delays in states under a mild assumption by using the graph-theoretic method under the framework of the semi-tensor product of matrices. First, the BCN under consideration is split into a finite number of subsystems with no time delays. Second, the observability of the BCN is verified by testing the observability of the so-called observability constructed path（a special subsystem without time delays） based on graph theory. These results extend the recent related results on the observability of BCNs. Examples are shown to illustrate the effectiveness of the results.

Graph Theory Methods for Decomposition w.r.t. Outputs of Boolean Control Networks

This paper focuses graph theory method for the problem of decomposition w.r.t. outputs for Boolean control networks(BCNs). First, by resorting to the semi-tensor product of matrices and the matrix expression of BCNs, the definition of decomposition w.r.t. outputs is introduced. Second, by referring to the graphical structure of BCNs, a necessary and sufficient condition for the decomposition w.r.t. outputs is obtained based on graph theory method. Third, an effective algorithm to realize the maximum decomposition w.r.t. outputs is proposed. Finally, some examples are addressed to validate the theoretical results.

Stability of Nonlinear Feedback Shift Registers with Periodic Input

The stability of Non-Linear Feedback Shift Registers(NFSRs)plays an important role in the cryptographic security.Due to the complexity of nonlinear systems and the lack of efficient algebraic tools,the theorems related to the stability of NFSRs are still not well-developed.In this paper,we view the NFSR with periodic inputs as a Boolean control network.Based on the mathematical tool of semi-tensor product(STP),the Boolean network can be mapped into an algebraic form.Through these basic theories,we analyze the state space of non-autonomous NFSRs,and discuss the stability of an NFSR with periodic inputs of limited length or unlimited length.The simulation results are provided to prove the efficiency of the model.Based on these works,we can provide a method to analyze the stability of the NFSR with periodic input,including limited length and unlimited length.By this,we can efficiently reduce the computational complexity,and its efficiency is demonstrated by applying the theorem in simulations dealing with the stability of a non-autonomous NFSR.

Morgan＇s problem of Boolean control networks

This paper investigates the Morgan＇s problem of Boolean control networks. Based on the matrix expression of logical functions, two key steps are proposed to solve the problem. First, the Boolean control network is converted into an output- decomposed form by constructing a set of consistent outputfriendly subspaces, and a necessary and sufficient condition for the existence of the consistent output-friendly subspaces is obtained. Secondly, a type of state feedback controllers are designed to solve the Morgan＇s problem if it is solvable. By solving a set of matrix equations, a necessary and sufficient condition for converting an output-decomposed form to an input-output decomposed form is given, and by verifying the output controllability matrix, the solvability of Morgan＇s problem is obtained.

Fundamental Boolean network modelling for childhood acute lymphoblastic leukaemia pathways

Background:A novel data-driven Boolean model,namely,the fundamental Boolean model(FBM),has been proposed to draw genetic regulatory insights into gene activation,inhibition,and protein decay,published in 2018.This novel Boolean model facilitates the analysis of the activation and inhibition pathways.However,the novel model does not handle the situation well,where genetic regulation might require more time steps to complete.Methods:Here,we propose extending the fundamental Boolean modelling to address the issue that some gene regulations might require more time steps to complete than others.We denoted this extension model as the temporal fundamental Boolean model(TFBM)and related networks as the temporal fundamental Boolean networks(TFBNs).The leukaemia microarray datasets downloaded from the National Centre for Biotechnology Information have been adopted to demonstrate the utility of the proposed TFBM and TFBNs.Results:We developed the TFBNs that contain 285 components and 2775 Boolean rules based on TFBM on the leukaemia microarray datasets,which are in the form of short-time series.The data contain gene expression measurements for 13 GC-sensitive children under therapy for acute lymphoblastic leukaemia,and each sample has three time points:0 hour(before GC treatment),6/8 hours(after GC treatment)and 24 hours(after GC treatment).Conclusion:We conclude that the proposed TFBM unlocks their predecessor’s limitation,Le.,FBM,that could help pharmaceutical agents identify any side effects on clinic-related data.New hypotheses could be identified by analysing the extracted fundamental Boolean networks and analysing their up-regulatory and down-regulatory pathways.