Stability and Guaranteed Cost Analysis of Time-Triggered Boolean Networks

Minimal Pinning Control for Oscillatority of Boolean Networks

In this article, minimal pinning control for oscillatority (i.e., instability) of Boolean networks (BNs) under algebraic state space representations method is studied. First, two criteria for oscillatority of BNs are obtained from the aspects of state transition matrix (STM) and network structure (NS) of BNs, respectively. A distributed pinning control (DPC) from these two aspects is proposed: one is called STM-based DPC and the other one is called NS-based DPC, both of which are only dependent on local in-neighbors. As for STM-based DPC, one arbitrary node can be chosen to be controlled, based on certain solvability of several equations, meanwhile a hybrid pinning control (HPC) combining DPC and conventional pinning control (CPC) is also proposed. In addition, as for NS-based DPC, pinning control nodes (PCNs) can be found using the information of NS, which efficiently reduces the high computational complexity. The proposed STM-based DPC and NS-based DPC in this article are shown to be simple and concise, which provide a new direction to dramatically reduce control costs and computational complexity. Finally, gene networks are simulated to discuss the effectiveness of theoretical results.

State Estimation for Probabilistic Boolean Networks via Outputs Observation

This article studies the state estimation for probabilistic Boolean networks via observing output sequences. Detectability describes the ability of an observer to uniquely estimate system states. By defining the probability of an observed output sequence, a new concept called detectability measure is proposed. The detectability measure is defined as the limit of the sum of probabilities of all detectable output sequences when the length of output sequences goes to infinity, and it can be regarded as a quantitative assessment of state estimation. A stochastic state estimator is designed by defining a corresponding nondeterministic stochastic finite automaton, which combines the information of state estimation and probability of output sequences. The proposed concept of detectability measure further performs the quantitative analysis on detectability. Furthermore, by defining a Markov chain, the calculation of detectability measure is converted to the calculation of the sum of probabilities of certain specific states in Markov chain. Finally, numerical examples are given to illustrate the obtained theoretical results.

Steady-State Design of Large-Dimensional Boolean Networks

Analysis and design of steady states representing cell types, such as cell death or unregulated growth, are of significant interest in modeling genetic regulatory networks. In this article, the steady-state design of large-dimensional Boolean networks (BNs) is studied via model reduction and pinning control. Compared with existing literature, the pinning control design in this article is based on the original node's connection, but not on the state-transition matrix of BNs. Hence, the computational complexity is dramatically reduced in this article from O(2ⁿ×2ⁿ) to O(2×2^r) , where n is the number of nodes in the large-dimensional BN and is the largest number of in-neighbors of the reduced BN. Finally, the proposed method is well demonstrated by a T-LGL survival signaling network with 18 nodes and a model of survival signaling in large granular lymphocyte leukemia with 29 nodes. Just as shown in the simulations, the model reduction method reduces 99.98% redundant states for the network with 18 nodes, and 99.99% redundant states for the network with 29 nodes.

On Optimal Time-Varying Feedback Controllability for Probabilistic Boolean Control Networks

This brief studies controllability for probabilistic Boolean control network (PBCN) with time-varying feedback control laws. The concept of feedback controllability with an arbitrary probability for PBCNs is formulated first, and a control problem to maximize the probability of time-varying feedback controllability is investigated afterward. By introducing semitensor product (STP) technique, an equivalent multistage decision problem is deduced, and then a novel optimization algorithm is proposed to obtain the maximum probability of controllability and the corresponding optimal feedback law simultaneously. The advantages of the time-varying optimal controller obtained by the proposed algorithm, compared to the time-invariant one, are illustrated by numerical simulations.

Output Feedback Control for Set Stabilization of Boolean Control Networks

In this paper, the output feedback set stabilization problem for Boolean control networks (BCNs) is investigated with the help of the semi-tensor product (STP) tool. The concept of output feedback control invariant (OFCI) subset is introduced, and novel methods are developed to obtain the OFCI subsets. Based on the OFCI subsets, a technique, named spanning tree method, is further introduced to calculate all possible output feedback set stabilizers. An example concerning lac operon for the bacterium Escherichia coli is given to illustrate the effectiveness of the proposed method. This technique can also be used to solve the state feedback (set) stabilization problem for BCNs. Compared with the existing results, our method can dramatically reduce the computational cost when designing all possible state feedback stabilizers for BCNs.

Stabilization of Mode-Dependent Impulsive Hybrid Systems Driven by DFA With Mixed-Mode Effects

This paper is concerned with mode-dependent impulsive hybrid systems driven by deterministic finite automaton (DFA) with mixed-mode effects. In the hybrid systems, a complex phenomenon called mixed mode, caused in time-varying delay switching systems, is considered explicitly. Furthermore, mode-dependent impulses, which can exist not only at the instants coinciding with mode switching but also at the instants when there is no system switching, are also taken into consideration. First, we establish a rigorous mathematical equation expression of this class of hybrid systems. Then, several criteria of stabilization of this class of hybrid systems are presented based on semi-tensor product (STP) techniques, multiple Lyapunov-Krasovskii functionals, as well as the average dwell time approach. Finally, an example is simulated to illustrate the effectiveness of the obtained results.

Tractable Learning and Inference for Large-Scale Probabilistic Boolean Networks

Probabilistic Boolean networks (PBNs) have previously been proposed so as to gain insights into complex dynamical systems. However, identification of large networks and their underlying discrete Markov chain which describes their temporal evolution still remains a challenge. In this paper, we introduce an equivalent representation for PBNs, the stochastic conjunctive normal form network (SCNFN), which enables a scalable learning algorithm and helps predict long-run dynamic behavior of large-scale systems. State-of-the-art methods turn out to be 400 times slower for middle-sized networks (i.e., containing 100 nodes) and incapable of terminating for large networks (i.e., containing 1000 nodes) compared to the SCNFN-based learning, when attempting to achieve comparable accuracy. In addition, in contrast to the currently used methods which introduce strict restrictions on the structure of the learned PBNs, the hypothesis space of our training paradigm is the set of all possible PBNs. Moreover, SCNFNs enable efficient sampling so as to statistically infer multistep transition probabilities which can provide information on the activity levels of individual nodes in the long run. Extensive experimental results showcase the scalability of the proposed approach both in terms of sample and runtime complexity. In addition, we provide examples to study large and complex cell signaling networks to show the potential of our model. Finally, we suggest several directions for future research on model variations, theoretical analysis, and potential applications of SCNFNs.

Fast-Time Stability of Temporal Boolean Networks

In real systems, most of the biological functionalities come from the fact that the connections are not active all the time. Based on the fact, temporal Boolean networks (TBNs) are proposed in this paper, and the fast-time stability is analyzed via semi-tensor product (STP) of matrices and incidence matrices. First, the algebraic form of a TBN is obtained based on the STP method, and one necessary and sufficient condition for global fast-time stability is presented. Moreover, incidence matrices are used to obtain several sufficient conditions, which reduce the computational complexity from O(n2ⁿ) (exponential type) to O(n⁴) (polynomial type) compared with the STP method. In addition, the global fast-time stabilization of TBNs is considered, and pinning controllers are designed based on the neighbors of controlled nodes rather than all the nodes. Finally, the local fast-time stability of TBNs is considered based on the incidence matrices as well. Several examples are provided to illustrate the effectiveness of the obtained results.

Function Perturbation Impact on Feedback Stabilization of Boolean Control Networks

Function perturbation analysis of Boolean networks is an important topic in the study of gene regulation due to gene mutation or immeasurable variables. This brief studies the function perturbation impact on feedback stabilization of Boolean control networks (BCNs) by using the algebraic state space representation approach. First, the state feedback stabilization control design of BCNs is recalled and the function perturbation problem is formulated. Second, given a state feedback stabilizer, it is robust to the considered function perturbation if one of the following three cases happens: 1) the block where the function perturbation occurs is different from the block which is affected by the state feedback stabilizer (Case 1); 2) when Case 1 does not happen, the perturbed column converges to the equilibrium faster than the original column (Case 2); 3) when Cases 1 and 2 do not happen, the perturbed column does not belong to the reachable set of the original column (Case 3). Third, when the perturbed column belongs to the reachable set of the original column, a constructive procedure is proposed to modify the given state feedback stabilizer to be robust to the function perturbation. Finally, the obtained new results are applied to the function perturbation analysis of lactose operon in Escherichia coli. The main novelty of this brief is to develop a new theoretical framework for the robustness of feedback controllers of BCNs with respect to function perturbation, which is not solved in the existing literature.