Probabilistic Boolean Networks Research Articles

A Feedback Vertex Set-Based Approach to Simplifying Probabilistic Boolean Networks

A Feedback Vertex Set-Based Approach to Simplifying Probabilistic Boolean Networks

Total-activity conservation of probabilistic Boolean networks and its applications

Conservation law is a basic property of biological networks, which exhibits the total-activity conservation in Boolean networks (BNs). This article analyzes the total-activity conservation of probabilistic Boolean networks (PBNs) based on the algebraic state space representation approach. The concept of total-activity conservation is proposed for PBNs, and a vector-based criterion is presented to verify the total-activity conservation of PBNs. Moreover, the impact of deterministic function perturbation on the total-activity conservation of PBNs is analyzed. Finally, as an application of total-activity conservation for PBNs, the robust total-activity conservation of BNs subject to stochastic function perturbation is explored.

Finite-time observability of probabilistic Boolean multiplex control networks

Finite-time observability of probabilistic Boolean multiplex control networks

Parameter Learning of Probabilistic Boolean Control Networks with Input-Output Data

Article Parameter Learning of Probabilistic Boolean Control Networks with Input-Output Data Hongwei Chen 1,*, Qi Chen 1, Bo Shen 1, and Yang Liu 2 1 College of Information Science and Technology, Donghua University, Shanghai 201620, China 2 School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China * Correspondence: hongwei@dhu.edu.cn Received: 23 September 2023 Accepted: 27 November 2023 Published: 26 March 2024 Abstract: This paper investigates the parameter learning problem for the probabilistic Boolean control networks (PBCNs) with input-output data. Firstly, an algebraic expression of the PBCNs is obtained by taking advantage of the semi-tensor product technique, and then, the parameter learning problem is transformed into an optimal problem to reveal the parameter matrices of a linear system in a computationally efficient way. Secondly, two recursive semi-tensor product based algorithms are designed to calculate the forward and backward probabilities. Thirdly, the expectation maximization algorithm is proposed as an elaborate technique to address the parameter learning problem. In addition, a useful index is introduced to describe the performance of the proposed parameter learning algorithm. Finally, two numerical examples are employed to demonstrate the reliability of the proposed parameter learning approach.

Set stabilization of logical control networks: A minimum node control approach

In network systems, control using minimum nodes or pinning control can be effectively used for stabilization problems to cut down the cost of control. In this paper, we investigate the set stabilization problem of logical control networks. In particular, we study the set stabilization problem of probabilistic Boolean networks (PBNs) and probabilistic Boolean control networks (PBCNs) via controlling minimal nodes. Firstly, an algorithm is given to search for the minimum index set of pinning nodes. Then, based on the analysis of its high computational complexity, we present optimized algorithms with lower computational complexity to ascertain the network control using minimum node sets. Moreover, some sufficient and necessary conditions are proposed to ensure the feasibility and effectiveness of the proposed algorithms. Furthermore, a theorem is presented for PBCNs to devise all state-feedback controllers corresponding to the set of pinning nodes. Finally, two models of gene regulatory networks are considered to show the efficacy of obtained results.

Development of a Probabilistic Boolean network (PBN) to model intraoperative blood pressure management

BackgroundBlood pressure is a vital sign for organ perfusion that anesthesiologists measure and modulate during surgery. However, current decision-making processes rely heavily on clinicians’ experience, which can lead to variability in treatment across surgeries. With the advent of machine learning, we can now create models to predict the outcomes of interventions and guide perioperative decision-making. The first step in this process involves translating the clinical decision-making process into a framework understood by an algorithm. Probabilistic Boolean networks (PBNs) provide an information-rich approach to this problem. A PBN trends toward a steady state, and its decisions are easily understood via its Boolean predictor functions. We hypothesize that a PBN can be developed that corrects hemodynamic instability in patients by selecting clinical interventions to maintain blood pressure within a given range. MethodsData on patients over the age of 65 undergoing surgery with general anesthesia from 2018 to 2020 were drawn from the UF Health PRECEDE data set with IRB approval (IRB201700747). Parameters examined included heart rate, blood pressure, and frequency of medications given 15 min after anesthetic induction and 15 min before awakening. The medication frequency data were truncated into a 66/33 split for the training and validation set used in the PBN. The model was coded using Python 3 and evaluated by comparing the frequency of medications chosen by the program to the values in the testing set via linear regression analysis. ResultsThe network developed successfully models a hemodynamically unstable patient and corrects the imbalance by administering medications. This is evidenced by the model achieving a stable, steady state matrix in all iterations. However, the model's ability to emulate clinical drug selection was variable. It was successful with its use of vasodilator selection but struggled with the appropriate selection of vasopressors. ConclusionsThe PBN has demonstrated the ability to choose appropriate interventions based on a patient's current vitals. Additional work must be done to have the network emulate the frequency at which drugs are selected from in clinical practice. In its current state, the model provides an understanding of how a PBN behaves in the context of correcting hemodynamic instability and can aid in developing more robust models in the future.

Stabilization of Probabilistic Boolean Networks via State-Flipped Control and Reinforcement Learning

Stabilization of Probabilistic Boolean Networks via State-Flipped Control and Reinforcement Learning

A Polynomial-Time Criterion for Stability in Distribution of Probabilistic Boolean Networks

A Polynomial-Time Criterion for Stability in Distribution of Probabilistic Boolean Networks

Finite-time set stabilization of probabilistic Boolean control networks via output-feedback control

This paper investigates the finite-time output-feedback set stabilization problem for probabilistic Boolean control networks (PBCNs), a crucial aspect of set convergence. Firstly, an algorithm is presented to effectively identify the largest output-feedback control invariant subset in a given set. Next, a sufficient condition is put forward to bridge set stabilization and the question at hand via the minimum transient period. Additionally, sufficient and necessary criteria are proposed, addressing the solvability of finite-time output-feedback set stabilization, followed by two corresponding algorithms for effectively designing all feasible output-feedback stabilizers. Finally, a biological example is successfully explored to demonstrate our theoretical results.

Detectability of Boolean networks: A finite-time convergent matrix approach

This paper investigates the detectability of Boolean networks (including Boolean control networks (BCNs) and probabilistic Boolean networks (PBNs)) and reveals the relationship between detectability and observability from the perspective of decompositions. Firstly, the single-experiment detectability and the arbitrary-experiment detectability of BCNs are converted into the consistent reachability and the any-input sequence reachability of a logica control network through an augmented approach. Then these two kinds of detectability are testified via two defined non-augmented matrix sequences. Moreover, some algorithms are given to obtain the current state and suitable input sequences satisfying corresponding detectability. Subsequently, the non-augmented approach is used to discuss the strong detectability of PBNs, and is generalized to analyze detectability and observability decompositions of BCNs. Finally, several examples are presented to illustrate these results.

Fault Detection and Isolation in Smart-Grid Networks of Intelligent Power Routers Modeled as Probabilistic Boolean Networks

A self-organizing complex-network modeling method, probabilistic Boolean networks, is presented as a model-based diagnostic system for detecting and isolating different types of faults, failures, and modes of operation in which a network of intelligent power routers is deployed over a standard power test case: the Western System Coordinating Council 9 Bus System. Such a system allows designers and engineering professionals to make educated decisions pertaining to the design of smart-grid systems endowed with intelligent power routers. There is a recurrent necessity to design reliable and fault-tolerant smart power systems, maintaining adequate operation and adherence to performance specifications, while keeping costs at the minimum. This diagnostics system will help achieve such goals: better design through thorough analysis of the conditions that lead to a fault on a smart grid, proper detection of these faults, and isolation of the respective assets.

Pinning Stabilizer Design for Probabilistic Boolean Control Networks via Condensation Digraph.

This article investigates the design of pinning controllers for state feedback stabilization of probabilistic Boolean control networks (PBCNs), based on the condensation digraph method. First, two effective algorithms are presented to achieve state feedback stabilization of the considered system from the perspective of condensation digraph. One is to find the desired matrix, and the other is to search for the minimum number of pinned nodes and specific pinned nodes. Then, all the mode-independent pinning controllers can be designed based on the desired matrix and pinned nodes. Several examples are delineated to illustrate the validity of the main results.

Stability analysis of Boolean networks: An eigenvalue approach

This article investigates the stability of probabilistic Boolean networks (PBNs) and switched Boolean networks (SBNs). To begin with, a unique Boolean network is presented with both stochastic signal and switching signal. It is then converted to algebraic form using semi-tensor product. An important concept named “existed path” is proposed, and further a novel method based on eigenvalue and trace-based method is used to derive some necessary and sufficient stability criteria for the first time. Moreover, these criteria are extended to include ordinary PBNs and SBNs. Finally, several examples are given to illustrate the main results.

Deep Reinforcement Learning for Stabilization of Large-Scale Probabilistic Boolean Networks

Deep Reinforcement Learning for Stabilization of Large-Scale Probabilistic Boolean Networks

Strong Structural Controllability of Boolean Networks: Polynomial-Time Criteria, Minimal Node Control, and Distributed Pinning Strategies

In this paper, we initiate the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">strong structural controllability</i> of Boolean networks (BNs), coping with the trouble in identifying full nodal dynamics in practice, and utilize it to build up some existing results about BNs. The derived criteria for the strong structural controllability of BNs are polynomial-time checkable. As an interesting feature, controllability is shown to be equivalent to fixed-time controllability in this regard. We further consider the minimal strong structural controllability problem of BNs which, reduced from the minimal vertex cover problem of graphs, is turned out to be NP-hard. Proceeding forward, strong structural controllability provides useful insights to exclude the deficiencies of existing approaches. First, the network aggregation subject to the controllability of BNs is built by showing that the predecessor aggregated blocks can be dropped in turn if they are strongly structurally controllable. Second, the distributed pinning controllers are carried out to render arbitrary BN controllable through node-to-node message exchange, while traditional results only check the controllability with the preassigned controllers. Notably, the time complexity to design such controllers is only exponential with the maximal in-degree of pinned nodes rather than the node number of BNs. Third, in scenarios of BNs encountering probability disturbances, a basic theorem is put forward to elaborate the equivalence between several types of asymptotic stability. Together with this theorem, the criteria of strong structural controllability also facilitate the design of distributed controllers to asymptotically stabilize probabilistic BNs in probability.

Design of reduced-order and pinning controllers for probabilistic Boolean networks using reinforcement learning

In this paper, we study a stabilization method for probabilistic Boolean networks (PBNs) using Q-learning, which is one of the typical methods in reinforcement learning. A PBN is a class of discrete-time stochastic logical systems in which update functions are randomly chosen from the set of the candidates of Boolean functions. In the existing methods using reinforcement learning, a design method of structured controllers has not been studied. In this paper, we propose reward design methods to derive reduced-order controllers and pinning controllers. The key idea is to adjust the structure of controllers by the reward for the structure of the Q-table. The advantage of the proposed method is that implementation is easy, because the proposed method can be embedded in the existing Q-learning-based stabilization algorithm. In design of pinning controllers, we can calculate not only controllers but also pinning nodes in which the control input is assigned. We demonstrate through numerical examples that the controller obtained from our proposed method is simpler than that from the existing method.

Set stabilizability of impulsive probabilistic Boolean networks via impulsive sequence design

Set stabilizability of impulsive probabilistic Boolean networks via impulsive sequence design

A unified transform method for general robust property of probabilistic Boolean control networks

This article proposes a unified transform method for general robust property of probabilistic Boolean control networks (PBCNs). Here, the general robust property of PBCNs means any given robust property of any given PBCN with probability one. Specifically, the method is that the investigation on any given robust property of any given PBCN with probability one can be equivalently transformed into the investigation on the given robust property of the corresponding Boolean control network (BCN). Note that the PBCN has time-varying probability distribution and the property can be set according to the needs of specific problems. As an application of the method, we study the robust set stabilization with probability one of PBCNs with positive time-varying probability distribution. We derive the results through a discussion of the robust set stabilization of the corresponding BCNs. Also, we characterize all of the time-optimal robust set stabilizers of the PBCNs. A reduced disturbed PBCN model of the lac operon in the Escherichia coli is provided to show the effectiveness of the achieved results.

Observability of Probabilistic Boolean Networks via Matrix Approach

In this article, the observability of probabilistic Boolean networks (PBNs) is investigated. First, the distinguishable state pairs are obtained by constructing a matrix sequence, based on which a novel criterion for examining the observability of PBNs is given. Then, an algorithm is proposed to verify the observability of PBNs. Finally, some numerical examples are presented to illustrate the proposed method.

Maximum-Likelihood State Estimators in Probabilistic Boolean Control Networks.

This study addresses state estimation problems for probabilistic Boolean control networks (PBCNs). Compared with deterministic Boolean networks, PBCNs have the stochastic switching in logical update functions in the state equation. Consequently, statistical analysis is required to estimate unavailable states, which induces an optimization problem called maximum-likelihood estimation. This article mainly focuses on two scenarios: 1) state estimation from partially measured state and 2) state estimation from output data, meaning observer design. The resulting optimization problems are solved using efficient algorithms based on dynamic programming. Concurrently, Dijkstra-type algorithms, which solve equivalent shortest path problems, are also proposed using best-first search. Furthermore, both the proposed algorithms derive novel observer design methods for PBCNs. The proposed algorithms are evaluated with practical estimation problems aiming to the sensor reduction and applied to gene regulatory networks of apoptosis and Lac operon.