Probabilistic Boolean Networks Research Articles

Local set stability and target control of probabilistic Boolean networks

This paper analyzes the local set stability of probabilistic Boolean networks (PBNs), including local finite-time set stability with probability one (FSSPO) and local asymptotic set stability with probability one (ASSPO). Firstly, a necessary and sufficient condition is proposed for the local FSSPO of PBNs. Then, a reachable set with probability one is constructed, based on which, the largest domain of attraction for the local FSSPO of PBNs is determined. Secondly, by constructing two parameterized sets, the transition probability matrix of PBNs is properly partitioned, and the largest domain of attraction for the ASSPO is depicted. Finally, for PBNs which are locally stable, the target control is introduced to achieve the global FSSPO and global ASSPO of PBNs, respectively.

New method for stability of probabilistic Boolean networks with inequality constraints

This paper studies the stability of probabilistic Boolean networks (PBNs) with state inequality constraints by establishing a new algebraic representation. On one hand, combining with the semi-tensor product (STP) of matrices and the law of total expectation, the algebraic representation of PBNs is established. On the other hand, based on the STP of matrices, an algorithm is proposed to determine the solution set of the state inequality constraints. Then, based on the algebraic representation and the inequality constrained solution set, a new form of transition probability matrix is constructed for PBNs. Finally, by the proposed transition probability matrix, new criteria are proposed for the finite-time stability with positive probability of PBNs. Several simulation examples are used to illustrate the effectiveness of the obtained results.

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

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

Relation coarsest partition method to observability of probabilistic Boolean networks

Using the relational coarsest partition (RCP) method, this article aims to study the observability problem of probabilistic Boolean networks (PBNs). The key step to solving this problem is to find the parallel cycle and the coarsest refinement partition of the given initial partition. Firstly, several definitions of observability for PBNs are proposed, and the connections between finite-time observability with positive probability (FTOPP), asymptotical observability (AO), and finite-time observability with probability one (FTOPO) are revealed. Secondly, by defining the appropriate transition relation, the RCP method is introduced into PBNs. In addition, based on the output signal mapping set, the initial partition is constructed. Thirdly, the concept of a parallel cycle for PBNs is proposed, and two algorithms are presented for computing the cycle and parallel cycle of PBNs. Moreover, with the help of the RCP method, a series of criteria are derived to solve the observability problem of PBNs, which works more efficiently. Fourth, when PBN is unobservable, an algorithm is proposed to design the state-flipping set with the smallest number of states, which can make the unobservable PBN observable. To show the validity and effectiveness of the obtained results, examples are finally given.

Minimum observability of probabilistic Boolean networks

This paper studies the minimum observability of probabilistic Boolean networks (PBNs), the main objective of which is to add the fewest measurements such that an unobservable PBN becomes observable. First of all, the algebraic form of a PBN is established with the help of semi-tensor product (STP) of matrices. By combining the algebraic forms of two identical PBNs into a parallel system, a method to search the states that need to be H-distinguishable is proposed based on the robust set reachability technique. Secondly, a necessary and sufficient condition is given to find the minimum measurements such that a given set can be H-distinguishable. Moreover, by comparing the numbers of measurements for all the feasible H-distinguishable state sets, the least measurements that make the system observable are gained. Finally, an example is given to verify the validity of the obtained results.

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

DEEP REINFORCEMENT LEARNING FOR SCALABLE CONTROL OF BOOLEAN MODELS IN THE CONTEXT OF CELLULAR REPROGRAMMING

Cellular reprogramming, understood as the artificial changing of one cell type into another, could potentially be used for the prevention or cure of complex diseases, such as cancer or neurodegenerative disorders. However, the identification of efficient reprogramming targets and strategies with classical wet-lab experiments is hindered by lengthy time commitments and high costs. To address these issues, in-silico methods are considered. In this study, we formulate a novel control problem in the context of cellular reprogramming for the Boolean network (BN) and Probabilistic Boolean Network (PBN) computational models of gene regulatory networks [1]. To solve this problem, we develop pbn-STAC - a novel computational framework based on deep reinforcement learning (DRL) [2] that facilitates the identification of reprogramming strategies for large BN/PBN models. To address the issue of the computational complexity of finding stable states, i.e., configurations which correspond to cell types or cellular phenotypic states, we introduce the notion of a pseudo-attractor and we devise a procedure for the identification of pseudo-attractor states during DRL training. Given source and target (pseudo-)attractor states, pbn-STAC finds proper control targets and strategies that drive the network from the source to the target by intervening only in intermediate attractor states that correspond to phenotypic cellular states observable in wet-lab practice. We evaluate the performance of pbn-STAC on a number of models of various sizes and a biological case study, i.e., a published BN model of immune response against B. bronchiseptica infection. We compare the results obtained with our approach with the exact, optimal solutions wherever possible and we show the effectiveness of PBN-STAC in identifying control targets and strategies. We consider our framework as a contribution towards the ultimate aim of developing scalable control methods for large models of gene regulatory networks.

Optimal Reconstruction of Probabilistic Boolean Networks.

In gene regulatory networks (GRNs), it is important to model gene regulation based on a priori information and experimental data. As a useful mathematical model, probabilistic Boolean networks (PBNs) have been widely applied in GRNs. This article addresses the optimal reconstruction problem of PBNs based on several priori Boolean functions and sampled data. When all candidate Boolean functions are known in advance, the optimal reconstruction problem is reformulated into an optimization problem. This problem can be well solved by a recurrent neural network approach which decreases the computational cost. When parts of candidate Boolean functions are known in advance, necessary and sufficient conditions are provided for the reconstruction of PBNs. In this case, two types of reconstruction problems are further proposed: one is aimed at minimizing the number of reconstructed Boolean functions, and the other one is aimed at maximizing the selection probability of the main dynamics under noises. At last, examples in GRNs are elaborated to demonstrate the effectiveness of the main results.

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