Boolean Dynamical Systems Research Articles

Bayesian Optimization for State and Parameter Estimation of Dynamic Networks with Binary Space.

This paper focuses on joint state and parameter estimation in partially observed Boolean dynamical systems (POBDS), a hidden Markov model tailored for modeling complex networks with binary state variables. The majority of current techniques for parameter estimation rely on computationally expensive gradient-based methods, which become intractable in most practical applications with large size of networks. We propose a gradient-free approach that uses Gaussian processes to model the expensive log-likelihood function and utilizes Bayesian optimization for efficient likelihood search over parameter space. Joint state estimation is also achieved alongside parameter estimation using the Boolean Kalman filter. The performance of the proposed method is demonstrated using gene regulatory networks observed through synthetic gene-expression data. The numerical results demonstrate the scalability and effectiveness of the proposed method in the joint estimation of the model parameters and genes' states.

A generalized uniqueness theorem for generalized Boolean dynamical systems

We characterize the canonical diagonal subalgebra of the C⁎-algebra associated with a generalized Boolean dynamical system. We also introduce a particular commutative subalgebra, which we call the abelian core, in our C⁎-algebra. We then establish a uniqueness theorem under the assumptions that B and L are countable, which says that a ⁎-homomorphism of our C⁎-algebra is injective if and only if its restriction to the abelian core is injective.

Kernel-Based Particle Filtering for Scalable Inference in Partially Observed Boolean Dynamical Systems

This paper addresses the inference challenges associated with a class of hidden Markov models with binary state variables, known as partially observed Boolean dynamical systems (POBDS). POBDS have demonstrated remarkable success in modeling the ON and OFF dynamics of genes, microbes, and bacteria in systems biology, as well as in network security to represent the propagation of attacks among interconnected elements. Despite existing optimal and approximate inference solutions for POBDS, scalability remains a significant issue due to the computational cost associated with likelihood evaluations and the exploration of extensive parameter spaces. To overcome these challenges, this paper proposes a kernel-based particle filtering approach for large-scale inference of POBDS. Our method employs a Gaussian process (GP) to efficiently represent the expensive-to-evaluate likelihood function across the parameter space. The likelihood evaluation is approximated using a particle filtering technique, enabling the GP to account for various sources of uncertainty, including limited likelihood evaluations. Leveraging the GP’s predictive behavior, a Bayesian optimization strategy is derived for effectively seeking parameters yielding the highest likelihood, minimizing the overall computational burden while balancing exploration and exploitation. The proposed method’s performance is demonstrated using two biological networks: the mammalian cell-cycle network and the T-cell large granular lymphocyte leukemia network.

Probabilistic edge weights fine-tune Boolean network dynamics.

Biological systems are noisy by nature. This aspect is reflected in our experimental measurements and should be reflected in the models we build to better understand these systems. Noise can be especially consequential when trying to interpret specific regulatory interactions, i.e. regulatory network edges. In this paper, we propose a method to explicitly encode edge-noise in Boolean dynamical systems by probabilistic edge-weight (PEW) operators. PEW operators have two important features: first, they introduce a form of edge-weight into Boolean models through the noise, second, the noise is dependent on the dynamical state of the system, which enables more biologically meaningful modeling choices. Moreover, we offer a simple-to-use implementation in the already well-established BooleanNet framework. In two application cases, we show how the introduction of just a few PEW operators in Boolean models can fine-tune the emergent dynamics and increase the accuracy of qualitative predictions. This includes fine-tuning interactions which cause non-biological behaviors when switching between asynchronous and synchronous update schemes in dynamical simulations. Moreover, PEW operators also open the way to encode more exotic cellular dynamics, such as cellular learning, and to implementing edge-weights for regulatory networks inferred from omics data.

Representation of gene regulation networks by hypothesis logic-based Boolean systems

Boolean Dynamical Systems (BDSs) are networks described by Boolean variables. A new representation of BDSs is presented in this article by using modal non-monotonic logic (\({\mathcal {H}}\)). This approach allows Boolean Networks to be represented by a set of modal formulas and therefore can be used to describe and learn their properties. The study of a BDS focuses in particular on the search of stable configurations, limit cycles and unstable cycles, which help to characterize a large type of Gene Networks. In this article is presented the identification of such asymptotic properties by introduction of a new concept, ghost extensions. Using ghost extensions, it is possible to translate BDSs in propositional calculus and consequently to use SAT algorithms.

$$C^*$$-algebras of generalized Boolean dynamical systems as partial crossed products

In this paper, we realize $$C^*$$ -algebras of generalized Boolean dynamical systems as partial crossed products. Reciprocally, we give some sufficient conditions for a partial crossed product to be isomorphic to a $$C^*$$ -algebra of a generalized Boolean dynamical system. As an application, we show that gauge-invariant ideals of $$C^*$$ -algebras of generalized Boolean dynamical systems are themselves $$C^*$$ -algebras of generalized Boolean dynamical systems.

Boundary path groupoids of generalized Boolean dynamical systems and their C⁎-algebras

In this paper, we provide two types of boundary path groupoids from a generalized Boolean dynamical system (B,L,θ,Iα). For the first groupoid, we associate an inverse semigroup to a generalized Boolean dynamical system and use the tight spectrum T as the unit space of a groupoid Γ(B,L,θ,Iα) that is isomorphic to the tight groupoid Gtight. The other one is defined as the Renault-Deaconu groupoid Γ(∂E,σE) arising from a topological correspondence E associated with a generalized Boolean dynamical system. We then prove that the tight spectrum T is homeomorphic to the boundary path space ∂E obtained from the topological correspondence. Using this result, we prove that the groupoid Γ(B,L,θ,Iα) equipped with the topology induced from the topology on Gtight is isomorphic to Γ(∂E,σE) as a topological groupoid. Finally, we show that their C⁎-algebras are isomorphic to the C⁎-algebra of the generalized Boolean dynamical system.

Computational approaches for direct cell reprogramming: from the bulk omics era to the single cell era.

Recent advances in direct cell reprogramming have made possible the conversion of one cell type to another cell type, offering a potential cell-based treatment to many major diseases. Despite much attention, substantial roadblocks remain including the inefficiency in the proportion of reprogrammed cells of current experiments, and the requirement of a significant amount of time and resources. To this end, several computational algorithms have been developed with the goal of guiding the hypotheses to be experimentally validated. These approaches can be broadly categorized into two main types: transcription factor identification methods which aim to identify candidate transcription factors for a desired cell conversion, and transcription factor perturbation methods which aim to simulate the effect of a transcription factor perturbation on a cell state. The transcription factor perturbation methods can be broken down into Boolean networks, dynamical systems and regression models. We summarize the contributions and limitations of each method and discuss the innovation that single cell technologies are bringing to these approaches and we provide a perspective on the future direction of this field.

Existence, coexistence and uniqueness of fixed points in parallel and sequential dynamical systems over directed graphs

In this work, we solve the fixed-point existence, coexistence and uniqueness problems in the context of homogeneous parallel and sequential dynamical systems on maxterm and minterm Boolean functions over directed dependency graphs. More specifically, we give characterizations for the existence of fixed points. Likewise, we provide a sufficient condition and a necessary one for the non-coexistence of other periods in the dynamics of the system. In addition, we state a necessary and sufficient condition for the uniqueness of a fixed point, what allows us to establish a Fixed-Point Theorem in the sense of Banach for this kind of systems. Finally, we give an advance in the analysis of the number of fixed points of such systems, finding a lower bound for this number in the case of the most simplest maxterm and minterm. Numerical examples are given to show the feasibility of the results. Thus, this work completes the study of the fixed-point problems for homogeneous Boolean finite dynamical systems on maxterm and minterm functions.

CONDITION (K) FOR BOOLEAN DYNAMICAL SYSTEMS

AbstractWe generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $({{\mathcal {B}}},{{\mathcal {L}}},\theta )$ with countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ satisfies Condition (K) if and only if every ideal of its $C^*$ -algebra is gauge-invariant, if and only if its $C^*$ -algebra has the (weak) ideal property, and if and only if its $C^*$ -algebra has topological dimension zero. As a corollary we prove that if the $C^*$ -algebra of a locally finite Boolean dynamical system with ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ countable either has real rank zero or is purely infinite, then $({{\mathcal {B}}}, {{\mathcal {L}}}, \theta )$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$ -algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ .

A discrete dynamics approach to interbank financial contagion

Abstract The purpose of this paper is to describe in terms of mathematical models and systems theory the dynamics of interbank financial contagion. Such a description gives rise to a model that can be studied with mathematical tools and will provide a new framework for the study of contagion dynamics complementary to research by simulation studied so far. It provides a better understanding of such financial networks and a unifying network for the research of financial contagion. The mathematical description we present is in terms of Boolean dynamical systems and a linear operator. We relate the properties of the dynamical systems to the properties of the operator.

Isometries of the hypercube: A tool for Boolean regulatory networks analysis

Boolean finite dynamical systems (FDS) are commonly used in systems biology to model the dynamics of intracellular regulatory networks and interpret the emergence of cellular behaviors. Given a Boolean FDS, we can compute the corresponding regulatory network, that is a directed signed graph representing all the interactions between components (genes), endowed with logical rules specifying the dynamical behavior of the system. We consider the asynchronous trajectories generated by this Boolean FDS, supported by the hypercube. The exploration and analysis of this dynamics is a challenging task because of the combinatorial explosion that we face. A way to approach this problem is to exploit the links between the regulatory graph and the dynamics.We use the isometries of the hypercube to define classes gathering all the isometric Boolean FDS. Thus, we classify the set of Boolean FDS on the basis of those isometries, and emphasize their common features through regulatory graphs and logical rules. We can then restrict the dynamical analysis of all the Boolean FDS to one representative per class, and thereby considerably improve the efficiency of analysis of all the Boolean FDS. Relying on invariants properties, we propose a constructive method to provide, given a FDS, a representative regulatory graph of its class.We illustrate the efficiency of the method in concrete situations. For instance, the motif analysis (Remy et al., 2003; Remy et al., 2016; Didier and Remy, 2012) is strongly improved thanks to this classification. We also revisit the negative Thomas’ rule (Remy and Ruet, 2008; Richard, 2010) by establishing a new demonstration.

Dynamical systems and interbank networks

A mathematical model is presented that describes the default contagion in an interbank network. The proposed description is in terms of Boolean dynamical systems. The developed framework utilizes the usage of mathematical tools to the study of interbank networks and the contagion dynamics. In particular, it contributes to the assessment of the robustness of the network and its institutions and their resilience to systemic risk. Furthermore, we exploit the theory of Koopman operators in order to study the dynamical system describing the default contagion.

Coexistence of Periods in Parallel and Sequential Boolean Graph Dynamical Systems over Directed Graphs

In this work, we solve the problem of the coexistence of periodic orbits in homogeneous Boolean graph dynamical systems that are induced by a maxterm or a minterm (Boolean) function, with a direct underlying dependency graph. Specifically, we show that periodic orbits of any period can coexist in both kinds of update schedules, parallel and sequential. This result contrasts with the properties of their counterparts over undirected graphs with the same evolution operators, where fixed points cannot coexist with periodic orbits of other different periods. These results complete the study of the periodic structure of homogeneous Boolean graph dynamical systems on maxterm and minterm functions.

Gauge-invariant ideals of C⁎-algebras of Boolean dynamical systems

We enlarge the class of C⁎-algebras of Boolean dynamical systems in order to include all weakly left-resolving normal labelled space C⁎-algebras in it. We prove a gauge-invariant uniqueness theorem and classify all gauge-invariant ideals of these C⁎-algebras of generalized Boolean dynamical systems and describe the corresponding quotients as C⁎-algebras of relative generalized Boolean dynamical systems.

Boolean Kalman filter and smoother under model uncertainty

Partially-observed Boolean dynamical systems (POBDS) are a general class of nonlinear state-space models that provide a rich framework for modeling many complex dynamical systems. The model consists of a hidden Boolean state process, observed through an arbitrary noisy mapping to a measurement space. The optimal minimum mean-square error (MMSE) POBDS state estimators are the Boolean Kalman Filter and Smoother. However, in many practical problems, the system parameters are not fully known and must be estimated. In this paper, for POBDS under model uncertainty, we derive an optimal Bayesian estimator for state and parameter estimation. The exact algorithms are derived for the case of discrete and finite parameter space, and for general parameter spaces, an approximate Markov-Chain Monte-Carlo (MCMC) implementation is introduced. We demonstrate the performance of the proposed methodology by means of numerical experiments with POBDS models of gene regulatory networks observed through noisy measurements.

Scalable optimal Bayesian classification of single-cell trajectories under regulatory model uncertainty

BackgroundSingle-cell gene expression measurements offer opportunities in deriving mechanistic understanding of complex diseases, including cancer. However, due to the complex regulatory machinery of the cell, gene regulatory network (GRN) model inference based on such data still manifests significant uncertainty.ResultsThe goal of this paper is to develop optimal classification of single-cell trajectories accounting for potential model uncertainty. Partially-observed Boolean dynamical systems (POBDS) are used for modeling gene regulatory networks observed through noisy gene-expression data. We derive the exact optimal Bayesian classifier (OBC) for binary classification of single-cell trajectories. The application of the OBC becomes impractical for large GRNs, due to computational and memory requirements. To address this, we introduce a particle-based single-cell classification method that is highly scalable for large GRNs with much lower complexity than the optimal solution.ConclusionThe performance of the proposed particle-based method is demonstrated through numerical experiments using a POBDS model of the well-known T-cell large granular lymphocyte (T-LGL) leukemia network with noisy time-series gene-expression data.

Adaptive Particle Filtering for Fault Detection in Partially-Observed Boolean Dynamical Systems

We propose a novel methodology for fault detection and diagnosis in partially-observed Boolean dynamical systems (POBDS). These are stochastic, highly nonlinear, and derivativeless systems, rendering difficult the application of classical fault detection and diagnosis methods. The methodology comprises two main approaches. The first addresses the case when the normal mode of operation is known but not the fault modes. It applies an innovations filter (IF) to detect deviations from the nominal normal mode of operation. The second approach is applicable when the set of possible fault models is finite and known, in which case we employ a multiple model adaptive estimation (MMAE) approach based on a likelihood-ratio (LR) statistic. Unknown system parameters are estimated by an adaptive expectation-maximization (EM) algorithm. Particle filtering techniques are used to reduce the computational complexity in the case of systems with large state-spaces. The efficacy of the proposed methodology is demonstrated by numerical experiments with a large gene regulatory network (GRN) with stuck-at faults observed through a single noisy time series of RNA-seq gene expression measurements.

On the Rank and Periodic Rank of Finite Dynamical Systems

A finite dynamical system is a function $f : A^n \to A^n$ where $A$ is a finite alphabet, used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. The rank of a finite dynamical system is the cardinality of its image; the periodic rank is the number of its periodic points. In this paper, we determine the maximum rank and the maximum periodic rank of a finite dynamical system with a given interaction graph over any non-Boolean alphabet. The rank and the maximum rank are both computable in polynomial time. We also obtain a similar result for Boolean finite dynamical systems (also known as Boolean networks) whose interaction graphs are contained in a given digraph. We then prove that the average rank is relatively close (as the size of the alphabet is large) to the maximum. The results mentioned above only deal with the parallel update schedule. We finally determine the maximum rank over all block-sequential update schedules and the supremum periodic rank over all complete update schedules.

Generalized predecessor existence problems for Boolean finite dynamical systems on directed graphs

A Boolean Finite Synchronous Dynamical System (BFDS, for short) consists of a finite number of objects that each maintains a boolean state, where after individually receiving state assignments, the objects update their state with respect to object-specific time-independent boolean functions synchronously in discrete time steps. The present paper studies the computational complexity of determining, given a boolean finite synchronous dynamical system, a configuration, which is a boolean vector representing the states of the objects, and a positive integer t, whether there exists another configuration from which the given configuration can be reached in t steps. It was previously shown that this problem, which we call the t-Predecessor Problem, is NP-complete even for t=1 if the update function of an object is either the conjunction of arbitrary fan-in or the disjunction of arbitrary fan-in.This paper studies the computational complexity of the t-Predecessor Problem for a variety of sets of permissible update functions as well as for polynomially bounded t. It also studies the t-Garden-Of-Eden Problem, a variant of the t-Predecessor Problem that asks whether a configuration has a t-predecessor, which itself has no predecessor. The paper obtains complexity theoretical characterizations of all but one of these problems.