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15 pages, 1574 KiB

Open AccessArticle

Effective Connectivity and Bias Entropy Improve Prediction of Dynamical Regime in Automata Networks

by Felipe Xavier Costa, Jordan C. Rozum, Austin M. Marcus and Luis M. Rocha

Entropy 2023, 25(2), 374; https://0-doi-org.brum.beds.ac.uk/10.3390/e25020374 - 18 Feb 2023

Cited by 3 | Viewed by 1883

Biomolecular network dynamics are thought to operate near the critical boundary between ordered and disordered regimes, where large perturbations to a small set of elements neither die out nor spread on average. A biomolecular automaton (e.g., gene, protein) typically has high regulatory redundancy, [...] Read more.

Biomolecular network dynamics are thought to operate near the critical boundary between ordered and disordered regimes, where large perturbations to a small set of elements neither die out nor spread on average. A biomolecular automaton (e.g., gene, protein) typically has high regulatory redundancy, where small subsets of regulators determine activation via collective canalization. Previous work has shown that effective connectivity, a measure of collective canalization, leads to improved dynamical regime prediction for homogeneous automata networks. We expand this by (i) studying random Boolean networks (RBNs) with heterogeneous in-degree distributions, (ii) considering additional experimentally validated automata network models of biomolecular processes, and (iii) considering new measures of heterogeneity in automata network logic. We found that effective connectivity improves dynamical regime prediction in the models considered; in RBNs, combining effective connectivity with bias entropy further improves the prediction. Our work yields a new understanding of criticality in biomolecular networks that accounts for collective canalization, redundancy, and heterogeneity in the connectivity and logic of their automata models. The strong link we demonstrate between criticality and regulatory redundancy provides a means to modulate the dynamical regime of biochemical networks. Full article

(This article belongs to the Special Issue The Principle of Dynamical Criticality)

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19 pages, 764 KiB

Open AccessArticle

Temporal, Structural, and Functional Heterogeneities Extend Criticality and Antifragility in Random Boolean Networks

by Amahury Jafet López-Díaz, Fernanda Sánchez-Puig and Carlos Gershenson

Entropy 2023, 25(2), 254; https://0-doi-org.brum.beds.ac.uk/10.3390/e25020254 - 31 Jan 2023

Cited by 3 | Viewed by 1948

Abstract

Most models of complex systems have been homogeneous, i.e., all elements have the same properties (spatial, temporal, structural, functional). However, most natural systems are heterogeneous: few elements are more relevant, larger, stronger, or faster than others. In homogeneous systems, criticality—a balance between change [...] Read more.

Most models of complex systems have been homogeneous, i.e., all elements have the same properties (spatial, temporal, structural, functional). However, most natural systems are heterogeneous: few elements are more relevant, larger, stronger, or faster than others. In homogeneous systems, criticality—a balance between change and stability, order and chaos—is usually found for a very narrow region in the parameter space, close to a phase transition. Using random Boolean networks—a general model of discrete dynamical systems—we show that heterogeneity—in time, structure, and function—can broaden additively the parameter region where criticality is found. Moreover, parameter regions where antifragility is found are also increased with heterogeneity. However, maximum antifragility is found for particular parameters in homogeneous networks. Our work suggests that the “optimal” balance between homogeneity and heterogeneity is non-trivial, context-dependent, and in some cases, dynamic. Full article

(This article belongs to the Special Issue The Principle of Dynamical Criticality)

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16 pages, 6122 KiB

Open AccessArticle

Emergent Criticality in Coupled Boolean Networks

by Chris Kang, Madelynn McElroy and Nikolaos K. Voulgarakis

Entropy 2023, 25(2), 235; https://0-doi-org.brum.beds.ac.uk/10.3390/e25020235 - 27 Jan 2023

Cited by 1 | Viewed by 1627

Abstract

Early embryonic development involves forming all specialized cells from a fluid-like mass of identical stem cells. The differentiation process consists of a series of symmetry-breaking events, starting from a high-symmetry state (stem cells) to a low-symmetry state (specialized cells). This scenario closely resembles [...] Read more.

Early embryonic development involves forming all specialized cells from a fluid-like mass of identical stem cells. The differentiation process consists of a series of symmetry-breaking events, starting from a high-symmetry state (stem cells) to a low-symmetry state (specialized cells). This scenario closely resembles phase transitions in statistical mechanics. To theoretically study this hypothesis, we model embryonic stem cell (ESC) populations through a coupled Boolean network (BN) model. The interaction is applied using a multilayer Ising model that considers paracrine and autocrine signaling, along with external interventions. It is demonstrated that cell-to-cell variability can be interpreted as a mixture of steady-state probability distributions. Simulations have revealed that such models can undergo a series of first- and second-order phase transitions as a function of the system parameters that describe gene expression noise and interaction strengths. These phase transitions result in spontaneous symmetry-breaking events that generate new types of cells characterized by various steady-state distributions. Coupled BNs have also been shown to self-organize in states that allow spontaneous cell differentiation. Full article

(This article belongs to the Special Issue The Principle of Dynamical Criticality)

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21 pages, 5423 KiB

Open AccessFeature PaperArticle

On the Criticality of Adaptive Boolean Network Robots

by Michele Braccini, Andrea Roli, Edoardo Barbieri and Stuart A. Kauffman

Entropy 2022, 24(10), 1368; https://0-doi-org.brum.beds.ac.uk/10.3390/e24101368 - 27 Sep 2022

Cited by 7 | Viewed by 2191

Abstract

Systems poised at a dynamical critical regime, between order and disorder, have been shown capable of exhibiting complex dynamics that balance robustness to external perturbations and rich repertoires of responses to inputs. This property has been exploited in artificial network classifiers, and preliminary [...] Read more.

Systems poised at a dynamical critical regime, between order and disorder, have been shown capable of exhibiting complex dynamics that balance robustness to external perturbations and rich repertoires of responses to inputs. This property has been exploited in artificial network classifiers, and preliminary results have also been attained in the context of robots controlled by Boolean networks. In this work, we investigate the role of dynamical criticality in robots undergoing online adaptation, i.e., robots that adapt some of their internal parameters to improve a performance metric over time during their activity. We study the behavior of robots controlled by random Boolean networks, which are either adapted in their coupling with robot sensors and actuators or in their structure or both. We observe that robots controlled by critical random Boolean networks have higher average and maximum performance than that of robots controlled by ordered and disordered nets. Notably, in general, adaptation by change of couplings produces robots with slightly higher performance than those adapted by changing their structure. Moreover, we observe that when adapted in their structure, ordered networks tend to move to the critical dynamical regime. These results provide further support to the conjecture that critical regimes favor adaptation and indicate the advantage of calibrating robot control systems at dynamical critical states. Full article

(This article belongs to the Special Issue The Principle of Dynamical Criticality)

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18 pages, 3940 KiB

Open AccessArticle

The Analysis of Mammalian Hearing Systems Supports the Hypothesis That Criticality Favors Neuronal Information Representation but Not Computation

by Ruedi Stoop and Florian Gomez

Entropy 2022, 24(4), 540; https://0-doi-org.brum.beds.ac.uk/10.3390/e24040540 - 12 Apr 2022

Cited by 1 | Viewed by 1548

Abstract

In the neighborhood of critical states, distinct materials exhibit the same physical behavior, expressed by common simple laws among measurable observables, hence rendering a more detailed analysis of the individual systems obsolete. It is a widespread view that critical states are fundamental to [...] Read more.

In the neighborhood of critical states, distinct materials exhibit the same physical behavior, expressed by common simple laws among measurable observables, hence rendering a more detailed analysis of the individual systems obsolete. It is a widespread view that critical states are fundamental to neuroscience and directly favor computation. We argue here that from an evolutionary point of view, critical points seem indeed to be a natural phenomenon. Using mammalian hearing as our example, we show, however, explicitly that criticality does not describe the proper computational process and thus is only indirectly related to the computation in neural systems. Full article

(This article belongs to the Special Issue The Principle of Dynamical Criticality)

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19 pages, 3314 KiB

Open AccessArticle

Tipping the Balance: A Criticality Perspective

by Indrani Bose

Entropy 2022, 24(3), 405; https://0-doi-org.brum.beds.ac.uk/10.3390/e24030405 - 14 Mar 2022

Cited by 2 | Viewed by 2019

Abstract

Cell populations are often characterised by phenotypic heterogeneity in the form of two distinct subpopulations. We consider a model of tumour cells consisting of two subpopulations: non-cancer promoting (NCP) and cancer-promoting (CP). Under steady state conditions, the model has similarities with a well-known [...] Read more.

Cell populations are often characterised by phenotypic heterogeneity in the form of two distinct subpopulations. We consider a model of tumour cells consisting of two subpopulations: non-cancer promoting (NCP) and cancer-promoting (CP). Under steady state conditions, the model has similarities with a well-known model of population genetics which exhibits a purely noise-induced transition from unimodality to bimodality at a critical value of the noise intensity

σ^{2}

. The noise is associated with the parameter λ representing the system-environment coupling. In the case of the tumour model, λ has a natural interpretation in terms of the tissue microenvironment which has considerable influence on the phenotypic composition of the tumour. Oncogenic transformations give rise to considerable fluctuations in the parameter. We compute the

λ - σ^{2}

phase diagram in a stochastic setting, drawing analogies between bifurcations and phase transitions. In the region of bimodality, a transition from a state of balance to a state of dominance, in terms of the competing subpopulations, occurs at λ = 0. Away from this point, the NCP (CP) subpopulation becomes dominant as λ changes towards positive (negative) values. The variance of the steady state probability density function as well as two entropic measures provide characteristic signatures at the transition point. Full article

(This article belongs to the Special Issue The Principle of Dynamical Criticality)

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27 pages, 7447 KiB

Open AccessFeature PaperArticle

Attractor-Specific and Common Expression Values in Random Boolean Network Models (with a Preliminary Look at Single-Cell Data)

by Marco Villani, Gianluca D’Addese, Stuart A. Kauffman and Roberto Serra

Entropy 2022, 24(3), 311; https://0-doi-org.brum.beds.ac.uk/10.3390/e24030311 - 22 Feb 2022

Cited by 1 | Viewed by 1871

Abstract

Random Boolean Networks (RBNs for short) are strongly simplified models of gene regulatory networks (GRNs), which have also been widely studied as abstract models of complex systems and have been used to simulate different phenomena. We define the “common sea” (CS) as the [...] Read more.

Random Boolean Networks (RBNs for short) are strongly simplified models of gene regulatory networks (GRNs), which have also been widely studied as abstract models of complex systems and have been used to simulate different phenomena. We define the “common sea” (CS) as the set of nodes that take the same value in all the attractors of a given network realization, and the “specific part” (SP) as the set of all the other nodes, and we study their properties in different ensembles, generated with different parameter values. Both the CS and of the SP can be composed of one or more weakly connected components, which are emergent intermediate-level structures. We show that the study of these sets provides very important information about the behavior of the model. The distribution of distances between attractors is also examined. Moreover, we show how the notion of a “common sea” of genes can be used to analyze data from single-cell experiments. Full article

(This article belongs to the Special Issue The Principle of Dynamical Criticality)

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Review

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18 pages, 3999 KiB

Open AccessReview

Robustness and Flexibility of Neural Function through Dynamical Criticality

by Marcelo O. Magnasco

Entropy 2022, 24(5), 591; https://0-doi-org.brum.beds.ac.uk/10.3390/e24050591 - 23 Apr 2022

Viewed by 2377

Abstract

In theoretical biology, robustness refers to the ability of a biological system to function properly even under perturbation of basic parameters (e.g., temperature or pH), which in mathematical models is reflected in not needing to fine-tune basic parameter constants; flexibility refers to the [...] Read more.

In theoretical biology, robustness refers to the ability of a biological system to function properly even under perturbation of basic parameters (e.g., temperature or pH), which in mathematical models is reflected in not needing to fine-tune basic parameter constants; flexibility refers to the ability of a system to switch functions or behaviors easily and effortlessly. While there are extensive explorations of the concept of robustness and what it requires mathematically, understanding flexibility has proven more elusive, as well as also elucidating the apparent opposition between what is required mathematically for models to implement either. In this paper we address a number of arguments in theoretical neuroscience showing that both robustness and flexibility can be attained by systems that poise themselves at the onset of a large number of dynamical bifurcations, or dynamical criticality, and how such poising can have a profound influence on integration of information processing and function. Finally, we examine critical map lattices, which are coupled map lattices where the coupling is dynamically critical in the sense of having purely imaginary eigenvalues. We show that these map lattices provide an explicit connection between dynamical criticality in the sense we have used and “edge of chaos” criticality. Full article

(This article belongs to the Special Issue The Principle of Dynamical Criticality)

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