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The Principle of Dynamical Criticality

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Entropy and Biology".

Deadline for manuscript submissions: closed (17 December 2022) | Viewed by 19684

Special Issue Editors


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Guest Editor
Institute for Systems Biology, Seattle, WA 98109, USA
Interests: developmental biology; combinatorial chemistry; molecular evolution; biocomplexity

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Guest Editor
Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, 41125 Modena, Italy
Interests: complex systems; protocells; origin of life; information theory

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Guest Editor
Institute for Systems Biology, Seattle, WA 98109, USA
Interests: systems biology; computational biology; cancer

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Guest Editor
Institute for Systems Biology, Seattle, WA 98109, USA
Interests: systems biology; cancer; stem cells; complexity

Special Issue Information

Dear Colleagues,

While life, as Darwin noted, displays “endless forms most beautiful” at a macroscopic scale, it appears much more uniform at a microscopic level, where all living systems share many common structural and functional features. There are, however, very few “operating principles” at a macroscale that seem to hold for all (or for large classes of) living systems. The most prominent such principle is of course biological evolution, but unfortunately there are not many others. They would be very welcome in an era that withstands an unprecedented deluge of biological data, while theory is lagging behind.

A very interesting candidate is the “criticality” principle (CP for short), which posits that evolution should have driven living beings towards critical states, since they are favored with respect to those that are either chaotic or ordered—a statement that is, in principle, amenable to experimental verification. Moreover, since dynamically critical states are endowed with peculiar computational properties, they are also interesting outside the domain of biology, e.g., in designing artificial systems.

There are some major points which need to be addressed concerning the CP, including (but not limited to)

  • understanding under which conditions (e.g., a changing external environment) dynamically critical states outperform ordered or disordered ones
  • analyzing new biological systems to determine whether they do follow the principle
  • refining the definition of criticality in the case of strongly non-ergodic systems, where it cannot be assumed that the system visits all of its possible states
  • exploring the usefulness of the concept in artificial domains (e.g., in robotics or learning systems)
  • building abstract models of the coevolution of a set of interacting systems to understand under which conditions they evolve towards criticality
  • clinical implications of criticality, particularly for neurological disorders such as epilepsy and neurodegenarative disease, as well as anesthesia, sleep medicine and psychiatry

We will be particularly interested in papers dealing with

  • the theory of criticality
  • the role of criticality in biological systems
  • the role of criticality in artificial systems
  • clinical applications of criticality

Prof. Dr. Stuart A. Kauffman
Prof. Dr. Roberto Serra
Prof. Dr. Ilya Shmulevich
Prof. Dr. Sui Huang
Guest Editors

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Published Papers (8 papers)

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Research

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15 pages, 1574 KiB  
Article
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://doi.org/10.3390/e25020374 - 18 Feb 2023
Cited by 3 | Viewed by 2059
Abstract
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  
Article
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://doi.org/10.3390/e25020254 - 31 Jan 2023
Cited by 3 | Viewed by 2146
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  
Article
Emergent Criticality in Coupled Boolean Networks
by Chris Kang, Madelynn McElroy and Nikolaos K. Voulgarakis
Entropy 2023, 25(2), 235; https://doi.org/10.3390/e25020235 - 27 Jan 2023
Cited by 2 | Viewed by 1792
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  
Article
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://doi.org/10.3390/e24101368 - 27 Sep 2022
Cited by 8 | Viewed by 2329
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  
Article
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://doi.org/10.3390/e24040540 - 12 Apr 2022
Cited by 1 | Viewed by 1773
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  
Article
Tipping the Balance: A Criticality Perspective
by Indrani Bose
Entropy 2022, 24(3), 405; https://doi.org/10.3390/e24030405 - 14 Mar 2022
Cited by 2 | Viewed by 2188
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 σ2. 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 λσ2 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  
Article
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://doi.org/10.3390/e24030311 - 22 Feb 2022
Cited by 2 | Viewed by 2015
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  
Review
Robustness and Flexibility of Neural Function through Dynamical Criticality
by Marcelo O. Magnasco
Entropy 2022, 24(5), 591; https://doi.org/10.3390/e24050591 - 23 Apr 2022
Viewed by 2556
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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