{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:44:43Z","timestamp":1760237083205,"version":"build-2065373602"},"reference-count":99,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,2,25]],"date-time":"2020-02-25T00:00:00Z","timestamp":1582588800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Genetic regulatory networks have evolved by complexifying their control systems with numerous effectors (inhibitors and activators). That is, for example, the case for the double inhibition by microRNAs and circular RNAs, which introduce a ubiquitous double brake control reducing in general the number of attractors of the complex genetic networks (e.g., by destroying positive regulation circuits), in which complexity indices are the number of nodes, their connectivity, the number of strong connected components and the size of their interaction graph. The stability and robustness of the networks correspond to their ability to respectively recover from dynamical and structural disturbances the same asymptotic trajectories, and hence the same number and nature of their attractors. The complexity of the dynamics is quantified here using the notion of attractor entropy: it describes the way the invariant measure of the dynamics is spread over the state space. The stability (robustness) is characterized by the rate at which the system returns to its equilibrium trajectories (invariant measure) after a dynamical (structural) perturbation. The mathematical relationships between the indices of complexity, stability and robustness are presented in case of Markov chains related to threshold Boolean random regulatory networks updated with a Hopfield-like rule. The entropy of the invariant measure of a network as well as the Kolmogorov-Sina\u00ef entropy of the Markov transition matrix ruling its random dynamics can be considered complexity, stability and robustness indices; and it is possible to exploit the links between these notions to characterize the resilience of a biological system with respect to endogenous or exogenous perturbations. The example of the genetic network controlling the kinin-kallikrein system involved in a pathology called angioedema shows the practical interest of the present approach of the complexity and robustness in two cases, its physiological normal and pathological, abnormal, dynamical behaviors.<\/jats:p>","DOI":"10.3390\/e22030260","type":"journal-article","created":{"date-parts":[[2020,2,25]],"date-time":"2020-02-25T08:12:22Z","timestamp":1582618342000},"page":"260","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Entropy as a Robustness Marker in Genetic Regulatory Networks"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9203-056X","authenticated-orcid":false,"given":"Mustapha","family":"Rachdi","sequence":"first","affiliation":[{"name":"Team AGIM (Autonomy, Gerontechnology, Imaging, Modelling &amp; Tools for e-Gnosis Medical), Laboratory AGEIS, EA 7407, University Grenoble Alpes, Faculty of Medicine, 38700 La Tronche, France"}]},{"given":"Jules","family":"Waku","sequence":"additional","affiliation":[{"name":"LIRIMA-UMMISCO, Universit\u00e9 de Yaound\u00e9, Facult\u00e9 des Sciences, BP 812 Yaound\u00e9, Cameroun"}]},{"given":"Hana","family":"Hazgui","sequence":"additional","affiliation":[{"name":"Team AGIM (Autonomy, Gerontechnology, Imaging, Modelling &amp; Tools for e-Gnosis Medical), Laboratory AGEIS, EA 7407, University Grenoble Alpes, Faculty of Medicine, 38700 La Tronche, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8335-9240","authenticated-orcid":false,"given":"Jacques","family":"Demongeot","sequence":"additional","affiliation":[{"name":"Team AGIM (Autonomy, Gerontechnology, Imaging, Modelling &amp; Tools for e-Gnosis Medical), Laboratory AGEIS, EA 7407, University Grenoble Alpes, Faculty of Medicine, 38700 La Tronche, France"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"962","DOI":"10.1016\/j.neunet.2008.04.005","article-title":"Asymptotic behavior and phase transition in regulatory networks. I. Theoretical results","volume":"21","author":"Demongeot","year":"2008","journal-title":"Neural Netw."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1016\/j.neunet.2008.04.003","article-title":"Asymptotic behavior and phase transition in regulatory networks. II. Simulations","volume":"21","author":"Demongeot","year":"2008","journal-title":"Neural Netw."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10441-008-9029-x","article-title":"Robustness in neural and genetic networks","volume":"56","author":"Demongeot","year":"2008","journal-title":"Acta Biotheor."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1016\/S0019-9958(80)90681-6","article-title":"Comportement it\u00e9ratif des fonctions \u00e0 multiseuil","volume":"45","author":"Goles","year":"1980","journal-title":"Inf. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1007\/978-3-642-81703-8_8","article-title":"Sequential iteration of threshold functions","volume":"9","author":"Goles","year":"1981","journal-title":"Springer Ser. Synerg."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1007\/BF02462279","article-title":"Specific roles of the different Boolean mappings in random networks","volume":"44","author":"Goles","year":"1982","journal-title":"Bull. Math. Biol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/0304-3975(85)90057-X","article-title":"Dynamics of positive automata networks","volume":"41","author":"Goles","year":"1985","journal-title":"Theor. Comp. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/0166-218X(85)90029-0","article-title":"Decreasing energy functions as a tool for studying threshold networks","volume":"12","author":"Goles","year":"1985","journal-title":"Discrete Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/0166-218X(88)90031-5","article-title":"Bifurcation structure of a discrete neuronal equation","volume":"21","author":"Cosnard","year":"1988","journal-title":"Discret. Appl. Math."},{"key":"ref_10","first-page":"699","article-title":"Confineurs: Une approche stochastique","volume":"309","author":"Demongeot","year":"1989","journal-title":"Comptes Rendus Acad. Sci. S\u00e9rie I"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1063\/1.166064","article-title":"\u201cDynamical confinement\u201d in neural networks and cell cycle","volume":"5","author":"Demongeot","year":"1995","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1070\/RM1970v025n01ABEH001254","article-title":"On small random perturbations of dynamical systems","volume":"25","author":"Wentzell","year":"1970","journal-title":"Russ. Math. Surv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/cpa.3160280102","article-title":"Asymptotic evaluation of certain Markov process expectations for large time","volume":"28","author":"Donsker","year":"1975","journal-title":"Comm. Pure Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Freidlin, M.I., and Wentzell, A.D. (1984). Random Perturbations of Dynamical Systems, Springer.","DOI":"10.1007\/978-1-4684-0176-9"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1007\/BF01009685","article-title":"Statistical mechanics and population biology","volume":"30","author":"Demetrius","year":"1983","journal-title":"J. Stat. Phys."},{"key":"ref_16","first-page":"259","article-title":"A thermodynamic approach in the modeling of the cellular cycle","volume":"40","author":"Demetrius","year":"1984","journal-title":"Biometrics"},{"key":"ref_17","first-page":"231","article-title":"La d\u00e9rive d\u00e9mographique et la s\u00e9lection naturelle: Etude empirique de la France (1850\u20131965)","volume":"2","author":"Demongeot","year":"1989","journal-title":"Population"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3491","DOI":"10.1073\/pnas.94.8.3491","article-title":"Directionality principles in thermodynamics and evolution","volume":"94","author":"Demetrius","year":"1997","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jtbi.2000.2106","article-title":"Thermodynamics and evolution","volume":"206","author":"Demetrius","year":"2000","journal-title":"J. Theor. Biol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.tpb.2003.12.002","article-title":"Complexity and demographic stability in population models","volume":"65","author":"Demetrius","year":"2004","journal-title":"Theor. Popul. Biol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1016\/j.physa.2004.07.011","article-title":"Robustness and network evolution-an entropic principle","volume":"346","author":"Demetrius","year":"2005","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1098\/rsif.2006.0140","article-title":"An entropic characterization of protein interaction networks and cellular robustness","volume":"3","author":"Manke","year":"2006","journal-title":"J. R. Soc. Interface"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1016\/j.tpb.2007.05.004","article-title":"Darwinian fitness","volume":"72","author":"Demetrius","year":"2007","journal-title":"Theor. Popul. Biol."},{"key":"ref_24","first-page":"1","article-title":"Exact endomorphisms of Lebesgue spaces","volume":"39","author":"Rochlin","year":"1964","journal-title":"Am. Math. Soc. Transl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1070\/RM1967v022n05ABEH001224","article-title":"Lectures on the theory of entropy of transformations with invariant measure","volume":"22","author":"Rochlin","year":"1967","journal-title":"Russ. Math. Surv."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Wichert, A., Moreira, C., and Peter Bruza, P. (2020). Balanced Quantum-Like Bayesian Networks. Entropy, 22.","DOI":"10.3390\/e22020170"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1214\/10-PS164","article-title":"Symbolic extensions of smooth interval maps","volume":"7","author":"Downarowicz","year":"2010","journal-title":"Probab. Surv."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/S0012-365X(98)00400-2","article-title":"Complexity of sequences and dynamical systems","volume":"206","author":"Ferenczi","year":"1999","journal-title":"Discret. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2315","DOI":"10.5802\/aif.2242","article-title":"Substitution dynamical systems on infinite alphabets","volume":"56","author":"Ferenczi","year":"2006","journal-title":"Ann. l\u2019Institut Fourier"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"180399","DOI":"10.1098\/rsos.180399","article-title":"Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity","volume":"5","author":"Kiani","year":"2018","journal-title":"R. Soc. Open Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1160","DOI":"10.1016\/j.isci.2019.07.043","article-title":"An Algorithmic Information Calculus for Causal Discovery and Reprogramming Systems","volume":"19","author":"Zenil","year":"2019","journal-title":"iScience"},{"key":"ref_32","first-page":"33","article-title":"Discussions, In: Unit\u00e9s biologiques dou\u00e9es de continuit\u00e9 g\u00e9n\u00e9tique","volume":"8","year":"1949","journal-title":"Colloques Int. CNRS"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1016\/0022-5193(73)90247-6","article-title":"Boolean formalization of genetic control circuits","volume":"42","author":"Thomas","year":"1973","journal-title":"J. Theor. Biol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1007\/978-3-642-81703-8_24","article-title":"On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations","volume":"9","author":"Thomas","year":"1981","journal-title":"Springer Ser. Synerg."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1073\/pnas.79.8.2554","article-title":"Neural networks and physical systems with emergent collective computational abilities","volume":"79","author":"Hopfield","year":"1982","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4437","DOI":"10.3390\/ijms10104437","article-title":"Robustness of regulatory networks. A generic approach with applications at different levels: Physiologic, metabolic and genetic","volume":"10","author":"Demongeot","year":"2009","journal-title":"Int. J. Mol. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Demongeot, J., Goles, E., Morvan, M., Noual, M., and Sen\u00e9, S. (2010). Attraction basins as gauges of environmental robustness against boundary conditions in biological complex systems. PLoS ONE, 5.","DOI":"10.1371\/journal.pone.0011793"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.jtbi.2011.03.023","article-title":"Immunetworks, intersecting circuits and dynamics","volume":"280","author":"Demongeot","year":"2011","journal-title":"J. Theor. Biol."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.crma.2012.01.003","article-title":"Robustness in biological regulatory networks. I\u2013IV","volume":"350","author":"Demongeot","year":"2012","journal-title":"Comptes Rendus Math\u00e9matique"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1016\/j.dam.2011.11.005","article-title":"Combinatorics of Boolean automata circuits dynamics","volume":"160","author":"Demongeot","year":"2012","journal-title":"Discret. Appl. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/978-3-642-38505-6_4","article-title":"MicroRNAs and Robustness in Biological Regulatory Networks. A Generic Approach with Applications at Different Levels: Physiologic, Metabolic, and Genetic","volume":"16","author":"Demongeot","year":"2013","journal-title":"Springer Ser. Biophys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s10441-014-9229-5","article-title":"Stability, complexity and robustness in population dynamics","volume":"62","author":"Demongeot","year":"2014","journal-title":"Acta Biotheor."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"40013","DOI":"10.1142\/S0218127415400131","article-title":"Complexity and Stability in Biological Systems","volume":"25","author":"Demongeot","year":"2015","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_44","unstructured":"Tran, Q.N., and Arabnia, H.R. (2015). Genetic regulatory networks: Focus on attractors of their dynamics. Computational Biology, Bioinformatics & Systems Biology, Elsevier."},{"key":"ref_45","unstructured":"Ahima, R.S. (2015). Social Networks and Obesity. Application to an interactive system for patient-centred therapeutic education. Metabolic Syndrome: A Comprehensive Textbook, Springer."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1080\/08898480.2017.1348718","article-title":"From Susceptibility to Frailty in social networks: The case of obesity","volume":"24","author":"Demongeot","year":"2017","journal-title":"Math. Pop. Studies"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Demongeot, J., Jelassi, M., Hazgui, H., Ben Miled, S., Bellamine Ben Saoud, N., and Taramasco, C. (2018). Biological Networks Entropies: Examples in Neural Memory Networks, Genetic Regulation Networks and Social Epidemic Networks. Entropy, 20.","DOI":"10.3390\/e20010036"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.aam.2018.03.003","article-title":"Phase transitions in stochastic non-linear threshold Boolean automata networks on \u21242: The boundary impact","volume":"98","author":"Demongeot","year":"2018","journal-title":"Adv. Appl. Math."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.jtbi.2019.01.019","article-title":"Memory in plants: Boolean modelling of the learning and store\/recall mnesic functions in response to environmental stimuli","volume":"467","author":"Demongeot","year":"2019","journal-title":"J. Theor. Biol."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Demongeot, J., and Sen\u00e9, S. (2020). About block-parallel Boolean networks: A position paper. Nat. Comput., 19.","DOI":"10.1007\/s11047-019-09779-x"},{"key":"ref_51","first-page":"290","article-title":"On random graphs","volume":"6","year":"1959","journal-title":"Pub. Math."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Barabasi","year":"1959","journal-title":"Science"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1038\/35019019","article-title":"Error and attack tolerance of complex networks","volume":"406","author":"Albert","year":"2000","journal-title":"Nature"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"3039","DOI":"10.1038\/srep03039","article-title":"Cellular network entropy as the energy potential in Waddington\u2019s differentiation landscape","volume":"3","author":"Banerji","year":"2013","journal-title":"Sci. Rep."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1038\/35075138","article-title":"Lethality and centrality in protein networks","volume":"411","author":"Jeong","year":"2001","journal-title":"Nature"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"065102","DOI":"10.1103\/PhysRevE.78.065102","article-title":"Entropy rate of diffusion processes on complex networks","volume":"78","author":"Latora","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"4177","DOI":"10.1016\/j.physleta.2008.03.061","article-title":"Network Entropy Based on Topology Configuration and Its Computation to Random Networks","volume":"25","author":"Li","year":"2008","journal-title":"Chin. Phys. Lett."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Teschendorff, A.E., and Severini, S. (2010). Increased entropy of signal transduction in the cancer metastasis phenotype. BMC Syst. Biol., 4.","DOI":"10.1186\/1752-0509-4-104"},{"key":"ref_59","first-page":"802","article-title":"Increased signaling entropy in cancer requires the scale-free property of protein interaction networks","volume":"2","author":"Teschendorff","year":"2014","journal-title":"Sci. Rep."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1016\/j.ymeth.2014.03.013","article-title":"Signalling entropy: A novel network theoretical framework for systems analysis and interpretation of functional omic data","volume":"67","author":"Teschendorff","year":"2014","journal-title":"Methods"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1038\/srep00802","article-title":"Differential network entropy reveals cancer hallmarks","volume":"2","author":"West","year":"2012","journal-title":"Sci. Rep."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"6801","DOI":"10.3390\/e17106801","article-title":"On the Calculation of System Entropy in Nonlinear Stochastic Biological Networks","volume":"17","author":"Chen","year":"2015","journal-title":"Entropy"},{"key":"ref_63","first-page":"551","article-title":"Attracteurs: Une approche d\u00e9terministe","volume":"300","author":"Cosnard","year":"1985","journal-title":"Comptes Rendus Acad. Sci. Maths. S\u00e9rie I"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/BFb0076414","article-title":"On the definitions of attractors","volume":"1163","author":"Cosnard","year":"1985","journal-title":"Lect. Notes Maths."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1070","DOI":"10.1103\/PhysRevE.50.1070","article-title":"Thermodynamics of attractor enlargement","volume":"50","year":"1994","journal-title":"Phys. Rev. E"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"12271","DOI":"10.1073\/pnas.0800579105","article-title":"Potential landscape and flux framework of nonequilibrium networks: Robustness, dissipation, and coherence of biochemical oscillations","volume":"105","author":"Wang","year":"2008","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1038\/nphys2516","article-title":"How basin stability complements the linear-stability paradigm","volume":"9","author":"Menck","year":"2013","journal-title":"Nat. Phys."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"e1352","DOI":"10.1002\/wsbm.1352","article-title":"Network dynamics: Quantitative analysis of complex behavior in metabolism, organelles, and cells, from experiments to models and back","volume":"9","author":"Kurz","year":"2017","journal-title":"Wiley Interdiscipl. Rev. Syst. Biol. Med."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/0022-0396(75)90065-0","article-title":"\u03c9-limit sets for Axiom A diffeomorphisms","volume":"18","author":"Bowen","year":"1978","journal-title":"J. Diff. Equ."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/BF02684369","article-title":"Expanding attractors","volume":"43","author":"Williams","year":"1974","journal-title":"Publ. Math. l\u2019IH\u00c9S"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/BF01206949","article-title":"Small random perturbations and the definition of attractors","volume":"82","author":"Ruelle","year":"1981","journal-title":"Comm. Math. Phys."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"1383","DOI":"10.1080\/03605308808820580","article-title":"Attractors of asymptotically compact processes and applications to nonlinear partial differential equations","volume":"13","author":"Haraux","year":"1988","journal-title":"Commun. Partial Differ. Equ."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1016\/j.crma.2006.11.014","article-title":"Li\u00e9nard systems and potential-Hamiltonian decomposition. III Applications in biology","volume":"344","author":"Glade","year":"2007","journal-title":"Comptes Rendus Math\u00e9matique"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"1213","DOI":"10.1016\/S0893-6080(99)00068-4","article-title":"Synchronization and desynchronization of neural oscillators: Comparison of two models","volume":"12","author":"Tonnelier","year":"1999","journal-title":"Neural Netw."},{"key":"ref_75","doi-asserted-by":"crossref","unstructured":"Robert, F. (1986). Discrete Iterations: A Metric Study, Springer Series in Computational Mathematics, Springer.","DOI":"10.1007\/978-3-642-61607-5"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s10441-010-9099-4","article-title":"The isochronal fibration: Characterization and implication in biology","volume":"58","author":"Glade","year":"2010","journal-title":"Acta Biotheor."},{"key":"ref_77","first-page":"69","article-title":"Positive feedback circuits and memory","volume":"323","author":"Demongeot","year":"2000","journal-title":"Comptes Rendus Acad. Sci. Paris Life Sci."},{"key":"ref_78","first-page":"115","article-title":"Theory of the frustration effect in spin glasses: I","volume":"2","author":"Toulouse","year":"1977","journal-title":"Commun. Phys."},{"key":"ref_79","unstructured":"Waddington, C.H. (1940). Organizers & Genes, Cambridge University Press."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"784","DOI":"10.1038\/228784a0","article-title":"Connectance of large dynamic systems: Critical values for stability","volume":"228","author":"Gardner","year":"1970","journal-title":"Nature"},{"key":"ref_81","unstructured":"Thom, R. (1972). Structural Stability and Morphogenesis, CRC Press."},{"key":"ref_82","unstructured":"Wagner, A. (2005). Robustness and Evolvability in Living Systems, Princeton University Press."},{"key":"ref_83","first-page":"581","article-title":"The robustness of a biochemical network can be inferred mathematically from its architecture","volume":"328","author":"Gunawardena","year":"2010","journal-title":"Biol. Syst. Theory"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1016\/j.crvi.2006.07.010","article-title":"Potential-Hamiltonian decomposition of cellular automata. Application to degeneracy of genetic code and cyclic codes III","volume":"329","author":"Demongeot","year":"2006","journal-title":"Comptes Rendus Biol."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1140\/epjb\/e2004-00111-4","article-title":"Betweenness centrality in large complex networks","volume":"38","year":"2004","journal-title":"Eur. Phys. J. B"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"12201","DOI":"10.1073\/pnas.1810452115","article-title":"Eigenvector centrality for characterization of protein allosteric pathways","volume":"115","author":"Negre","year":"2018","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"103","DOI":"10.2147\/BTT.S27566","article-title":"Hereditary angioedema: Epidemiology, management, and role of icatibant","volume":"7","author":"Ghazi","year":"2013","journal-title":"Biol. Targets Ther."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1016\/j.anai.2018.05.031","article-title":"SERPING1 and F12 combined variants in a hereditary angioedema family","volume":"121","author":"Charignon","year":"2018","journal-title":"Ann. Allergy Asthma Immunol."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1006\/jtbi.2002.2544","article-title":"Positive and negative feedback: Striking a balance between necessary antagonists","volume":"216","author":"Cinquin","year":"2002","journal-title":"J. Theor. Biol."},{"key":"ref_90","doi-asserted-by":"crossref","unstructured":"Fat\u00e8s, N. (2020). A tutorial on elementary cellular automata with fully asynchronous updating. Nat. Comput., 19.","DOI":"10.1007\/s11047-020-09782-7"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1049\/iet-syb:20060038","article-title":"Graph theory and networks in Biology","volume":"1","author":"Mason","year":"2007","journal-title":"IET Syst. Biol."},{"key":"ref_92","first-page":"0645","article-title":"Network science of biological systems at different scales: A review","volume":"1571","author":"Gosak","year":"2017","journal-title":"Phys. Life Rev."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1056\/NEJMsa066082","article-title":"The Spread of Obesity in a Large Social Network over 32 Years","volume":"357","author":"Christakis","year":"2007","journal-title":"N. Engl. J. Med."},{"key":"ref_94","unstructured":"Federer, C., and Zylberberg, J. (2017). A self-organizing memory network. BioRxiv."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1016\/j.neuron.2006.11.008","article-title":"Biological Pattern Generation: The Cellular and Computational Logic of Networks in Motion","volume":"52","author":"Grillner","year":"2006","journal-title":"Neuron"},{"key":"ref_96","first-page":"31","article-title":"Ciliary Movement and Its Control in Paramecium","volume":"31","author":"Naitoh","year":"2007","journal-title":"J. Eukaryot. Microbiol."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"16570","DOI":"10.1038\/s41598-019-53118-0","article-title":"Flexible control of movement in plants","volume":"9","author":"Guerra","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_98","first-page":"525","article-title":"Some large deviation results for dynamical systems","volume":"318","author":"Young","year":"1990","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_99","first-page":"112","article-title":"Modeling regulatory networks with weight matrices","volume":"4","author":"Weaver","year":"1999","journal-title":"Pac. Symp. Biocomput."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/260\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:01:26Z","timestamp":1760173286000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/260"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,25]]},"references-count":99,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["e22030260"],"URL":"https:\/\/doi.org\/10.3390\/e22030260","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,2,25]]}}}