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Konishi K, Yoshida K, Sugitani Y, Hara N. Delay-induced amplitude death in multiplex oscillator network with frequency-mismatched layers. Phys Rev E 2024; 109:014220. [PMID: 38366515 DOI: 10.1103/physreve.109.014220] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2023] [Accepted: 12/01/2023] [Indexed: 02/18/2024]
Abstract
The present paper analytically investigates the stability of amplitude death in a multiplex Stuart-Landau oscillator network with a delayed interlayer connection. The network consists of two frequency-mismatched layers, and all oscillators in each layer have identical frequencies. We show that, if the matrices describing the network topologies of each layer commute, then the characteristic equation governing the stability can be reduced to a simple form. This form reveals that the stability of amplitude death in the multiplex network is equally or more conservative than that in a pair of frequency-mismatched oscillators coupled by a delayed connection. In addition, we provide a procedure for designing the delayed interlayer connection such that amplitude death is stable for any commuting matrices and for any intralayer coupling strength. These analytical results are verified through numerical examples. Moreover, we numerically discuss the results for the case in which the commutative property does not hold.
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Affiliation(s)
- Keiji Konishi
- Department of Electrical and Electronic Systems Engineering, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
| | - Koki Yoshida
- National Institute of Technology, Toyama College, 13 Hongo-machi, Toyama city, Toyama 939-8630, Japan
| | - Yoshiki Sugitani
- Department of Electrical and Electronic Systems Engineering, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
| | - Naoyuki Hara
- Department of Electrical and Electronic Systems Engineering, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
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2
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Winkler M, Dumont G, Schöll E, Gutkin B. Phase response approaches to neural activity models with distributed delay. BIOLOGICAL CYBERNETICS 2022; 116:191-203. [PMID: 34853889 DOI: 10.1007/s00422-021-00910-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2021] [Accepted: 10/31/2021] [Indexed: 06/13/2023]
Abstract
In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson-Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks.
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Affiliation(s)
- Marius Winkler
- Group for Neural Theory, LNC INSERM U960, DEC, Ecole Normale Supérieure PSL* University, 24 rue Lhomond, 75005, Paris, France
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623, Berlin, Germany
| | - Grégory Dumont
- Group for Neural Theory, LNC INSERM U960, DEC, Ecole Normale Supérieure PSL* University, 24 rue Lhomond, 75005, Paris, France
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623, Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität, Philippstraße 13, 10115, Berlin, Germany
- Potsdam Institute for Climate Impact Research, Telegrafenberg A 31, 14473, Potsdam, Germany
| | - Boris Gutkin
- Group for Neural Theory, LNC INSERM U960, DEC, Ecole Normale Supérieure PSL* University, 24 rue Lhomond, 75005, Paris, France.
- Center for Cognition and Decision Making, Institue for Cognitive Neuroscience, NRU Higher School of Economics, Krivokolenniy sidewalk 3, 101000, Moscow, Russia.
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3
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Mizukami S, Konishi K, Sugitani Y, Kouda T, Hara N. Effects of frequency mismatch on amplitude death in delay-coupled oscillators. Phys Rev E 2021; 104:054207. [PMID: 34942770 DOI: 10.1103/physreve.104.054207] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2021] [Accepted: 10/18/2021] [Indexed: 11/07/2022]
Abstract
The present paper analytically reveals the effects of frequency mismatch on the stability of an equilibrium point within a pair of Stuart-Landau oscillators coupled by a delay connection. By analyzing the roots of the characteristic function governing the stability, we find that there exist four types of boundary curves of stability in a coupling parameters space. These four types depend only on the frequency mismatch. The analytical results allow us to design coupling parameters and frequency mismatch such that the equilibrium point is locally stable. We show that, if we choose appropriate frequency mismatches and delay times, then it is possible to induce amplitude death with strong stability, even by weak coupling. In addition, we show that parts of these analytical results are valid for oscillator networks with complete bipartite topologies.
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Affiliation(s)
- Shinsuke Mizukami
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
| | - Keiji Konishi
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
| | - Yoshiki Sugitani
- Department of Electrical and Electronic Systems Engineering, Ibaraki University, 4-12-1 Nakanarusawa, Hitachi, Ibaraki 316-8511, Japan
| | - Takahiro Kouda
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
| | - Naoyuki Hara
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
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4
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Ross A, Kyrychko SN, Blyuss KB, Kyrychko YN. Dynamics of coupled Kuramoto oscillators with distributed delays. CHAOS (WOODBURY, N.Y.) 2021; 31:103107. [PMID: 34717313 DOI: 10.1063/5.0055467] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/29/2021] [Accepted: 09/07/2021] [Indexed: 06/13/2023]
Abstract
This paper studies the effects of two different types of distributed-delay coupling in the system of two mutually coupled Kuramoto oscillators: one where the delay distribution is considered inside the coupling function and the other where the distribution enters outside the coupling function. In both cases, the existence and stability of phase-locked solutions is analyzed for uniform and gamma distribution kernels. The results show that while having the distribution inside the coupling function only changes parameter regions where phase-locked solutions exist, when the distribution is taken outside the coupling function, it affects both the existence, as well as stability properties of in- and anti-phase states. For both distribution types, various branches of phase-locked solutions are computed, and regions of their stability are identified for uniform, weak, and strong gamma distributions.
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Affiliation(s)
- A Ross
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
| | - S N Kyrychko
- Polyakov Institute of Geotechnical Mechanics, National Academy of Sciences of Ukraine, Simferopolska str. 2a, Dnipro 49005, Ukraine
| | - K B Blyuss
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
| | - Y N Kyrychko
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
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Cortez MJ, Hong H, Choi B, Kim JK, Josić K. Hierarchical Bayesian models of transcriptional and translational regulation processes with delays. Bioinformatics 2021; 38:187-195. [PMID: 34450624 PMCID: PMC8696106 DOI: 10.1093/bioinformatics/btab618] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2021] [Revised: 08/19/2021] [Accepted: 08/25/2021] [Indexed: 02/05/2023] Open
Abstract
MOTIVATION Simultaneous recordings of gene network dynamics across large populations have revealed that cell characteristics vary considerably even in clonal lines. Inferring the variability of parameters that determine gene dynamics is key to understanding cellular behavior. However, this is complicated by the fact that the outcomes and effects of many reactions are not observable directly. Unobserved reactions can be replaced with time delays to reduce model dimensionality and simplify inference. However, the resulting models are non-Markovian, and require the development of new inference techniques. RESULTS We propose a non-Markovian, hierarchical Bayesian inference framework for quantifying the variability of cellular processes within and across cells in a population. We illustrate our approach using a delayed birth-death process. In general, a distributed delay model, rather than a popular fixed delay model, is needed for inference, even if only mean reaction delays are of interest. Using in silico and experimental data we show that the proposed hierarchical framework is robust and leads to improved estimates compared to its non-hierarchical counterpart. We apply our method to data obtained using time-lapse microscopy and infer the parameters that describe the dynamics of protein production at the single cell and population level. The mean delays in protein production are larger than previously reported, have a coefficient of variation of around 0.2 across the population, and are not strongly correlated with protein production or growth rates. AVAILABILITY AND IMPLEMENTATION Accompanying code in Python is available at https://github.com/mvcortez/Bayesian-Inference. CONTACT kresimir.josic@gmail.com or jaekkim@kaist.ac.kr or cbskust@korea.ac.kr. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.
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Affiliation(s)
- Mark Jayson Cortez
- Department of Mathematics, University of Houston, Houston, TX 77204, USA,Institute of Mathematical Sciences and Physics, University of the Philippines Los Baños, Laguna 4031, Philippines
| | - Hyukpyo Hong
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea,Biomedical Mathematics Group, Institute for Basic Science, Daejeon 34126, Korea
| | - Boseung Choi
- To whom correspondence should be addressed. or or
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It's about time: Analysing simplifying assumptions for modelling multi-step pathways in systems biology. PLoS Comput Biol 2020; 16:e1007982. [PMID: 32598362 PMCID: PMC7351226 DOI: 10.1371/journal.pcbi.1007982] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2019] [Revised: 07/10/2020] [Accepted: 05/27/2020] [Indexed: 11/19/2022] Open
Abstract
Thoughtful use of simplifying assumptions is crucial to make systems biology models tractable while still representative of the underlying biology. A useful simplification can elucidate the core dynamics of a system. A poorly chosen assumption can, however, either render a model too complicated for making conclusions or it can prevent an otherwise accurate model from describing experimentally observed dynamics. Here, we perform a computational investigation of sequential multi-step pathway models that contain fewer pathway steps than the system they are designed to emulate. We demonstrate when such models will fail to reproduce data and how detrimental truncation of a pathway leads to detectable signatures in model dynamics and its optimised parameters. An alternative assumption is suggested for simplifying such pathways. Rather than assuming a truncated number of pathway steps, we propose to use the assumption that the rates of information propagation along the pathway is homogeneous and, instead, letting the length of the pathway be a free parameter. We first focus on linear pathways that are sequential and have first-order kinetics, and we show how this assumption results in a three-parameter model that consistently outperforms its truncated rival and a delay differential equation alternative in recapitulating observed dynamics. We then show how the proposed assumption allows for similarly terse and effective models of non-linear pathways. Our results provide a foundation for well-informed decision making during model simplifications.
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7
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Rahman B, Kyrychko YN, Blyuss KB. Dynamics of unidirectionally-coupled ring neural network with discrete and distributed delays. J Math Biol 2020; 80:1617-1653. [PMID: 32002658 DOI: 10.1007/s00285-020-01475-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2019] [Revised: 01/13/2020] [Indexed: 11/27/2022]
Abstract
In this paper, we consider a ring neural network with one-way distributed-delay coupling between the neurons and a discrete delayed self-feedback. In the general case of the distribution kernels, we are able to find a subset of the amplitude death regions depending on even (odd) number of neurons in the network. Furthermore, in order to show the full region of the amplitude death, we use particular delay distributions, including Dirac delta function and gamma distribution. Stability conditions for the trivial steady state are found in parameter spaces consisting of the synaptic weight of the self-feedback and the coupling strength between the neurons, as well as the delayed self-feedback and the coupling strength between the neurons. It is shown that both Hopf and steady-state bifurcations may occur when the steady state loses stability. We also perform numerical simulations of the fully nonlinear system to confirm theoretical findings.
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Affiliation(s)
- Bootan Rahman
- Mathematics Unit, School of Science and Engineering, University of Kurdistan Hewlêr (UKH), Erbil, Kurdistan Region, Iraq.
| | - Yuliya N Kyrychko
- Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, UK
| | - Konstantin B Blyuss
- Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, UK
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8
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Kyrychko YN, Schwartz IB. Enhancing noise-induced switching times in systems with distributed delays. CHAOS (WOODBURY, N.Y.) 2018; 28:063106. [PMID: 29960399 DOI: 10.1063/1.5034106] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
The paper addresses the problem of calculating the noise-induced switching rates in systems with delay-distributed kernels and Gaussian noise. A general variational formulation for the switching rate is derived for any distribution kernel, and the obtained equations of motion and boundary conditions represent the most probable, or optimal, path, which maximizes the probability of escape. Explicit analytical results for the switching rates for small mean time delays are obtained for the uniform and bi-modal (or two-peak) distributions. They suggest that increasing the width of the distribution leads to an increase in the switching times even for longer values of mean time delays for both examples of the distribution kernel, and the increase is higher in the case of the two-peak distribution. Analytical predictions are compared to the direct numerical simulations and show excellent agreement between theory and numerical experiment.
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Affiliation(s)
- Y N Kyrychko
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
| | - I B Schwartz
- US Naval Research Laboratory, Code 6792, Nonlinear System Dynamics Section, Plasma Physics Division, Washington, DC 20375, USA
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9
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Sugitani Y, Konishi K. Design of coupling parameters for inducing amplitude death in Cartesian product networks of delayed coupled oscillators. Phys Rev E 2018; 96:042216. [PMID: 29347511 DOI: 10.1103/physreve.96.042216] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2017] [Indexed: 11/07/2022]
Abstract
The present study investigates amplitude death in Cartesian product networks of two subnetworks, where each subnetwork has a different coupling delay. The property of the Cartesian product helps us to analyze the stability of amplitude death. Our analysis reveals that amplitude death can occur for long coupling delays if there is a suitable difference in the coupling delays in the two subnetworks. Furthermore, based on the edge theorem in robust control theory, we propose two design procedures of coupling parameters for inducing amplitude death in the Cartesian product networks. Our procedures do not require any information of topologies of the subnetworks. The validity of these procedures is numerically confirmed.
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Affiliation(s)
- Yoshiki Sugitani
- Department of Electrical and Electronic Engineering, Ibaraki University 4-12-1 Nakanarusawa, Hitachi, Ibaraki 316-8511, Japan
| | - Keiji Konishi
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
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10
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Rahman B, Blyuss KB, Kyrychko YN. Aging transition in systems of oscillators with global distributed-delay coupling. Phys Rev E 2017; 96:032203. [PMID: 29347035 DOI: 10.1103/physreve.96.032203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2016] [Indexed: 06/07/2023]
Abstract
We consider a globally coupled network of active (oscillatory) and inactive (nonoscillatory) oscillators with distributed-delay coupling. Conditions for aging transition, associated with suppression of oscillations, are derived for uniform and gamma delay distributions in terms of coupling parameters and the proportion of inactive oscillators. The results suggest that for the uniform distribution increasing the width of distribution for the same mean delay allows aging transition to happen for a smaller coupling strength and a smaller proportion of inactive elements. For gamma distribution with sufficiently large mean time delay, it may be possible to achieve aging transition for an arbitrary proportion of inactive oscillators, as long as the coupling strength lies in a certain range.
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Affiliation(s)
- B Rahman
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, England, United Kingdom
| | - K B Blyuss
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, England, United Kingdom
| | - Y N Kyrychko
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, England, United Kingdom
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12
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Gjurchinovski A, Schöll E, Zakharova A. Control of amplitude chimeras by time delay in oscillator networks. Phys Rev E 2017; 95:042218. [PMID: 28505829 DOI: 10.1103/physreve.95.042218] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2017] [Indexed: 06/07/2023]
Abstract
We investigate the influence of time-delayed coupling in a ring network of nonlocally coupled Stuart-Landau oscillators upon chimera states, i.e., space-time patterns with coexisting partially coherent and partially incoherent domains. We focus on amplitude chimeras, which exhibit incoherent behavior with respect to the amplitude rather than the phase and are transient patterns, and we show that their lifetime can be significantly enhanced by coupling delay. To characterize their transition to phase-lag synchronization (coherent traveling waves) and other coherent structures, we generalize the Kuramoto order parameter. Contrasting the results for instantaneous coupling with those for constant coupling delay, for time-varying delay, and for distributed-delay coupling, we demonstrate that the lifetime of amplitude chimera states and related partially incoherent states can be controlled, i.e., deliberately reduced or increased, depending upon the type of coupling delay.
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Affiliation(s)
- Aleksandar Gjurchinovski
- Institute of Physics, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, P.O. Box 162, 1000 Skopje, Macedonia
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
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Zakharova A, Kapeller M, Schöll E. Amplitude chimeras and chimera death in dynamical networks. ACTA ACUST UNITED AC 2016. [DOI: 10.1088/1742-6596/727/1/012018] [Citation(s) in RCA: 36] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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14
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Time-Delayed Models of Gene Regulatory Networks. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2015; 2015:347273. [PMID: 26576197 PMCID: PMC4632181 DOI: 10.1155/2015/347273] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/14/2015] [Revised: 08/31/2015] [Accepted: 09/14/2015] [Indexed: 11/17/2022]
Abstract
We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternative modelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems.
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15
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Wille C, Lehnert J, Schöll E. Synchronization-desynchronization transitions in complex networks: an interplay of distributed time delay and inhibitory nodes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032908. [PMID: 25314505 DOI: 10.1103/physreve.90.032908] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2014] [Indexed: 05/26/2023]
Abstract
We investigate the combined effects of distributed delay and the balance between excitatory and inhibitory nodes on the stability of synchronous oscillations in a network of coupled Stuart-Landau oscillators. To this end a symmetric network model is proposed for which the stability can be investigated analytically. It is found that beyond a critical inhibition ratio, synchronization tends to be unstable. However, increasing distributional widths can counteract this trend, leading to multiple resynchronization transitions at relatively high inhibition ratios. The extended applicability of the results is confirmed by numerical studies on asymmetrically perturbed network topologies. All investigations are performed on two distribution types, a uniform distribution and a Γ distribution.
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Affiliation(s)
- Carolin Wille
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Judith Lehnert
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
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16
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Blyuss KB, Nicholson LB. Understanding the roles of activation threshold and infections in the dynamics of autoimmune disease. J Theor Biol 2014; 375:13-20. [PMID: 25150457 DOI: 10.1016/j.jtbi.2014.08.019] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2014] [Revised: 06/30/2014] [Accepted: 08/11/2014] [Indexed: 12/21/2022]
Abstract
Onset and development of autoimmunity have been attributed to a number of factors, including genetic predisposition, age and different environmental factors. In this paper we discuss mathematical models of autoimmunity with an emphasis on two particular aspects of immune dynamics: breakdown of immune tolerance in response to an infection with a pathogen, and interactions between T cells with different activation thresholds. We illustrate how the explicit account of T cells with different activation thresholds provides a viable model of immune dynamics able to reproduce several types of immune behaviour, including normal clearance of infection, emergence of a chronic state, and development of a recurrent infection with autoimmunity. We discuss a number of open research problems that can be addressed within the same modelling framework.
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Affiliation(s)
- K B Blyuss
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UK.
| | - L B Nicholson
- School of Cellular and Molecular Medicine & School of Clinical Sciences, University of Bristol, University Walk, Bristol BS8 1TD, UK.
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17
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Zakharova A, Kapeller M, Schöll E. Chimera death: symmetry breaking in dynamical networks. PHYSICAL REVIEW LETTERS 2014; 112:154101. [PMID: 24785041 DOI: 10.1103/physrevlett.112.154101] [Citation(s) in RCA: 152] [Impact Index Per Article: 15.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2014] [Indexed: 05/03/2023]
Abstract
For a network of generic oscillators with nonlocal topology and symmetry-breaking coupling we establish novel partially coherent inhomogeneous spatial patterns, which combine the features of chimera states (coexisting incongruous coherent and incoherent domains) and oscillation death (oscillation suppression), which we call "chimera death". We show that due to the interplay of nonlocality and breaking of rotational symmetry by the coupling, two distinct scenarios from oscillatory behavior to a stationary state regime are possible: a transition from an amplitude chimera to chimera death via in-phase synchronized oscillations and a direct abrupt transition for larger coupling strength.
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Affiliation(s)
- Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Marie Kapeller
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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18
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Gjurchinovski A, Zakharova A, Schöll E. Amplitude death in oscillator networks with variable-delay coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:032915. [PMID: 24730921 DOI: 10.1103/physreve.89.032915] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/2013] [Indexed: 06/03/2023]
Abstract
We study the conditions of amplitude death in a network of delay-coupled limit cycle oscillators by including time-varying delay in the coupling and self-feedback. By generalizing the master stability function formalism to include variable-delay connections with high-frequency delay modulations (i.e., the distributed-delay limit), we analyze the regimes of amplitude death in a ring network of Stuart-Landau oscillators and demonstrate the superiority of the proposed method with respect to the constant delay case. The possibility of stabilizing the steady state is restricted by the odd-number property of the local node dynamics independently of the network topology and the coupling parameters.
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Affiliation(s)
- Aleksandar Gjurchinovski
- Institute of Physics, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, P. O. Box 162, 1000 Skopje, Macedonia
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
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19
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Choe CU, Kim RS, Jang H, Hövel P, Schöll E. Delayed-feedback control: arbitrary and distributed delay-time and noninvasive control of synchrony in networks with heterogeneous delays. ACTA ACUST UNITED AC 2014. [DOI: 10.1007/s40435-013-0049-2] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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20
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Flunkert V, Fischer I, Schöll E. Dynamics, control and information in delay-coupled systems: an overview. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2013; 371:20120465. [PMID: 23960223 PMCID: PMC3758166 DOI: 10.1098/rsta.2012.0465] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Affiliation(s)
- Valentin Flunkert
- IFISC (UIB-CSIC), Universitat de les Illes Balears, 07122 Palma de Mallorca, Spain.
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Gjurchinovski A, Jüngling T, Urumov V, Schöll E. Delayed feedback control of unstable steady states with high-frequency modulation of the delay. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:032912. [PMID: 24125330 DOI: 10.1103/physreve.88.032912] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2013] [Indexed: 06/02/2023]
Abstract
We analyze the stabilization of unstable steady states by delayed feedback control with a periodic time-varying delay in the regime of a high-frequency modulation of the delay. The average effect of the delayed feedback term in the control force is equivalent to a distributed delay in the interval of the modulation, and the obtained distribution depends on the type of the modulation. In our analysis we use a simple generic normal form of an unstable focus, and investigate the effects of phase-dependent coupling and the influence of the control loop latency on the controllability. In addition, we have explored the influence of the modulation of the delays in multiple delay feedback schemes consisting of two independent delay lines of Pyragas type. A main advantage of the variable delay is the considerably larger domain of stabilization in parameter space.
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Affiliation(s)
- Aleksandar Gjurchinovski
- Institute of Physics, Faculty of Natural Sciences and Mathematics, Saints Cyril and Methodius University, P.O. Box 162, 1000 Skopje, Macedonia
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