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Klinshov VV, Smelov PS, Kirillov SY. Constructive role of shot noise in the collective dynamics of neural networks. CHAOS (WOODBURY, N.Y.) 2023; 33:2894498. [PMID: 37276575 DOI: 10.1063/5.0147409] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2023] [Accepted: 05/18/2023] [Indexed: 06/07/2023]
Abstract
Finite-size effects may significantly influence the collective dynamics of large populations of neurons. Recently, we have shown that in globally coupled networks these effects can be interpreted as additional common noise term, the so-called shot noise, to the macroscopic dynamics unfolding in the thermodynamic limit. Here, we continue to explore the role of the shot noise in the collective dynamics of globally coupled neural networks. Namely, we study the noise-induced switching between different macroscopic regimes. We show that shot noise can turn attractors of the infinitely large network into metastable states whose lifetimes smoothly depend on the system parameters. A surprising effect is that the shot noise modifies the region where a certain macroscopic regime exists compared to the thermodynamic limit. This may be interpreted as a constructive role of the shot noise since a certain macroscopic state appears in a parameter region where it does not exist in an infinite network.
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Affiliation(s)
- V V Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, Ulyanova Street 46, 603950 Nizhny Novgorod, Russia
- National Research University Higher School of Economics, 25/12 Bol'shaya Pecherskaya Street, Nizhny Novgorod 603155, Russia
| | - P S Smelov
- Institute of Applied Physics of the Russian Academy of Sciences, Ulyanova Street 46, 603950 Nizhny Novgorod, Russia
| | - S Yu Kirillov
- Institute of Applied Physics of the Russian Academy of Sciences, Ulyanova Street 46, 603950 Nizhny Novgorod, Russia
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2
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Klinshov VV, D'Huys O. Noise-induced switching in an oscillator with pulse delayed feedback: A discrete stochastic modeling approach. CHAOS (WOODBURY, N.Y.) 2022; 32:093141. [PMID: 36182395 DOI: 10.1063/5.0100698] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/26/2022] [Accepted: 08/22/2022] [Indexed: 06/16/2023]
Abstract
We study the dynamics of an oscillatory system with pulse delayed feedback and noise of two types: (i) phase noise acting on the oscillator and (ii) stochastic fluctuations of the feedback delay. Using an event-based approach, we reduce the system dynamics to a stochastic discrete map. For weak noise, we find that the oscillator fluctuates around a deterministic state, and we derive an autoregressive model describing the system dynamics. For stronger noise, the oscillator demonstrates noise-induced switching between various deterministic states; our theory provides a good estimate of the switching statistics in the linear limit. We show that the robustness of the system toward this switching is strikingly different depending on the type of noise. We compare the analytical results for linear coupling to numerical simulations of nonlinear coupling and find that the linear model also provides a qualitative explanation for the differences in robustness to both types of noise. Moreover, phase noise drives the system toward higher frequencies, while stochastic delays do not, and we relate this effect to our theoretical results.
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Affiliation(s)
- Vladimir V Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, Nizhny Novgorod 603950, Russia
| | - Otti D'Huys
- Department of Applied Computing Sciences, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands
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3
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Noise-to-State Stability in Probability for Random Complex Dynamical Systems on Networks. MATHEMATICS 2022. [DOI: 10.3390/math10122096] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/14/2023]
Abstract
This paper studies noise-to-state stability in probability (NSSP) for random complex dynamical systems on networks (RCDSN). On the basis of Kirchhoff’s matrix theorem in graph theory, an appropriate Lyapunov function which combines with every subsystem for RCDSN is established. Moreover, some sufficient criteria closely related to the topological structure of RCDSN are given to guarantee RCDSN to meet NSSP by means of the Lyapunov method and stochastic analysis techniques. Finally, to show the usefulness and feasibility of theoretical findings, we apply them to random coupled oscillators on networks (RCON), and some numerical tests are given.
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Klinshov V, Shchapin D, D'Huys O. Mode Hopping in Oscillating Systems with Stochastic Delays. PHYSICAL REVIEW LETTERS 2020; 125:034101. [PMID: 32745403 DOI: 10.1103/physrevlett.125.034101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/24/2020] [Revised: 05/06/2020] [Accepted: 06/08/2020] [Indexed: 06/11/2023]
Abstract
We study a noisy oscillator with pulse delayed feedback, theoretically and in an electronic experimental implementation. Without noise, this system has multiple stable periodic regimes. We consider two types of noise: (i) phase noise acting on the oscillator state variable and (ii) stochastic fluctuations of the coupling delay. For both types of stochastic perturbations the system hops between the deterministic regimes, but it shows dramatically different scaling properties for different types of noise. The robustness to conventional phase noise increases with coupling strength. However for stochastic variations in the coupling delay, the lifetimes decrease exponentially with the coupling strength. We provide an analytic explanation for these scaling properties in a linearized model. Our findings thus indicate that the robustness of a system to stochastic perturbations strongly depends on the nature of these perturbations.
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Affiliation(s)
- Vladimir Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
| | - Dmitry Shchapin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
| | - Otti D'Huys
- Department of Mathematics, Aston University, B4 7ET Birmingham, United Kingdom
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Jörg DJ, Morelli LG, Jülicher F. Chemical event chain model of coupled genetic oscillators. Phys Rev E 2018; 97:032409. [PMID: 29776186 DOI: 10.1103/physreve.97.032409] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2017] [Indexed: 06/08/2023]
Abstract
We introduce a stochastic model of coupled genetic oscillators in which chains of chemical events involved in gene regulation and expression are represented as sequences of Poisson processes. We characterize steady states by their frequency, their quality factor, and their synchrony by the oscillator cross correlation. The steady state is determined by coupling and exhibits stochastic transitions between different modes. The interplay of stochasticity and nonlinearity leads to isolated regions in parameter space in which the coupled system works best as a biological pacemaker. Key features of the stochastic oscillations can be captured by an effective model for phase oscillators that are coupled by signals with distributed delays.
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Affiliation(s)
- David J Jörg
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany
| | - Luis G Morelli
- Instituto de Investigación en Biomedicina de Buenos Aires (IBioBA)-CONICET-Partner Institute of the Max Planck Society, Polo Científico Tecnológico, Godoy Cruz 2390, C1425FQD, Buenos Aires, Argentina
- Departamento de Física, FCEyN UBA, Ciudad Universitaria, 1428 Buenos Aires, Argentina
- Max Planck Institute for Molecular Physiology, Department of Systemic Cell Biology, Otto-Hahn-Str. 11, 44227 Dortmund, Germany
| | - Frank Jülicher
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany
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Schwartz IB, Szwaykowska K, Carr TW. Asymmetric noise-induced large fluctuations in coupled systems. Phys Rev E 2018; 96:042151. [PMID: 29347603 DOI: 10.1103/physreve.96.042151] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2017] [Indexed: 11/07/2022]
Abstract
Networks of interacting, communicating subsystems are common in many fields, from ecology, biology, and epidemiology to engineering and robotics. In the presence of noise and uncertainty, interactions between the individual components can lead to unexpected complex system-wide behaviors. In this paper, we consider a generic model of two weakly coupled dynamical systems, and we show how noise in one part of the system is transmitted through the coupling interface. Working synergistically with the coupling, the noise on one system drives a large fluctuation in the other, even when there is no noise in the second system. Moreover, the large fluctuation happens while the first system exhibits only small random oscillations. Uncertainty effects are quantified by showing how characteristic time scales of noise-induced switching scale as a function of the coupling between the two coupled parts of the experiment. In addition, our results show that the probability of switching in the noise-free system scales inversely as the square of reduced noise intensity amplitude, rendering the virtual probability of switching an extremely rare event. Our results showing the interplay between transmitted noise and coupling are also confirmed through simulations, which agree quite well with analytic theory.
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Affiliation(s)
- Ira B Schwartz
- U.S. Naval Research Laboratory Code 6792, Plasma Physics Division, Nonlinear Systems Dynamics Section, Washington, D.C. 20375, USA
| | - Klimka Szwaykowska
- U.S. Naval Research Laboratory Code 6792, Plasma Physics Division, Nonlinear Systems Dynamics Section, Washington, D.C. 20375, USA
| | - Thomas W Carr
- Department of Mathematics, Southern Methodist University, Dallas, Texas 75275, USA
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Ruschel S, Yanchuk S. Chaotic bursting in semiconductor lasers. CHAOS (WOODBURY, N.Y.) 2017; 27:114313. [PMID: 29195313 DOI: 10.1063/1.5007876] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We investigate the dynamic mechanisms for low frequency fluctuations in semiconductor lasers subjected to delayed optical feedback, using the Lang-Kobayashi model. This system of delay differential equations displays pronounced envelope dynamics, ranging from erratic, so called low frequency fluctuations to regular pulse packages, if the time scales of fast oscillations and envelope dynamics are well separated. We investigate the parameter regions where low frequency fluctuations occur and compute their Lyapunov spectra. Using the geometric singular perturbation theory, we study this intermittent chaotic behavior and characterize these solutions as bursting slow-fast oscillations.
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Affiliation(s)
- Stefan Ruschel
- Institute of Mathematics, Technical University of Berlin, Berlin, Germany
| | - Serhiy Yanchuk
- Institute of Mathematics, Technical University of Berlin, Berlin, Germany
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Klinshov V, Shchapin D, Yanchuk S, Wolfrum M, D'Huys O, Nekorkin V. Embedding the dynamics of a single delay system into a feed-forward ring. Phys Rev E 2017; 96:042217. [PMID: 29347517 DOI: 10.1103/physreve.96.042217] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2017] [Indexed: 11/07/2022]
Abstract
We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example, we demonstrate how the complex bifurcation scenario of simultaneously emerging multijittering solutions can be transferred from a single oscillator with delayed pulse feedback to multijittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.
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Affiliation(s)
- Vladimir Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia
| | - Dmitry Shchapin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia
| | - Serhiy Yanchuk
- Technical University of Berlin, Institute of Mathematics, Straße des 17. Juni 136, 10623 Berlin, Germany
| | - Matthias Wolfrum
- Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany
| | - Otti D'Huys
- Aston University, Department of Mathematics, B4 7ET Birmingham, United Kingdom
| | - Vladimir Nekorkin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia
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Bolhasani E, Azizi Y, Valizadeh A, Perc M. Synchronization of oscillators through time-shifted common inputs. Phys Rev E 2017; 95:032207. [PMID: 28415346 DOI: 10.1103/physreve.95.032207] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2016] [Indexed: 06/07/2023]
Abstract
Shared upstream dynamical processes are frequently the source of common inputs in various physical and biological systems. However, due to finite signal transmission speeds and differences in the distance to the source, time shifts between otherwise common inputs are unavoidable. Since common inputs can be a source of correlation between the elements of multi-unit dynamical systems, regardless of whether these elements are directly connected with one another or not, it is of importance to understand their impact on synchronization. As a canonical model that is representative for a variety of different dynamical systems, we study limit-cycle oscillators that are driven by stochastic time-shifted common inputs. We show that if the oscillators are coupled, time shifts in stochastic common inputs do not simply shift the distribution of the phase differences, but rather the distribution actually changes as a result. The best synchronization is therefore achieved at a precise intermediate value of the time shift, which is due to a resonance-like effect with the most probable phase difference that is determined by the deterministic dynamics.
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Affiliation(s)
- Ehsan Bolhasani
- Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran
- School of Cognitive Science, Institute for Research in Fundamental Sciences (IPM), P.O. Box 1954851167, Tehran, Iran
| | - Yousef Azizi
- Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran
| | - Alireza Valizadeh
- Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran
- School of Cognitive Science, Institute for Research in Fundamental Sciences (IPM), P.O. Box 1954851167, Tehran, Iran
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia
- CAMTP-Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia
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Stich M, Chattopadhyay AK. Noise-induced standing waves in oscillatory systems with time-delayed feedback. Phys Rev E 2016; 93:052221. [PMID: 27300894 DOI: 10.1103/physreve.93.052221] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2016] [Indexed: 11/07/2022]
Abstract
In oscillatory reaction-diffusion systems, time-delay feedback can lead to the instability of uniform oscillations with respect to formation of standing waves. Here, we investigate how the presence of additive, Gaussian white noise can induce the appearance of standing waves. Combining analytical solutions of the model with spatiotemporal simulations, we find that noise can promote standing waves in regimes where the deterministic uniform oscillatory modes are stabilized. As the deterministic phase boundary is approached, the spatiotemporal correlations become stronger, such that even small noise can induce standing waves in this parameter regime. With larger noise strengths, standing waves could be induced at finite distances from the (deterministic) phase boundary. The overall dynamics is defined through the interplay of noisy forcing with the inherent reaction-diffusion dynamics.
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Affiliation(s)
- Michael Stich
- Non-linearity and Complexity Research Group, Systems Analytics Research Institute, School of Engineering and Applied Science, Aston University, Aston Triangle, Birmingham, B4 7ET, United Kingdom
| | - Amit K Chattopadhyay
- Non-linearity and Complexity Research Group, Systems Analytics Research Institute, School of Engineering and Applied Science, Aston University, Aston Triangle, Birmingham, B4 7ET, United Kingdom
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Jüngling T, D'Huys O, Kinzel W. The transition between strong and weak chaos in delay systems: Stochastic modeling approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062918. [PMID: 26172783 DOI: 10.1103/physreve.91.062918] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2015] [Indexed: 06/04/2023]
Abstract
We investigate the scaling behavior of the maximal Lyapunov exponent in chaotic systems with time delay. In the large-delay limit, it is known that one can distinguish between strong and weak chaos depending on the delay scaling, analogously to strong and weak instabilities for steady states and periodic orbits. Here we show that the Lyapunov exponent of chaotic systems shows significant differences in its scaling behavior compared to constant or periodic dynamics due to fluctuations in the linearized equations of motion. We reproduce the chaotic scaling properties with a linear delay system with multiplicative noise. We further derive analytic limit cases for the stochastic model illustrating the mechanisms of the emerging scaling laws.
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Affiliation(s)
- Thomas Jüngling
- Institute for Cross-Disciplinary Physics and Complex Systems, IFISC (UIB-CSIC), Campus University of the Balearic Islands, 07122 Palma de Mallorca, Spain
| | - Otti D'Huys
- Institute for Theoretical Physics, University of Würzburg, Am Hubland, 97074 Würzburg, Germany
- Department of Physics, Duke University, 120 Science Dr., Durham, North Carolina 27708, USA
| | - Wolfgang Kinzel
- Institute for Theoretical Physics, University of Würzburg, Am Hubland, 97074 Würzburg, Germany
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