1
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Liu C, Wu ZX, Wang CY, Yang HX, Guan JY. Double resonance induced by group coupling with quenched disorder. CHAOS (WOODBURY, N.Y.) 2023; 33:013114. [PMID: 36725631 DOI: 10.1063/5.0132107] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/26/2022] [Accepted: 12/13/2022] [Indexed: 06/18/2023]
Abstract
Results show that the astrocytes can not only listen to the talk of large assemble of neurons but also give advice to the conversations and are significant sources of heterogeneous couplings as well. In the present work, we focus on such regulation character of astrocytes and explore the role of heterogeneous couplings among interacted neuron-astrocyte components in a signal response. We consider reduced dynamics in which the listening and advising processes of astrocytes are mapped into the form of group coupling, where the couplings are normally distributed. In both globally coupled overdamped bistable oscillators and an excitable FitzHugh-Nagumo (FHN) neuron model, we numerically and analytically demonstrate that two types of bell-shaped collective response curves can be obtained as the ensemble coupling strength or the heterogeneity of group coupling rise, respectively, which can be seen as a new type of double resonance. Furthermore, through the bifurcation analysis, we verify that these resonant signal responses stem from the competition between dispersion and aggregation induced by heterogeneous group and positive pairwise couplings, respectively. Our results contribute to a better understanding of the signal propagation in coupled systems with quenched disorder.
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Affiliation(s)
- Cong Liu
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Zhi-Xi Wu
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Chong-Yang Wang
- Institute of Computational Physics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Han-Xin Yang
- Department of Physics, Fuzhou University, Fuzhou 350108, People's Republic of China
| | - Jian-Yue Guan
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
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2
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Pedergnana T, Noiray N. Exact potentials in multivariate Langevin equations. CHAOS (WOODBURY, N.Y.) 2022; 32:123146. [PMID: 36587313 DOI: 10.1063/5.0124031] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2022] [Accepted: 12/01/2022] [Indexed: 06/17/2023]
Abstract
Systems governed by a multivariate Langevin equation featuring an exact potential exhibit straightforward dynamics but are often difficult to recognize because, after a general coordinate change, the gradient flow becomes obscured by the Jacobian matrix of the mapping. In this work, a detailed analysis of the transformation rules for Langevin equations under general nonlinear mappings is presented. We show how to identify systems with exact potentials by understanding their differential-geometric properties. To demonstrate the power of our method, we use it to derive exact potentials for broadly studied models of nonlinear deterministic and stochastic oscillations. In selected examples, we visualize the identified potentials. Our results imply a broad class of exactly solvable stochastic models, which can be self-consistently defined from given deterministic gradient systems.
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Affiliation(s)
- Tiemo Pedergnana
- CAPS Laboratory, Department of Mechanical and Process Engineering, ETH Zürich, Sonneggstrasse 3, 8092 Zürich, Switzerland
| | - Nicolas Noiray
- CAPS Laboratory, Department of Mechanical and Process Engineering, ETH Zürich, Sonneggstrasse 3, 8092 Zürich, Switzerland
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3
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Qiao L, Zhang ZB, Zhao W, Wei P, Zhang L. Network design principle for robust oscillatory behaviors with respect to biological noise. eLife 2022; 11:76188. [PMID: 36125857 PMCID: PMC9489215 DOI: 10.7554/elife.76188] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2021] [Accepted: 08/22/2022] [Indexed: 11/13/2022] Open
Abstract
Oscillatory behaviors, which are ubiquitous in transcriptional regulatory networks, are often subject to inevitable biological noise. Thus, a natural question is how transcriptional regulatory networks can robustly achieve accurate oscillation in the presence of biological noise. Here, we search all two- and three-node transcriptional regulatory network topologies for those robustly capable of accurate oscillation against the parameter variability (extrinsic noise) or stochasticity of chemical reactions (intrinsic noise). We find that, no matter what source of the noise is applied, the topologies containing the repressilator with positive autoregulation show higher robustness of accurate oscillation than those containing the activator-inhibitor oscillator, and additional positive autoregulation enhances the robustness against noise. Nevertheless, the attenuation of different sources of noise is governed by distinct mechanisms: the parameter variability is buffered by the long period, while the stochasticity of chemical reactions is filtered by the high amplitude. Furthermore, we analyze the noise of a synthetic human nuclear factor κB (NF-κB) signaling network by varying three different topologies and verify that the addition of a repressilator to the activator-inhibitor oscillator, which leads to the emergence of high-robustness motif—the repressilator with positive autoregulation—improves the oscillation accuracy in comparison to the topology with only an activator-inhibitor oscillator. These design principles may be applicable to other oscillatory circuits.
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Affiliation(s)
- Lingxia Qiao
- Beijing International Center for Mathematical Research, Peking University, Beijing, China
| | - Zhi-Bo Zhang
- Center for Quantitative Biology, Peking University, Beijing, China.,Peking-Tsinghua Joint Center for Life Sciences, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing, China
| | - Wei Zhao
- Center for Quantitative Biology, Peking University, Beijing, China
| | - Ping Wei
- Center for Quantitative Biology, Peking University, Beijing, China.,Center for Cell and Gene Circuit Design, CAS Key Laboratory of Quantitative Engineering Biology, Shenzhen Institute of Synthetic Biology, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China
| | - Lei Zhang
- Beijing International Center for Mathematical Research, Peking University, Beijing, China.,Center for Quantitative Biology, Peking University, Beijing, China
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4
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Liu C, Wang CY, Wu ZX, Yang HX, Guan JY. Diversity-induced resonance in a globally coupled bistable system with diversely distributed heterogeneity. CHAOS (WOODBURY, N.Y.) 2022; 32:083112. [PMID: 36049925 DOI: 10.1063/5.0094685] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2022] [Accepted: 07/05/2022] [Indexed: 06/15/2023]
Abstract
A moderate degree of diversity, in form of quenched noise or intrinsic heterogeneity, can significantly strengthen the collective response of coupled extended systems. As yet, related discoveries on diversity-induced resonance are mainly concentrated on symmetrically distributed heterogeneity, e.g., the Gaussian or uniform distributions with zero-mean. The necessary conditions that guarantee the arise of resonance phenomenon in heterogeneous oscillators remain largely unknown. In this work, we show that the standard deviation and the ratio of negative entities of a given distribution jointly modulate diversity-induced resonance and the concomitance of negative and positive entities is the prerequisite for this resonant behavior emerging in diverse symmetrical and asymmetrical distributions. Particularly, for a proper degree of diversity of a given distribution, the collective signal response behaves like a bell-shaped curve as the ratio of negative oscillator increases, which can be termed negative-oscillator-ratio induced resonance. Furthermore, we analytically reveal that the ratio of negative oscillators plays a gating role in the resonance phenomenon on the basis of a reduced equation. Finally, we examine the robustness of these results in globally coupled bistable elements with asymmetrical potential functions. Our results suggest that the phenomenon of diversity-induced resonance can arise in arbitrarily distributed heterogeneous bistable oscillators by regulating the ratio of negative entities appropriately.
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Affiliation(s)
- Cong Liu
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Chong-Yang Wang
- Institute of Computational Physics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Zhi-Xi Wu
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Han-Xin Yang
- Department of Physics, Fuzhou University, Fuzhou 350116, People's Republic of China
| | - Jian-Yue Guan
- Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
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5
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Garain K, Sarathi Mandal P. Stochastic sensitivity analysis and early warning signals of critical transitions in a tri-stable prey-predator system with noise. CHAOS (WOODBURY, N.Y.) 2022; 32:033115. [PMID: 35364818 DOI: 10.1063/5.0074242] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2021] [Accepted: 02/24/2022] [Indexed: 06/14/2023]
Abstract
Near a tipping point, small changes in a certain parameter cause an irreversible shift in the behavior of a system, called critical transitions. Critical transitions can be observed in a variety of complex dynamical systems, ranging from ecology to financial markets, climate change, molecular bio-systems, health, and disease. As critical transitions can occur suddenly and are hard to manage, it is important to predict their occurrence. Although it is very tough to predict such critical transitions, various recent works suggest that generic early warning signals can detect the situation when systems approach a critical point. The most important indicator that predicts the risk of an upcoming critical transition is critical slowing down (CSD). CSD indicates a slow recovery rate from external perturbations of the stable state close to a bifurcation point. In this contribution, we study a two dimensional prey-predator model. Without any noise, the prey-predator model shows bistability and tri-stability due to the Allee effect in predators. We explore the critical transitions when external noise is added to the prey-predator system. We investigate early warning indicators, e.g., recovery rate, lag-1 autocorrelation, variance, and skewness to predict the critical transition. We explore the confidence domain method using the stochastic sensitivity function (SSF) technique near a stable equilibrium point to find a threshold value of noise intensity for a transition. The SSF technique in a two stage transition through confidence ellipse is described. We also show that the possibility of a transition to the predator-free state is independent of initial conditions. Our result may serve as a paradigm to understand and predict the critical transition in a two dimensional system.
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Affiliation(s)
- Koushik Garain
- Department of Mathematics, NIT Patna, Patna, Bihar 800005, India
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6
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Baspinar E, Schülen L, Olmi S, Zakharova A. Coherence resonance in neuronal populations: Mean-field versus network model. Phys Rev E 2021; 103:032308. [PMID: 33862689 DOI: 10.1103/physreve.103.032308] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2020] [Accepted: 02/22/2021] [Indexed: 01/17/2023]
Abstract
The counterintuitive phenomenon of coherence resonance describes a nonmonotonic behavior of the regularity of noise-induced oscillations in the excitable regime, leading to an optimal response in terms of regularity of the excited oscillations for an intermediate noise intensity. We study this phenomenon in populations of FitzHugh-Nagumo (FHN) neurons with different coupling architectures. For networks of FHN systems in an excitable regime, coherence resonance has been previously analyzed numerically. Here we focus on an analytical approach studying the mean-field limits of the globally and locally coupled populations. The mean-field limit refers to an averaged behavior of a complex network as the number of elements goes to infinity. We apply the mean-field approach to the globally coupled FHN network. Further, we derive a mean-field limit approximating the locally coupled FHN network with low noise intensities. We study the effects of the coupling strength and noise intensity on coherence resonance for both the network and the mean-field models. We compare the results of the mean-field and network frameworks and find good agreement in the globally coupled case, where the correspondence between the two approaches is sufficiently good to capture the emergence of coherence resonance, as well as of anticoherence resonance.
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Affiliation(s)
- Emre Baspinar
- Inria Sophia Antipolis Méditerranée Research Centre, 2004 Route des Lucioles, 06902 Valbonne, France
| | - Leonhard Schülen
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Simona Olmi
- Inria Sophia Antipolis Méditerranée Research Centre, 2004 Route des Lucioles, 06902 Valbonne, France.,CNR - Consiglio Nazionale delle Ricerche - Istituto dei Sistemi complessi, 50019, Sesto Fiorentino, Italy.,Joint Senior Authorship
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany.,Joint Senior Authorship
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7
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Ryashko L. Noise-induced complex oscillatory dynamics in the Zeldovich-Semenov model of a continuous stirred tank reactor. CHAOS (WOODBURY, N.Y.) 2021; 31:013105. [PMID: 33754765 DOI: 10.1063/5.0030511] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/22/2020] [Accepted: 12/10/2020] [Indexed: 06/12/2023]
Abstract
Noise-induced variability of thermochemical processes in a continuous stirred tank reactor is studied on the basis of the Zeldovich-Semenov dynamical model. For the deterministic variant of this model, mono- and bistability parametric zones as well as local and global bifurcations are determined. Noise-induced transitions between coexisting attractors (equilibria and cycles) and stochastic excitement with spike oscillations are investigated by direct numerical simulation and the analytical approach based on the stochastic sensitivity technique. For the stochastic model, the phenomenon of coherence resonance is discovered and studied.
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Affiliation(s)
- Lev Ryashko
- Ural Mathematical Center, Ural Federal University, Lenina, 51, 620000 Ekaterinburg, Russia
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8
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Kades L, Pawlowski JM. Discrete Langevin machine: Bridging the gap between thermodynamic and neuromorphic systems. Phys Rev E 2020; 101:063304. [PMID: 32688507 DOI: 10.1103/physreve.101.063304] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/04/2020] [Accepted: 04/08/2020] [Indexed: 01/09/2023]
Abstract
A formulation of Langevin dynamics for discrete systems is derived as a class of generic stochastic processes. The dynamics simplify for a two-state system and suggest a network architecture which is implemented by the Langevin machine. The Langevin machine represents a promising approach to compute successfully quantitative exact results of Boltzmann distributed systems by LIF neurons. Besides a detailed introduction of the dynamics, different simplified models of a neuromorphic hardware system are studied with respect to a control of emerging sources of errors.
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Affiliation(s)
- Lukas Kades
- Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
| | - Jan M Pawlowski
- Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
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9
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Yuan S, Wu D, Lan G, Wang H. Noise-Induced Transitions in a Nonsmooth Producer-Grazer Model with Stoichiometric Constraints. Bull Math Biol 2020; 82:55. [PMID: 32350614 PMCID: PMC7190610 DOI: 10.1007/s11538-020-00733-y] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2019] [Accepted: 04/09/2020] [Indexed: 12/03/2022]
Abstract
Stoichiometric producer-grazer models are nonsmooth due to the Liebig's Law of Minimum and can generate new dynamics such as bistability for producer-grazer interactions. Environmental noises can be extremely important and change dynamical behaviors of a stoichiometric producer-grazer model. In this paper, we consider a stochastically forced producer-grazer model and study the phenomena of noise-induced state switching between two stochastic attractors in the bistable zone. Namely, there is a frequent random hopping of phase trajectories between attracting basins of the attractors. In addition, by applying the stochastic sensitivity function technique, we construct the confidence ellipse and confidence band to find the configurational arrangement of equilibria and a limit cycle, respectively.
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Affiliation(s)
- Sanling Yuan
- College of Science, University of Shanghai for Science and Technology, Shanghai, 200093 China
| | - Dongmei Wu
- College of Science, University of Shanghai for Science and Technology, Shanghai, 200093 China
| | - Guijie Lan
- College of Science, University of Shanghai for Science and Technology, Shanghai, 200093 China
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 Canada
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10
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Ratas I, Pyragas K. Noise-induced macroscopic oscillations in a network of synaptically coupled quadratic integrate-and-fire neurons. Phys Rev E 2019; 100:052211. [PMID: 31869871 DOI: 10.1103/physreve.100.052211] [Citation(s) in RCA: 17] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2019] [Indexed: 05/26/2023]
Abstract
We consider the effect of small independent local noise on a network of quadratic integrate-and-fire neurons, globally coupled via synaptic pulses of finite width. The Fokker-Planck equation for a network of infinite size is reduced to a low-dimensional system of ordinary differential equations using the recently proposed perturbation theory based on circular cumulants. A bifurcation analysis of the reduced equations is performed, and areas in the parameter space, where the noise causes macroscopic oscillations of the network, are determined. The validity of the reduced equations is verified by comparing their solutions with "exact" solutions of the Fokker-Planck equation, as well as with the results of direct simulation of stochastic microscopic dynamics of a finite-size network.
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Affiliation(s)
- Irmantas Ratas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
| | - Kestutis Pyragas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
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11
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Falcke M, Moein M, Tilūnaitė A, Thul R, Skupin A. On the phase space structure of IP 3 induced Ca 2+ signalling and concepts for predictive modeling. CHAOS (WOODBURY, N.Y.) 2018; 28:045115. [PMID: 31906671 DOI: 10.1063/1.5021073] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
The correspondence between mathematical structures and experimental systems is the basis of the generalizability of results found with specific systems and is the basis of the predictive power of theoretical physics. While physicists have confidence in this correspondence, it is less recognized in cellular biophysics. On the one hand, the complex organization of cellular dynamics involving a plethora of interacting molecules and the basic observation of cell variability seem to question its possibility. The practical difficulties of deriving the equations describing cellular behaviour from first principles support these doubts. On the other hand, ignoring such a correspondence would severely limit the possibility of predictive quantitative theory in biophysics. Additionally, the existence of functional modules (like pathways) across cell types suggests also the existence of mathematical structures with comparable universality. Only a few cellular systems have been sufficiently investigated in a variety of cell types to follow up these basic questions. IP3 induced Ca2+signalling is one of them, and the mathematical structure corresponding to it is subject of ongoing discussion. We review the system's general properties observed in a variety of cell types. They are captured by a reaction diffusion system. We discuss the phase space structure of its local dynamics. The spiking regime corresponds to noisy excitability. Models focussing on different aspects can be derived starting from this phase space structure. We discuss how the initial assumptions on the set of stochastic variables and phase space structure shape the predictions of parameter dependencies of the mathematical models resulting from the derivation.
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Affiliation(s)
- Martin Falcke
- Max Delbrück Centre for Molecular Medicine, Robert Rössler Strasse 10, 13125 Berlin, Germany and Department of Physics, Humboldt University, Newtonstr. 15, 12489 Berlin, Germany
| | - Mahsa Moein
- Luxembourg Centre for Systems Biomedicine, University of Luxembourg, 7, Rue de Swing, Belval L-4367, Luxembourg
| | - Agne Tilūnaitė
- Systems Biology Laboratory, School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia
| | - Rüdiger Thul
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - Alexander Skupin
- Luxembourg Centre for Systems Biomedicine, University of Luxembourg, 7, Rue de Swing, Belval L-4367, Luxembourg
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12
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Franović I, Klinshov V. Clustering promotes switching dynamics in networks of noisy neurons. CHAOS (WOODBURY, N.Y.) 2018; 28:023111. [PMID: 29495663 DOI: 10.1063/1.5017822] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Macroscopic variability is an emergent property of neural networks, typically manifested in spontaneous switching between the episodes of elevated neuronal activity and the quiescent episodes. We investigate the conditions that facilitate switching dynamics, focusing on the interplay between the different sources of noise and heterogeneity of the network topology. We consider clustered networks of rate-based neurons subjected to external and intrinsic noise and derive an effective model where the network dynamics is described by a set of coupled second-order stochastic mean-field systems representing each of the clusters. The model provides an insight into the different contributions to effective macroscopic noise and qualitatively indicates the parameter domains where switching dynamics may occur. By analyzing the mean-field model in the thermodynamic limit, we demonstrate that clustering promotes multistability, which gives rise to switching dynamics in a considerably wider parameter region compared to the case of a non-clustered network with sparse random connection topology.
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Affiliation(s)
- Igor Franović
- Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
| | - Vladimir Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanov Street, 603950 Nizhny Novgorod, Russia
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13
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Franović I, Maslennikov OV, Bačić I, Nekorkin VI. Mean-field dynamics of a population of stochastic map neurons. Phys Rev E 2018; 96:012226. [PMID: 29347187 DOI: 10.1103/physreve.96.012226] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2017] [Indexed: 11/07/2022]
Abstract
We analyze the emergent regimes and the stimulus-response relationship of a population of noisy map neurons by means of a mean-field model, derived within the framework of cumulant approach complemented by the Gaussian closure hypothesis. It is demonstrated that the mean-field model can qualitatively account for stability and bifurcations of the exact system, capturing all the generic forms of collective behavior, including macroscopic excitability, subthreshold oscillations, periodic or chaotic spiking, and chaotic bursting dynamics. Apart from qualitative analogies, we find a substantial quantitative agreement between the exact and the approximate system, as reflected in matching of the parameter domains admitting the different dynamical regimes, as well as the characteristic properties of the associated time series. The effective model is further shown to reproduce with sufficient accuracy the phase response curves of the exact system and the assembly's response to external stimulation of finite amplitude and duration.
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Affiliation(s)
- Igor Franović
- Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
| | - Oleg V Maslennikov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanov Street, 603950 Nizhny Novgorod, Russia
| | - Iva Bačić
- Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
| | - Vladimir I Nekorkin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanov Street, 603950 Nizhny Novgorod, Russia
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14
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Semenov VV. Noise-induced transitions in a double-well excitable oscillator. Phys Rev E 2017; 95:052205. [PMID: 28618461 DOI: 10.1103/physreve.95.052205] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2017] [Indexed: 06/07/2023]
Abstract
The model of a double-well oscillator with nonlinear dissipation is studied. The self-sustained oscillation regime and the excitable one are described. The first regime consists of the coexistence of two stable limit cycles in the phase space, which correspond to self-sustained oscillations of the point mass in either potential well. The self-sustained oscillations do not occur in a noise-free system in the excitable regime, but appropriate conditions for coherence resonance in either potential well can be achieved. The stochastic dynamics in both regimes is researched by using numerical simulation and electronic circuit implementation of the considered system. Multiple qualitative changes of the probability density function caused by noise intensity varying are explained by using the phase-space structure of the deterministic system.
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Affiliation(s)
- Vladimir V Semenov
- Department of Physics, Saratov State University, Astrakhanskaya str., 83, 410012 Saratov, Russia
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15
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Kawamura Y. Collective phase reduction of globally coupled noisy dynamical elements. Phys Rev E 2017; 95:032225. [PMID: 28415367 DOI: 10.1103/physreve.95.032225] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2017] [Indexed: 06/07/2023]
Abstract
We formulate a theory for the collective phase reduction of globally coupled noisy dynamical elements exhibiting macroscopic rhythms. We first transform the Langevin-type equation that represents a group of globally coupled noisy dynamical elements into the corresponding nonlinear Fokker-Planck equation and then develop the phase reduction method for limit-cycle solutions to the nonlinear Fokker-Planck equation. The theory enables us to describe the collective dynamics of a group of globally coupled noisy dynamical elements by a single degree of freedom called the collective phase. As long as the group collectively exhibits macroscopic rhythms, the theory is applicable even when the coupling and noise are strong; it is also independent of the assumption that each element of the group is a self-sustained oscillator. We also provide a simple and accurate numerical algorithm for the collective phase description method and numerically illustrate the theory using a group of globally coupled noisy FitzHugh-Nagumo elements.
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Affiliation(s)
- Yoji Kawamura
- Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan and Research and Development Center for Marine Biosciences, Japan Agency for Marine-Earth Science and Technology, Yokosuka 237-0061, Japan
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16
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Zhu J, Kong C, Liu X. Subthreshold and suprathreshold vibrational resonance in the FitzHugh-Nagumo neuron model. Phys Rev E 2016; 94:032208. [PMID: 27739746 DOI: 10.1103/physreve.94.032208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/11/2016] [Indexed: 06/06/2023]
Abstract
We study the subthreshold and suprathreshold vibrational resonance in the FitzHugh-Nagumo neuron model. For the subthreshold situation, two cases where the stationary states are equilibrium point and limit cycle are considered, where different natures of vibrational resonance are observed via theoretical and numerical methods. Especially when the frequency of the high-frequency driving force is near the so-called canard-resonance frequency, the firing rate can be significantly enhanced at the presence of noise. For the suprathreshold situation, we show that the local maxima of the response amplitude are located at the transition boundaries of different phase-locking patterns. The minimal required forcing amplitudes of high-frequency signal of the firing onset are just multiples of the spiking frequency. Furthermore, phase portraits and time series show that the presence of the global maxima of the response results from not only the suprathreshold but also the subthreshold phase-locking modes. In spite of the distinct characteristics for two stationary states on subthreshold oscillation, the suprathreshold vibrational resonance showed no qualitative difference between the two cases.
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Affiliation(s)
- Jinjie Zhu
- State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
| | - Chen Kong
- State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
| | - Xianbin Liu
- State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
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17
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Bielievtsov D, Ladenbauer J, Obermayer K. Controlling statistical moments of stochastic dynamical networks. Phys Rev E 2016; 94:012306. [PMID: 27575147 DOI: 10.1103/physreve.94.012306] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2015] [Indexed: 06/06/2023]
Abstract
We consider a general class of stochastic networks and ask which network nodes need to be controlled, and how, to stabilize and switch between desired metastable (target) states in terms of the first and second statistical moments of the system. We first show that it is sufficient to directly interfere with a subset of nodes which can be identified using information about the graph of the network only. Then we develop a suitable method for feedback control which acts on that subset of nodes and preserves the covariance structure of the desired target state. Finally, we demonstrate our theoretical results using a stochastic Hopfield network and a global brain model. Our results are applicable to a variety of (model) networks and further our understanding of the relationship between network structure and collective dynamics for the benefit of effective control.
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Affiliation(s)
- Dmytro Bielievtsov
- Bernstein Center for Computational Neuroscience Berlin, Philippstraße 13, 10115 Berlin, Germany
| | - Josef Ladenbauer
- Bernstein Center for Computational Neuroscience Berlin, Philippstraße 13, 10115 Berlin, Germany
- Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berlin, Marchstraße 23, 10587 Berlin, Germany
| | - Klaus Obermayer
- Bernstein Center for Computational Neuroscience Berlin, Philippstraße 13, 10115 Berlin, Germany
- Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berlin, Marchstraße 23, 10587 Berlin, Germany
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Semenov VV, Neiman AB, Vadivasova TE, Anishchenko VS. Noise-induced transitions in a double-well oscillator with nonlinear dissipation. Phys Rev E 2016; 93:052210. [PMID: 27300883 DOI: 10.1103/physreve.93.052210] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2016] [Indexed: 06/06/2023]
Abstract
We develop a model of bistable oscillator with nonlinear dissipation. Using a numerical simulation and an electronic circuit realization of this system we study its response to additive noise excitations. We show that depending on noise intensity the system undergoes multiple qualitative changes in the structure of its steady-state probability density function (PDF). In particular, the PDF exhibits two pitchfork bifurcations versus noise intensity, which we describe using an effective potential and corresponding normal form of the bifurcation. These stochastic effects are explained by the partition of the phase space by the nullclines of the deterministic oscillator.
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Affiliation(s)
- Vladimir V Semenov
- Department of Physics, Saratov State University, Astrakhanskaya str., 83, 410012, Saratov, Russia
| | - Alexander B Neiman
- Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA
| | - Tatyana E Vadivasova
- Department of Physics, Saratov State University, Astrakhanskaya str., 83, 410012, Saratov, Russia
| | - Vadim S Anishchenko
- Department of Physics, Saratov State University, Astrakhanskaya str., 83, 410012, Saratov, Russia
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Franović I, Perc M, Todorović K, Kostić S, Burić N. Activation process in excitable systems with multiple noise sources: Large number of units. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062912. [PMID: 26764779 DOI: 10.1103/physreve.92.062912] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2014] [Indexed: 06/05/2023]
Abstract
We study the activation process in large assemblies of type II excitable units whose dynamics is influenced by two independent noise terms. The mean-field approach is applied to explicitly demonstrate that the assembly of excitable units can itself exhibit macroscopic excitable behavior. In order to facilitate the comparison between the excitable dynamics of a single unit and an assembly, we introduce three distinct formulations of the assembly activation event. Each formulation treats different aspects of the relevant phenomena, including the thresholdlike behavior and the role of coherence of individual spikes. Statistical properties of the assembly activation process, such as the mean time-to-first pulse and the associated coefficient of variation, are found to be qualitatively analogous for all three formulations, as well as to resemble the results for a single unit. These analogies are shown to derive from the fact that global variables undergo a stochastic bifurcation from the stochastically stable fixed point to continuous oscillations. Local activation processes are analyzed in the light of the competition between the noise-led and the relaxation-driven dynamics. We also briefly report on a system-size antiresonant effect displayed by the mean time-to-first pulse.
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Affiliation(s)
- Igor Franović
- Scientific Computing Laboratory, Institute of Physics, University of Belgrade, P. O. Box 68, 11080 Beograd-Zemun, Serbia
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška Cesta 160, SI-2000 Maribor, Slovenia
- Department of Physics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia
| | - Kristina Todorović
- Department of Physics and Mathematics, Faculty of Pharmacy, University of Belgrade, Vojvode Stepe 450, Belgrade, Serbia
| | - Srdjan Kostić
- Institute for the Development of Water Resources "Jaroslav Černi," Jaroslava Černog 80, 11226 Belgrade, Serbia
| | - Nikola Burić
- Scientific Computing Laboratory, Institute of Physics, University of Beograd, P. O. Box 68, 11080 Beograd-Zemun, Serbia
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Franović I, Todorović K, Vasović N, Burić N. Persistence and failure of mean-field approximations adapted to a class of systems of delay-coupled excitable units. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:022926. [PMID: 25353564 DOI: 10.1103/physreve.89.022926] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2013] [Indexed: 06/04/2023]
Abstract
We consider the approximations behind the typical mean-field model derived for a class of systems made up of type II excitable units influenced by noise and coupling delays. The formulation of the two approximations, referred to as the Gaussian and the quasi-independence approximation, as well as the fashion in which their validity is verified, are adapted to reflect the essential properties of the underlying system. It is demonstrated that the failure of the mean-field model associated with the breakdown of the quasi-independence approximation can be predicted by the noise-induced bistability in the dynamics of the mean-field system. As for the Gaussian approximation, its violation is related to the increase of noise intensity, but the actual condition for failure can be cast in qualitative, rather than quantitative terms. We also discuss how the fulfillment of the mean-field approximations affects the statistics of the first return times for the local and global variables, further exploring the link between the fulfillment of the quasi-independence approximation and certain forms of synchronization between the individual units.
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Affiliation(s)
- Igor Franović
- Faculty of Physics, University of Belgrade, P. O. Box 44, 11001 Belgrade, Serbia
| | - Kristina Todorović
- Department of Physics and Mathematics, Faculty of Pharmacy, University of Belgrade, Vojvode Stepe 450, Belgrade, Serbia
| | - Nebojša Vasović
- Department of Applied Mathematics, Faculty of Mining and Geology, University of Belgrade, P. O. Box 162, Belgrade, Serbia
| | - Nikola Burić
- Scientific Computing Laboratory, Institute of Physics, University of Beograd, P. O. Box 68, 11080 Beograd-Zemun, Serbia
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21
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Lavrova AI, Vanag VK. Two pulse-coupled non-identical, frequency-different BZ oscillators with time delay. Phys Chem Chem Phys 2014; 16:6764-72. [DOI: 10.1039/c3cp54373k] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Affiliation(s)
- Anastasia I Lavrova
- Immanuel Kant Baltic Federal University, A.Nevskogo str. 14A, Kaliningrad, 236041, Russia.
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Ansmann G, Karnatak R, Lehnertz K, Feudel U. Extreme events in excitable systems and mechanisms of their generation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:052911. [PMID: 24329335 DOI: 10.1103/physreve.88.052911] [Citation(s) in RCA: 38] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2013] [Indexed: 06/03/2023]
Abstract
We study deterministic systems, composed of excitable units of FitzHugh-Nagumo type, that are capable of self-generating and self-terminating strong deviations from their regular dynamics without the influence of noise or parameter change. These deviations are rare, short-lasting, and recurrent and can therefore be regarded as extreme events. Employing a range of methods we analyze dynamical properties of the systems, identifying features in the systems' dynamics that may qualify as precursors to extreme events. We investigate these features and elucidate mechanisms that may be responsible for the generation of the extreme events.
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Affiliation(s)
- Gerrit Ansmann
- Department of Epileptology, University of Bonn, Sigmund-Freud-Straße 25, 53105 Bonn, Germany and Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany and Interdisciplinary Center for Complex Systems, University of Bonn, Brühler Straße 7, 53175 Bonn, Germany
| | - Rajat Karnatak
- Theoretical Physics/Complex Systems, ICBM, Carl von Ossietzky University of Oldenburg, Carl-von-Ossietzky-Straße 9-11, Box 2503, 26111 Oldenburg, Germany
| | - Klaus Lehnertz
- Department of Epileptology, University of Bonn, Sigmund-Freud-Straße 25, 53105 Bonn, Germany and Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany and Interdisciplinary Center for Complex Systems, University of Bonn, Brühler Straße 7, 53175 Bonn, Germany
| | - Ulrike Feudel
- Theoretical Physics/Complex Systems, ICBM, Carl von Ossietzky University of Oldenburg, Carl-von-Ossietzky-Straße 9-11, Box 2503, 26111 Oldenburg, Germany and Research Center Neurosensory Science, Carl von Ossietzky University of Oldenburg, Carl-von-Ossietzky-Straße 9-11, 26111 Oldenburg, Germany and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742-2431, USA
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Sonnenschein B, Schimansky-Geier L. Approximate solution to the stochastic Kuramoto model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:052111. [PMID: 24329218 DOI: 10.1103/physreve.88.052111] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2013] [Indexed: 06/03/2023]
Abstract
We study Kuramoto phase oscillators with temporal fluctuations in the frequencies. The infinite-dimensional system can be reduced in a Gaussian approximation to two first-order differential equations. This yields a solution for the time-dependent order parameter, which characterizes the synchronization between the oscillators. The known critical coupling strength is exactly recovered by the Gaussian theory. Extensive numerical experiments further show that the analytical results are very accurate below and sufficiently above the critical value. We obtain the asymptotic order parameter in closed form, which suggests a tighter upper bound for the corresponding scaling. As a last point, we elaborate the Gaussian approximation in complex networks with distributed degrees.
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Affiliation(s)
- Bernard Sonnenschein
- Department of Physics, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany and Bernstein Center for Computational Neuroscience Berlin, Philippstrasse 13, 10115 Berlin, Germany
| | - Lutz Schimansky-Geier
- Department of Physics, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany and Bernstein Center for Computational Neuroscience Berlin, Philippstrasse 13, 10115 Berlin, Germany
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Abbasian AH, Fallah H, Razvan MR. Symmetric bursting behaviors in the generalized FitzHugh-Nagumo model. BIOLOGICAL CYBERNETICS 2013; 107:465-476. [PMID: 23801268 DOI: 10.1007/s00422-013-0559-1] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2012] [Accepted: 05/16/2013] [Indexed: 06/02/2023]
Abstract
In the current paper, we have investigated the generalized FitzHugh-Nagumo model. We have shown that symmetric bursting behaviors of different types could be observed in this model with an appropriate recovery term. A modified version of this system is used to construct bursting activities. Furthermore, we have shown some numerical examples of delayed Hopf bifurcation and canard phenomenon in the symmetric bursting of super-Hopf/homoclinic type near its super-Hopf and homoclinic bifurcations, respectively.
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Affiliation(s)
- A H Abbasian
- School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran
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25
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Franović I, Todorović K, Vasović N, Burić N. Mean-field approximation of two coupled populations of excitable units. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:012922. [PMID: 23410419 DOI: 10.1103/physreve.87.012922] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2012] [Indexed: 06/01/2023]
Abstract
The analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations composed of N stochastic excitable units each is performed by studying an approximate system, obtained by replacing each population with the corresponding mean-field model. In the exact system, one has the units within an ensemble communicating via the time-delayed linear couplings, whereas the interensemble terms involve the nonlinear time-delayed interaction mediated by the appropriate global variables. The aim is to demonstrate that the bifurcations affecting the stability of the stationary state of the original system, governed by a set of 4N stochastic delay-differential equations for the microscopic dynamics, can accurately be reproduced by a flow containing just four deterministic delay-differential equations which describe the evolution of the mean-field based variables. In particular, the considered issues include determining the parameter domains where the stationary state is stable, the scenarios for the onset, and the time-delay induced suppression of the collective mode, as well as the parameter domains admitting bistability between the equilibrium and the oscillatory state. We show how analytically tractable bifurcations occurring in the approximate model can be used to identify the characteristic mechanisms by which the stationary state is destabilized under different system configurations, like those with symmetrical or asymmetrical interpopulation couplings.
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Affiliation(s)
- Igor Franović
- Faculty of Physics, University of Belgrade, P. O. Box 44, 11001 Belgrade, Serbia
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Patriarca M, Postnova S, Braun HA, Hernández-García E, Toral R. Diversity and noise effects in a model of homeostatic regulation of the sleep-wake cycle. PLoS Comput Biol 2012; 8:e1002650. [PMID: 22927806 PMCID: PMC3426568 DOI: 10.1371/journal.pcbi.1002650] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2012] [Accepted: 06/20/2012] [Indexed: 11/18/2022] Open
Abstract
Recent advances in sleep neurobiology have allowed development of physiologically based mathematical models of sleep regulation that account for the neuronal dynamics responsible for the regulation of sleep-wake cycles and allow detailed examination of the underlying mechanisms. Neuronal systems in general, and those involved in sleep regulation in particular, are noisy and heterogeneous by their nature. It has been shown in various systems that certain levels of noise and diversity can significantly improve signal encoding. However, these phenomena, especially the effects of diversity, are rarely considered in the models of sleep regulation. The present paper is focused on a neuron-based physiologically motivated model of sleep-wake cycles that proposes a novel mechanism of the homeostatic regulation of sleep based on the dynamics of a wake-promoting neuropeptide orexin. Here this model is generalized by the introduction of intrinsic diversity and noise in the orexin-producing neurons, in order to study the effect of their presence on the sleep-wake cycle. A simple quantitative measure of the quality of a sleep-wake cycle is introduced and used to systematically study the generalized model for different levels of noise and diversity. The model is shown to exhibit a clear diversity-induced resonance: that is, the best wake-sleep cycle turns out to correspond to an intermediate level of diversity at the synapses of the orexin-producing neurons. On the other hand, only a mild evidence of stochastic resonance is found, when the level of noise is varied. These results show that disorder, especially in the form of quenched diversity, can be a key-element for an efficient or optimal functioning of the homeostatic regulation of the sleep-wake cycle. Furthermore, this study provides an example of a constructive role of diversity in a neuronal system that can be extended beyond the system studied here.
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Affiliation(s)
- Marco Patriarca
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos-CSIC-UIB, Campus Universitat de les Illes Balears, Palma de Mallorca, Spain.
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Hermann G, Touboul J. Heterogeneous connections induce oscillations in large-scale networks. PHYSICAL REVIEW LETTERS 2012; 109:018702. [PMID: 23031137 DOI: 10.1103/physrevlett.109.018702] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2012] [Indexed: 06/01/2023]
Abstract
Realistic large-scale networks display a heterogeneous distribution of connectivity weights that might also randomly vary in time. We show that, depending on the level of heterogeneity in the connectivity coefficients, different qualitative macroscopic and microscopic regimes emerge. We evidence, in particular, generic transitions from stationary to perfectly periodic phase-locked regimes as the disorder parameter is increased, both in a simple model treated analytically and in a biologically relevant network made of excitable cells.
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Lavrova AI, Zaks MA, Schimansky-Geier L. Modeling rhythmic patterns in the hippocampus. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:041922. [PMID: 22680513 DOI: 10.1103/physreve.85.041922] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/11/2011] [Revised: 02/10/2012] [Indexed: 06/01/2023]
Abstract
We investigate different dynamical regimes of the neuronal network in the CA3 area of the hippocampus. The proposed neuronal circuit includes two fast- and two slowly spiking cells which are interconnected by means of dynamical synapses. On the individual level, each neuron is modeled by FitzHugh-Nagumo equations. Three basic rhythmic patterns are observed: the gamma rhythm in which the fast neurons are uniformly spiking, the theta rhythm in which the individual spikes are separated by quiet epochs, and the theta-gamma rhythm with repeated patches of spikes. We analyze the influence of asymmetry of synaptic strengths on the synchronization in the network and demonstrate that strong asymmetry reduces the variety of available dynamical states. The model network exhibits multistability; this results in the occurrence of hysteresis in dependence on the conductances of individual connections. We show that switching between different rhythmic patterns in the network depends on the degree of synchronization between the slow cells.
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Affiliation(s)
- A I Lavrova
- Institute of Physics, Humboldt-University at Berlin, Newtonstrasse 15, 12489 Berlin, Germany.
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Lefebvre J, Perkins TJ. Neural population densities shape network correlations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:021914. [PMID: 22463251 DOI: 10.1103/physreve.85.021914] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2011] [Revised: 11/08/2011] [Indexed: 05/31/2023]
Abstract
The way sensory microcircuits manage cellular response correlations is a crucial question in understanding how such systems integrate external stimuli and encode information. Most sensory systems exhibit heterogeneities in terms of population sizes and features, which all impact their dynamics. This work addresses how correlations between the dynamics of neural ensembles depend on the relative size or density of excitatory and inhibitory populations. To do so, we study an apparently symmetric system of coupled stochastic differential equations that model the evolution of the populations' activities. Excitatory and inhibitory populations are connected by reciprocal recurrent connections, and both receive different stimuli exhibiting a certain level of correlation with each other. A stability analysis is performed, which reveals an intrinsic asymmetry in the distribution of the fixed points with respect to the threshold of the nonlinearities. Based on this, we show how the cross correlation between the population responses depends on the density of the inhibitory population, and that a specific ratio between both population sizes leads to a state of zero correlation. We show that this so-called asynchronous state subsists, despite the presence of stimulus correlation, and most importantly, that it occurs only in asymmetrical systems where one population outnumbers the other. Using linear approximations, we derive analytical expressions for the root of the cross-correlation function and study how the asynchronous state is impacted by the model's parameters. This work suggests a possible explanation for why inhibitory cells outnumber excitatory cells in the visual system.
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Affiliation(s)
- Jérémie Lefebvre
- Ottawa Hospital Research Institute, 501 Smyth Road, Ottawa, Ontario K1H 8L6, Canada.
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Bashkirtseva I, Ryashko L. Sensitivity analysis of stochastic attractors and noise-induced transitions for population model with Allee effect. CHAOS (WOODBURY, N.Y.) 2011; 21:047514. [PMID: 22225388 DOI: 10.1063/1.3647316] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We study a stochastically forced predator-prey model with Allee effect. In the deterministic case, this model exhibits non-trivial stable equilibrium or limit cycle corresponding to the coexistence of both species. Computational methods based on the stochastic sensitivity functions technique are suggested for the analysis of the dispersion of random states in stochastic attractors. Our method allows to construct confidence domains and estimate the threshold value of the intensity for noise generating a transition from the coexistence to the extinction.
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Franović I, Todorović K, Vasović N, Burić N. Stability, bifurcations, and dynamics of global variables of a system of bursting neurons. CHAOS (WOODBURY, N.Y.) 2011; 21:033109. [PMID: 21974644 DOI: 10.1063/1.3619293] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
An approximate mean field model of an ensemble of delayed coupled stochastic Hindmarsh-Rose bursting neurons is constructed and analyzed. Bifurcation analysis of the approximate system is performed using numerical continuation. It is demonstrated that the stability domains in the parameter space of the large exact systems are correctly estimated using the much simpler approximate model.
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Affiliation(s)
- Igor Franović
- Faculty of Physics, University of Belgrade, P.O. Box 44, 11001 Belgrade, Serbia
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Bashkirtseva I, Ryashko L. Analysis of excitability for the FitzHugh-Nagumo model via a stochastic sensitivity function technique. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:061109. [PMID: 21797304 DOI: 10.1103/physreve.83.061109] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2011] [Revised: 03/10/2011] [Indexed: 05/31/2023]
Abstract
We study excitability phenomena for the stochastically forced FitzHugh-Nagumo system modeling a neural activity. Noise-induced changes in the dynamics of this model can be explained by the high stochastic sensitivity of its attractors. Computational methods based on the stochastic sensitivity functions technique are suggested for the analysis of these attractors. Our method allows us to construct confidence ellipses and estimate a threshold value of a noise intensity corresponding to the neuron excitement. On the basis of the proposed technique, a supersensitive limit cycle is found for the FitzHugh-Nagumo model.
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Kaluza P, Strege C, Meyer-Ortmanns H. Noise as control parameter in networks of excitable media: Role of the network topology. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:036104. [PMID: 21230136 DOI: 10.1103/physreve.82.036104] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/01/2010] [Revised: 06/03/2010] [Indexed: 05/30/2023]
Abstract
We analyze coupled FitzHugh-Nagumo oscillators on various network topologies, in particular random diluted and scale-free topologies, under the influence of noise. Similarly to globally coupled excitable units, noise acts as control parameter: changing monotonically its strength, the collective dynamical behavior varies from stable equilibrium solutions to coherent firing of a large fraction, and for even stronger noise to incoherent firing leading to chaotic behavior of the excitable elements. For strong noise the system is less sensitive to the network topology. The specific topology enters via the degree of nodes and determines the average number of spikes. Apart from bifurcation regions, it is the ratio of noise intensity to size that determines the dynamical behavior of average values. Specific behavior such as limit cycles may then be realized for strong noise and large systems or for low noise and small systems. Within bifurcation regions, the actual values of noise intensity and system-size matter independently. Here we analyze in more detail phase portraits of small systems. For a given noise intensity and network topology we have studied the regularity of signals as a function of time. We observe the phenomenon of system-size resonance for a whole interval of noise intensities as long as the degree distribution is homogeneous, so that no fine tuning of the noise is needed. Therefore it is plausible that natural systems make actually use of noise when noise is unavoidably present.
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Bosl WJ, Li R. The role of noise and positive feedback in the onset of autosomal dominant diseases. BMC SYSTEMS BIOLOGY 2010; 4:93. [PMID: 20587063 PMCID: PMC2902440 DOI: 10.1186/1752-0509-4-93] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/28/2009] [Accepted: 06/29/2010] [Indexed: 01/26/2023]
Abstract
Background Autosomal dominant (AD) diseases result when a single mutant or non-functioning gene is present on an autosomal chromosome. These diseases often do not emerge at birth. There are presently two prevailing theories explaining the expression of AD diseases. One explanation originates from the Knudson two-hit theory of hereditary cancers, where loss of heterozygosity or occurrence of somatic mutations impairs the function of the wild-type copy. While these somatic second hits may be sufficient for stable disease states, it is often difficult to determine if their occurrence necessarily marks the initiation of disease progression. A more direct consequence of a heterozygous genetic background is haploinsufficiency, referring to a lack of sufficient gene function due to reduced wild-type gene copy number; however, haploinsufficiency can involve a variety of additional mechanisms, such as noise in gene expression or protein levels, injury and second hit mutations in other genes. In this study, we explore the possible contribution to the onset of autosomal dominant diseases from intrinsic factors, such as those determined by the structure of the molecular networks governing normal cellular physiology. Results First, simple models of single gene insufficiency using the positive feedback loops that may be derived from a three-component network were studied by computer simulation using Bionet software. The network structure is shown to affect the dynamics considerably; some networks are relatively stable even when large stochastic variations in are present, while others exhibit switch-like dynamics. In the latter cases, once the network switches over to the disease state it remains in that state permanently. Model pathways for two autosomal dominant diseases, AD polycystic kidney disease and mature onset diabetes of youth (MODY) were simulated and the results are compared to known disease characteristics. Conclusions By identifying the intrinsic mechanisms involved in the onset of AD diseases, it may be possible to better assess risk factors as well as lead to potential new drug targets. To illustrate the applicability of this study of pathway dynamics, we simulated the primary pathways involved in two autosomal dominant diseases, Polycystic Kidney Disease (PKD) and mature onset diabetes of youth (MODY). Simulations demonstrate that some of the primary disease characteristics are consistent with the positive feedback - stochastic variation theory presented here. This has implications for new drug targets to control these diseases by blocking the positive feedback loop in the relevant pathways.
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Leonhardt H, Zaks MA, Falcke M, Schimansky-Geier L. Stochastic hierarchical systems: excitable dynamics. J Biol Phys 2009; 34:521-38. [PMID: 19669511 DOI: 10.1007/s10867-008-9112-1] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/14/2008] [Accepted: 08/05/2008] [Indexed: 11/29/2022] Open
Abstract
We present a discrete model of stochastic excitability by a low-dimensional set of delayed integral equations governing the probability in the rest state, the excited state, and the refractory state. The process is a random walk with discrete states and nonexponential waiting time distributions, which lead to the incorporation of memory kernels in the integral equations. We extend the equations of a single unit to the system of equations for an ensemble of globally coupled oscillators, derive the mean field equations, and investigate bifurcations of steady states. Conditions of destabilization are found, which imply oscillations of the mean fields in the stochastic ensemble. The relation between the mean field equations and the paradigmatic Kuramoto model is shown.
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Affiliation(s)
- Helmar Leonhardt
- Institute of Physics, Humboldt University at Berlin, Newtonstr. 15, D-12489, Berlin, Germany,
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Bashkirtseva I, Ryashko L. Constructive analysis of noise-induced transitions for coexisting periodic attractors of the Lorenz model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:041106. [PMID: 19518172 DOI: 10.1103/physreve.79.041106] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2008] [Indexed: 05/27/2023]
Abstract
We study the stochastically forced Lorenz model in the parameter zone admitting two coexisting limit cycles under the transition to chaos via period-doubling bifurcations. Noise-induced transitions between both different parts of the single attractor and two coexisting separate attractors are demonstrated. The effects of structural stabilization and noise symmetrization are discussed. We suggest a stochastic sensitivity function technique for the analysis of noise-induced transitions between two coexisting limit cycles. This approach allows us to construct the dispersion ellipses of random trajectories for any Poincare sections. Possibilities of our descriptive-geometric method for a detailed analysis of noise-induced transitions between two periodic attractors of Lorenz model are demonstrated.
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Mikikian M, Cavarroc M, Couëdel L, Tessier Y, Boufendi L. Mixed-mode oscillations in complex-plasma instabilities. PHYSICAL REVIEW LETTERS 2008; 100:225005. [PMID: 18643427 DOI: 10.1103/physrevlett.100.225005] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/11/2008] [Indexed: 05/26/2023]
Abstract
Instabilities in dusty plasmas are frequent phenomena. We show that some instabilities can be described by mixed-mode oscillations often encountered in chemical systems or neuronal dynamics and studied through dynamical system theories. The time evolution of these instabilities is studied through the change in the associated waveform. Frequency and interspike interval are analyzed and compared to results obtained in other scientific fields concerned by mixed-mode oscillations.
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Affiliation(s)
- Maxime Mikikian
- GREMI, Groupe de Recherches sur l'Energétique des Milieux Ionisés, UMR6606, CNRS/Université d'Orléans, 14 rue d'Issoudun, BP6744, 45067 Orléans Cedex 2, France.
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38
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Li X, Wang J, Hu W. Effects of chemical synapses on the enhancement of signal propagation in coupled neurons near the canard regime. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:041902. [PMID: 17995021 DOI: 10.1103/physreve.76.041902] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2007] [Revised: 07/27/2007] [Indexed: 05/25/2023]
Abstract
The response of three coupled FitzHugh-Nagumo neurons, under Gaussian white noise, to a subthreshold periodic signal is studied in this paper. By combining the canard dynamics, chemical coupling, and stochastic resonance together, the information transfer in this neural system is investigated. We find that chemical synaptic coupling is more efficient than the well-known linear coupling (gap junction) for local signal input, i.e., only one of the three neurons is subject to the periodic signal. This weak and local input is common in biological systems for the sake of low energy consumption.
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Affiliation(s)
- Xiumin Li
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People's Republic of China.
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Riaz SS, Sharma R, Bhattacharyya SP, Ray DS. Instability and pattern formation in reaction-diffusion systems: A higher order analysis. J Chem Phys 2007; 127:064503. [PMID: 17705608 DOI: 10.1063/1.2759212] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We analyze the condition for instability and pattern formation in reaction-diffusion systems beyond the usual linear regime. The approach is based on taking into account perturbations of higher orders. Our analysis reveals that nonlinearity present in the system can be instrumental in determining the stability of a system, even to the extent of destabilizing one in a linearly stable parameter regime. The analysis is also successful to account for the observed effect of additive noise in modifying the instability threshold of a system. The analytical study is corroborated by numerical simulation in a standard reaction-diffusion system.
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Affiliation(s)
- Syed Shahed Riaz
- Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India
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Kostur M, Schindler M, Talkner P, Hänggi P. Neuron firing in driven nonlinear integrate-and-fire models. Math Biosci 2006; 207:302-11. [PMID: 17011592 DOI: 10.1016/j.mbs.2006.08.014] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/22/2006] [Revised: 08/11/2006] [Accepted: 08/21/2006] [Indexed: 10/24/2022]
Abstract
Statistical properties of neuron firing are studied in the framework of a nonlinear leaky integrate-and-fire model that is driven by a slow periodic subthreshold signal. The firing events are characterized by first passage time densities. The experimentally better accessible interspike interval density generally depends on the sojourn times in a refractory state of the neuron. This aspect is not part of the integrate-and-fire model and must be modelled additionally. For a large class of refractory dynamics, a general expression for the interspike interval density is given and further evaluated for the two cases with an instantaneous resetting (i.e. no refractory state) and a refractory state possessing a deterministic lifetime. First passage time densities and interspike interval densities following from the proposed theory compare favorably with precise numerical simulations.
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Affiliation(s)
- Marcin Kostur
- Institut für Physik, Universität Augsburg, D-86135 Augsburg, Germany
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Aradian A, Cates ME. Minimal model for chaotic shear banding in shear thickening fluids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:041508. [PMID: 16711810 DOI: 10.1103/physreve.73.041508] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/2005] [Revised: 02/13/2006] [Indexed: 05/09/2023]
Abstract
We present a minimal model for spatiotemporal oscillation and rheochaos in shear thickening complex fluids at zero Reynolds number. In the model, a tendency towards inhomogeneous flows in the form of shear bands combines with a slow structural dynamics, modeled by delayed stress relaxation. Using Fourier-space numerics, we study the nonequilibrium "phase diagram" of the fluid as a function of a steady mean (spatially averaged) stress, and of the relaxation time for structural relaxation. We find several distinct regions of periodic behavior (oscillating bands, traveling bands, and more complex oscillations) and also regions of spatiotemporal rheochaos. A low-dimensional truncation of the model retains the important physical features of the full model (including rheochaos) despite the suppression of sharply defined interfaces between shear bands. Our model maps onto the FitzHugh-Nagumo model for neural network dynamics, with an unusual form of long-range coupling.
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Affiliation(s)
- A Aradian
- SUPA, School of Physics, University of Edinburgh, JCMB Kings Buildings, Edinburgh EH9 3JZ, United Kingdom.
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Hänggi P, Marchesoni F. Introduction: 100 years of Brownian motion. CHAOS (WOODBURY, N.Y.) 2005; 15:26101. [PMID: 16035903 DOI: 10.1063/1.1895505] [Citation(s) in RCA: 77] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Affiliation(s)
- Peter Hänggi
- Institut für Physik, Universität Augsburg, 86135 Augsburg, Germany
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