scholarly journals Multistability in a star network of Kuramoto-type oscillators with synaptic plasticity

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Irmantas Ratas ◽  
Kestutis Pyragas ◽  
Peter A. Tass
Keyword(s):  
Synaptic Plasticity ◽  
Time Scales ◽  
Phase Difference ◽  
Coupling Strength ◽  
Coupled Oscillators ◽  
Analytical Approach ◽  
Boundary Function ◽  
Phase Oscillators ◽  
Star Network ◽  
Small Parameters

AbstractWe analyze multistability in a star-type network of phase oscillators with coupling weights governed by phase-difference-dependent plasticity. It is shown that a network with N leaves can evolve into $$2^N$$ 2 N various asymptotic states, characterized by different values of the coupling strength between the hub and the leaves. Starting from the simple case of two coupled oscillators, we develop an analytical approach based on two small parameters $$\varepsilon$$ ε and $$\mu$$ μ , where $$\varepsilon$$ ε is the ratio of the time scales of the phase variables and synaptic weights, and $$\mu$$ μ defines the sharpness of the plasticity boundary function. The limit $$\mu \rightarrow 0$$ μ → 0 corresponds to a hard boundary. The analytical results obtained on the model of two oscillators are generalized for multi-leaf star networks. Multistability with $$2^N$$ 2 N various asymptotic states is numerically demonstrated for one-, two-, three- and nine-leaf star-type networks.

TWOFOLD IMPACT OF DELAYED FEEDBACK ON COUPLED OSCILLATORS

2007 ◽  
Vol 17 (07) ◽  
pp. 2517-2530 ◽  
Author(s):  
OLEKSANDR V. POPOVYCH ◽  
VALERII KRACHKOVSKYI ◽  
PETER A. TASS
Keyword(s):  
Time Delay ◽  
Coupling Strength ◽  
Coupled Oscillators ◽  
Phase Synchronization ◽  
Threshold Value ◽  
Phase Oscillators ◽  
Intermediate Coupling ◽  
Rich Variety ◽  
Dynamical Regimes ◽  
The Impact

We present a detailed bifurcation analysis of desynchronization transitions in a system of two coupled phase oscillators with delay. The coupling between the oscillators combines a delayed self-feedback of each oscillator with an instantaneous mutual interaction. The delayed self-feedback leads to a rich variety of dynamical regimes, ranging from phase-locked and periodically modulated synchronized states to chaotic phase synchronization and desynchronization. We show that an increase of the coupling strength between oscillators may lead to a loss of synchronization. Intriguingly, the delay has a twofold influence on the oscillations: synchronizing for small and intermediate coupling strength and desynchronizing if the coupling strength exceeds a certain threshold value. We show that the desynchronization transition has the form of a crisis bifurcation of a chaotic attractor of chaotic phase synchronization. This study contributes to a better understanding of the impact of time delay on interacting oscillators.

scholarly journals Synchronization of Nonidentical Coupled Phase Oscillators in the Presence of Time Delay and Noise

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Somrita Ray ◽  
Monoj Kumar Sen ◽  
Alendu Baura ◽  
Bidhan Chandra Bag
Keyword(s):  
Time Delay ◽  
Phase Difference ◽  
Coupling Strength ◽  
Deterministic System ◽  
Critical Coupling ◽  
Noise Strength ◽  
Phase Oscillators ◽  
Intrinsic Correlation ◽  
Non Gaussian ◽  
Coupled Phase Oscillators

We have studied in this paper the dynamics of globally coupled phase oscillators having the Lorentzian frequency distribution with zero mean in the presence of both time delay and noise. Noise may be Gaussian or non-Gaussian in characteristics. In the limit of zero noise strength, we find that the critical coupling strength (CCS) increases linearly as a function of time delay. Thus the role of time delay in the dynamics for the deterministic system is qualitatively equivalent to the effect of frequency fluctuations of the phase oscillators by additive white noise in absence of time delay. But for the stochastic model, the critical coupling strength grows nonlinearly with the increase of the time delay. The linear dependence of the critical coupling strength on the noise intensity also changes to become nonlinear due to creation of additional phase difference among the oscillators by the time delay. We find that the creation of phase difference plays an important role in the dynamics of the system when the intrinsic correlation induced by the finite correlation time of the noise is small. We also find that the critical coupling is higher for the non-Gaussian noise compared to the Gaussian one due to higher effective noise strength.

FREQUENCY SYNCHRONIZATION IN NETWORKS OF COUPLED OSCILLATORS, A MONOTONE DYNAMICAL SYSTEMS APPROACH

2009 ◽  
Vol 19 (12) ◽  
pp. 4107-4116 ◽  
Author(s):  
WEN-XIN QIN
Keyword(s):  
Dynamical Systems ◽  
Coupling Strength ◽  
Coupled Oscillators ◽  
Systems Approach ◽  
System Size ◽  
Coupling Scheme ◽  
Frequency Synchronization ◽  
Nonlinear Coupling ◽  
New Approach ◽  
Monotone Dynamical Systems

We propose a new approach to investigate the frequency synchronization in networks of coupled oscillators. By making use of the theory of monotone dynamical systems, we show that frequency synchronization occurs in networks of coupled oscillators, provided the coupling scheme is symmetric, connected, and strongly cooperative. Our criterion is independent of the system size, the coupling strength and the details of the connections, and applies also to nonlinear coupling schemes.

scholarly journals Mathematical Framework for Breathing Chimera States

2022 ◽  
Vol 32 (2) ◽  
Author(s):  
O. E. Omel’chenko
Keyword(s):  
Coupled Oscillators ◽  
Analytic Solution ◽  
Coarse Grained ◽  
Mathematical Framework ◽  
Intensive Study ◽  
Phase Oscillators ◽  
Chimera States ◽  
The Subject ◽  
Self Consistency

AbstractAbout two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence–incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott–Antonsen reduction technique.

scholarly journals Behavioral Time Scale Plasticity of Place Fields: Mathematical Analysis

2021 ◽  
Vol 15 ◽  
Author(s):  
Ian Cone ◽  
Harel Z. Shouval
Keyword(s):  
Synaptic Plasticity ◽  
Fixed Points ◽  
Time Scales ◽  
Convergence Time ◽  
Simple Mathematical Model ◽  
System Parameters ◽  
Temporal Correlations ◽  
Postsynaptic Activity ◽  
Naturally Occurring ◽  
Place Fields

Traditional synaptic plasticity experiments and models depend on tight temporal correlations between pre- and postsynaptic activity. These tight temporal correlations, on the order of tens of milliseconds, are incompatible with significantly longer behavioral time scales, and as such might not be able to account for plasticity induced by behavior. Indeed, recent findings in hippocampus suggest that rapid, bidirectional synaptic plasticity which modifies place fields in CA1 operates at behavioral time scales. These experimental results suggest that presynaptic activity generates synaptic eligibility traces both for potentiation and depression, which last on the order of seconds. These traces can be converted to changes in synaptic efficacies by the activation of an instructive signal that depends on naturally occurring or experimentally induced plateau potentials. We have developed a simple mathematical model that is consistent with these observations. This model can be fully analyzed to find the fixed points of induced place fields and how these fixed points depend on system parameters such as the size and shape of presynaptic place fields, the animal's velocity during induction, and the parameters of the plasticity rule. We also make predictions about the convergence time to these fixed points, both for induced and pre-existing place fields.

The Neuromolecular Brain

Neuro ◽  
2013 ◽  
Author(s):  
Nikolas Rose ◽  
Joelle M. Abi-Rached
Keyword(s):  
Gene Expression ◽  
Synaptic Plasticity ◽  
Learning And Memory ◽  
Time Scales ◽  
Molecular Level ◽  
Human Life ◽  
Neuronal Circuits ◽  
Molecular Processes ◽  
Nerve Development ◽  
The Brain

This chapter examines the neuromolecular and plastic brain. Ideas about plasticity and the openness of brains to environment influences, from initial evidence about nerve development, through the recognition that synaptic plasticity was the very basis of learning and memory, to evidence about the influence of environment on gene expression and the persistence throughout life of the capacity to make new neurons—all this made the neuromolecular brain seem exquisitely open to its milieu, with changes at the molecular level occurring throughout the course of a human life and thus shaping the growth, organization, and regeneration of neurons and neuronal circuits at time scales from the millisecond to the decade. This was an opportunity to explore the myriad ways in which the milieu got “under the skin,” implying an openness of these molecular processes of the brain to biography, sociality, and culture, and hence perhaps even to history and politics.

PHASE SYNCHRONIZATION OF LINEARLY AND NONLINEARLY COUPLED OSCILLATORS WITH INTERNAL RESONANCE

2009 ◽  
Vol 23 (30) ◽  
pp. 5715-5726
Author(s):  
YONG LIU
Keyword(s):  
Coupling Strength ◽  
Coupled Oscillators ◽  
Internal Resonance ◽  
Natural Frequencies ◽  
Phase Synchronization ◽  
Coupled Systems ◽  
Nonlinear Coupling ◽  
Linear Coupling ◽  
Phase Dynamics ◽  
Diffuse Clouds

Phase synchronization between linearly and nonlinearly coupled systems with internal resonance is investigated in this paper. By introducing the conception of phase for a chaotic motion, it demonstrates that the detuning parameter σ between the two natural frequencies ω1and ω2affects phase dynamics, and with the increase in the linear coupling strength, the effect of phase synchronization between two sub-systems was enhanced, while increased firstly, and then decayed as nonlinear coupling strength increases. Further investigation reveals that the transition of phase states between the two oscillators are related to the critical changes of the Lyapunov exponents, which can also be explained by the diffuse clouds.

scholarly journals Phase Transition and Hysteresis in an Ensemble of Stochastic Spiking Neurons

2007 ◽  
Vol 19 (11) ◽  
pp. 3011-3050 ◽  
Author(s):  
Andreas Kaltenbrunner ◽  
Vicenç Gómez ◽  
Vicente López
Keyword(s):  
Phase Transition ◽  
Coupling Strength ◽  
Coupled Oscillators ◽  
Stochastic Dynamics ◽  
Upper And Lower Bounds ◽  
Critical Coupling ◽  
Sustained Activity ◽  
Large Ensembles ◽  
Coupling Strengths ◽  
Pulse Coupled Oscillators

An ensemble of stochastic nonleaky integrate-and-fire neurons with global, delayed, and excitatory coupling and a small refractory period is analyzed. Simulations with adiabatic changes of the coupling strength indicate the presence of a phase transition accompanied by a hysteresis around a critical coupling strength. Below the critical coupling production of spikes in the ensemble is governed by the stochastic dynamics, whereas for coupling greater than the critical value, the stochastic dynamics loses its influence and the units organize into several clusters with self-sustained activity. All units within one cluster spike in unison, and the clusters themselves are phase-locked. Theoretical analysis leads to upper and lower bounds for the average interspike interval of the ensemble valid for all possible coupling strengths. The bounds allow calculating the limit behavior for large ensembles and characterize the phase transition analytically. These results may be extensible to pulse-coupled oscillators.

scholarly journals Cascades of Multiheaded Chimera States for Coupled Phase Oscillators

2014 ◽  
Vol 24 (08) ◽  
pp. 1440014 ◽  
Author(s):  
Yuri L. Maistrenko ◽  
Anna Vasylenko ◽  
Oleksandr Sudakov ◽  
Roman Levchenko ◽  
Volodymyr L. Maistrenko
Keyword(s):  
Periodic Orbit ◽  
Coupled Oscillators ◽  
Phase Oscillators ◽  
Invariant Circle ◽  
Saddle Node Bifurcation ◽  
Chimera States ◽  
Chimera State ◽  
Self Organized ◽  
Saddle Node ◽  
Different Types

Chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states with increasing number of intervals of irregularity, so-called chimera's heads. We report three scenarios for the chimera birth: (1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER, (2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and (3) via homoclinic transition with complex multistable dynamics including an "eight-like" limit cycle resulting eventually in a chimera state.

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