Phase reduction and synchronization of a network of coupled dynamical elements exhibiting collective oscillations

Nakao, Hiroya; Yasui, Sho; Ota, Masahiro; Arai, Kohei; Kawamura, Y.

doi:10.1063/1.5009669

Cited by 21 publications

(29 citation statements)

References 35 publications

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“…The sensitivity functions of the individual elements, characterizing how tiny perturbations applied to each element affect the collective phase of the network, have been derived. In [41], the theory has further been generalized to arbitrary networks of coupled heterogeneous dynamical elements, where the dynamics of individual elements can also be arbitrary. As long as the network exhibits stable collective limit-cycle oscillations as a whole, the network dynamics can be reduced to a single phase equation for the collective phase and, for example, synchronization between a pair of such networks can be analysed by using the same classical methods as those for ordinary low-dimensional oscillators.…”

Section: Discussionmentioning

confidence: 99%

On the concept of dynamical reduction: the case of coupled oscillators

Kuramoto

Nakao

2019

Phil. Trans. R. Soc. A.

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An overview is given on two representative methods of dynamical reduction known as centermanifold reduction and phase reduction. These theories are presented in a somewhat more unified fashion than the theories in the past. The target systems of reduction are coupled limit-cycle oscillators. Particular emphasis is placed on the remarkable structural similarity existing between these theories. While the two basic principles, i.e. (i) reduction of dynamical degrees of freedom and (ii) transformation of reduced evolution equation to a canonical form, are shared commonly by reduction methods in general, it is shown how these principles are incorporated into the above two reduction theories in a coherent manner. Regarding the phase reduction, a new formulation of perturbative expansion is presented for discrete populations of oscillators. The style of description is intended to be so informal that one may digest, without being bothered with technicalities, what has been done after all under the word reduction. arXiv:1910.13775v1 [nlin.AO] 30 Oct 2019 II. CENTER-MANIFOLD REDUCTIONWe begin with the center-manifold reduction for a single oscillator, and then proceed to multi-oscillator systems. It will be seen how the two elements of reduction, namely, the reduction of dynamical degrees of freedom and transformation of the evolution equation to a canonical form, are incorporated into a systematic perturbation theory. A. Reduction of a single free oscillatorThe situation of our concern is the neighborhood of the Hopf bifurcation of an n-dimensional dynamical systeṁ X = F(X) with fixed point X = 0. We separate F into an unperturbed part and perturbation, where the former

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Section: Discussionmentioning

confidence: 99%

On the concept of dynamical reduction: the case of coupled oscillators

Kuramoto

Nakao

2019

Phil. Trans. R. Soc. A.

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“…As a second example, we consider the network of electrically coupled FHN neurons introduced in Ref. [45]. The network size N = 10 is two times larger than in The effective PRC is z = z4 + z5.…”

Section: B a Network Of Electrically Coupled Fhn Neuronsmentioning

confidence: 99%

“…In this paper, we apply this approach to a network of interacting neurons exhibiting collective periodic oscillations. We use the results of a recently developed phase reduction theory for arbitrary networks of coupled heterogeneous dynamical elements [45]. We also apply our approach to a large-scale heterogeneous network of globally coupled quadratic integrate-and-fire (QIF) neurons, which can be reduced to an exact low-dimensional macroscopic model in the infinite-size limit [46,47].…”

Section: Introductionmentioning

confidence: 99%

Optimal waveform for entrainment of a spiking neuron with minimum stimulating charge

et al. 2018

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Periodic pulse train stimulation is generically used to study the function of the nervous system and to counteract disease-related neuronal activity, e.g., collective periodic neuronal oscillations. The efficient control of neuronal dynamics without compromising brain tissue is key to research and clinical purposes. We here adapt the minimum charge control theory, recently developed for a single neuron, to a network of interacting neurons exhibiting collective periodic oscillations. We present a general expression for the optimal waveform, which provides an entrainment of a neural network to the stimulation frequency with a minimum absolute value of the stimulating current. As in the case of a single neuron, the optimal waveform is of bang-off-bang type, but its parameters are now determined by the parameters of the effective phase response curve of the entire network, rather than of a single neuron. The theoretical results are confirmed by three specific examples: two small-scale networks of FitzHugh-Nagumo neurons with synaptic and electric couplings, as well as a large-scale network of synaptically coupled quadratic integrate-and-fire neurons.

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“…The model reduction for collective dynamics has been intensively investigated for analyzing the macroscopic synchronization properties between each network that shows collective dynamics [45][46][47][48][49][50][51]. Moreover, several theoretical frameworks to reduce the collective dynamics to a single phase variable have been developed [52][53][54][55][56][57][58][59]. Using these methods, thus, we can investigate a macroscopic phase sensitivity function and a macroscopic phase coupling function between networks.…”

Section: Introductionmentioning

confidence: 99%

Extracting phase coupling functions between collectively oscillating networks directly from time-series data

Arai¹,

Kawamura²,

Aoyagi³

2021

Preprint

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Many real-world systems are often regarded as weakly coupled limit-cycle oscillators, in which each oscillator corresponds to a dynamical system with a large number of degrees of freedom exhibiting collective oscillations. One of the most practical methods for investigating the synchronization properties of such a rhythmic system is to statistically extract phase coupling functions between limit-cycle oscillators directly from observed time-series data. Particularly, using the method that combines phase reduction theory and Bayesian inference, the phase coupling functions can be extracted even from the time-series data of only one variable in each oscillatory dynamical system with many degrees of freedom. However, it remains unclear how the choice of the observed variables affects the statistical inference for the phase coupling functions. In this study, we examine the influence of observed variable types on the extraction of phase coupling functions using some typical dynamical elements under various conditions. Thus, we demonstrate that our method can consistently extract the macroscopic phase coupling functions between two phases representing collective oscillations regardless of the observed variable types, e.g., even when using one variable of any element in one system and the mean-field value over all the elements in another system.

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Phase reduction and synchronization of a network of coupled dynamical elements exhibiting collective oscillations

Cited by 21 publications

References 35 publications

On the concept of dynamical reduction: the case of coupled oscillators

On the concept of dynamical reduction: the case of coupled oscillators

Optimal waveform for entrainment of a spiking neuron with minimum stimulating charge

Extracting phase coupling functions between collectively oscillating networks directly from time-series data

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