Analysis of a solvable model of a phase oscillator network on a circle with infinite-range Mexican-hat-type interaction

Uezu, Tatsuya; Kimoto, Tsunenobu; Okada, Masato

doi:10.1103/physreve.88.032918

Cited by 3 publications

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“…In our local coupling, the systems are positioned in space so that near-by systems excite each other whereas systems farther away inhibit each other, a so-called Mexican Hat coupling. This type of local coupling has been studied for deterministic Kuramoto phase oscillators in several papers, which we will discuss in relation to our results in the Discussion section [13,21,32,38,39]. Here we ask: can a spatial pattern of stochastic phase synchronization result from such a local coupling?…”

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Rapidly forming, slowly evolving, spatial patterns from quasi-cycle Mexican Hat coupling

Greenwood

Ward

2019

Mathematical Biosciences and Engineering

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A lattice-indexed family of stochastic processes has quasi-cycle oscillations if its otherwise-damped oscillations are sustained by noise. Such a family performs the reaction part of a discrete stochastic reaction-diffusion system when we insert a local Mexican Hat-type, difference of Gaussians, coupling on a one-dimensional and on a two-dimensional lattice. Quasi-cycles are a proposed mechanism for the production of neural oscillations, and Mexican Hat coupling is ubiquitous in the brain. Thus this combination might provide insight into the function of neural oscillations in the brain. Importantly, we study this system only in the transient case, on time intervals before saturation occurs. In one dimension, for weak coupling, we find that the phases of the coupled quasi-cycles synchronize (establish a relatively constant relationship, or phase lock) rapidly at coupling strengths lower than those required to produce spatial patterns of their amplitudes. In two dimensions the amplitude patterns form more quickly, but there remain parameter regimes in which phase synchronization patterns form without being accompanied by clear amplitude patterns. At higher coupling strengths we find patterns both of phase synchronization and of amplitude (resembling Turing patterns) corresponding to the patterns of phase synchronization. Specific properties of these patterns are controlled by the parameters of the reaction and of the Mexican Hat coupling.

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“…Solvable models of deterministic oscillators arranged in a ring with infinite Mexican Hat coupling, and corresponding simulations, appear in [38,39]. A paper that extends Kuramoto's result of an attracting invariant manifold for the phases of heterogeneous oscillators to higher dimensional space is [13].…”

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confidence: 97%

Rapidly forming, slowly evolving, spatial patterns from quasi-cycle Mexican Hat coupling

Greenwood

Ward

2019

Mathematical Biosciences and Engineering

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Correspondence between Phase Oscillator Network and Classical XY Model with the Same Infinite-Range Interaction in Statics

et al. 2015

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We study the phase oscillator networks with distributed natural frequencies and classical XY models both of which have a class of infinite-range interactions in common. We find that the integral kernel of the self-consistent equations (SCEs) for oscillator networks correspond to that of the saddle point equations (SPEs) for XY models, and that the quenched randomness (distributed natural frequencies) corresponds to thermal noise. We find a sufficient condition that the probability density of natural frequency distributions is one-humped in order that the kernel in the oscillator network is strictly decreasing as that in the XY model. Furthermore, taking the uniform and Mexican-hat type interactions, we prove the one to one correspondence between the solutions of the SCEs and SPEs. As an application of the correspondence, we study the associative memory type interaction. In the XY model with this interaction, there exists a peculiar one-parameter family of solutions. For the oscillator network, we find a non-trivial solution, i.e., a limit cycle oscillation.

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Properties of Coupled Oscillator Model for Bidirectional Associative Memory

Kawaguchi

2016

J. Phys. Soc. Jpn.

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Analysis of a solvable model of a phase oscillator network on a circle with infinite-range Mexican-hat-type interaction

Cited by 3 publications

References 35 publications

Rapidly forming, slowly evolving, spatial patterns from quasi-cycle Mexican Hat coupling

Rapidly forming, slowly evolving, spatial patterns from quasi-cycle Mexican Hat coupling

Correspondence between Phase Oscillator Network and Classical XY Model with the Same Infinite-Range Interaction in Statics

Properties of Coupled Oscillator Model for Bidirectional Associative Memory

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