Reliability of coupled oscillators

Kevin K. Lin; Eric Shea-Brown; Lai Sang Young

doi:10.1007/s00332-009-9042-5

Reliability of coupled oscillators

Kevin K. Lin, Eric Shea-Brown, Lai Sang Young

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study the reliability of phase oscillator networks in response to fluctuating inputs. Reliability means that an input elicits essentially identical responses upon repeated presentations, regardless of the network's initial condition. Single oscillators are well known to be reliable. We show in this paper that unreliable behavior can occur in a network as small as a coupled oscillator pair in which the signal is received by the first oscillator and relayed to the second with feedback. A geometric explanation based on shear-induced chaos at the onset of phase-locking is proposed. We treat larger networks as decomposed into modules connected by acyclic graphs, and give a mathematical analysis of the acyclic parts. Moreover, for networks in this class, we show how the source of unreliability can be localized, and address questions concerning downstream propagation of unreliability once it is produced.

Original language	English (US)
Pages (from-to)	497-545
Number of pages	49
Journal	Journal of Nonlinear Science
Volume	19
Issue number	5
DOIs	https://doi.org/10.1007/s00332-009-9042-5
State	Published - Oct 2009

Keywords

Coupled oscillators
Neural network dynamics
Random dynamical systems

ASJC Scopus subject areas

Modeling and Simulation
Engineering(all)
Applied Mathematics

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title = "Reliability of coupled oscillators",

abstract = "We study the reliability of phase oscillator networks in response to fluctuating inputs. Reliability means that an input elicits essentially identical responses upon repeated presentations, regardless of the network's initial condition. Single oscillators are well known to be reliable. We show in this paper that unreliable behavior can occur in a network as small as a coupled oscillator pair in which the signal is received by the first oscillator and relayed to the second with feedback. A geometric explanation based on shear-induced chaos at the onset of phase-locking is proposed. We treat larger networks as decomposed into modules connected by acyclic graphs, and give a mathematical analysis of the acyclic parts. Moreover, for networks in this class, we show how the source of unreliability can be localized, and address questions concerning downstream propagation of unreliability once it is produced.",

keywords = "Coupled oscillators, Neural network dynamics, Random dynamical systems",

author = "Lin, {Kevin K.} and Eric Shea-Brown and Young, {Lai Sang}",

note = "Funding Information: This subsection reviews a number of ideas surrounding the mechanism above. This mechanism is known to occur in many different dynamical settings. We have elected to introduce the ideas in the context of discrete-time kicking of limit cycles instead of the continuous-time forcing in (6) because the geometry of discrete-time kicks is more transparent, and many of the results have been shown numerically to carry over with relatively minor modifications. Extensions to relevant settings are discussed later on in this subsection. A part of this body of work is supported by rigorous analysis. Specifically, theorems on shear-induced chaos for periodic kicks of limit cycles are proved in Wang and Young (2001, 2002, 2003, 2009); it is from these articles that many of the ideas reviewed here have originated. Numerical studies extending the ideas in Wang and Young (2002, 2003) to other types of underlying dynamics and forcing are carried out in Lin and Young (2008). For readers who wish to see a more",

year = "2009",

month = oct,

doi = "10.1007/s00332-009-9042-5",

language = "English (US)",

volume = "19",

pages = "497--545",

journal = "Journal of Nonlinear Science",

issn = "0938-8974",

publisher = "Springer New York",

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AU - Shea-Brown, Eric

AU - Young, Lai Sang

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PY - 2009/10

Y1 - 2009/10

N2 - We study the reliability of phase oscillator networks in response to fluctuating inputs. Reliability means that an input elicits essentially identical responses upon repeated presentations, regardless of the network's initial condition. Single oscillators are well known to be reliable. We show in this paper that unreliable behavior can occur in a network as small as a coupled oscillator pair in which the signal is received by the first oscillator and relayed to the second with feedback. A geometric explanation based on shear-induced chaos at the onset of phase-locking is proposed. We treat larger networks as decomposed into modules connected by acyclic graphs, and give a mathematical analysis of the acyclic parts. Moreover, for networks in this class, we show how the source of unreliability can be localized, and address questions concerning downstream propagation of unreliability once it is produced.

AB - We study the reliability of phase oscillator networks in response to fluctuating inputs. Reliability means that an input elicits essentially identical responses upon repeated presentations, regardless of the network's initial condition. Single oscillators are well known to be reliable. We show in this paper that unreliable behavior can occur in a network as small as a coupled oscillator pair in which the signal is received by the first oscillator and relayed to the second with feedback. A geometric explanation based on shear-induced chaos at the onset of phase-locking is proposed. We treat larger networks as decomposed into modules connected by acyclic graphs, and give a mathematical analysis of the acyclic parts. Moreover, for networks in this class, we show how the source of unreliability can be localized, and address questions concerning downstream propagation of unreliability once it is produced.

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Reliability of coupled oscillators

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