On the Phase Reduction and Response Dynamics of Neural Oscillator Populations

Brown, Eric; Moehlis, Jeff; Holmes, Philip

doi:10.1162/089976604322860668

Cited by 489 publications

(694 citation statements)

References 56 publications

(110 reference statements)

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“…In general, the coupling functions h ij depend on the iPRC and may not be sinusoidal. Hence, the iPRC serves as a natural analysis (Sacré and Sepulchre, 2014;Sacré, 2013;Brown et al, 2004) and design Wang and Doyle, 2012) tool for general limit-cycle oscillator networks.…”

Section: Canonical Coupled Oscillator Modelmentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

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The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research.

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Section: Canonical Coupled Oscillator Modelmentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

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“…The most heavily studied systems, such as the Kuramoto model, assume that interactions among oscillators are sinusoidal (Kuramoto 1984;Strogatz 2000) or that they perturb state velocities directly (Brown et al 2004). In circadian clock models, a signal sent to an oscillator ultimately manifests as the manipulation of a single parameter.…”

Section: Theoretical Backgroundmentioning

confidence: 99%

Synchrony and entrainment properties of robust circadian oscillators

Bagheri

Taylor

Meeker

et al. 2008

J. R. Soc. Interface.

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Systems theoretic tools (i.e. mathematical modelling, control, and feedback design) advance the understanding of robust performance in complex biological networks. We highlight phase entrainment as a key performance measure used to investigate dynamics of a single deterministic circadian oscillator for the purpose of generating insight into the behaviour of a population of (synchronized) oscillators. More specifically, the analysis of phase characteristics may facilitate the identification of appropriate coupling mechanisms for the ensemble of noisy (stochastic) circadian clocks. Phase also serves as a critical control objective to correct mismatch between the biological clock and its environment. Thus, we introduce methods of investigating synchrony and entrainment in both stochastic and deterministic frameworks, and as a property of a single oscillator or population of coupled oscillators.

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“…This is because near this onset the neural dynamics can be assimilated to their normal forms, which depend only on the bifurcation structure. In Brown et al (2004) where that phase variable goes from 0 to 2 and the constant H can be derived from the original equations of the model. It is important to remark that in the second case the iPRC of the full model will approach to the result of (15.13) only if the radius of the oscillation near the bifurcation is small (Brown et al 2004).…”

Section: Examples Of Iprc Of Neuronsmentioning

confidence: 99%

“…In Brown et al (2004) where that phase variable goes from 0 to 2 and the constant H can be derived from the original equations of the model. It is important to remark that in the second case the iPRC of the full model will approach to the result of (15.13) only if the radius of the oscillation near the bifurcation is small (Brown et al 2004). This is not the case in the HH model, therefore the relation between the iPRC of the full model and the iPRC of the normal form is only qualitative.…”

Section: Examples Of Iprc Of Neuronsmentioning

confidence: 99%

“…1 or in the limit of small firing rate, the QIF model reduces to the theta model (Ermentrout and Kopell 1986;Ermentrout 1996). This model describes the normal form of the oscillatory dynamics appearing after a saddle-node bifurcation (Brown et al 2004;Ermentrout 1996) that characterizes Type I neurons (see Hansel and Mato 2003 for the details of the derivation).…”

Section: Iprc Of Integrate-and-fire Neuronsmentioning

confidence: 99%

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The Role of Intrinsic Cell Properties in Synchrony of Neurons Interacting via Electrical Synapses

Hansel

Mato

Pfeuty

2011

Phase Response Curves in Neuroscience

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The study of the synchronization of neural activity has to deal with the existence of a great diversity of neuronal and synaptic properties, which both may influence the collective behavior of neurons. In this chapter, we present an approach that allows to disentangle the respective role of neuronal and synaptic properties in synchronization. It relies on the theory of weakly coupled oscillators and on the concept of infinitesimal Phase Response Curve (iPRC) which is relevant for systems where neurons fire periodically. Applying this theory to networks of neurons coupled with electrical synapses provides a unifying framework explaining how their synchronization behavior depends on the shape of the iPRC. It reveals that synchronization mediated by electrical synapses is more versatile than previously considered. It depends on the intrinsic currents and morphology of neurons as well as on their combination with inhibitory synapses.

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On the Phase Reduction and Response Dynamics of Neural Oscillator Populations

Cited by 489 publications

References 56 publications

Synchronization in complex networks of phase oscillators: A survey

Synchronization in complex networks of phase oscillators: A survey

Synchrony and entrainment properties of robust circadian oscillators

The Role of Intrinsic Cell Properties in Synchrony of Neurons Interacting via Electrical Synapses

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