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Local stability results for the collective behaviors of infinite populations of pulse-coupled oscillators

Alexandre Mauroy, Rodolphe Sepulchre
2011 IEEE Conference on Decision and Control and European Control Conference  
In addition to recovering known stability results of finite populations, we also obtain new stability results for infinite populations.  ...  Because the stability analysis of finite populations is intricate, we investigate stability results in the approximation of infinite populations.  ...  PRELIMINARIES We present a brief overview of pulse-coupled oscillators and the phase density model obtained in the case of infinite populations. A.  ... 
doi:10.1109/cdc.2011.6160621 dblp:conf/cdc/MauroyS11 fatcat:scxx2bkqrzcalpoiewkbfxnbnq

Birth and destruction of collective oscillations in a network of two populations of coupled type 1 neurons [article]

Benjamin Jüttner and Christian Henriksen and Erik A. Martens
2020 arXiv   pre-print
Here, we explore how collective oscillations emerge for two identical neuron populations in the limit of an infinite number of neurons.  ...  , where the firing activity of one population exhibits quiescent but the other exhibits spiking behavior.  ...  ACKNOWLEDGEMENTS The authors would like to thank C. Bick and B. Pietras for helpful discussions, and A. Torcini and P. So for helpful correspondence.  ... 
arXiv:2009.11589v1 fatcat:wnn7zlxhqnberb7gdm4qpmrbh4

Global Analysis of a Continuum Model for Monotone Pulse-Coupled Oscillators

Alexandre Mauroy, Rodolphe J. Sepulchre
2013 IEEE Transactions on Automatic Control  
In addition, they provide a novel approach for the (global) analysis of pulse-coupled oscillators.  ...  In contrast with previous studies on related pulse-coupled models, the stability results obtained in the continuum limit are global.  ...  ACKNOWLEDGMENT The authors are grateful to Prof. Jean-Michel Coron for numerous valuable comments and useful discussions and to Prof. Igor Mezić for his help in revising the manuscript.  ... 
doi:10.1109/tac.2012.2229811 fatcat:tqr73qnhfngxdidc4q643fxdz4

Dynamics of a population of oscillatory and excitable elements

Kevin P. O'Keeffe, Steven H. Strogatz
2016 Physical review. E  
We present the stability diagram and calculate several of its bifurcation curves analytically, for both excitatory and inhibitory coupling.  ...  Using the Ott-Antonsen ansatz, we reduce the behavior of the population to a two-dimensional dynamical system with three parameters.  ...  When K = 0, however, the excitable elements feel the pulses of the self-oscillators, and the collective dynamics are no longer as clear.  ... 
doi:10.1103/physreve.93.062203 pmid:27415251 fatcat:izxubibdmbchlnzum6fsrzikm4

Synchronization Behavior in a Novel Phase Model [article]

Noah DeTal, Hossein Taheri, Kurt Wiesenfeld
2019 arXiv   pre-print
An infinite-oscillator limit of the model is derived and solved to provide theoretical predictions for the observed transitions.  ...  These results are compared with the phenomenology of the Kuramoto model which exhibits starkly different behavior.  ...  In Section III, we present the results of numerical simulations showing the transition behavior for different choices of disorder, and compare this with the corresponding behavior of the Kuramoto model  ... 
arXiv:1903.09930v1 fatcat:di6isyboovbo3dujql3guyojja

Kick synchronization versus diffusive synchronization [article]

Alexandre Mauroy, Pierre Sacré, Rodolphe Sepulchre
2013 arXiv   pre-print
, and the (hybrid) kick model, that accounts for most popular examples of synchronization, but for which only few theoretical results exist.  ...  coupling.  ...  Acknowledgments This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by  ... 
arXiv:1209.4970v3 fatcat:lj2hla3iqjgrzkhkufecljvnu4

Phase Diagram for the Winfree Model of Coupled Nonlinear Oscillators

Joel T. Ariaratnam, Steven H. Strogatz
2001 Physical Review Letters  
The system displays rich collective dynamics as a function of the coupling strength and the spread of natural frequencies.  ...  In 1967 Winfree proposed a mean-field model for the spontaneous synchronization of chorusing crickets, flashing fireflies, circadian pacemaker cells, or other large populations of biological oscillators  ...  With these choices, Eq. (1) becomes an idealized model for a population of pulse-coupled biological oscillators.  ... 
doi:10.1103/physrevlett.86.4278 pmid:11328154 fatcat:mwka3xm4jfg6xkwf6gcc2ybkc4

Kick synchronization versus diffusive synchronization

Alexandre Mauroy, Pierre Sacre, Rodolphe Sepulchre
2012 2012 IEEE 51st IEEE Conference on Decision and Control (CDC)  
, and the (hybrid) kick model, that accounts for most popular examples of synchronization, but for which only few theoretical results exist.  ...  coupling.  ...  Global stability results and collective behaviors In the weak coupling limit, the collective behaviors of diffusively coupled and impulsively coupled oscillators depend on the (global) stability properties  ... 
doi:10.1109/cdc.2012.6425821 dblp:conf/cdc/MauroySS12 fatcat:rbwgvlbllbf4nlumzddp5m67we

Emergence of one- and two-cluster states in populations of globally pulse-coupled oscillators [article]

Leonhard Lücken, Serhiy Yanchuk
2011 arXiv   pre-print
The subject of this paper is a system of phase-oscillators, which are globally pulse-coupled via excitatory interaction.  ...  The appearance and stability of one- and two-cluster-states is investigated for a family of unimodal phase-response-curves (PRC).  ...  In the neuroscience, for instance, there is a need for modeling large populations of coupled neurons in order to approach problems connected with the synchronization of neural cells or other types of collective  ... 
arXiv:1101.4487v2 fatcat:s6jrys2q5rcz5b4w33z3fjx47u

Suppression of synchronous spiking in two interacting populations of excitatory and inhibitory quadratic integrate-and-fire neurons [article]

Kestutis Pyragas, Augustinas P. Fedaravičius, Tatjana Pyragienė
2021 arXiv   pre-print
We show that in the limit cycle mode, high-frequency stimulation of an inhibitory population can stabilize an unstable resting state and effectively suppress collective oscillations.  ...  Collective oscillations and their suppression by external stimulation are analyzed in a large-scale neural network consisting of two interacting populations of excitatory and inhibitory quadratic integrate-and-fire  ...  For simplicity, we do not consider recurrent excitation within the E population, since it is not essential for the emergence of collective oscillations.  ... 
arXiv:2103.00867v2 fatcat:tsjwns5phffajgvaw6jiyepv5y

Contraction of monotone phase-coupled oscillators

A. Mauroy, R. Sepulchre
2012 Systems & control letters (Print)  
This paper establishes a global contraction property for networks of phase-coupled oscillators characterized by a monotone coupling function. The contraction measure is a total variation distance.  ...  The contraction property determines the asymptotic behavior of the network, which is either finite-time synchronization or asymptotic convergence to a splay state.  ...  Acknowledgments This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by  ... 
doi:10.1016/j.sysconle.2012.08.007 fatcat:br6ixi5rkfdznoriag7g7dw3ra

Symmetry breaking in two interacting populations of quadratic integrate-and-fire neurons

Irmantas Ratas, Kestutis Pyragas
2017 Physical review. E  
The coupling within and between the populations is global via synapses that take into account the finite width of synaptic pulses.  ...  We analyze the dynamics of two coupled identical populations of quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold.  ...  In Fig. 1 , we show the results of the above stability analysis in the plain of parameters (J ex , J in ).  ... 
doi:10.1103/physreve.96.042212 pmid:29347512 fatcat:wj5fomu4ejfybfn56e2ouz7gai

Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review

Christian Bick, Marc Goodfellow, Carlo R. Laing, Erik A. Martens
2020 Journal of Mathematical Neuroscience  
Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction.  ...  Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only.  ...  This article is part of the research activity (EAM, CL) of the Advanced Study Group 2017 "From Microscopic to Collective Dynamics in Neural Circuits" held at Max Planck Institute for the Physics of Complex  ... 
doi:10.1186/s13408-020-00086-9 pmid:32462281 fatcat:r35intdfsvcn7li6spbupxwyoa

Self-Sustained Irregular Activity in an Ensemble of Neural Oscillators

Ekkehard Ullner, Antonio Politi
2016 Physical Review X  
An ensemble of pulse-coupled phase-oscillators is thoroughly analysed in the presence of a mean-field coupling and a dispersion of their natural frequencies.  ...  The complex dynamics appears to be the result of the non-perturbative action of a suitably shaped phase-response curve.  ...  ϕ i ∈ ½0; 1 is the local phase, ω i the bare oscillator frequency (i.e., in the absence of coupling), g the coupling strength, and N the system size.  ... 
doi:10.1103/physrevx.6.011015 fatcat:yo4wzfoi3va2nicn2zdgdnlirq

Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review [article]

Christian Bick, Marc Goodfellow, Carlo R. Laing, Erik Andreas Martens
2020 arXiv   pre-print
Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction.  ...  Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.  ...  This article is part of the research activity (EAM, CL) of the Advanced Study Group 2017 "From Microscopic to Collective Dynamics in Neural Circuits" held at Max Planck Institute for the Physics of Complex  ... 
arXiv:1902.05307v3 fatcat:u5x2duovcnhrrg4wjvyt452dde
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