Phase models and clustering in networks of oscillators with delayed coupling

Campbell, Sue Ann; Wang, Zhen

doi:10.1016/j.physd.2017.09.004

Cited by 16 publications

(33 citation statements)

References 60 publications

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“…Systems of discrete coupled oscillators with and without delay show a wealth of different types of behavior. 27,29,30 Some of this behavior, such as locking and antiphase oscillations, is also seen in models where the coupling is indirect, such as through a bath. 31 An avenue for future research is thus to map our spatially extended system to a system of discrete oscillators where coupling strength and time delay depend on d and D and compare their behavior.…”

Section: Many Pacemakersmentioning

confidence: 93%

Synchronization in reaction–diffusion systems with multiple pacemakers

Nolet

Rombouts

Gelens

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Spatially extended oscillatory systems can be entrained by pacemakers, regions which oscillate with a higher frequency than the rest of the medium. Entrainment happens through waves originating at a pacemaker. Typically, biological and chemical media can contain multiple pacemaker regions which compete with each other. In this paper we perform a detailed numerical analysis of how wave propagation and synchronization of the medium depend on the properties of these pacemakers. We discuss the influence of the size and intrinsic frequency of pacemakers on the synchronization properties. We also study a system in which the pacemakers are embedded in a medium without any local dynamics. In this case, synchronization occurs if the coupling determined by distance and diffusion is strong enough. The transition to synchronization is reminiscent of systems of discrete coupled oscillators.

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Section: Many Pacemakersmentioning

confidence: 93%

Synchronization in reaction–diffusion systems with multiple pacemakers

Nolet

Rombouts

Gelens

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…3.2.1. When k = 1, however, there is only one zero eigenvalue which corresponds to motion along the solutions [7]. Thus for k = 1 the condition (10) does give asymptotic stability.…”

Section: Existence and Stabilitymentioning

confidence: 97%

“…In neural network models, the formation of neural assemblies has been analyzed by identifying cluster solutions in networks of intrinsically oscillating neurons [16,17,28,15,22,31,7]. Clustering defines a type of solution where the network of oscillators breaks into subgroups.…”

Section: Introductionmentioning

confidence: 99%

“…In previous work [31,7], we used the phase model approach to determine existence and stability conditions of cluster solutions of networks with neurons arranged in a 1-dimensional ring, of arbitrary size, with various connectivity schemes. As in many other studies [5,33], our work focused on cluster solutions where the phase difference between any two adjacent neurons in the network is the same.…”

Section: Introductionmentioning

confidence: 99%

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Spatially localized cluster solutions in inhibitory neural networks

Ryu

Miller

Teymuroglu

et al. 2021

Mathematical Biosciences

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Neurons in the inhibitory network of the striatum display cell assembly firing patterns which recent results suggest may consist of spatially compact neural clusters. Previous computational modeling of striatal neural networks has indicated that non-monotonic, distance-dependent coupling may promote spatially localized cluster firing. Here, we identify conditions for the existence and stability of cluster firing solutions in which clusters consist of spatially adjacent neurons in inhibitory neural networks. We consider simple non-monotonic, distance-dependent connectivity schemes in weakly coupled 1-D networks where cells make strong connections with their k th nearest neighbors on each side. Using the phase model reduction of the network system, we prove the existence of cluster solutions where neurons that are spatially close together are also synchronized in the same cluster, and find stability conditions for these solutions. Our analysis predicts the long-term behavior for networks of neurons, and we confirm our results by numerical simulations of biophysical neuron network models. Additionally, we add weaker coupling between closer neighbors as a perturbation to our network connectivity. We analyze the existence and stability of cluster solutions of the perturbed network and validate our results with numerical simulations. Our results demonstrate that an inhibitory network with non-monotonic, distance-dependent connectivity can exhibit cluster

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“…Several different approaches to this problem can be taken, depending on the context. Model networks of neurons that are intrinsically oscillatory can be studied using a phase model, phase resetting curve or a Poincaré map approach [16,5,6,36,13], but other approaches exist [43,59]. Continuum models have been useful for studying delay induced wave-propagation in large scale networks [27,28,47].…”

Section: Introductionmentioning

confidence: 99%

Effect of time delay on the synchronization of excitatory-inhibitory neural networks

Ryu

Campbell

2020

Preprint

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We study a model for a network of synaptically coupled, excitable neurons to identify the role of coupling delays in generating different network behaviors. The network consists of two distinct populations, each of which contains one excitatory-inhibitory neuron pair. The two pairs are coupled via delayed synaptic coupling between the excitatory neurons, while each inhibitory neuron is connected only to the corresponding excitatory neuron in the same population. We show that multiple equilibria can exist depending on the strength of the excitatory coupling between the populations. We conduct linear stability analysis of the equilibria and derive necessary conditions for delay-induced Hopf bifurcation. We show that these can induce two qualitatively different phase-locked behaviors, with the type of behavior determined by the sizes of the coupling delays. Numerical bifurcation analysis and simulations supplement and confirm our analytical results. Our work shows that the resting equilibrium point is unaffected by the coupling, thus the network exhibits bistability between a rest state and an oscillatory state. This may help understand how rhythms spontaneously arise neuronal networks.

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Phase models and clustering in networks of oscillators with delayed coupling

Cited by 16 publications

References 60 publications

Synchronization in reaction–diffusion systems with multiple pacemakers

Synchronization in reaction–diffusion systems with multiple pacemakers

Spatially localized cluster solutions in inhibitory neural networks

Effect of time delay on the synchronization of excitatory-inhibitory neural networks

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