The dynamics ofn weakly coupled identical oscillators

Ashwin, Peter; Swift, James W.

doi:10.1007/bf02429852

Cited by 221 publications

(283 citation statements)

References 27 publications

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“…As has been observed, these systems show very rich dynamics, both in physical and numerical experiments [ 1,21,17,5,4 ]. In particular, they admit trajectories and attractors with different symmetries.…”

Section: Introductionmentioning

confidence: 84%

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Detecting symmetry breaking bifurcations in the system describing the dynamics of coupled arrays of Josephson junctions

Tchistiakov

1996

Physica D: Nonlinear Phenomena

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Symmetry breaking and symmetry increasing bifurcations in a system of globally coupled Josephson junctions which possesses Sn symmetry have been investigated by constructing a detective for the group S,, the permutation group of n elements. In addition, with help of this detective several bifurcations are followed numerically in two parameters.An algorithm to obtain a detective for any finite subgroup of O(n) is given.

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Section: Introductionmentioning

confidence: 84%

“…While their technological interest has been the primary motivation for research over the past two decades, Josephson junction arrays have more recently become an archetype in the field of dynamical systems theory. Recent papers [ 1,21,17,22,5,4] have mainly been concerned with properties and bifurcations of symmetric solutions.…”

Section: Introductionmentioning

confidence: 99%

Detecting symmetry breaking bifurcations in the system describing the dynamics of coupled arrays of Josephson junctions

Tchistiakov

1996

Physica D: Nonlinear Phenomena

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“…For globally coupled networks with g i j = g maximally symmetric solutions describing synchronous, asynchronous, and cluster states are expected to be generic [2]. Here we shall focus on asynchronous states defined by φ i = i/N.…”

Section: Gap Junction Couplingmentioning

confidence: 99%

Gap Junctions and Emergent Rhythms

Coombes

Zachariou²

2009

Coherent Behavior in Neuronal Networks

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Gap junction coupling is ubiquitous in the brain, particularly between the dendritic trees of inhibitory interneurons. Such direct non-synaptic interaction allows for direct electrical communication between cells. Unlike spike-time driven synaptic neural network models, which are event based, any model with gap junctions must necessarily involve a single neuron model that can represent the shape of an action potential. Indeed, not only do neurons communicating via gaps feel super-threshold spikes, but they also experience, and respond to, sub-threshold voltage signals. In this chapter we show that the so-called absolute integrate-and-fire model is ideally suited to such studies. At the single neuron level voltage traces for the model may be obtained in closed form, and are shown to mimic those of fastspiking inhibitory neurons. Interestingly in the presence of a slow spike adaptation current the model is shown to support periodic bursting oscillations. For both tonic and bursting modes the phase response curve can be calculated in closed form. At the network level we focus on global gap junction coupling and show how to analyze the asynchronous firing state in large networks. Importantly, we are able to determine the emergence of non-trivial network rhythms due to strong coupling instabilities. To illustrate the use of our theoretical techniques (particularly the phase-density formalism used to determine stability) we focus on a spike adaptation induced transition from asynchronous tonic activity to synchronous bursting in a gap-junction coupled network.

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“…One of the aims is to establish under what conditions synchronous or asynchronous oscillations naturally arise. It is known from the theory of coupled oscillators (Winfree, 1980;Ashwin and Swift, 1992;Murray, 1993;Sole and Gamarra 1998;Lloyd and May, 1999) that synchronized oscillations through phase locking are a generic phenomenon. I will therefore concentrate on the occurrence of asynchronous dynamics and the conditions under which these arise from the interplay of the local population dynamics and the spatial interactions between locally oscillating populations.…”

Section: Introductionmentioning

confidence: 99%

The Dynamics of Two Diffusively Coupled Predator–Prey Populations

Jansen¹

2001

Theoretical Population Biology

153

110

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I analyze the dynamics of predator and prey populations living in two patches. Within a patch the prey grow logistically and the predators have a Holling type II functional response. The two patches are coupled through predator migration. The system can be interpreted as a simple predator prey metapopulation or as a spatially explicit predator prey system. Asynchronous local dynamics are presumed by metapopulation theory. The main question I address is when synchronous and when asynchronous dynamics arise. Contrary to biological intuition, for very small migration rates the oscillations always synchronize. For intermediate migration rates the synchronous oscillations are unstable and I found periodic, quasi-periodic, and intermittently chaotic attractors with asynchronous dynamics. For large predator migration rates, attractors in the form of equilibria or limit cycles exist in which one of the patches contains no prey. The dynamical behavior of the system is described using bifurcation diagrams. The model shows that spatial predator prey populations can be regulated through the interplay of local dynamics and migration. ]

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The dynamics ofn weakly coupled identical oscillators

Cited by 221 publications

References 27 publications

Detecting symmetry breaking bifurcations in the system describing the dynamics of coupled arrays of Josephson junctions

Detecting symmetry breaking bifurcations in the system describing the dynamics of coupled arrays of Josephson junctions

Gap Junctions and Emergent Rhythms

The Dynamics of Two Diffusively Coupled Predator–Prey Populations

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