[11] The mathematics of biological oscillators

Gb, Ermentrout

doi:10.1016/s0076-6879(94)40050-4

Cited by 12 publications

(13 citation statements)

References 22 publications

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“…Our simulations have been carried out with the XP-PAUT [54] package with a minimum of 201 globally coupled discrete oscillators and the results have been confirmed to remain unchanged with larger number of oscillators. The initial conditions for most of the simulations have been a splay state which is unstable in the parameter space of our simulations.…”

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confidence: 72%

Chimera States: The Existence Criteria Revisited

2014

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Chimera states, representing a spontaneous break-up of a population of identical oscillators that are identically coupled, into sub-populations displaying synchronized and desynchronized behaviour, have traditionally been found to exist in weakly coupled systems and with some form of nonlocal coupling between the oscillators. Here we show that neither the weak-coupling approximation nor nonlocal coupling are essential conditions for their existence. We obtain for the first time amplitudemediated chimera states in a system of globally coupled complex Ginzburg-Landau oscillators. We delineate the dynamical origins for the formation of such states from a bifurcation analysis of a reduced model equation and also discuss the practical implications of our discovery of this broader class of chimera states.PACS numbers: 05.45. Ra, 05.45.Xt, The spontaneous break-up of a system of identical oscillators, that are identically coupled, into sub-groups of oscillators with different synchronous properties is a fascinating collective phenomenon that was first reported by Kuramoto and Battogtokh[1] and has since been the subject of many investigations . This spatio-temporal pattern of co-existing synchronous and de-synchronous oscillations, named as a chimera state by Abrams and Strogatz [3], has also been experimentally demonstrated in a number of laboratory systems [31][32][33][34][35][36]. The natural manifestation of this state can be seen in such phenomena as unihemispherical sleep in many animals [37,38] where the awake side of the brain shows desynchronized electrical activity, whereas the sleeping side is highly synchronized [8] or in the human brain when in certain regions the neuronal activity gets highly synchronized during epileptic seizures [39] or damage due to Parkinson's disease [40]. In model studies mentioned above chimera states have been found in phase only oscillator systems and in the presence of a nonlocal coupling between the oscillators. This has given rise to a general notion that a weak coupling approximation (implying phase only oscillators) and nonlocal coupling are two essential ingredients for the existence of a chimera state. In a recent work [41] we have demonstrated that the weak coupling approximation is not critical and a more generalized version of the chimera state that includes amplitude effects can be a collective state of the nonlocal complex GinzburgLandau equation (NLCGLE). These amplitude-mediated chimeras (AMCs) can exist as stationary or travelling patterns and show intermittent emergence and decay of amplitude dips in the phase incoherent regions. The next question that naturally arises is whether the nonlocality in the coupling can also be relaxed and whether chimera states can form through other forms of coupling in a system of oscillators. In this paper we address this issue and show for the first time that the amplitude-mediated * e-mail: gautam.sethia@gmail.com chimera state can emerge even in a globally coupled system of oscillators. We discover these states from a numerical...

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confidence: 72%

Chimera States: The Existence Criteria Revisited

2014

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“…Saddle-node of (unstable) limit cycles 3 and 6 We used XPPAUT [35,36] to draw the bifurcation diagram ( Fig. 3) and then MATCONT [37] to draw several important trajectories (Figs.…”

Section: Numerical Simulations and Bifurcation Diagramsmentioning

confidence: 99%

Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network

Curtu

2010

Physica D: Nonlinear Phenomena

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“…To illustrate system (2.1)'s dynamics under the escape and release mechanisms we run numerical simulations in XPPAUT [7,9] with S as in (2.2), r = 10, θ = 0.2, and parameters β = 1.1, g = 0.5, τ = 100 as in Figure 3F. Then in the fast plane (u 1 , u 2 ) we obtain the trajectory of the point on the rivalry limit cycle: the point is drawn as a black thick dot; the u 1 -nullcline is colored in red, and the u 2 -nullcline is colored in blue.…”

Section: Appendix C Additional Materialsmentioning

confidence: 99%

Mechanisms for Frequency Control in Neuronal Competition Models

Curtu

Shpiro

Rubin³

et al. 2008

SIAM J. Appl. Dyn. Syst.

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We investigate analytically a firing rate model for a two-population network based on mutual inhibition and slow negative feedback in the form of spike frequency adaptation. Both neuronal populations receive external constant input whose strength determines the system's dynamical state -a steady state of identical activity levels or periodic oscillations or a winner-take-all state of bistability. We prove that oscillations appear in the system through supercritical Hopf bifurcations and that they are antiphase. The period of oscillations depends on the input strength in a nonmonotonic fashion, and we show that the increasing branch of the period versus input curve corresponds to a release mechanism and the decreasing branch to an escape mechanism. In the limiting case of infinitely slow feedback we characterize the conditions for release, escape, and occurrence of the winner-take-all behavior. Some extensions of the model are also discussed.

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[11] The mathematics of biological oscillators

Cited by 12 publications

References 22 publications

Chimera States: The Existence Criteria Revisited

Chimera States: The Existence Criteria Revisited

Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network

Mechanisms for Frequency Control in Neuronal Competition Models

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