Low-dimensional dynamics and bifurcations in oscillator networks via bi-orthogonal spectral decomposition

Schwarz, Jürgen; Bräuer, K.; Dangelmayr, Gerhard; Stevens, Andreas

doi:10.1088/0305-4470/33/18/303

Cited by 4 publications

(2 citation statements)

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“…The results can easily be extended to other systems of oscillators, such as for example bistable van der Pol oscillators (Defontaines et al 1990, Schwarz et al 2000a. Although this analytical study is for two coupled bursters without external forcing, it can serve to explain some of the signi® cant features observed in larger networks of bistable oscillators under external periodic stimulation (Schwarz et al 2000a, b).…”

Section: Discussionmentioning

confidence: 90%

Burst and spike synchronization of coupled neural oscillators

Schwarz¹,

Dangelmayr²,

Stevens³

et al. 2001

Dynamics & Stability Sys.

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Neural excitability and the bifurcations involved in transitions from quiescence to oscillations largely determine the neuro-computationa l properties of neurons. Neurons near Hopf bifurcation ® re in a small frequency range, respond preferentially to resonant excitation, and are easily synchronized. In the present paper we study the interaction of coupled elliptic bursters with non-resonant spike frequencies. Bursting behaviour arises from recurrent transitions between a quiescent state and repetitive ® ring, i.e. the rapid oscillatory behaviour is modulated by a slowly varying dynamical process. Bursting is referred to as elliptic bursting when the rest state loses stability via a Hopf bifurcation and the repetitive ® ring disappears via another Hopf bifurcation or a double limit cycle bifurcation. By studying the fast subsystem of two coupled bursters we obtain a reduced system of equations, allowing the description of synchronized and non-synchronized oscillations depending on frequency detuning and mutual coupling strength. We show that a certain`overall' coupling constant must exceed a critical value depending on the detuning and the attraction rates in order that burst and spike synchronization can take place. The reduced system allows an analytical study of the bifurcation structure up to codimension-3 revealing a variety of stationary and periodic bifurcations which will be analysed in detail. Finally, the implications of the bifurcation structure for burst and spike synchronization are discussed.

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