Burst and spike synchronization of coupled neural oscillators

Abstract

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.

A model for the collective synchronization of flashing in Photinus carolinus

“…Interestingly, one aspect of the EB dynamics that does depend on the details of the full model used is the synchronization of events within bursts. Although our model maintains the synchronization of fireflies’ flashes as long as their bursts overlap, past work has shown that other models can be tuned in or out of this regime [43,44]. Hence, the empirical observation of the maintenance of flash synchrony within bursts represents a useful constraint on which models may be relevant.…”