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-computational properties of neurons. Neurons near Hopf bifurcation fire 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 firing, 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 firing 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.