Burst and spike synchronization of coupled neural oscillators

Neural excitability and the bifurcations involved in transitions from quies cence 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 recurr ent transitions between a quiescent state and repetitive firing, i.e. the r apid oscillatory behaviour is modulated by a slowly varying dynamical proce ss. Bursting is referred to as elliptic bursting when the rest slate loses stability via a Hopf bifurcation and the repetitive firing disappears via a nother Hopf bifurcation or a double limit cycle bifurcation. By studying th e fast subsystem of two coupled bursters we obtain a reduced system of equa tions, allowing the description of synchronized and non-synchronized oscill ations depending on frequency detuning and mutual coupling strength. We sho w that a certain 'overall' coupling constant must exceed a critical value d epending on the detuning and the attraction rates in order that burst and s pike synchronization can lake place. The reduced system allows an analytica l study of the bifurcation structure up to co-dimension-3 revealing a varie ty of stationary and periodic bifurcations which will be analysed in detail . Finally, the implications of the bifurcation structure for burst and spik e synchronization are discussed.