A mechanism for elliptic-like bursting and synchronization of bursts in a map-based neuron network.

A system consisting of two Rulkov map-based neurons coupled through reciprocal electrical synapses as a simple phenomenological example is discussed. When the electrical coupling is excitatory, the square-wave bursting can be well predicted by the bifurcation analysis of a comparatively simple low-dimensional subsystem embedded in the invariant manifold. While, when the synapses are inhibitory due to the artificial electrical coupling, a fast-slow analysis is carried out by treating the two slow variables as two different bifurcation parameters. The main result of this paper is to present a mechanism for the occurrence of a kind of special elliptic bursting. The mechanism for this kind of elliptic-like bursting is due to the interaction between two chaotic oscillations with different amplitudes. Moreover, the generation of antiphase synchronization of networks lies in the different switching orders between two pairs of different chaotic oscillations corresponding to the first neuron and the second neuron, respectively.