A Numerical Mechanism for Square-Wave or Elliptic Bursting of Bursts in a Map-Based Neuron Network
A network of two identical Rulkov map-based neurons coupled by reciprocal excitatory or inhibitory electrical synapses as a phenomenological example is investigated. This is because on the one hand, this network can exhibit many regular and irregular bursting oscillations, and such behaviors can reflect many functional roles in real neuron assemblies, especially when the information transmission and processing of biological neurons are concerned. On the other hand, this is motivated by experimental studies where the pyloric central pattern generators of the lobster stomatogastric ganglion are coupled by an artificial dynamical current clamp device [Phys. Rev. Lett. 81.5692, 1998]. So it is worthwhile to make a detailed study even for this simple map-based network. Our results demonstrate that there exist multiple cooperative behaviors of bursts. Moreover they can be well explained and predicted by two kinds of different strategies by using a fast-slow dynamics technique and bifurcation analysis. When the electrical coupling is excitatory or inhibitory due to the artificial electrical coupling, separately, a fast-slow analysis is carried out by treating the two slow variables as two different bifurcation parameters. The main contribution of this paper is to present a numerical mechanism for the occurrence of square-wave or elliptic bursting, which is due to the interaction between multiple stable branches of fixed points of the fast subsystem or two chaotic oscillations with different amplitudes. Particularly, 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.
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