Bursting near Bautin bifurcation in a neural network with delay coupling

Bursting behavior is one of the most important firing activities of neural system and plays an important role in signal encoding and transmission. In the present paper, a neural network with delay coupling is modeled to investigate the generation mechanism of bursting behavior. The Andronov-Hopf bifurcation is firstly studied and then the degenerated Andronov-Hopf bifurcation, namely Bautin bifurcation, is analyzed with the external input varying. Classifying dynamics in the neighborhood of the Bautin bifurcation, we obtain the bifurcation sets where the supercritical/subcritical Andronov-Hopf, or the fold limit cycle bifurcation may happen in the system under consideration. For a periodic disturbance occurring in the neighborhood of the Bautin bifurcation, it is seen that the Andronov-Hopf bifurcation and fold limit cycle bifurcation may lead to the transition from quiescent state to firing activities. Complex bursting phenomena, including Hopf/Hopf bursting, Hopf/Fold cycle bursting, SubHopf/Hopf bursting and SubHopf/Fold cycle bursting are found in the firing area. The results show that the dynamical properties of different burstings are related to the dynamical behaviors derived from the bifurcations of the system. Finally, it is seen that the bursting disappears but the periodic spiking appears in the delayed neural network for large values of delay.