Coexistence of Multiple Stable States and Bursting Oscillations in a 4D Hopfield Neural Network

Neurons are regarded as basic, structural and functional units of the central nervous system. They play an active role in the collection, storing and transferring of the information during signal processing in the brain. In this paper, we investigate the dynamics of a model of a 4D autonomous Hopfield neural network (HNN). Our analyses highlight complex phenomena such as chaotic oscillations, periodic windows, hysteretic dynamics, the coexistence of bifurcations and bursting oscillations. More importantly, it has been found several sets of synaptic weight for which the proposed HNN displays multiple coexisting stable states including three disconnected attractors. Besides the phenomenon of coexistence of attractors, the bursting phenomenon characterized by homoclinic/Hopf cycle-cycle bursting via homoclinic/fold hysteresis loop is observed. This contribution represents the first case where the later phenomenon (bursting oscillations) occurs in an autonomous HNN. Also, PSpice simulations are used to support the results of the previous analyses.