Nowadays, research, modeling, simulation and realization of brain-like systems to reproduce brain behaviors have become urgent requirements. In this paper, neural bursting and synchronization are imitated by modeling two neural network models based on the Hopfield neural network (HNN). The first neural network model consists of four neurons, which correspond to realizing neural bursting firings. Theoretical analysis and numerical simulation show that the simple neural network can generate abundant bursting dynamics including multiple periodic bursting firings with different spikes per burst, multiple coexisting bursting firings, as well as multiple chaotic bursting firings with different amplitudes. The second neural network model simulates neural synchronization using a coupling neural network composed of two above small neural networks. The synchronization dynamics of the coupling neural network is theoretically proved based on the Lyapunov stability theory. Extensive simulation results show that the coupling neural network can produce different types of synchronous behaviors dependent on synaptic coupling strength, such as anti-phase bursting synchronization, anti-phase spiking synchronization, and complete bursting synchronization. Finally, two neural network circuits are designed and implemented to show the effectiveness and potential of the constructed neural networks.
Due to the existence of membrane potential differences, the electromagnetic induction flows can be induced in the interconnected neurons of Hopfield neural network (HNN). To express the induction flows, this paper presents a unified memristive HNN model using hyperbolic-type memristors to link neurons. By employing theoretical analysis along with multiple numerical methods, we explore the electromagnetic induction effects on the memristive HNN with three neurons. Three cases are classified and discussed. When using one memristor to link two neurons bidirectionally, the coexisting bifurcation behaviors are disclosed with respect to the memristor coupling strength. When using two memristors to link three neurons, the antimonotonicity phenomena of periodic and chaotic bubbles are yielded, and initial-related multistable patterns are emerged. When using three memristors to link three neurons end to end, the extreme event owning riddled basin of attraction is demonstrated. In addition, we develop the printed circuit board (PCB)-based hardware experiments by synthesizing the memristive HNN and the experimental results well confirm the memristive electromagnetic induction effects.
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