Dynamic behaviors analysis of fraction-order neural network under memristive electromagnetic induction

Abstract

The dynamic behaviors of coupled neurons with different mathematical representations have received more and more attention in recent years. The coupling among heterogeneous neurons can show richer dynamic phenomena, which is of great significance in understanding the function of the human brain. In this paper, we present a fraction-order heterogeneous network with three neurons, which is built by coupling an FN neuron with two HR neurons. Complex electromagnetic surroundings have meaningful physical influence on the electrical activities of neurons. To imitate the effects of electromagnetic induction on the three-neuron heterogeneous network, we introduce a fraction-order locally active memristor in the neural network. The characteristics of this memristor are carefully analyzed by pinched hysteresis loops and its locally active characteristic is proved by the power-off plot and the DC <i>v-i</i> plot. Then, the parameter-dependent dynamic activities are investigated numerically by using several dynamical analysis methods, such as the phase diagrams, bifurcation diagrams, Lyapunov exponent spectra, and attraction basins. In addition, the network also reveals rich dynamic behaviors, including coexisting activities, anti-monotonicity phenomena, transient chaos and firing patterns, providing support for further investigating the firing patterns of the human brain. In particular, complex dynamics, including coexisting attractors, anti-monotonicity, and firing patterns, can be influenced by the order and strength of electrical synaptic coupling and electromagnetic induction. The control of the bistable state can be realized through the time feedback control method, so that the bistable state can be transformed into an ideal monostable state. The study of the fraction-order memristive neural network may expand the field of view for understanding the collective behaviors of neurons. Finally, based on the ARM platform, we give a digital implementation of the fraction-order memristive neural network, which can verify the consistency with the numerical simulation results. In the future, we will explore more interesting memristive neural networks and study different types of methods to control the firing behaviors of the networks.