Nowadays, there is a lot of study on memristorbased systems with multistability. However, there is no study on memristor with multistability. This brief constructs a mathematical memristor model with multistability. The origin of the multi-stable dynamics is revealed using standard nonlinear theory as well as circuit and system theory. Moreover, the multi-stable memristor is applied to simulate a synaptic connection in a Hopfield neural network. The memristive neural network successfully generates infinitely many coexisting chaotic attractors unobserved in previous Hopfield-type neural networks. The results are also confirmed in analog circuits based on commercially available electronic elements.

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The theoretical, numerical and experimental demonstrations of firing dynamics in isolated neuron are of great significance for the understanding of neural function in human brain. In this paper, a new type of locally active and non-volatile memristor with three stable pinched hysteresis loops is presented. Then a novel locally active memristive neuron model is established by using the locally active memristor as a connecting autapse, both firing patterns and multistability in this neuronal system are investigated. We have confirmed that, on the one hand, the construced neuron can generate multiple firing patterns like periodic bursting, periodic spiking, chaotic bursting, chaotic spiking, stochastic bursting, transient chaotic bursting and transient stochastic bursting. On the other hand, the phenomenon of firing multistability with coexisting four kinds of firing patterns can be observed via changing its initial states. It is worth noting that the proposed neuron exhibits such firing multistability previously unobserved in single neuron model. Finally, an electric neuron is designed and implemented, which is extremely useful for the practical scientific and engineering applications. The results captured from neuron hardware experiments match well with the theoretical and numerical simulation results.

Neural networks have been widely and deeply studied in the field of computational neurodynamics. However, coupled neural networks and their brain-like chaotic dynamics have not been noticed yet. This paper focuses on the coupled neural network-based brain-like initial boosting coexisting hyperchaos and its application in biomedical image encryption. We first construct a memristive coupled neural network (MCNN) model based on two sub-neural networks and one multistable memristor synapse. Then we investigate its coupling strength-related dynamical behaviors, initial states-related dynamical behaviors, and initial-boosted coexisting hyperchaos using bifurcation diagrams, phase portraits, Lyapunov exponents and attraction basins. The numerical results demonstrate that the proposed MCNN can not only generate hyperchaotic attractors with high complexity but also boost the attractor positions by switching their initial states. This makes the MCNN more suitable for many chaos-based engineering applications. Moreover, we design a biomedical image encryption scheme to explore the application of the MCNN. Performance evaluations show that the designed cryptosystem has several advantages in the keyspace, information entropy, and key sensitivity. Finally, we develop a field-programmable gate array (FPGA) test platform to verify the practicability of the presented MCNN and the designed medical image cryptosystem.

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