Memristors can be employed to mimic biological neural synapses or to describe electromagnetic induction effects. To exhibit the threshold effect of electromagnetic induction, this paper presents a threshold flux-controlled memristor and examines its frequency-dependent pinched hysteresis loops. Using an electromagnetic induction current generated by the threshold memristor to replace the external current in 2-D Hindmarsh-Rose (HR) neuron model, a 3-D memristive HR (mHR) neuron model with global hidden oscillations is established and the corresponding numerical simulations are performed. It is found that due to no equilibrium point, the obtained mHR neuron model always operates in hidden bursting firing patterns, including coexisting hidden bursting firing patterns with bistability also. In addition, the model exhibits complex dynamics of the actual neuron electrical activities, which acts like the 3-D HR neuron model, indicating its feasibility. In particular, by constructing the fold and Hopf bifurcation sets of the fast-scale subsystem, the bifurcation mechanisms of hidden bursting firings are expounded. Finally, circuit experiments on hardware breadboards are deployed and the captured results well match with the numerical results, validating the physical mechanism of biological neuron and the reliability of electronic neuron.

Continuous memristor has been widely used in chaotic oscillating circuits and neuromorphic computing systems. However, discrete memristor and its coupling discrete map haven't been noticed yet. This paper presents a discrete memristor and constructs a general two-dimensional memristive map model by coupling the discrete memristor with an existing discrete map. The pinched hysteresis loops of the discrete memristor are demonstrated. Four examples of memristive discrete maps are provided, and their coupling strength-relied and memristor initial-boosted complex dynamics are investigated using numerical measures. The evaluation results manifest that the discrete memristor can enhance chaos complexity and its coupling maps can generate hyperchaos. Particularly, the hyperchaotic sequences can non-destructively be controlled by memristor initial state and the initial-controlled hyperchaos is robust, which are applicable to many chaos-based applications. Additionally, we develop a hardware platform to implement the memristive maps and acquire the four-channel hyperchaotic sequences. We also apply the memristive maps to the application of secure communication and the experiments show that the memristive maps display better performance than some existing discrete maps.

The magnetic induction effects have been emulated by various continuous memristive models but they have not been successfully described by a discrete memristive model yet. To address this issue, this article first constructs a discrete memristor and then presents a discrete memristive Rulkov (m-Rulkov) neuron model. The bifurcation routes of the m-Rulkov model are declared by detecting the eigenvalue loci. Using numerical measures, we investigate the complex dynamics shown in the m-Rulkov model, including regime transition behaviors, transient chaotic bursting regimes, and hyperchaotic firing behaviors, all of which are closely relied on the memristor parameter. Consequently, the involvement of memristor can be used to simulate the magnetic induction effects in such a discrete neuron model. Besides, we elaborate a hardware platform for implementing the m-Rulkov model and acquire diverse spiking-bursting sequences. These results show that the presented model is viable to better characterize the actual firing activities in biological neurons than the Rulkov model when biophysical memory effect is supplied.