In order to enrich the dynamic behaviors of discrete neuron models and more effectively mimic biological neural networks, this paper proposes a bistable locally active discrete memristor (LADM) model to mimic synapses. We explored the dynamic behaviors of neural networks by introducing the LADM into two identical Rulkov neurons. Based on numerical simulation, the neural network manifested multistability and new firing behaviors under different system parameters and initial values. In addition, the phase synchronization between the neurons was explored. Additionally, it is worth mentioning that the Rulkov neurons showed synchronization transition behavior; that is, anti-phase synchronization changed to in-phase synchronization with the change in the coupling strength. In particular, the anti-phase synchronization of different firing patterns in the neural network was investigated. This can characterize the different firing behaviors of coupled homogeneous neurons in the different functional areas of the brain, which is helpful to understand the formation of functional areas. This paper has a potential research value and lays the foundation for biological neuron experiments and neuron-based engineering applications.

Continuous memristors have been widely studied in recent years; however, there are few studies on discrete memristors in the field of neural networks. In this paper, a four-stable locally active discrete memristor (LADM) is proposed as a synapse, which is used to connect a two-dimensional Chialvo neuron and a three-dimensional KTZ neuron, and construct a simple heterogeneous discrete neural network (HDNN). Through a bifurcation diagram and Lyapunov exponents diagram, the period and chaotic regions of the discrete neural network model are shown. Through numerical analysis, it was found that the chaotic region and periodic region of the neural network based on DLAM are significantly improved. In addition, coexisting chaos and chaos attractors, coexisting periodic and chaotic attractors, and coexisting periodic and periodic attractors will appear when the initial value of the LADM is changed. Coupled by a LADM synapse, two heterogeneous discrete neurons are gradually synchronized by changing the coupling strength. This paper lays a good foundation for the future analysis of LADMs and the related research of discrete neural networks coupled by LADMs.