The processing and transmission of biological neural information are realized via firing activities of neurons in different brain regions. Memristors, due to their nanoscale size, non-volatility and synapse-like plasticity, are regarded as ideal devices for emulating biological synapses. Hence, investigating firing modes of memristor-coupled heterogeneous neurons is significant. This paper focuses on modelling, firing modes and chaos synchronization of a memristor-coupled heterogeneous neuron. First, a novel locally active memristor is proposed, and its frequency characteristics, local activity and non-volatility are analyzed. Then, the novel locally active memristor is introduced into the two-dimensional HR neuron and the two-dimensional FHN neuron to construct a novel memristor-coupled heterogeneous neuron model. In numerical simulations, by changing a coupling strength, it is found that the model exhibits the periodic spiking firing mode, the chaotic spiking firing mode, the periodic bursting firing mode and the random bursting firing mode. Besides, the dynamic behavior of the novel memristor-coupled heterogeneous neuron can switch between periodic and chaotic behaviors by changing the initial state. Finally, based on the Lyapunov stability theory and the predefined-time stability theory, a novel predefined-time synchronization strategy is proposed and applied to achieve chaos synchronization of the novel memristor-coupled heterogeneous neuron. The results show that compared with a finite-time synchronization strategy, a fixed-time synchronization strategy and a traditional predefined-time synchronization strategy, the actual convergence time of the novel predefined-time synchronization strategy is the shortest. Studying the firing modes and chaos synchronization of the novel memristor-coupled heterogeneous neuron is helpful to explore the neural functions of the brain and has important significances in the neural signal processing and secure communication fields.
