Neuromorphic computing plays an important role in brain-inspired computers, which can solve complex problems with higher efficiency and lower energy. As a promising candidate of neuromorphic devices, a locally active memristor has the ability to emulate complex neuromorphic functions, which has the potential to implement large-scale hardware spiking neural networks to unleash the efficiency and intelligence of the brain. In this paper, we analyze the characteristics of the locally active 4-lobe Chua corsage memristor (CCM) using small-signal equivalent circuits. In addition, we design a CCM-based third-order neuron circuit and illustrate its rich neuromorphic behaviors near the edge of chaos, such as self-sustained oscillation, periodic spiking, chaos, transient chaos, resting states, burst-number adaption, spike latency, and refractory period. Furthermore, collective behaviors such as exotic chimera states (e.g. coherent and incoherent patterns) have been observed in the ring neural network made of third-order memristive neurons with linear and nonlocal coupling.