Many quadruped robots have been developed to imitate their biological counterparts, several of which show excellent performance. However, the biological neural control mechanisms responsible for self-organized adaptive quadruped locomotion remain elusive. By drawing lessons from biological findings and using an artificial neural approach, we simulated a mammal-like quadruped robot and used it as our simulation platform to investigate and develop neural control mechanisms. In this study, we proposed an adaptive neural control network that can autonomously generate self-organized emergent locomotion with adaptability for the robot. The control network consists of three main components: Decoupled neural central pattern generator circuits (one for each leg), sensory feedback adaptation with dual-rate learning, and multiple neural reflex mechanisms. Simulation results show that the robot can perform quadruped-like gaits in a self-organized manner and adapt its gait to negotiate an obstacle. In addition, this work also suggests that the tight combination of the body-environment interaction and adaptive neural control, guided by sensory feedback adaptation and neural reflexes, is a powerful approach to better understand and solve self-organized adaptive coordination problems in quadruped locomotion.