We aim to design a neuromorphic controller for the locomotion of a quadruped robot with muscle-driven leg mechanisms. To this end, we use a simulated cat model; each leg of the model is equipped with three joints driven by six muscle models incorporating two-joint muscles. For each leg, we use a two-level central pattern generator consisting of a rhythm generation part to produce basic rhythms and a pattern formation part to synergistically activate a different set of muscles in each of the four sequential phases (swing, touchdown, stance, and liftoff). Conventionally, it was difficult for a quadruped model with such realistic neural systems and muscle-driven leg mechanisms to walk even on flat terrain, but because of our improved neural and mechanical components, our quadruped model succeeds in reproducing motoneuron activations and leg trajectories similar to those in cats and achieves stable three-dimensional locomotion at a variety of speeds. Moreover, the quadruped is capable of walking upslope and over irregular terrains and adapting to perturbations, even without adjusting the parameters.