We present an experimental system of networks of coupled non-linear nanoliter-scale chemical reactors containing the Belousov-Zhabotinsky (BZ) reaction, which we theoretically model within a reaction-diffusion framework. Microfluidic fabrication techniques are developed that provide the ability to vary the network topology, the reactor coupling strength and offer the freedom to choose whether an arbitrary reactor is inhibitory or excitatory coupled to its neighbor. This versatile experimental and theoretical framework can be used to create a wide variety of chemical networks. Here we design, construct and characterize chemical networks that achieve the complexity of Central Pattern Generators, which are found in the autonomic nervous system of a variety of organisms. We envision that this artificial nervous system can serve as the controller of a synthetic musculature comprised of chemomechanical gels coupled to the BZ layer in order to create soft robots capable of autonomous activity.