Biological neurons exhibit rich and complex nonlinear dynamics, which are computationally expensive and power-hungry for hardware implementation. This paper demonstrates the design and development of a hardware-friendly nonlinear neuron model based on an intuitive control theory perspective. The neuron consists of a mixed-feedback system operating at multiple timescales to exhibit a variety of modalities that resemble the biophysical mechanisms found in neurophysiology. The single neuron dynamics emerges from four voltage-controlled current sources and features spiking and bursting output modes that can be controlled using tunable parameters. The bifurcation structures of the neuron, modeled as a 4D dynamical system, illustrate the roles of sources acting on different timescales in shaping the neural dynamics. For the first time, a neural network test chip consisting of 6 nonlinear bio-mimetic neurons and 10 tunable synapses was designed on 180nm CMOS technology. A 4-neuron network with inhibitory synapses of increasing strength was verified to achieve coupled rhythms. The test chip has an area of 0.6mm x 2mm and consumes 0.753mW of total average power.