Neuronal sensitivity and selectivity to our environment arise from the transformation of external inputs by the cortical circuitry. We dissect the circuitry for this emergence using a combination of large-scale simultaneous measurements from single cells in awake animals and computational models. In area MT, an area of the neocortex that processes visual motion, even weak motion signals evoke selective responses in the presence of noise. We also observe tuned patterns of activity in the absence of visual motion that suggest feature-selective amplification within the circuit. In addition, responses in MT have fast dynamics, posing critical constraints on the nature of the underlying amplification mechanisms. We identified different regimes in computational models capable of generating high amplification. These computational regimes exhibit distinct activity signatures, including the speed of amplification dynamics, the response to sudden shifts in input, as well as the structure and statistics of spontaneous activity. We examined the activity of large populations of primate MT neurons across the same conditions and compared the results with the computational models. We find that our recorded responses from MT network match a regime where amplification arises from separate excitatory and inhibitory populations operating in a balanced regime with tuned recurrent interactions. This allows the internal circuitry of the sensory cortex to strong amplify incoming inputs, while maintaining sensitivity to and rapid tracking of input changes because of strong inhibitory contributions that quench amplification (Murphy and Miller, Neuron, 2009). This balanced amplification regime only emerges in models composed of segregated excitatory and inhibitory populations. Our discovery provides a potential explanation for the specialization of neurons into distinct excitatory and inhibitory populations: the fundamental asymmetry that arises from coupling these populations is essential to the generation of large but rapid amplification without response persistence.