Neuronal network connectivity demonstrates sparsity on multiple spatial scales and natural stimuli also possess sparse representations in numerous domains. In this talk, we underline the role of sparsity in the efficient encoding of network connectivity and inputs through nonlinear neuronal network dynamics. Addressing the fundamental challenge of recovering the structural connectivity of large-scale neuronal networks, we leverage properties of the balanced dynamical regime and compressive sensing theory to develop a theoretical framework for efficiently reconstructing sparse network connections through measurements of the network response to a relatively small ensemble of random stimuli. We further utilize sparse recovery ideas to probe the neural correlates of binocular rivalry through dynamic percept reconstructions based on the activity of a two-layer network model with competing downstream pools driven by disparate image stimuli. The resultant model dynamics agree with key experimental observations and give insights into the excitation/inhibition hypothesis for autism.