By using compressive sensing (CS) theory, a broad class of static signals can be reconstructed through a sequence of very few measurements in the framework of a linear system. For networks with nonlinear and time-evolving dynamics, is it similarly possible to recover an unknown input signal from only a small number of network output measurements? We address this question for pulse-coupled networks, and investigate the network dynamics necessary for successful input signal recovery. Determining the specific network characteristics that correspond to a minimal input re- construction error, we are able to achieve high quality signal reconstructions with few measurements of network output. Using various measures to characterize dynamical properties of network output, we determine that networks with highly variable and aperiodic output can successfully encode net- work input information with high fidelity and achieve the most accurate CS input reconstructions. For time-varying inputs, we also find that high-quality reconstructions are achievable by measuring network output over a relatively short time-window. Even when network inputs change with time, the same optimal choice of network characteristics and corresponding dynamics apply as in the case of static inputs.
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This work was partly supported by the National Science Foundation through grant DMS-0636358.
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