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The nonvolatile memory property of a memristor enables the realization of new methods for a variety of computational engines ranging from innovative memristive-based neuromorphic circuitry through to advanced memory applications. The nanometer-scale feature of the device creates a new opportunity for realization of innovative circuits that in some cases are not possible or have inefficient realization in the present and established design domain. The nature of the boundary, the complexity of the ionic transport and tunneling mechanism, and the nanoscale feature of the memristor introduces challenges in modeling, characterization, and simulation of future circuits and systems. Here, a deeper insight is gained in understanding the device operation, leading to the development of practical models that can be implemented in current computer-aided design (CAD) tools.