The memristor is a two-terminal semiconductor device that is able to mimic the conductance response of synapses and can be utilized in next-generation computing platforms that will compute similarly to the mammalian brain. The initial memristor implementation is operated by the digital formation and dissolution of a highly conductive filament. However, an analog memristor is necessary to mimic analog synapses in the mammalian brain. To understand the mechanisms of operation and impact of different device designs, analog memristors were fabricated, modeled, and characterized. To realize analog memristors, lithiated transition metal oxides were grown by molecular beam epitaxy, RF sputtering, and liquid phase electro-epitaxy. Analog memristors were modeled using a finite element model simulation and characterized with X-ray absorption spectroscopy, impedance spectroscopy, and other electrical methods. It was shown that lithium movement facilitates analog memristance and nanoscopic ionic-electronic memristors with ion-soluble electrodes can be key enabling devices for brain-inspired computing.