Replicating the computational functionalities and performances of the brain remains one of the biggest challenges for the future of information and communication technologies. Such an ambitious goal requires research efforts from the architecture level to the basic device level (i.e., investigating the opportunities offered by emerging nanotechnologies to build such systems). Nanodevices, or, more precisely, memory or memristive devices, have been proposed for the implementation of synaptic functions, offering the required features and integration in a single component. In this paper, we demonstrate that the basic physics involved in the filamentary switching of electrochemical metallization cells can reproduce important biological synaptic functions that are key mechanisms for information processing and storage. The transition from short- to long-term plasticity has been reported as a direct consequence of filament growth (i.e., increased conductance) in filamentary memory devices. In this paper, we show that a more complex filament shape, such as dendritic paths of variable density and width, can permit the short- and long-term processes to be controlled independently. Our solid-state device is strongly analogous to biological synapses, as indicated by the interpretation of the results from the framework of a phenomenological model developed for biological synapses. We describe a single memristive element containing a rich panel of features, which will be of benefit to future neuromorphic hardware systems.