Biological neural networks do not only include long-term memory and weight multiplication capabilities, as commonly assumed in artificial neural networks, but also more complex functions such as short-term memory, short-term plasticity, and meta-plasticity - all collocated within each synapse. Here, we demonstrate memristive nano-devices based on SrTiO3 that inherently emulate all these synaptic functions. These memristors operate in a non-filamentary, low conductance regime, which enables stable and energy efficient operation. They can act as multi-functional hardware synapses in a class of bio-inspired deep neural networks (DNN) that make use of both long- and short-term synaptic dynamics and are capable of meta-learning or "learning-to-learn". The resulting bio-inspired DNN is then trained to play the video game Atari Pong, a complex reinforcement learning task in a dynamic environment. Our analysis shows that the energy consumption of the DNN with multi-functional memristive synapses decreases by about two orders of magnitude as compared to a pure GPU implementation. Based on this finding, we infer that memristive devices with a better emulation of the synaptic functionalities do not only broaden the applicability of neuromorphic computing, but could also improve the performance and energy costs of certain artificial intelligence applications.