Biological synapses are behaving as dynamically-rich nonlinear elements, participating in complicated computing tasks through their adaptation due to external stimuli. Such adaptivity constitutes an intrinsic property of non-volatile memristor devices, which are also able to maintain their internal state, under zero input, enabling novel bio-inspired learning operations. In this work, synaptic element based on the memristive bridge is studied, aiming to investigate complex memristor-based topologies that may result in rich synaptic dynamics. The memristive bridge allows the realization of both positive and negative synaptic weights, while an asymmetric tuning of a weight, stemming from memristor's features, is demonstrated. In particular, by properly selecting memristor's position and polarity within the bridge, different tuning behaviors have been observed, showcasing versatile learning properties of the topology. Along with the synaptic weight tuning, the read overall process of the synaptic weight, necessary for inference operations, is also investigated. We explore the dynamics of the bridge via numerical simulations.