In this study, we explore spintronic synapses composed of several Magnetic Tunnel Junctions (MTJs), leveraging their attractive characteristics such as endurance, nonvolatility, stochasticity, and energy efficiency for hardware implementation of unsupervised neuromorphic systems. Spiking Neural Networks (SNNs) running on dedicated hardware are suitable for edge computing and IoT devices where continuous online learning and energy efficiency are important characteristics. We focus in this work on synaptic plasticity by conducting comprehensive electrical simulations to optimize the MTJ-based synapse design and find the accurate neuronal pulses that are responsible for the Spike Timing Dependent Plasticity (STDP) behavior. Most proposals in the literature are based on hardware-independent algorithms that require the network to store the spiking history to be able to update the weights accordingly. In this work, we developed a new learning rule, the Bi-Sigmoid STDP (B2STDP), which originates from the physical properties of MTJs. This rule enables immediate synaptic plasticity based on neuronal activity, leveraging in-memory computing. Finally, the integration of this learning approach within an SNN framework leads to a 91.71% accuracy in unsupervised image classification, demonstrating the potential of MTJ-based synapses for effective online learning in hardware-implemented SNNs.