The Spike-Time-Dependent-Plasticity (STDP) learning rule is frequently associated with the memristor structures in the neuromorphic studies, recently. In this study, after two HR neurons are coupled with a memristor based synapse structure as a unidirectional coupling; it is aimed to correlate the STDP learning rule and the memristor based synapse structure. In line of this motivation, two HR neurons (the transmitter and the receiver neurons) have been coupled with a memductance structure. After the current at the output of this memristor based synapse has been applied to the receiver neuron, a phase shift has been induced on the membrane potential of this receiver neuron. Additionally, this phase shift can be controlled by changing the ’Ron’ resistance that is a parameter of the windowing function definition of the memristor device similar to the synaptic coupling weight parameter. Since the firing times of these coupled HR neurons can be controlled by changing this ’Ron’ resistance, the STDP learning rule, which is based on the instantaneous firing time difference between the coupled neurons, has been validated as an alternative view point in here. The similarity between the pattern of the STDP learning rule and the pattern of two HR neurons coupled with the chemical and the memristor based synapses has been confirmed by comparing the numerical simulation results and it has been seen that the memristor device can be used as an alternative synapse definition in the neuromorphic studies. Finally, two HR neurons coupled with the memristor based synapse have been successfully realized by using the FPGA equipment, so an alternative neuromorphic study, which is associated with the STDP learning rule, memristor synapse and HR neurons, has been presented to the literature with this implementation.