We investigate spike-timing dependent plasticity (STPD) in the case of a synapse connecting two neural cells. We develop a theoretical analysis of several STDP rules using Markovian theory. In this context there are two different timescales, fast neural activity and slower synaptic weight updates. Exploiting this timescale separation, we derive the long-time limits of a single synaptic weight subject to STDP. We show that the pairing model of presynaptic and postsynaptic spikes controls the synaptic weight dynamics for small external input, on an excitatory synapse. This result implies in particular that mean-field analysis of plasticity may miss some important properties of STDP. Anti-Hebbian STDP seems to favor the emergence of a stable synaptic weight, but only for high external input. In the case of inhibitory synapse the pairing schemes matter less, and we observe convergence of the synaptic weight to a non-null value only for Hebbian STDP. We extensively study different asymptotic regimes for STDP rules, raising interesting questions for future works on adaptative neural networks and, more generally, on adaptive systems.