  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.
