  In neuroscience, learning and memory are usually associated to long-term
changes of neuronal connectivity. In this context, synaptic plasticity refers
to the set of mechanisms driving the dynamics of neuronal connections, called
{\em synapses} and represented by a scalar value, the synaptic weight.
Spike-Timing Dependent Plasticity (STDP) is a biologically-based model
representing the time evolution of the synaptic weight as a functional of the
past spiking activity of adjacent neurons.
  If numerous models of neuronal cells have been proposed in the mathematical
literature, few of them include a variable for the time-varying strength of the
connection. A new, general, mathematical framework is introduced to study
synaptic plasticity associated to different STDP rules. The system composed of
two neurons connected by a single synapse is investigated and a stochastic
process describing its dynamical behavior is presented and analyzed. The notion
of plasticity kernel is introduced as a key component of plastic neural
networks models, generalizing a notion used for pair-based models. We show that
a large number of STDP rules from neuroscience and physics can be represented
by this formalism. Several aspects of these models are discussed and compared
to canonical models of computational neuroscience. An important sub-class of
plasticity kernels with a Markovian formulation is also defined and
investigated. In these models, the time evolution of cellular processes such as
the neuronal membrane potential and the concentrations of chemical components
created/suppressed by spiking activity has the Markov property.
