Stochastic perturbationmethods for spike-timing-dependent plasticity

Online machine learning rules and many biological spike-timingdependent plasticity (STDP) learning rules generate jump process Markovchains for the synaptic weights. We give aperturbation expansion for the dynamics that, unlike the usual approximation by a Fokker-Planck equation (FPE), iswell justified. Our approach extends the related system size expansion by giving an expansion for the probability density as well as its moments. We apply the approach to two observed STDP learning rules and show that in regimes where the FPE breaks down, the new perturbation expansion agrees well with Monte Carlo simulations. The methods are also applicable to the dynamics of stochastic neural activity. Like previous ensemble analyses of STDP, we focus on equilibrium solutions, although the methods can in principle be applied to transients as well.