Online machine learning rules and many biological spike-timing-dependent plasticity (STDP) learning rules generate jump process Markov chains for the synaptic weights. We give a perturbation expansion for the dynamics that, unlike the usual approximation by a Fokker-Planck equation (FPE), is well 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.

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On-line machine learning and biological spiketiming-dependent plasticity (STDP) rules both generate Markov chains for the synaptic weights. We give a perturbation expansion (in powers of the learning rate) for the dynamics that, unlike the usual approximation by a Fokker-Planck equation (FPE), is rigorous. Our approach extends the related system size expansion by giving an expansion for the probability density as well as its moments. Applied to two observed STDP learning rules, our approach provides better agreement with MonteCarlo simulations than either the FPE or a simple linearized theory. The approach is also applicable to stochastic neural dynamics.

SUMMARYFour new methods are presented a method of accelerating the truncation error convergence, a fourth and fifth order difference scheme for discretizing the normal partial derivatives, high order quadrature formulae for integrating a stream function and a third order implicit scheme for treating the streamwise partial derivatives.These are seen to be effective in finding solutions to the boundary layer equations in which the step sizes are adaptively altered to meet an error bound.

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