October 2008

Document Type


Degree Name



Dept. of Science & Engineering


Oregon Health & Science University


Under the influence of input stimuli, synaptic strength changes based on the relative timing of pre- and postsynaptic events. This mechanism is called Spike-Timing-Dependent Plasticity (STDP) and is recognized as a basis of neural plasticity in biological systems. Changes in synaptic strength (or weight) are de-scribed mathematically using a stochastic learning rule that generates a Markov process over the weights. This process determines a master equation whose solution provides the time evolution of the probability density function for the weights. The master equation has an expansion in a perturbation-like series (the Kramers-Moyal expansion), which when truncated after the second term gives a Fokker-Planck equation. Solving this equation provides an approximation to the probability density. Van Rossum et al. [1] use this approach to analytically predict the equilibrium distribution of synaptic weights governed by anti-symmetric spike-timing-dependent learning rules of the type observed by Bi and Poo [2]. However, the use of the Fokker-Planck equation is ill-advised and does not always lead to an accurate approximation, as was shown by Heskes and Kappen [3, 4] in the context of machine learning. We show that if for all k≤K, the k[superscript]th jump moment is a polynomial of order less than or equal to k in the weights, the Kramers-Moyal expansion produces a recurrence in the first K moments that can be solved in closed form. The model of van Rossum et al. [1] has this property and we find an exact solution for the the equilibrium moments of the probability density. Our simulations validate this result across a broad range of model parameters for the antisymmetric STDP model described in [1].




Div. of Biomedical Computer Science


School of Medicine



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