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.  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 . 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 kâ¤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.  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 .
Div. of Biomedical Computer Science
School of Medicine
Adrian, Frank Alan, "Exact ensemble dynamics for spike-timing-dependent plasticity" (2008). Scholar Archive. 342.