Random walks for spike-timing-dependent plasticity.

Random walk methods are used to calculate the moments of negative image equilibrium distributions in synaptic weight dynamics governed by spike-timing-dependent plasticity. The neural architecture of the model is based on the electrosensory lateral line lobe of mormyrid electric fish, which forms a negative image of the reafferent signal from the fish's own electric discharge to optimize detection of sensory electric fields. Of particular behavioral importance to the fish is the variance of the equilibrium postsynaptic potential in the presence of noise, which is determined by the variance of the equilibrium weight distribution. Recurrence relations are derived for the moments of the equilibrium weight distribution, for arbitrary postsynaptic potential functions and arbitrary learning rules. For the case of homogeneous network parameters, explicit closed form solutions are developed for the covariances of the synaptic weight and postsynaptic potential distributions.