Hebbian learning and spiking neurons

A correlation-based ("Hebbian") learning rule at a spike level with millisecond resolution is formulated, mathematically analyzed, and compared with learning in a firing-rate description. The relative timing of presynaptic and postsynaptic spikes influences synaptic weights via an asymmetric "learning window." A differential equation for the learning dynamics is derived under the assumption that the time scales of learning and neuronal spike dynamics can be separated. The differential equation is solved for a Poissonian neuron model with stochastic spike arrival. It is shown that correlations between input and output spikes tend to stabilize structure formation. With an appropriate choice of parameters, learning leads to an intrinsic normalization of the average weight and the output firing rate. Noise generates diffusion-like spreading of synaptic weights.