**
Richard Kempter and Wulfram Gerstner and Leo van Hemmen, paper
**
### **Hebbian Learning and Spiking Neurons.**

** **

** R. Kempter, W. Gerstner and J.L. van Hemmen (1999)**

Physical Review E, 59:4498-4514

A correlation-based (``Hebbian'') learning rule at the spike level
is formulated, mathematically analyzed, and compared
with learning in a firing-rate description.
As for spike coding, we take advantage of a
``learning window'' that describes the effect of
timing of pre- and postsynaptic spikes on synaptic weights.
A differential equation for the learning dynamics
is derived under the assumption that the time
scales of learning and spiking dynamics can be
separated. Formation of structured synapses is
analyzed for a Poissonian neuron model which
receives time-dependent stochastic input. 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 rates. Noise
generates diffusion-like spreading of synaptic
weights.