  The fundamental `plasticity' of the nervous system (i.e high adaptability at
different structural levels) is primarily based on Hebbian learning mechanisms
that modify the synaptic connections. The modifications rely on neural activity
and assign a special dynamic behavior to the neural networks. Another striking
feature of the nervous system is that spike based information transmission,
which is supposed to be robust against noise, is noisy in itself: the variance
of the spiking of the individual neurons is surprisingly large which may
deteriorate the adequate functioning of the Hebbian mechanisms. In this paper
we focus on networks in which Hebbian-like adaptation is induced only by
external random noise and study spike-timing dependent synaptic plasticity. We
show that such `HebbNets' are able to develop a broad range of network
structures, including scale-free small-world networks. The development of such
network structures may provide an explanation of the role of noise and its
interplay with Hebbian plasticity. We also argue that this model can be seen as
a unification of the famous Watts-Strogatz and preferential attachment models
of small-world nets.
