  Learning in neural networks poses peculiar challenges when using discretized
rather then continuous synaptic states. The choice of discrete synapses is
motivated by biological reasoning and experiments, and possibly by hardware
implementation considerations as well. In this paper we extend a previous large
deviations analysis which unveiled the existence of peculiar dense regions in
the space of synaptic states which accounts for the possibility of learning
efficiently in networks with binary synapses. We extend the analysis to
synapses with multiple states and generally more plausible biological features.
The results clearly indicate that the overall qualitative picture is unchanged
with respect to the binary case, and very robust to variation of the details of
the model. We also provide quantitative results which suggest that the
advantages of increasing the synaptic precision (i.e.~the number of internal
synaptic states) rapidly vanish after the first few bits, and therefore that,
for practical applications, only few bits may be needed for near-optimal
performance, consistently with recent biological findings. Finally, we
demonstrate how the theoretical analysis can be exploited to design efficient
algorithmic search strategies.
