  Characterizing the relation between weight structure and input/output
statistics is fundamental for understanding the computational capabilities of
neural circuits. In this work, I study the problem of storing associations
between analog signals in the presence of correlations, using methods from
statistical mechanics. I characterize the typical learning performance in terms
of the power spectrum of random input and output processes. I show that optimal
synaptic weight configurations reach a capacity of 0.5 for any fraction of
excitatory to inhibitory weights and have a peculiar synaptic distribution with
a finite fraction of silent synapses. I further provide a link between typical
learning performance and principal components analysis in single cases. These
results may shed light on the synaptic profile of brain circuits, such as
cerebellar structures, that are thought to engage in processing time-dependent
signals and performing on-line prediction.
