A recent article (Stanton and Sejnowski 1989) on long-term synaptic depression in the hippocampus has reopened the issue of the computational efficiency of particular synaptic learning rules (Hebb 1949; Palm 1988a; Morris and Willshaw 1989) — homosynaptic versus heterosynaptic and monotonic versus nonmonotonic changes in synaptic efficacy. We have addressed these questions by calculating and maximizing the signal-to-noise ratio, a measure of the potential fidelity of recall, in a class of associative matrix memories. Up to a multiplicative constant, there are three optimal rules, each providing for synaptic depression such that positive and negative changes in synaptic efficacy balance out. For one rule, which is found to be the Stent-Singer rule (Stent 1973; Rauschecker and Singer 1979), the depression is purely heterosynaptic; for another (Stanton and Sejnowski 1989), the depression is purely homosynaptic; for the third, which is a generalization of the first two, and has a higher signal-to-noise ratio, it is both heterosynaptic and homosynaptic. The third rule takes the form of a covariance rule (Sejnowski 1977a,b) and includes, as a special case, the prescription due to Hopfield (1982) and others (Willshaw 1971; Kohonen 1972).