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The optical implementation of weighted interconnections is investigated and basic relationship are derived between the number of neurons, the number of connections and methods for selecting the positions of the neurons to achieve the maximum density of independent connections are presented. The connectivity of a neural network (number of synapses per neuron) is related to the complexity of the problems it can handle. For a network that learns a problem from examples using a local learning rule, it is proved that the entropy of the problem becomes a lower bound for the connectivity of the network.