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An upper bound for the number of metastable states in the Hopfield model is calculated as a function of the Hamming fraction from an input pattern. For all finite values of alpha , the ratio of number of patterns to nodes, the hamming fraction from the input pattern to the nearest metastable state is infinite. When alpha <0.113, the bound also implies that there is a gap between a set of states close to the input pattern and another set centred around the Hamming fraction 0.5 from it.
E. Gardner (Tue,) studied this question.
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