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A Hopfield network is an auto-associative, distributive model of neuralmemory storage and retrieval. A form of error-correcting code, the Hopfieldnetwork can learn a set of patterns as stable points of the network dynamic,and retrieve them from noisy inputs -- thus Hopfield networks are their owndecoders. Unlike in coding theory, where the information rate of a good code(in the Shannon sense) is finite but the cost of decoding does not play a rolein the rate, the information rate of Hopfield networks trained withstate-of-the-art learning algorithms is of the order ${\log(n)}/{n}$, aquantity that tends to zero asymptotically with $n$, the number of neurons inthe network. For specially constructed networks, the best information ratecurrently achieved is of order ${1}/{\sqrt{n}}$. In this work, we design simplebinary Hopfield networks that have asymptotically vanishing error rates at aninformation rate of ${1}/{\log(n)}$. These networks can be added as thedecoders of any neural code with noisy neurons. As an example, we apply ournetwork to a binary neural decoder of the grid cell code to attain informationrate ${1}/{\log(n)}$.

A binary Hopfield network with $1/\log(n)$ information rate and applications to grid cell decoding