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To understand cognitive reasoning in the brain, it has been proposed thatsymbols and compositions of symbols are represented by activity patterns(vectors) in a large population of neurons. Formal models implementing thisidea [Plate 2003], [Kanerva 2009], [Gayler 2003], [Eliasmith 2012] include areversible superposition operation for representing with a single vector anentire set of symbols or an ordered sequence of symbols. If the representationspace is high-dimensional, large sets of symbols can be superposed andindividually retrieved. However, crosstalk noise limits the accuracy ofretrieval and information capacity. To understand information processing in thebrain and to design artificial neural systems for cognitive reasoning, a theoryof this superposition operation is essential. Here, such a theory is presented.The superposition operations in different existing models are mapped to linearneural networks with unitary recurrent matrices, in which retrieval accuracycan be analyzed by a single equation. We show that networks representinginformation in superposition can achieve a channel capacity of about half a bitper neuron, a significant fraction of the total available entropy. Going beyondexisting models, superposition operations with recency effects are proposedthat avoid catastrophic forgetting when representing the history of infinitedata streams. These novel models correspond to recurrent networks withnon-unitary matrices or with nonlinear neurons, and can be analyzed andoptimized with an extension of our theory.

Theory of the superposition principle for randomized connectionist representations in neural networks