Redundant representations are required to disambiguate simultaneously presented complex stimuli

An individual observing a barking dog and purring cat together in a field has distinct pairs of representations of the two animals in their visual and auditory systems. Without prior knowledge, how does the observer infer that the dog barks and the cat purrs? This integration of distributed representations is called the assignment problem, and it must be solved to integrate distinct representations across but also within sensory modalities. Here, we identify and analyze a solution to the assignment problem: the representation of one or more common stimulus features in pairs of relevant brain regions - for example, estimates of the spatial position of both the cat and the dog represented in both the visual and auditory systems. We characterize how the reliability of this solution depends on different features of the stimulus set (e.g., the size of the set and the complexity of the stimuli) and the details of the split representations (e.g., the precision of each stimulus representation and the amount of overlapping information). Next, we implement this solution in a biologically plausible receptive field code and show how constraints on the number of neurons and spikes used by the code force the brain to navigate a tradeoff between local and catastrophic errors. We show that, when many spikes and neurons are available, representing stimuli from a single sensory modality can be done more reliably across multiple brain regions, despite the risk of assignment errors. Finally, we show that a feedforward neural network can learn the optimal solution to the assignment problem. We also discuss relevant results on assignment errors from the human working memory literature and show that several key predictions of our theory already have support.