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Empirical estimates of the dimensionality of neural population activity areoften much lower than the population size. Similar phenomena are also observedin trained and designed neural network models. These experimental andcomputational results suggest that mapping low-dimensional dynamics tohigh-dimensional neural space is a common feature of cortical computation.Despite the ubiquity of this observation, the constraints arising from suchmapping are poorly understood. Here we consider a specific example of mappinglow-dimensional dynamics to high-dimensional neural activity -- the neuralengineering framework. We analytically solve the framework for the classic ringmodel -- a neural network encoding a static or dynamic angular variable. Ourresults provide a complete characterization of the success and failure modesfor this model. Based on similarities between this and other frameworks, wespeculate that these results could apply to more general scenarios.

Mapping low-dimensional dynamics to high-dimensional neural activity: A derivation of the ring model from the neural engineering framework