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