  Continuous attractors have been used to understand recent neuroscience
experiments where persistent activity patterns encode internal representations
of external attributes like head direction or spatial location. However, the
conditions under which the emergent bump of neural activity in such networks
can be manipulated by space and time-dependent external sensory or motor
signals are not understood. Here, we find fundamental limits on how rapidly
internal representations encoded along continuous attractors can be updated by
an external signal. We apply these results to place cell networks to derive a
velocity-dependent non-equilibrium memory capacity in neural networks.
