  Persistent activity in neuronal populations has been shown to represent the
spatial position of remembered stimuli. Networks that support bump attractors
are often used to model such persistent activity. Such models usually exhibit
translational symmetry. Thus activity bumps are neutrally stable, and
perturbations in position do not decay away. We extend previous work on bump
attractors by constructing model networks capable of encoding the certainty or
salience of a stimulus stored in memory. Such networks support bumps that are
not only neutrally stable to perturbations in position, but also perturbations
in amplitude. Possible bump solutions then lie on a two-dimensional attractor,
determined by a continuum of positions and amplitudes. Such an attractor
requires precisely balancing the strength of recurrent synaptic connections.
The amplitude of activity bumps represents certainty, and is determined by the
initial input to the system. Moreover, bumps with larger amplitudes are more
robust to noise, and over time provide a more faithful representation of the
stored stimulus. In networks with separate excitatory and inhibitory
populations, generating bumps with a continuum of possible amplitudes, requires
tuning the strength of inhibition to precisely cancel background excitation.
