  Localized persistent neural activity can encode delayed estimates of
continuous variables. Common experiments require that subjects store and report
the feature value (e.g., orientation) of a particular cue (e.g., oriented bar
on a screen) after a delay. Visualizing recorded activity of neurons along
their feature tuning reveals activity bumps whose centers wander
stochastically, degrading the estimate over time. Bump position therefore
represents the remembered estimate. Recent work suggests bump amplitude may
represent estimate certainty reflecting a probabilistic population code for a
Bayesian posterior. Idealized models of this type are fragile due to the fine
tuning common to constructed continuum attractors in dynamical systems. Here we
propose an alternative metastable model for robustly supporting multiple bump
amplitudes by extending neural circuit models to include quantized
nonlinearities. Asymptotic projections of circuit activity produce
low-dimensional evolution equations for the amplitude and position of bump
solutions in response to external stimuli and noise perturbations. Analysis of
reduced equations accurately characterizes phase variance and the dynamics of
amplitude transitions between stable discrete values. More salient cues
generate bumps of higher amplitude which wander less, consistent with the
experimental finding that greater certainty correlates with more accurate
memories.
