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We study the effects of additive noise on stationary bump solutions tospatially extended neural fields near a saddle-node bifurcation. The integralterms of these evolution equations have a weight kernel describing synapticinteractions between neurons at different locations of the network. Excitedregions of the neural field correspond to parts of the domain whose fraction ofactive neurons exceeds a sharp threshold of a firing rate nonlinearity. Forsufficiently low firing threshold, a stable bump coexists with an unstable bumpand a homogeneous quiescent state. As the threshold is increased, the stableand unstable branch of bump solutions annihilate in a saddle node bifurcation.Near this criticality, we derive a quadratic amplitude equation that describesthe slow evolution of the even mode (bump contractions) as it depends on thedistance from the bifurcation. Beyond the bifurcation, bumps eventually becomeextinct, and the time it takes for this to occur increases for systems nearerthe bifurcation. When noise is incorporated, a stochastic amplitude equationfor the even mode can be derived, which can be analyzed to reveal bumpextinction time both below and above the saddle-node.

Ghosts of bump attractors in stochastic neural fields: Bottlenecks and extinction