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

Zachary P.  Kilpatrick

doi:10.3934/dcdsb.2016044

Article Contents

2016, Volume 21, Issue 7: 2211-2231. Doi: 10.3934/dcdsb.2016044

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

Zachary P. Kilpatrick^1,

1.
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309

Received: June 30, 2015

Revised: July 31, 2015

Published: August 2016

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Abstract

We study the dynamics of stationary bumps in continuum neural field equations near a saddle-node bifurcation. The integral terms of these evolution equations have a weight kernel describing synaptic interactions between neurons at different locations of the network. Excited regions of the neural field correspond to parts of the domain whose fraction of active neurons exceeds a sharp threshold of a firing rate nonlinearity. For sufficiently low firing threshold, a stable bump coexists with an unstable bump and a homogeneous quiescent state. As the threshold is increased, the stable and unstable branch of bump solutions annihilate in a saddle-node bifurcation. Near this criticality, we derive a quadratic amplitude equation that describes the slow evolution of the even mode (bump contractions) as it depends on the distance from the bifurcation. Beyond the bifurcation, bumps eventually become extinct, and the lifetime of bumps increases for systems nearer the bifurcation. When noise is incorporated, a stochastic amplitude equation for the even mode can be derived, which can be analyzed to quantify bump extinction time both below and above the saddle-node.

Keywords:

Stochastic partial differential equations,
Langevin equation,
perturbation theory,
amplitude equations,
saddle-node bifurcation.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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