Wandering bumps in a stochastic neural field: a variational approach
File(s)
Accepted version
Author(s)
MacLaurin, James N
Bressloff, Paul C
Type
Journal Article
Abstract
We develop a generalized variational method for analyzing wandering bumps in
a stochastic neural field model defined on some domain U. For concreteness,
we take U = S
1 and consider a stochastic ring model. First, we decompose the
stochastic neural field into a phase-shifted deterministic bump solution and a
small error term, which is assumed to be valid up to some exponentially large
stopping time. An exact, implicit stochastic differential equation (SDE) for the
phase of the bump is derived by minimizing the error term with respect to a
weighted L
2
(U, ρ) norm. The positive weight ρ is chosen so that the error term
consists of fast transverse fluctuations of the bump profile. We then carry out a
perturbation series expansion of the exact variational phase equation in powers
of the noise strength √
to obtain an explicit nonlinear SDE for the phase
that decouples from the error term. Solving the corresponding steady-state
Fokker-Planck equation up to O( ), we determine a leading-order expression
for the long-time distribution of the position of the bump. Finally, we use
the variational formulation to obtain rigorous exponential bounds on the error
term, demonstrating that with very high probability the system stays in a small
neighborhood of the bump for times of order exp(C −1
).
a stochastic neural field model defined on some domain U. For concreteness,
we take U = S
1 and consider a stochastic ring model. First, we decompose the
stochastic neural field into a phase-shifted deterministic bump solution and a
small error term, which is assumed to be valid up to some exponentially large
stopping time. An exact, implicit stochastic differential equation (SDE) for the
phase of the bump is derived by minimizing the error term with respect to a
weighted L
2
(U, ρ) norm. The positive weight ρ is chosen so that the error term
consists of fast transverse fluctuations of the bump profile. We then carry out a
perturbation series expansion of the exact variational phase equation in powers
of the noise strength √
to obtain an explicit nonlinear SDE for the phase
that decouples from the error term. Solving the corresponding steady-state
Fokker-Planck equation up to O( ), we determine a leading-order expression
for the long-time distribution of the position of the bump. Finally, we use
the variational formulation to obtain rigorous exponential bounds on the error
term, demonstrating that with very high probability the system stays in a small
neighborhood of the bump for times of order exp(C −1
).
Date Issued
2020-05
Date Acceptance
2020-02-07
Citation
Physica D: Nonlinear Phenomena, 2020, 406
ISSN
0167-2789
Publisher
Elsevier
Journal / Book Title
Physica D: Nonlinear Phenomena
Volume
406
Copyright Statement
Copyright © Elsevier Ltd. All rights reserved. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/
Identifier
http://dx.doi.org/10.1016/j.physd.2020.132403
Publication Status
Published
Article Number
132403
Date Publish Online
2020-02-12