Stochastic neural field theory of wandering bumps on a sphere
File(s)
Accepted version
Author(s)
Bressloff, Paul C
Type
Journal Article
Abstract
We use a combination of group theoretic and perturbation methods to analyze the stochastic wandering of bump solutions in a neural field model on the
sphere S
2
. We first construct an explicit bump solution in the absence of external inputs and noise, by taking the synaptic weight distribution to be the sum
of first-order spherical harmonics. The corresponding neural field equation is
equivariant under the action of the special orthogonal group SO(3), which implies that the bump is marginally stable with respect to rotations of the sphere.
We then carry out an amplitude-phase decomposition of the solution in the presence of a weakly biased external input and weak noise, and use this to derive
a pair of stochastic differential equations for the wandering of the bump, expressed in terms of angular coordinates on the sphere. The stochastic dynamics
is a non-trivial generalization of the corresponding phase dynamics describing
the wandering of a bump on a ring network with SO(2) symmetry, since SO(3)
is non-abelian and S
2
is a curved manifold.
sphere S
2
. We first construct an explicit bump solution in the absence of external inputs and noise, by taking the synaptic weight distribution to be the sum
of first-order spherical harmonics. The corresponding neural field equation is
equivariant under the action of the special orthogonal group SO(3), which implies that the bump is marginally stable with respect to rotations of the sphere.
We then carry out an amplitude-phase decomposition of the solution in the presence of a weakly biased external input and weak noise, and use this to derive
a pair of stochastic differential equations for the wandering of the bump, expressed in terms of angular coordinates on the sphere. The stochastic dynamics
is a non-trivial generalization of the corresponding phase dynamics describing
the wandering of a bump on a ring network with SO(2) symmetry, since SO(3)
is non-abelian and S
2
is a curved manifold.
Date Issued
2019-12-01
Date Acceptance
2019-04-23
Citation
Physica D: Nonlinear Phenomena, 2019, 399, pp.138-152
ISSN
0167-2789
Publisher
Elsevier
Start Page
138
End Page
152
Journal / Book Title
Physica D: Nonlinear Phenomena
Volume
399
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.2019.04.010
Publication Status
Published
Date Publish Online
2019-05-17