Stability of bumps in piecewise smooth neural fields with nonlinear adaptation

Handle URI:
http://hdl.handle.net/10754/599710
Title:
Stability of bumps in piecewise smooth neural fields with nonlinear adaptation
Authors:
Kilpatrick, Zachary P.; Bressloff, Paul C.
Abstract:
We study the linear stability of stationary bumps in piecewise smooth neural fields with local negative feedback in the form of synaptic depression or spike frequency adaptation. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Discontinuities in the adaptation variable associated with a bump solution means that bump stability cannot be analyzed by constructing the Evans function for a network with a sigmoidal gain function and then taking the high-gain limit. In the case of synaptic depression, we show that linear stability can be formulated in terms of solutions to a system of pseudo-linear equations. We thus establish that sufficiently strong synaptic depression can destabilize a bump that is stable in the absence of depression. These instabilities are dominated by shift perturbations that evolve into traveling pulses. In the case of spike frequency adaptation, we show that for a wide class of perturbations the activity and adaptation variables decouple in the linear regime, thus allowing us to explicitly determine stability in terms of the spectrum of a smooth linear operator. We find that bumps are always unstable with respect to this class of perturbations, and destabilization of a bump can result in either a traveling pulse or a spatially localized breather. © 2010 Elsevier B.V. All rights reserved.
Citation:
Kilpatrick ZP, Bressloff PC (2010) Stability of bumps in piecewise smooth neural fields with nonlinear adaptation. Physica D: Nonlinear Phenomena 239: 1048–1060. Available: http://dx.doi.org/10.1016/j.physd.2010.02.016.
Publisher:
Elsevier BV
Journal:
Physica D: Nonlinear Phenomena
KAUST Grant Number:
KUK-C1-013-4
Issue Date:
Jun-2010
DOI:
10.1016/j.physd.2010.02.016
Type:
Article
ISSN:
0167-2789
Sponsors:
We thank Steve Coombes and Markus Owen for helpful comments clarifying the specifics of their papers [23,25]. This publication was based on work supported in part by the National Science Foundation (DMS-0813677) and by Award No. KUK-C1-013-4 granted by King Abdullah University of Science and Technology (KAUST). PCB was also partially supported by the Royal Society Wolfson Foundation.
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Full metadata record

DC FieldValue Language
dc.contributor.authorKilpatrick, Zachary P.en
dc.contributor.authorBressloff, Paul C.en
dc.date.accessioned2016-02-28T06:08:04Zen
dc.date.available2016-02-28T06:08:04Zen
dc.date.issued2010-06en
dc.identifier.citationKilpatrick ZP, Bressloff PC (2010) Stability of bumps in piecewise smooth neural fields with nonlinear adaptation. Physica D: Nonlinear Phenomena 239: 1048–1060. Available: http://dx.doi.org/10.1016/j.physd.2010.02.016.en
dc.identifier.issn0167-2789en
dc.identifier.doi10.1016/j.physd.2010.02.016en
dc.identifier.urihttp://hdl.handle.net/10754/599710en
dc.description.abstractWe study the linear stability of stationary bumps in piecewise smooth neural fields with local negative feedback in the form of synaptic depression or spike frequency adaptation. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Discontinuities in the adaptation variable associated with a bump solution means that bump stability cannot be analyzed by constructing the Evans function for a network with a sigmoidal gain function and then taking the high-gain limit. In the case of synaptic depression, we show that linear stability can be formulated in terms of solutions to a system of pseudo-linear equations. We thus establish that sufficiently strong synaptic depression can destabilize a bump that is stable in the absence of depression. These instabilities are dominated by shift perturbations that evolve into traveling pulses. In the case of spike frequency adaptation, we show that for a wide class of perturbations the activity and adaptation variables decouple in the linear regime, thus allowing us to explicitly determine stability in terms of the spectrum of a smooth linear operator. We find that bumps are always unstable with respect to this class of perturbations, and destabilization of a bump can result in either a traveling pulse or a spatially localized breather. © 2010 Elsevier B.V. All rights reserved.en
dc.description.sponsorshipWe thank Steve Coombes and Markus Owen for helpful comments clarifying the specifics of their papers [23,25]. This publication was based on work supported in part by the National Science Foundation (DMS-0813677) and by Award No. KUK-C1-013-4 granted by King Abdullah University of Science and Technology (KAUST). PCB was also partially supported by the Royal Society Wolfson Foundation.en
dc.publisherElsevier BVen
dc.subjectNeural networken
dc.subjectPiecewise smooth dynamicsen
dc.subjectSpike frequency adaptationen
dc.subjectSynaptic depressionen
dc.titleStability of bumps in piecewise smooth neural fields with nonlinear adaptationen
dc.typeArticleen
dc.identifier.journalPhysica D: Nonlinear Phenomenaen
dc.contributor.institutionUniversity of Utah, Salt Lake City, United Statesen
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
kaust.grant.numberKUK-C1-013-4en
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