Stability of bumps in piecewise smooth neural fields with nonlinear adaptation
KAUST Grant NumberKUK-C1-013-4
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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.
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.
SponsorsWe 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.
JournalPhysica D: Nonlinear Phenomena