The Mathematical Institute, University of Oxford, Eprints Archive

Stability of bumps in piecewise smooth neuralelds with nonlinear adaptation

Kilpatrick,, Zachary P. and Bressloff, P. C. (2010) Stability of bumps in piecewise smooth neuralelds with nonlinear adaptation. Stability of bumps in piecewise smooth neural elds with nonlinear adaptation . (Submitted)



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 the spectrum of a piecewise linear operator. We analyze the real part of the spectrum and show 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 travelling 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 travelling pulse or a spatially localized breather.

Item Type:Article
Subjects:D - G > General
Research Groups:Oxford Centre for Collaborative Applied Mathematics
ID Code:956
Deposited By: Ruby Hawkins
Deposited On:02 Sep 2010 09:44
Last Modified:29 May 2015 18:38

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