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On the spectra of certain integro-differential-delay problems with applications in neurodynamics

Grindrod, P; Pinotsis, DA; (2011) On the spectra of certain integro-differential-delay problems with applications in neurodynamics. PHYSICA D , 240 (1) 13 - 20. 10.1016/j.physd.2010.08.002.

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Abstract

We investigate the spectrum of certain integro-differential-delay equations (IDDEs) which arise naturally within spatially distributed, nonlocal, pattern formation problems. Our approach is based on the reformulation of the relevant dispersion relations with the use of the Lambert function. As a particular application of this approach, we consider the case of the Amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal. We show that if the kernel appearing in this equation is symmetric around some point a not equal 0 or consists of a sum of such terms, then the relevant dispersion relation yields spectra with an infinite number of branches, as opposed to finite sets of eigenvalues considered in previous works. Also, in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation. Here, we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications. Indeed, the Amari IDDE acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes, thereby amplifying such inputs and dampening others. Finally, we discuss how these results can be generalised to the case of systems of IDDEs. (C) 2010 Elsevier B.V. All rights reserved.

Type: Article
Title: On the spectra of certain integro-differential-delay problems with applications in neurodynamics
DOI: 10.1016/j.physd.2010.08.002
Keywords: Neural field equation, Spectra, Mathematical neuroscience, Integrodifferential equations, Delay equations, NEURAL FIELDS, NEURONAL NETWORKS, PATTERN-FORMATION, TRAVELING-WAVES, STANDING PULSES, STABILITY, BUMPS, DYNAMICS, MODELS, BIFURCATIONS
UCL classification: UCL > School of Life and Medical Sciences
UCL > School of Life and Medical Sciences > Faculty of Brain Sciences
UCL > School of Life and Medical Sciences > Faculty of Brain Sciences > Institute of Neurology
UCL > School of Life and Medical Sciences > Faculty of Brain Sciences > Institute of Neurology > Imaging Neuroscience
URI: http://discovery.ucl.ac.uk/id/eprint/98484
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