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

Grindrod, P. and Pinotsis, D. (2010) On the spectra of certain integro-differential-delay problems with applications in neurodynamics. Physica D: Nonlinear Phenomena, 240 (1). pp. 13-20. ISSN 0167-2789 doi: 10.1016/j.physd.2010.08.002

Abstract/Summary

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= 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.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/15743
Identification Number/DOI	10.1016/j.physd.2010.08.002
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics > Centre for the Mathematics of Human Behaviour (CMOHB)
Uncontrolled Keywords	Neural field equation; Spectra; Mathematical neuroscience; Integrodifferential equations; Delay equations
Publisher	Elsevier
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