Exploiting the Hamiltonian structure of a neural field model
type: Item
instance of: scholarly article
title: Exploiting the Hamiltonian structure of a neural field model
zbMATH Open document ID: 1203.37034
author: A. J. Elvin
author: M. G. Roberts
author: Carlo Laing
author: Robert I. McLachlan
published in: Physica D
publication date: 2010-04-21T00:00:00Z
review text: This work is concerned with the following model \[ \frac{\partial u(x,t)}{\partial t} \, = \, - u(x,t) + \int_{-\infty}^{+\infty} w(x-y) f[u(y,t)] dy, \] where \(u(x,t)\) is the average voltage, or activity level, of a neuronal population of spatial position \(x\) and time \(t\). The coupling function \(w(x)\) is the distance-dependent strength of connectivity between neuronal elements and is given by \[ w(x) \, = \, e^{-b|x|} \left( b \sin |x| + \cos x \right), \] with \(b\) a parameter which governs the rate at which oscillations in \(w\) decay with distance. The firing rate function \(f(u)\) models neurons firing once threshold is reached and tends to a maximal limit as the stimulus is increased. It is given by: \[ f(u) \, = \, 2 \, \exp \left[-r/(u-\theta)^2 \right] \Theta(u-\theta), \] where the parameter \(\theta\) is the firing threshold, \(r\) is the steepness parameter and \(\Theta\) is the Heaviside step function. We recall that the Heaviside step function is defined by \[ \Theta(x) \, = \, \int_{-\infty}^{x} \delta(t) \, dt, \] where \(\delta(t)\) is the Dirac delta function. \(\Theta(x)\) is zero for \(x<0\) and it is \(1\) for \(x>0\). This model has been studied in [\textit{C. R. Laing} et al., SIAM J. Appl. Math., 63, No.1, 62--97 (2002; Zbl 1017.45006)] and [\textit{C. R. Laing, W.C. Troy}, SIAM J. Appl. Dyn. Syst., 2, No. 3, 487--516 (2003; Zbl 1088.34011)]. The authors consider time-independent solutions of the equation which are stationary and spatially-localised, that is, \[ \lim_{|x| \to \infty} (u,u',u'',u''') \, = \, (0,0,0,0). \] These solutions satisfy the integral equation \[ u(x) \, = \, \int_{-\infty}^{\infty} w(x-y) f[u(y)] dy, \] which can be transformed to the following differential equation by use of Fourier transforms: \[ u'''' - 2(b^2-1) u'' + (b^2+1)^2 u \, = \, 4b(b^2+1) f(u). \tag{eq9} \] This equation can be written as a system of four first-order ODEs with the following properties: 1) the linearisation of the system at the orig…
reviewed by: Maite Grau
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 92C20
zbMATH DE Number: 5696538
zbMATH Keywords: pattern formation
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: bifurcation
zbMATH Keywords: homoclinic
describes a project that uses: AUTO
describes a project that uses: HomCont
MaRDI profile type: Publication
full work available at URL: j.physd.2009.08.004
OpenAlex ID: W2166442352
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DOI: 10.1016/J.PHYSD.2009.08.004
description: scientific article; zbMATH DE number 5696538