Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Waves in random neural media

Pages: 2951 - 2970, Volume 32, Issue 8, August 2012      doi:10.3934/dcds.2012.32.2951

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Stephen Coombes - School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom (email)
Helmut Schmidt - School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom (email)
Carlo R. Laing - Institute of Information and Mathematical Sciences, Massey University, Private Bag 102-904, North Shore Mail Centre, Auckland, New Zealand (email)
Nils Svanstedt - Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, S-412 96 Göteborg, Sweden (email)
John A. Wyller - Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, P. O. Box 5003, NO-1432 Ås, Norway (email)

Abstract: Translationally invariant integro-differential equations are a common choice of model in neuroscience for describing the coarse-grained dynamics of cortical tissue. Here we analyse the propagation of travelling wavefronts in models of neural media that incorporate some form of modulation or randomness such that translational invariance is broken. We begin with a study of neural architectures in which there is a periodic modulation of the neuronal connections. Recent techniques from two-scale convergence analysis are used to construct a homogenized model in the limit that the spatial modulation is rapid compared with the scale of the long range connections. For the special case that the neuronal firing rate is a Heaviside we calculate the speed of a travelling homogenized front exactly and find how the wave speed changes as a function of the amplitude of the modulation. For this special case we further show how to obtain more accurate results about wave speed and the conditions for propagation failure by using an interface dynamics approach that circumvents the requirement of fast modulation. Next we turn our attention to forms of disorder that arise via the variation of firing rate properties across the tissue. To model this we draw parameters of the firing rate function from a distribution with prescribed spatial correlations and analyse the corresponding fluctuations in the wave speed. Finally we consider generalisations of the model to incorporate adaptation and stochastic forcing and show how recent numerical techniques developed for stochastic partial differential equations can be used to determine the wave speed by minimising the $L^2$ norm of a travelling disordered activity profile against a fixed test function.

Keywords:  Integro-differential equations, neural field models, homogenization, random media.
Mathematics Subject Classification:  Primary: 45J05; Secondary: 92C20.

Received: April 2011;      Revised: July 2011;      Available Online: March 2012.