Abstract
The Nunez model for the generation of electroencephalogram (EEG) signals is naturally described as a neural field model on a sphere with space-dependent delays. For simplicity, dynamical realisations of this model either as a damped wave equation or an integro-differential equation, have typically been studied in idealised one dimensional or planar settings. Here we revisit the original Nunez model to specifically address the role of spherical topology on spatio-temporal pattern generation. We do this using a mixture of Turing instability analysis, symmetric bifurcation theory, centre manifold reduction and direct simulations with a bespoke numerical scheme. In particular we examine standing and travelling wave solutions using normal form computation of primary and secondary bifurcations from a steady state. Interestingly, we observe spatio-temporal patterns which have counterparts seen in the EEG patterns of both epileptic and schizophrenic brain conditions.
Highlights
Modern neuroimaging methodologies give us a window on the activity of the brain that may reveal both structure and function
As a result of this, many of the mathematical techniques developed for the analysis of ODEs and partial differential equation (PDE), such as Turing analysis, symmetric bifurcation theory, and center manifold reduction can be adapted for use in the delayed integro-differential equation we study in this article
An onset delay has been shown to lead to dynamics reminiscent of those seen in simulations of large-scale spiking networks [41], and its physiological interpretation can be connected to the relaxation time-scale for which spiking networks can reasonably allow for a firing rate description
Summary
Modern neuroimaging methodologies give us a window on the activity of the brain that may reveal both structure and function. Paul Nunez was one of the first to realise the importance of modelling the long range cortico-cortico connections for generating the all important α-rhythm of EEG (an 8 − 13 Hz frequency) [2] He recognised that because the cortical white matter system is topologically close to a sphere that a model that respected this (with periodic boundaries) should naturally produce standing waves (via interference) [3, 4]. [7, 8, 9, 10] Understandably this facilitates both mathematical and numerical analysis, though the results have less relevance to the application of standing waves seen in EEG One exception to this is the numerical study of Jirsa et al [11], though even here analysis and simulations are performed by using the less general partial differential equation (PDE) formulation of the model. In this paper we undertake a first step along this path
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