June 16, 2009 22:15 World Scientific Review Volume - 9in x 6in liley˙v1 Chapter 1 Approaches to modelling the dynamical activity of brain function based on the electroencephalogram David T. J. Liley ∗ and Federico Frascoli † Brain Sciences Institute, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia The brain is arguably the quintessential complex system as indicated by the patterns of behaviour it produces. Despite many decades of concen- trated research efforts we remain largely ignorant regarding the essen- tial processes that regulate and define its function. While advances in functional neuroimaging have provided welcome windows into the coarse organisation of the neuronal networks that underlie a range of cognitive functions, they have also largely ignored the fact that behaviour, and by inference brain function, unfolds dynamically. Modelling the brain’s dynamics is therefore a critical step towards understanding the under- lying mechanisms of its functioning. To date, models have concentrated on describing the sequential organisation of either abstract mental states (functionalism, hard AI) or the objectively measurable manifestations of the brain’s ongoing activity (rCBF, EEG, MEG). While the former types of modelling approach may seem to better characterise brain function, they do so at the expense of not making a definite connection with the actual physical brain. Of the latter, only models of the EEG (or MEG) offer a temporal resolution well matched to the anticipated temporal scales of brain (mental processes) function. This chapter will outline the most pertinent of these modelling approaches, and illustrate, using the electrocortical model of Liley et al, how the detailed application of the methods of nonlinear dynamics and bifurcation theory is central in exploring and characterising their various dynamical features. The rich repertoire of dynamics revealed by such dynamical systems approaches arguably represents a critical step towards an understanding of the com- plexity of brain function. * dliley@swin.edu.au † ffrascoli@swin.edu.au 1