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This paper investigates the global dynamics of a mean field model of theelectroencephalogram developed by Liley et al., 2002.  The model is presented as a system of coupled ordinary and partialdifferential equations with periodic boundary conditions.  Existence, uniqueness, and regularity of weak and strong solutions of themodel are established in appropriate function spaces, and the associatedinitial-boundary value problems are proved to be well-posed.  Sufficient conditions are developed for the phase spaces of the model toensure nonnegativity of certain quantities in the model, as required by theirbiophysical interpretation.  It is shown that the semigroups of weak and strong solution operators possessbounded absorbing sets for the entire range of biophysical values of theparameters of the model.  Challenges towards establishing a global attractor for the model arediscussed and it is shown that there exist parameter values for which theconstructed semidynamical systems do not possess a compact global attractor dueto the lack of the asymptotic compactness property.  Finally, using the theoretical results of the paper, instructive insights areprovided into the complexity of the behavior of the model and computationalanalysis of the model.

On the Global Dynamics of an Electroencephalographic Mean Field Model of the Neocortex