  This paper investigates the global dynamics of a mean field model of the
electroencephalogram developed by Liley et al., 2002.
  The model is presented as a system of coupled ordinary and partial
differential equations with periodic boundary conditions.
  Existence, uniqueness, and regularity of weak and strong solutions of the
model are established in appropriate function spaces, and the associated
initial-boundary value problems are proved to be well-posed.
  Sufficient conditions are developed for the phase spaces of the model to
ensure nonnegativity of certain quantities in the model, as required by their
biophysical interpretation.
  It is shown that the semigroups of weak and strong solution operators possess
bounded absorbing sets for the entire range of biophysical values of the
parameters of the model.
  Challenges towards establishing a global attractor for the model are
discussed and it is shown that there exist parameter values for which the
constructed semidynamical systems do not possess a compact global attractor due
to the lack of the asymptotic compactness property.
  Finally, using the theoretical results of the paper, instructive insights are
provided into the complexity of the behavior of the model and computational
analysis of the model.
