Mathematical analysis of a mean field model of electroencephalographic activity in the neocortex
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Electroencephalographic recordings from the scalp provide essential measures of mesoscopic electrical activity in the neocortex. The rhythmic patterns of variations observed in the electroencephalogram result from the dynamic activity that occurs, possibly heterogeneously, in a wide area of the neocortex. Such spatio-temporal electrical activity can be effectively modeled using mean field theory. The mesoscopic model of the electroencephalographic activity in the neocortex developed by Liley, Cadusch, and Dafilis [Network, 13 (2002), pp. 67-113] is a mean field model that has been widely used in the literature to study different patterns of rhythmic activity in conscious and unconscious states of the brain. This model is presented as a system of coupled ordinary and partial differential equations with periodic boundary conditions. In this dissertation, a mathematical analysis of this mean field model is provided using infinite-dimensional dynamical systems theory and the theory of partial differential equations. Specifically, 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. Moreover, sufficient conditions are developed for the phase spaces of the model to ensure nonnegativity of certain quantities, as required by their biophysical interpretation. To analyze the global dynamics of the model, semidynamical system frameworks are established and the semigroups of weak and strong solution operators are proved to possess bounded absorbing sets for the entire range of biophysical values of the parameters of the model. Moreover, challenges involved in establishing a global attractor for the model are discussed, and in particular, it is shown that there exist sets of parameter values for which the constructed semidynamical systems do not possess a global attractor due to the lack of the compactness property. To demonstrate an application of this model to problems of computational neuroscience, the emergence of rhythmic activity in the neocortex is studied using bifurcation theory. The results predicted by the bifurcation analysis are verified by numerically solving the equations of the model using COMSOL Multiphysics®. Finally, using the analytical and computational results developed in this dissertation, instructive insights are provided into the complexity of the behavior of the model, and suggestions are made for future research.