Ph.D. Thesis Proposal by
Advisors: Professor Wassim M. Haddad and Professor Rafael de la Llave
Mathematical Analysis of a Mean Field Model of Electroencephalographic Activity in the Neocortex
4:30 PM, Tuesday, September 19, 2017
Montgomery Knight, Room 317
The electroencephalographic recordings from the scalp are essential measures of mesoscopic electrical activity in the neocortex. Such spatio-temporal electrical activity can effectively be modeled using the mean field theory. The mean field model of the electroencephalogram developed by Liley et al., 2002, is one of these models that has been widely used in the literature to study different patterns of rhythmic activity in the 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 doctoral thesis, this model is mathematically analyzed using the theory of partial differential equations and infinite-dimensional dynamical systems. 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. Semidynamical system frameworks are established for the model and it is proved 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. A bifurcation analysis is performed on a finite-dimensional approximation of the model with respect to variations in critical parameters of the model. The various emerging behaviors at each set of parameter values are qualitatively studied by numerically solving the equations of the model using finite element software packages. Finally, using the theoretical results developed in this thesis, instructive insights are provided into the complexity of the behavior of the model and computational analysis of the model.
Professor Wassim M. Haddad, School of Aerospace Engineering, Georgia Institute of Technology
Professor Rafael de la Llave, School of Mathematics, Georgia Institute of Technology
Professor John-Paul B Clarke, School of Aerospace Engineering, Georgia Institute of Technology