PhD Defense by Farshad Shirani

Event Details
  • Date/Time:
    • Monday March 26, 2018
      10:00 am - 12:00 pm
  • Location: Montgomery Knight, Room 317
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Summaries

Summary Sentence: Mathematical Analysis of a Mean Field Model of Electroencephalographic Activity in the Neocortex

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Ph.D. Thesis Defense by

Farshad Shirani

(Advisors: Professor Wassim M. Haddad and Professor Rafael de la Llave)

“Mathematical Analysis of a Mean Field Model of Electroencephalographic Activity in the Neocortex”

Monday, March 26, 2018 @ 10:00 a.m.

Montgomery Knight, Room 317

Abstract

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 involved in 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 compactness property. Moreover, an application of the model in computational studies of the alpha- and gamma-band rhythmic electrical activity in the brain is demonstrated, using bifurcation analysis of a space-homogenous version of the model, followed by numerical computation of the solutions of the model for critical sets of parameter values. Finally, using the theoretical results developed in this thesis, instructive insights are provided into the complexity of the behavior of the model and suggestions are made for possible future research.

 

Committee:

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

Professor Dewey H. Hodges, School of Aerospace Engineering, Georgia Institute of Technology

Professor Andrzej Swiech, School of Mathematics, Georgia Institute of Technology

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Phd Defense
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  • Created By: Tatianna Richardson
  • Workflow Status: Published
  • Created On: Mar 19, 2018 - 1:41pm
  • Last Updated: Mar 19, 2018 - 1:41pm