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Oscillations in Neuronal Dynamical Systems

Fallah Elamloo, Hanieh | 2010

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 40446 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Razvan, Mohammad Reza
  7. Abstract:
  8. Periodic orbits are one of the basic objects of interest in Dynamical systems. And methods for detecting, counting and determining the periods of periodic orbits are of great interest. In spite of this interest, periodic orbits remain difficult to detect. The most celebrated existence result is the Poincare-Bendixson theorem. Unfortunately, this theorem is false in general for higher dimensional systems. The success of fixed point theory inspired the hope that similar topological techniques could be of use in proving the existence of periodic orbits. In this Thesis we have analyzed some topological and geometrical theories which could be of use in proving the existence of periodic orbits. The privilege of these theories is that they could be of use in higher dimensional systems. One of these methods is the Geometrical singular perturbation theory. In this theory, the transversal intersection of invariant manifolds in singular case is used to prove the existence of periodic orbits in nonsingular case. The canard solutions and relaxation oscillations in three dimensional systems is analyzed geometrically. The blow-up method is used to analyze the dynamics near the fold- curves. The other method is the Conley index theory. Conley index theory consists of topological and algebraic tools for understanding the global dynamics of flows and maps on compact invariant sets. It is useful for proving the existence of various objects and properties, such as equilibria, periodic orbits, connecting orbits, travelling waves and chaotic dynamics. This theory is used to prove the existence of a nonempty attractor in a bursting model of pancreatic cells.
  9. Keywords:
  10. Conley Index ; Singular Perturbation ; Bursting Phenomenon ; Canard Solusions ; Relaxation Oscillation

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