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Nonlinear Oscillations

Nonlinear Oscillations. Seminar talk based on Chapter 8 of the book “Spikes, Decisions, and Actions” by Hugh R. Wilson April 13, 2011 Christoph Mathys. Outline. The point of nonlinearity Oscillations and limit cycles N – S = 1 in the plane (Poincaré) Poincaré-Bendixson

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Nonlinear Oscillations

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  1. Nonlinear Oscillations Seminar talk based on Chapter 8 of the book “Spikes, Decisions, and Actions” by Hugh R. Wilson April 13, 2011 Christoph Mathys

  2. Outline Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys • The point of nonlinearity • Oscillations and limit cycles • N – S = 1 in the plane (Poincaré) • Poincaré-Bendixson • Wilson-Cowan network oscillator • FitzHugh-Nagumo equations • Hopf bifurcations • Van der Pol equation • Delayed negative feedback • Adaptation and perceptual reversals

  3. The point of nonlinearity Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys • In linear systems, there are infinitely many periodic solutions within any sufficiently small neighborhood of a given oscillation. • Nonlinear systems, however, can generate isolated oscillations that are surrounded by open, non-oscillatory trajectories that either spiral toward or away from the oscillation. • Therefore, nonlinear systems capture biological reality better.

  4. Oscillations and limit cycles Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  5. Oscillations and limit cycles Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  6. N – S = 1 in the plane (Poincaré) Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  7. N – S = 1 in the plane (Poincaré) Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  8. Poincaré-Bendixson Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  9. Poincaré-Bendixson Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  10. Wilson-Cowan network oscillator Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  11. Wilson-Cowan network oscillator Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  12. FitzHugh-Nagumo equations Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  13. Hopf bifurcations Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  14. Hopf bifurcations Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  15. Van der Pol equation (heart rhythm) Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  16. Van der Pol equation (heart rhythm) Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  17. Delayed negative feedback Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  18. Adaptation and perceptual reversals Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  19. Adaptation and perceptual reversals Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  20. Adaptation and perceptual reversals Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

  21. Adaptation and perceptual reversals Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys

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