1. Laboratory for Social and Neural Systems Research (SNS Lab) Page 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 April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 2 –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 April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 3 –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 April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 4

5. Oscillations and limit cycles April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 5

6. N – S = 1 in the plane (Poincaré) April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 6

7. N – S = 1 in the plane (Poincaré) April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 7

8. Poincaré-Bendixson April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 8

9. Poincaré-Bendixson April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 9

10. Wilson-Cowan network oscillator April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 10

11. Wilson-Cowan network oscillator April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 11

12. FitzHugh-Nagumo equations April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 12

13. Hopf bifurcations April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 13

14. Hopf bifurcations April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 14

15. Van der Pol equation (heart rhythm) April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 15

16. Van der Pol equation (heart rhythm) April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 16

17. Delayed negative feedback April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 17

18. Adaptation and perceptual reversals April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 18

19. Adaptation and perceptual reversals April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 19

20. Adaptation and perceptual reversals April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 20

21. Adaptation and perceptual reversals April 13, 2011Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph MathysPage 21