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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 - PowerPoint PPT Presentation
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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
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Outline
April 13, 2011 Page 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
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
The point of nonlinearity
April 13, 2011 Page 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.
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Oscillations and limit cycles
April 13, 2011 Page 4
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Oscillations and limit cycles
April 13, 2011 Page 5
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
N – S = 1 in the plane (Poincaré)
April 13, 2011 Page 6
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
N – S = 1 in the plane (Poincaré)
April 13, 2011 Page 7
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Poincaré-Bendixson
April 13, 2011 Page 8
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Poincaré-Bendixson
April 13, 2011 Page 9
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Wilson-Cowan network oscillator
April 13, 2011 Page 10
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Wilson-Cowan network oscillator
April 13, 2011 Page 11
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
FitzHugh-Nagumo equations
April 13, 2011 Page 12
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Hopf bifurcations
April 13, 2011 Page 13
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Hopf bifurcations
April 13, 2011 Page 14
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Van der Pol equation (heart rhythm)
April 13, 2011 Page 15
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Van der Pol equation (heart rhythm)
April 13, 2011 Page 16
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Delayed negative feedback
April 13, 2011 Page 17
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Adaptation and perceptual reversals
April 13, 2011 Page 18
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Adaptation and perceptual reversals
April 13, 2011 Page 19
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Adaptation and perceptual reversals
April 13, 2011 Page 20
Spikes, Decisions and Actions - Nonlinear Oscillations (Chapter 8) Christoph Mathys
Adaptation and perceptual reversals
April 13, 2011 Page 21
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