[R. M. May. Simple mathematical models with very complicated dynamics. Nature, 261:459, 1976]
Hénon map
• Introduced by Michel Hénon as a simplified model of the Poincarésection of the Lorenz model • One of the most studied examples of dynamical systems that exhibit chaotic behavior
Linear measures Introduction to non-linear dynamics Non-linear measures - Introduction to phase space reconstruction - Lyapunov exponent
[Acknowledgement: K. Lehnertz, University of Bonn, Germany]
Degree of asymmetry of distribution (relative to normal distribution) < 0 - asymmetric, more negative tails = 0 - symmetric > 0 - asymmetric, more positive tails
Advantages: - Localized in both frequency and time
- Mother wavelet can be selected according to the feature of interest
Power
Further applications:
- Filtering - Denoising - Compression
[M. Hénon. A two-dimensional mapping with a strange attractor. Commun. Math. Phys., 50:69, 1976]
Rössler system
• designed in 1976, for purely theoretical reasons • later found to be useful in modeling equilibrium in chemical reactions
EEG frequency bands
Description of brain rhythms • Delta: • Theta: 0.5 – 4 – 4 Hz 8 Hz
• Alpha:
• Beta: • Gamma:
8
12
– 12 Hz
– 30 Hz
> 30 Hz
[Buzsáki. Rhythms of the brain. Oxford University Press, 2006]
Autocorrelation
1 N ' ' xn xn C XX ( ) N n 1 C XX ( )
0 0
CXX (0) CXX ( )

Autocorrelation: Examples
periodic stochastic memory
Data acquisition
System / Object Sampling Sensor
Amplifier
Filter
AD-Converter
Computer
Dynamical system
Control parameter
Today’s lecture
Non-linear model systems
(Very preliminary) Schedule
•
• •
Lecture 1: Example (Epilepsy & spike train synchrony), Data acquisition, Dynamical systems
Lecture 2: Linear measures, Introduction to non-linear dynamics Lecture 3: Non-linear measures
Non-linear model systems
Non-linear model systems
Discrete maps Continuous Flows • Rössler system • Lorenz system
• Logistic map
• Hénon map
Logistic map
[Latka et al. Wavelet mJPP, 2005]
Introduction to non-linear dynamics
[O. E. Rössler. An equation for continuous chaos. Phys. Lett. A, 57:397, 1976]
Lorenz system
• Developed in 1963 as a simplified mathematical model for atmospheric convection • Arise in simplified models for lasers, dynamos, electric circuits, and chemical reactions
Discrete Fourier transform
Fourier series (sines and cosines):
Fourier coefficients:
Fourier series (complex exponentials): Fourier coefficients: Condition:
Lecture series: Data analysis
Thomas Kreuz, ISC, CNR
thomas.kreuz@cnr.it r.it/users/thomas.kreuz/
Lectures: Each Tuesday at 16:00
(First lecture: May 21, last lecture: June 25)
r - Control parameter
• Model of population dynamics • Classical example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations
•
• •
Lecture 4: Measures of continuous synchronization (EEG)
Lecture 5: Application to non-linear model systems and to epileptic seizure prediction, Surrogates Lecture 6: Measures of (multi-neuron) spike train synchrony
[E. N. Lorenz. Deterministic non-periodic flow. J. Atmos. Sci., 20:130, 1963]
Linear measures
Linearity
Overview
• Static measures - Moments of amplitude distribution (1st – 4th)
Example: White noise
Example: Rössler system
Example: Lorenz system
Example: Hénon map
Example: Inter-ictal EEG
Example: Ictal EEG
Time-frequency representation
Power spectrum
Wiener-Khinchin theorem:
Parseval’s theorem: Overall power:
Tapering: Window functions
Fourier transform assumes periodicity Edge effect Solution: Tapering (zeros at the edges)
First moment: Mean
Average of distribution
Second moment: Variance
Width of distribution (Variability, dispersion)
Standard deviation
Third moment: Skewness
Accessible brain time series: EEG (standard) and neuronal spike trains (recent)
Does a pre-ictal state exist (ictus = seizure)? Do characterizing measures allow a reliable detection of this state? Specific example for prediction of extreme events
Wavelet analysis
Basis functions with finite support
Example: complex Morlet wavelet
Implementation via filter banks (cascaded lowpass & highpass):