Lecture series: Data analysis

1. Lecture series: Data analysis Thomas Kreuz, ISC, CNR thomas.kreuz@cnr.it http://www.fi.isc.cnr.it/users/thomas.kreuz/ Lectures: Each Tuesday at 16:00 (First lecture: May 21, last lecture: June 25)

2. (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 • 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

3. Last lecture • Example: Epileptic seizure prediction • Data acquisition • Introduction to dynamical systems

4. Epileptic seizure prediction • Epilepsy results from abnormal, hypersynchronous neuronal activity in the brain • 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

5. Data acquisition AD-Converter Amplifier Sampling Sensor System / Object Computer Filter

6. Dynamical system • Described by time-dependent states • Evolution of state • - continuous (flow) • - discrete (map) • can be both be linear or non-linear • Example: sufficient sampling of sine wave (2 sampling values per cycle) Control parameter

7. Today’s lecture • Non-linear model systems • Linear measures • Introduction to non-linear dynamics • Non-linear measures • - Introduction to phase space reconstruction • - Lyapunov exponent [Acknowledgement: K. Lehnertz, University of Bonn, Germany]

8. Non-linear model systems

9. Non-linear model systems • Discrete maps • Logistic map • Hénonmap • Continuous Flows • Rössler system • Lorenz system

10. Logistic map • r - Control parameter • Model of population dynamics • Classical example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations [R. M. May. Simple mathematical models with very complicated dynamics. Nature, 261:459, 1976]

11. Hénonmap • Introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model • One of the most studied examples of dynamical systems that exhibit chaotic behavior [M. Hénon. A two-dimensional mapping with a strange attractor. Commun. Math. Phys., 50:69, 1976]

12. Rössler system • designed in 1976,for purely theoretical reasons • later found to be useful in modeling equilibrium in chemical reactions [O. E. Rössler. An equation for continuous chaos. Phys. Lett. A, 57:397, 1976]

13. 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 [E. N. Lorenz. Deterministic non-periodic flow. J. Atmos. Sci., 20:130, 1963]

14. Linear measures

15. Linearity • Dynamic of system (and thus of any time series measured from the system) is linear if: • H describes the dynamics and two state vectors • Superposition: • Homogeneity: scalar •  Linearity:

16. Overview • Static measures • - Moments of amplitude distribution (1st – 4th) • Dynamic measures • Autocorrelation • Fourier spectrum • Wavelet spectrum

17. Static measures • Based on analysis of distributions (e.g. amplitudes) • Do not contain any information about dynamics • Example: Moments of a distribution • - First moment: Mean • - Second moment: Variance • - Third moment: Skewness • - Fourth moment: Kurtosis

18. First moment: Mean • Average of distribution

19. Second moment: Variance • Width of distribution • (Variability, dispersion) • Standard deviation

20. Third moment: Skewness Degree of asymmetry of distribution (relative to normal distribution) < 0 - asymmetric, more negative tails Skewness = 0 - symmetric > 0 - asymmetric, more positive tails

21. Fourth moment: Kurtosis • Degree of flatness / steepness of distribution • (relative to normal distribution) • < 0 - platykurtic (flat) • Kurtosis = 0 - mesokurtic (normal) • > 0 - leptokurtic (peaked)

22. Dynamic measures Autocorrelation Fourier spectrum [ Cross correlation Covariance ] Physical phenomenon Time series Time domain Frequency domain x (t) Amplitude Fx() Frequency amplitude Complex number  Phase

23. Autocorrelation One signal with (Normalized to zero mean and unit variance) Time domain:Dependence on time lag

24. Autocorrelation: Examples periodic stochastic memory

25. Discrete Fourier transform Fourier series (sines and cosines): Fourier coefficients: Fourier series (complex exponentials): Fourier coefficients: Condition:

26. Power spectrum = Wiener-Khinchin theorem: Parseval’s theorem: Overall power:

27. Tapering: Window functions Fourier transform assumes periodicity Edge effect Solution: Tapering (zeros at the edges)

28. EEG frequency bands • Description of brain rhythms • Delta: 0.5 – 4 Hz • Theta: 4 – 8 Hz • Alpha: 8 – 12 Hz • Beta: 12 – 30 Hz • Gamma: > 30 Hz [Buzsáki. Rhythms of the brain. Oxford University Press, 2006]

29. Example: White noise

30. Example: Rössler system

31. Example: Lorenz system

32. Example: Hénon map

33. Example: Inter-ictal EEG

34. Example: Ictal EEG

35. Time-frequency representation

36. Wavelet analysis Basis functions with finite support Example: complex Morlet wavelet – scaling; – shift / translation (Mother wavelet: , ) Implementation via filter banks (cascaded lowpass & highpass): – lowpass (approximation) – highpass (detail)

37. Wavelet analysis: Example • Advantages: • - Localized in both • frequency and time • - Mother wavelet can • be selected according • to the feature of interest • Further applications: • Filtering • Denoising • Compression Power [Latka et al. Wavelet mapping of sleep splindles in epilepsy, JPP, 2005]

38. Introduction to non-linear dynamics

39. Linear systems • Weak causality • identical causes have the same effect • (strong idealization, not realistic in experimental situations) • Strong causality • similar causes have similar effects • (includes weak causality • applicable to experimental situations, small deviations in • initial conditions; external disturbances)

40. Non-linear systems Violation of strong causality Similar causes can have different effects Sensitive dependence on initial conditions (Deterministic chaos)

41. Linearity / Non-linearity • Linear systems • have simple solutions • Changes of parameters and initial conditions lead to proportional effects • Non-linear systems • can have complicated solutions • Changes of parameters and initial conditions lead to non-proportional effects • Non-linear systems are the rule, linear system is special case!

42. Phase space example: Pendulum Time series: Position x(t) t Velocity v(t) State space:

43. Phase space example: Pendulum Ideal world: Real world:

44. Phase space Phase space: space in which all possible states of a system are represented, with each possible system state corresponding to one unique point in a d dimensional cartesian space (d - number of system variables) Pendulum: d = 2 (position, velocity) Trajectory: time-ordered set of states of a dynamical system, movement in phase space (continuous for flows, discrete for maps)

45. Vector fields in phase space Dynamical system described by time-dependent states  – d-dimensional phase space – Vector field (assignment of a vector to each point in a subset of Euclidean space) Examples: - Speed and direction of a moving fluid -Strength and direction of a magnetic force Here: Flow in phase space Initial conditionTrajectory(t)

46. Divergence • Rate of change of an infinitesimal volume around a given point of a vector field: • Source: outgoing flow ( with , expansion) • Sink: incoming flow ( with , contraction)

47. System classification via divergence Liouville’s theorem: Temporal evolution of an infinitesimal volume: conservative (Hamiltonian) systems dissipative systems instable systems

48. Dynamical systems in the real world • In the real world internal and external frictionleads to • dissipation • Impossibility of perpetuummobile • (without continuous driving / energy input, the motion stops) • When disturbed, a system, after some initial transients, • settles on its typical behavior (stationary dynamics) • Attractor: Part of the phase space of the dynamical system • corresponding to the typical behavior.

49. Attractor • Subset X of phase space which satisfies three conditions: • Xis forward invariant under f: • If xis an element of X, then so is f(t,x) for all t > 0. • There exists a neighborhood of X, called the basin of attractionB(X), which consists of all points b that "enter X in the limit t → ∞". • There is no proper subset of Xhaving the first two properties.

50. Attractor classification Fixed point: point that is mapped to itself Limit cycle: periodic orbit of the system that is isolated (i.e., has its own basin of attraction) Limit torus: quasi-periodic motion defined by n incommensurate frequencies (n-torus) Strange attractor: Attractor with a fractal structure (2-torus)