Title: Tutorial for ASSC9 24 June 2005

1. Title: Tutorial for ASSC9 24 June 2005 Tutorial for ASSC9 on new developments in EEG research Walter J Freeman University of California at Berkeley http://sulcus.berkeley.edu

2. Oliver Sacks • “A movie, with its taut stream of thematically connected images, its visual narrative integrated by the viewpoint and values of its director, is not at all a bad metaphor for the stream of consciousness itself. .... Themechanism of our ordinary knowledge is of a cinematographical kind.” • New York Review, 2004 Oliver Sacks

3. Questions regarding cinematographic frames • If the cinematographic hypothesis is valid: • • How many screens are there? • • Where are they, and how large? • • What are the frame rates, durations, sizes? • • How do frames form and sequence? • • What is the structure of their contents? • •How can EEG be used to answer the questions?

4. Outline OUTLINE Part 1. Temporal analysis Introduction to basic concepts Temporal Fourier transform Gabor and Morlet wavelets Hilbert transform Part 2. Spatial analysis Design of electrode arrays, 1-D, 2-D Spatial Fourier transform Spatial patterns of phase Spatial patterns of amplitude

5. Basic concepts Introduction to basic concepts: Linearity Stationarity Gaussianity Measurement Decomposition Brains are neither linear, stationary nor Gaussian. EEGs are usually linear, stationary and Gaussian. This discrepancy must be dealt with explicitly.

6. Superposition Linearity: Superposition Additivity 1. Outputs for multiple inputs are additive. Proportionality 2. Outputs are proportional to inputs. Testing is by paired-shock using single-shock electrical stimuli and averaging the cortical evoked potentials.

7. Biedenbach paired shock Biedenbach & Freeman, 1965

8. Gaussianity Accompanying properties: 1. Output frequencies are the same as the input frequencies - no harmonics. 2. Gaussian amplitude distributions of input give Gaussian output amplitude distributions. By this test, EEG has robust near-linear domains, provided that the amplitudes of evoked potentials do not exceed the maxima of background EEG.

9. Amplitude histograms normal density function alpha Gaussianity demonstrated by use of amplitude histograms.

10. Stationarity Stationarity: Frequencies don’t change with time. Major state transitions reveal nonstationarity of cortical dynamics: • waking vs. sleeping • seizure onset vs. offset EEGs recorded during attentional shifts and cognitive activity reveal drifts and jumps in frequencies that reflect nonstationarity.

11. Measurement Measurement expresses some quantity of interest in numbers. It requires a unit of the selected dimension: • Time - sec • Space - cm • Magnitude - microvolt • Phase - radian (1 rad = 360°/2p = 57.3° ) • Wave form - basis function

12. Decomposition Decomposition Upon measurement, the wave form of interest is said to be “decomposed” into the matching sum of the selected basis functions. The simplest basis function is the digitizing step. A string of square waves is defined by the latency, the duration, and the number of squares we add to match an EEG at each time step.

13. Family of linear basis functions When EEGs conform to linearity and stationarity, they can be decomposed for measurement with linear basis functions: Sines, Cosines, Exponentials…. These basis functions are the solutions to linear differential equations. Linear, time-invariant neurodynamics!

14. Fourier Transform: The inner product

15. Brain oscillations decomposed in a semi-log plot of power spectral density (PSD) with classic bands.

16. Gabor Transform - Gaussian envelope, fixed duration, selected frequencies, moving window

17. Effect of the window duration on display of seizure onset

18. Morlet Wavelets - Gaussian envelope with fixed frequencies and fixed duration of moving window 5.5 cycles Clin. Neurophysiol. 110: 643-654 (1999) Brain Research Protocols 8: 16-24 (2001)

19. Multiresolution decomposition - calculation at discrete time steps and frequency scales, implemented by a recursive filter bank - faster than the FFT!

20. Event-related alpha responses by decomposing visual EP in an oddball paradigm Clin. Neurophysiol. 110: 643-654 (1999)

21. Tools for linear analysis Linear analysis is easy to implement, and it tolerates wide deviations from linearity and stationarity. Examples: FFT, ARMA, Discriminant analysis, SVD, Karhunen-Loéve, PCA, Factor analysis (Bayesian), ICA, Laplacian operators, ERP, AER … Why do these techniques work so well? Where and why do they fail?

22. An introduction to PSD in log-log coordinates Decomposition by FFT and wavelets requires the assumption that frequencies are discrete. We can relax this assumption by using a coordinate transform to re-plot the power spectral density (PSD): log power vs. log frequency The log-log plot indicates the existence of continuous distributions of frequencies. Walter J Freeman University of California at Berkeley

23. Temporal spectra from frontal scalp From Freeman et al. 2003

24. EEG awake and asleep: intracranial recording from right superior temporal gyrus in epileptic patient.

25. delta PSDt of EEG Slope = -3.46

26. alpha EEG awake intracranial beta Slope = -2.21

27. Histogram of power-law exponents - asleep

28. Histogram of power-law exponents - awake

29. EEG(t) and EMG(t) 100 uv 500 ms Human scalp recordings, right paracentral

30. Eyes closed Frontal scalp Resting Slope = -2 EEG PSDt in spatial spectral bands Decomposition of temporal spectrum by spatial pass band

31. Eyes closed Frontal scalp EMG PSDt in spatial spectral bands EMG power covers all parts of the spectrum: white noise

32. Occipital scalp Eyes closed EEG with strong alpha

33. EMG with persisting alpha Occipital scalp Eyes closed

34. An introduction to 1/f PSDt and PSDx The EEG spectrum tends to 1/f for amplitude, 1/f2 for power, except in sleep and before seizure. The EMG spectrum tends to be flat - white noise. The parameters of EEG appear to be fractal. EEG frequencies are not fixed; they vary over a spectral continuum. The Hilbert transform can address that property. Walter J Freeman University of California at Berkeley

35. An introduction to the Hilbert transform The Hilbert transform At each time step the EEG gives the real part, Re(t), of a complex number. The Hilbert transform gives the imaginary part, Im(t). The complex number gives the analytic amplitude: A(t) = [ Re(t)2 + Im(t)2 ] 0.5 and the analytic phase: P(t) = atan [ Im(t) / Re(t) ] The HT serves to decompose an EEG signal into independent functions of amplitude and phase. Walter J Freeman University of California at Berkeley

36. Simulated EEG - 20 Hz cosine with phase slip

37. Polar plot, simulated EEG analytic amplitude: A(t) = [ Re(t)2 + Im(t)2 ] 0.5

38. Calculate analytic phase and differences analytic phase: P(t) = atan [ Im(t) / Re(t) ]

39. Calculate analytic phase and differences “ Instantaneous” frequency is calculated by dividing each frequency difference by the digitizing step in s to give radians/s. Division by 2p gives frequency in Hz.

40. 8x8 recording in waking rabbit EEG from 8x8 pial array, rabbit auditory cortex. Spacing: 0.79 mm 1st component PCA: 94% Digitizing step: 2 ms Nyquist frequency: 250 Hz Walter J Freeman University of California at Berkeley

41. PSDt in waking rabbit theta gamma Subtract channel means; normalize globally to unit SD; FFT 64 EEGs in 500 ms; average PSDt Walter J Freeman University of California at Berkeley

42. Time series Rabbit auditory cortex; CS+ given at 400 ms

43. Polar plot Elapsed time is shown by counterclockwise rotation.

44. Analytic amplitude A(t) = [ Re(t)2 + Im(t)2 ] 0.5

45. Analytic phase - p(t) + P(t) P(t) = atan [ Im(t) / Re(t) ]; p(t) = P(t) + p

46. Coordinated analytic phase differences (CAPD) Time is on left abscissa, channel order is on right

47. Relation of amplitude to phase dispersion Maximal amplitude occurs when the phase stabilizes.

48. Evidence for stationarity on average • This result can explain the success of Fourier analysis. Most of the time the rate of change in phase (frequency) is quite low, implying stationarity. • But, epochs of phase constancy [frames] are bracketed by phase instabilities and discontinuities: phase slip. • The successive frames have different average rates of change in phase (frequencies), but they are similar owing to band pass filtering and low rates of change. • The EEG can appear to be stationary, but it is not; the • analytic amplitude varies with the phase discontinuities.

49. Interim Conclusions • Interim Conclusions • The Hilbert transform reveals ‘phase slips’ within empirical EEG beta and gamma ranges. • These discontinuities in phase may demarcate state transitions by which cinematographic frames form. • Further evidence is needed to support the • cinematographic hypothesis. • That evidence is in spatial analysis of the analytic amplitude of EEGs from multielectrode arrays: • Part 2. Spatial analysis

50. Kohler