Tutorial for ASSC9 on new developments in

EEG research

Walter J FreemanUniversity of California at Berkeley

http://sulcus.berkeley.edu

“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. .... The mechanism of our ordinary knowledge is of a cinematographical kind.”

New York Review, 2004Oliver Sacks

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?

Part 1. Temporal analysisIntroduction to basic conceptsTemporal Fourier transform Gabor and Morlet wavelets

Hilbert transform

Part 2. Spatial analysis Design of electrode arrays, 1-D, 2-D

Spatial Fourier transformSpatial patterns of phase

Spatial patterns of amplitude

OUTLINE

Introduction to basic concepts:

LinearityStationarityGaussianity

Measurement Decomposition

Brains are neither linear, stationary nor Gaussian.

EEGs are usually linear, stationary and Gaussian.

This discrepancy must be dealt with explicitly.

Basic concepts

Linearity: Superposition

Additivity1. Outputs for multiple inputs are additive.

Proportionality2. Outputs are proportional to inputs.

Testing is by paired-shockusing single-shock electrical stimuli and averaging the cortical evoked potentials.

Superposition

Biedenbach & Freeman, 1965

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.

Gaussianity

alpha

Gaussianity demonstrated by use of amplitude histograms.

normal density function

Amplitude histograms

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.

Stationarity

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°/2= 57.3° )

• Wave form - basis function

Measurement

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.

Decomposition

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!

Family of linear basis functions

Fourier Transform: The inner product

Rodrigo Quian QuirogaSloan-Swartz Center for Theoretical Neurobiology

California Institute of Technologyhttp://www.vis.caltech.edu/~rodri

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

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

Effect of the window duration on display of seizure onset

Morlet Wavelets - Gaussian envelope with fixed frequencies and fixed duration of moving window

Clin. Neurophysiol. 110: 643-654 (1999)Brain Research Protocols 8: 16-24 (2001)5.5 cycles

Multiresolution decomposition - calculation at discrete time steps and frequency scales, implemented by a

recursive filter bank - faster than the FFT!

Event-related alpha responses by decomposing visual EP in an oddball paradigm

Clin. Neurophysiol. 110: 643-654 (1999)

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?

Tools for linear analysis

Walter J Freeman University of California at Berkeley

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.

An introduction to PSD in log-log coordinates

Temporal spectra from frontal scalp

From Freeman et al. 2003

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

PSDt of EEG

Slope = -3.46

delta

EEG awake intracranial

Slope = -2.21

alpha

beta

Histogram of power-law exponents - asleep

Histogram of power-law exponents - awake

EEG(t) and EMG(t)

500 ms

100 uv

Human scalp recordings, right paracentral

EEG PSDt in spatial spectral bands

Frontal scalpEyes closed

Decomposition of temporal spectrum by spatial pass band

Resting Slope = -2

EMG PSDt in spatial spectral bands

Frontal scalpEyes closed

EMG power covers all parts of the spectrum: white noise

EEG with strong alpha

Occipital scalp

Eyes closed

EMG with persisting alpha

Occipital scalp

Eyes closed

Walter J Freeman University of California at Berkeley

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

The Hilbert transformAn introduction to 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.

Simulated EEG - 20 Hz cosine with phase slip

Polar plot, simulated EEG

analytic amplitude:A(t) = [ Re(t)2 + Im(t)2 ] 0.5

Calculate analytic phase and differences

analytic phase: P(t) = atan [ Im(t) / Re(t) ]

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 2 gives frequency in Hz.

Walter J Freeman University of California at Berkeley

EEG from 8x8 pial array, rabbit auditory cortex.Spacing: 0.79 mm 1st component PCA: 94%

Digitizing step: 2 ms Nyquist frequency: 250 Hz

8x8 recording in waking rabbit

Walter J Freeman University of California at Berkeley

Subtract channel means; normalize globally to unit SD; FFT 64 EEGs in 500 ms; average PSDt

PSDt in waking rabbit

theta gamma

Time series

Rabbit auditory cortex; CS+ given at 400 ms

Polar plot

Elapsed time is shown by counterclockwise rotation.

Analytic amplitude

A(t) = [ Re(t)2 + Im(t)2 ] 0.5

Analytic phase

P(t) = atan [ Im(t) / Re(t) ]; p(t) = P(t) +

P(t)

p(t)+

-

Coordinated analytic phase differences (CAPD)

Time is on left abscissa, channel order is on right

Relation of amplitude to phase dispersion

Maximal amplitude occurs when the phase stabilizes.

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.

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

Kohler

Kohler’s Category Error

Roger Sperry

Disproof:Roger Sperry inserted silver needles or strips of mica into the visual

cortex of cats and monkeys that distorted

the electric fields. Visual perception was undiminished.

But – EEG fields are

epiphenomenal – not

Köhler’s Gestalt fields.