Tutorial for ASSC9 on new developments in EEG research Walter J FreemanUniversity of California at Berkeley

http://sulcus.berkeley.eduTitle: Tutorial for ASSC9 24 June 2005

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. .... The mechanism of our ordinary knowledge is of a cinematographical kind. New York Review, 2004Oliver Sacks

Questions regarding cinematographic framesIf 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?

OutlinePart 1. Temporal analysisIntroduction to basic conceptsTemporal Fourier transform Gabor and Morlet wavelets Hilbert transformPart 2. Spatial analysis Design of electrode arrays, 1-D, 2-DSpatial Fourier transformSpatial patterns of phaseSpatial patterns of amplitudeOUTLINE

Introduction to basic concepts:

LinearityStationarityGaussianityMeasurement 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, 1965Biedenbach paired shock

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

alphaGaussianity demonstrated by use of amplitude histograms.normal density functionAmplitude histograms

Stationarity:

Frequencies dont 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 = 360/2p = 57.3 ) Wave form - basis functionMeasurement

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 Quiroga

Sloan-Swartz Center for Theoretical Neurobiology

California Institute of Technology

http://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

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 paradigmClin. 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-Love,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 BerkeleyDecomposition 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 scalpFrom Freeman et al. 2003

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

PSDt of EEGSlope = -3.46delta

EEG awake intracranialSlope = -2.21 alphabeta

Histogram of power-law exponents - asleep

Histogram of power-law exponents - awake

EEG(t) and EMG(t)500 ms100 uvHuman scalp recordings, right paracentral

EEG PSDt in spatial spectral bandsFrontal scalpEyes closedDecomposition of temporal spectrum by spatial pass bandRestingSlope = -2

EMG PSDt in spatial spectral bandsFrontal scalpEyes closedEMG power covers all parts of the spectrum: white noise

EEG with strong alphaOccipital scalpEyes closed

EMG with persisting alphaOccipital scalpEyes closed

*Walter J Freeman University of California at BerkeleyAn introduction to 1/f PSDt and PSDxThe 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 BerkeleyThe Hilbert transformAn introduction to the Hilbert transformAt 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 ] 0.5and 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 EEGanalytic amplitude:A(t) = [ Re(t)2 + Im(t)2 ] 0.5

Calculate analytic phase and differencesanalytic 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 2p gives frequency in Hz.

*Walter J Freeman University of California at BerkeleyEEG from 8x8 pial array, rabbit auditory cortex.Spacing: 0.79 mm 1st component PCA: 94%Digitizing step: 2 ms Nyquist frequency: 250 Hz8x8 recording in waking rabbit

*Walter J Freeman University of California at BerkeleySubtract channel means; normalize globally to unit SD; FFT 64 EEGs in 500 ms; average PSDtPSDt in waking rabbitthetagamma

Time seriesRabbit auditory cortex; CS+ given at 400 ms

Polar plotElapsed time is shown by counterclockwise rotation.

Analytic amplitudeA(t) = [ Re(t)2 + Im(t)2 ] 0.5

Analytic phaseP(t) = atan [ Im(t) / Re(t) ]; p(t) = P(t) + pP(t)p(t)+-

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

Relation of amplitude to phase dispersionMaximal amplitude occurs when the phase stabilizes.

Evidence for stationarity on averageThis 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; theanalytic amplitude varies with the phase discontinuities.

Interim ConclusionsInterim 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

Kohlers Category Error

Roger SperryDisproof: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 Khlers Gestalt fields.

Tutorial in spatial EEG analysisTutorial given at the 9th meeting of the Association for the Scientific Study of Consciousness at Cal Tech in Pasadena, California on 24 June 2005. Tutorial in spatial EEG analysisThe cinematographic hypothesis of brain function and consciousness has been discussed ever since the invention of the cinema over a century ago, particularly by William James in his description of the stream of consciousness whether it is a continuous or intermittent flow. The EEG is optimal for testing the cintematographic hypothesis owing to its superior temporal resolution and the potential for improvements in it spatial resolution in comparison to MEG and fMRI. Tutorial in spatial EEG analysisQuestions raised by the cinematographic hypothesis can be answered by EEG studies. Tutorial in spatial EEG analysisAbstract (continued) The findings from animals guide our noninvasive search for hemisphere-wide spatial patterns in human brain function. Subjects are at rest with eyes closed or open. A curvilinear array of 64 electrodes is placed on the scalp, 19 cm in length with 3 mm spacing to maximize spatial resolution. The 64 EEGs are digitized at 200/s, amplified with a Nicolet BMSI 5000 and analyzed in frames of 1000 bins (5 s). The time derivative of the instantaneous phase is approximated by the temporal difference on each channel. The results show, as in animals, repeated periods of broad spatial synchrony with phase stability that are bracketed by abrupt changes in phase. The phase jumps are simultaneous across varying distances ranging from a few cm to the entire array within each hemisphere, indicating a brain state of self-organizing criticality. Rates of recurrence are in the alpha range with eyes closed and tend to shift into the theta range with eyes open or with other intentional behavior. From our findings we predict that space-time patterns will be found in human brains that can be studied with our methods, including cognitive tasks with subjects reports of their experiences to correlate with EEG.

Tutorial in spatial EEG analysisFundamental concepts in EEG dynamicsTutorial in spatial EEG analysisThe classic example of testing for linearity is administration of electric pulses to nerve axon, demonstrating the all-or-none response of the axon, which is nonlinear. However, outside of the refractory periods, axons have a broad linear domain of additivity. Tutorial in spatial EEG analysisDemonstration of a linear domain for EEG of the prepyriform cortex of cat. Tutorial in spatial EEG analysisHarmonics are evidence for nonlinearity, which for EEG is the inhibition of neurons below their threshold for firing. Inhibition beyond the threshold is not expressed in the output. This property is the key to understanding the stability of cortical networks, because it provides the limit to the amplitude of the background EEG. Tutorial in spatial EEG analysisAmplitude histograms reveal how robust is the normal density distribution as a model for EEG consisting of white noise passed through a linear band pass filter, which is provided by the negative and positive feedback loops in the forebrain. Tutorial in spatial EEG analysisThe definition of stationarity is based in the invariance of the frequency distributions of the system under study. Tutorial in spatial EEG analysisMeasurement is the expression of an object or event in numbers. Tutorial in spatial EEG analysisThe square wave basis function is the basis for all analog-to-digital conversion and vice versa. Tutorial in spatial EEG analysisPiece-wise linear approximation is a powerful method for modeling the nonlinear dynamics of cortex in the near linear range. Freeman WJ. Mass Action in the Nervous System. Academic Press, New York, 1975. Reprinted 2004: http://sulcus.berkeley.edu/MANSWWW/MANSWWW.html Tutorial in spatial EEG analysisWe introduce the first method of analysis of EEGs, the Fourier transform. It can be seen as the inner product of the signal with sinusoids of different frequencies. This gives a representation of the signal in the frequency domain. The problem of Fourier is that it has no time resolution. Moreover it assumes stationarity.Tutorial in spatial EEG analysisThe Fourier transform is still the most used method for the analysis of EEG signals. In particular, the power spectrum given by the Fourier transform is divided in frequency bands, which have been correlated to different functions, states, pathologies, etc.Tutorial in spatial EEG analysisThe Fourier transform gives no information of time. Time-frequency resolution is achieved with the Gabor (or Short Time Fourier) Transform. The Gabor Transform is the inner product of the signal with sinusoids tapered with a Gaussian function. This allows a time localization of the frequency spectra. A problem (to be later described) is how large to choose the window length (i.e. the width of the Gaussian tapering function).We will further apply the Gabor transform to the study of tonic-clonic seizures.Tutorial in spatial EEG analysisOn top we have a 3 minutes scalp EEG recording of the tonic clonic seizure. The bottom plots are the time-frequency decompositions of the signal taking different window sizes. Large windows gives high frequency resolution but not temporal resolution and short windows give high temporal but not frequency resolution. An optimal compromise between time and frequency resolution is to take (in this case) a window of 5 sec. There is a limit to the maximum simultaneous resolution that can be achieved both in time and frequency given by the uncertainty principle of signal analysis.Tutorial in spatial EEG analysisThe main problem of the Gabor Transform is how to choose the window size. In principle, one would like to increase the time resolution for the high frequencies and to decrease it for the low ones. Wavelets offer a variable window size that is appropriate for all frequencies. Moreover, instead of using tapered sinusoidal functions, we have a dictionary of wavelets functions to choose from according to the problem.The wavelet function shown here is a biorthogonal cubic B-Spline. B-Splines are very suitable for the analysis of EEG due to its compact support (i.e. they are short), smoothness, time-frequency resolution (close to the best possible) and in the particular case of evoked potentials, due to their similarity with the evoked responses.Tutorial in spatial EEG analysisThe wavelet transform can be calculated at discrete times and scales giving a complete and non-redundant time-scale representation of the data. This so called multiresolution decomposition can be implemented via a recursive filter bank, therefore being very fast (order N, faster than the FFT!). The levels approx. correspond to the EEG frequency bands (determined by the sampling rate of the data). In the following we will show two examples of event-related oscillations, one in the alpha band (level 4) and the second one in the gamma band (level 2).Tutorial in spatial EEG analysisThis are the responses to pattern visual evoked potentials (upper plot), and non-target and target responses with an oddball paradigm (also with pattern VEP). Note alpha enhancements best localized in occipital electrodes.Tutorial in spatial EEG analysisThe major tools in current use for analysis of EEG are derived on the basis of the assumption of linearity. Tutorial in spatial EEG analysisWhat is at issue here is the finding that neural activity in the entire cerebral hemisphere of rabbits, cats and humans is globally organized into frames, and that each frame has a unique spatial pattern of amplitude and phase modulation of the beta-gamma oscillations. The oscillation in each from is a common waveform with the same instantaneous frequency and phase. With each new frame there is a phase-resetting that is revealed by the HT. Tutorial in spatial EEG analysisThe occurrence of wave sequences in the EEG time series is accompanied by peaks in the PSDt that provide the basis for labeling the waves in time series. The 1/fa form of the spectra is revealed in log log coordinates. The slopes of the spectra are highly variable between subjects and areas of scalp. These are spectra from the frontal area of nine subjects at rest with eyes closed. Intracranial spectra have the same form but steeper slopes. Tutorial in spatial EEG analysisA sample is shown of 8 adjacent channels selected as representative from 64 electrodes in an 8x8 array placed on the pial surface. The recordings were made between seizures when the patient was in normal behavioral states. Tutorial in spatial EEG analysisThis is the average of the temporal power spectral density of 1000 time points (5 seconds at 200 Hz sampling at 5 msec intervals) over 64 channels of EEG. Tutorial in spatial EEG analysisThe same sampling was used as in sleep. Tutorial in spatial EEG analysisThe slope of the PSDt of the EEG is a significant parameter for modeling the EEG dynamics in terms of power-law distributions, with the possibility that the measures of EEG are fractal and therefore do not conform to the central limit theorem or the laws of large numbers. Tutorial in spatial EEG analysisThe slope varies widely about the mean, which differs significantly from the slope in sleep. Tutorial in spatial EEG analysisA sample of 8 channels from 64 channels of scalp EEG with the subject at rest or deliberately tensing the scalp muscles in order to produce a modest overlay of electromyographic (EMG) muscle potentials. Tutorial in spatial EEG analysisThe EEG from a 1x64 lcurvilinear 1-D array was spatially filtered with a FIR band pass filter into bands as shown on the right abscissa. The alpha peak shows that there is no structure at high spatial frequency over the frontal area in contrast to the high texture over the occipital area (see two frames later). Tutorial in spatial EEG analysisEMG contributes power in all parts of the PSDt and spatial PSDx, indicating that EMG is closer to white noise with a flat spectrum than it is to the 1/f spectrum of EEG. Tutorial in spatial EEG analysisThe Alpha peak shows high texture over the occipital area, whereas the beta peaks do not, showing the seemingly paradoxical phenomenon that the low temporal frequency component has high local structure, whereas the higher frequency component does not. Tutorial in spatial EEG analysisThere is no filter that can removed EMG, but patients can be instructed to minimize it using biofeedback techniques based on detection through the PSDt.

Tutorial in spatial EEG analysisWhat is at issue here is the finding that neural activity in the entire cerebral hemisphere of rabbits, cats and humans is globally organized into frames, and that each frame has a unique spatial pattern of amplitude and phase modulation of the beta-gamma oscillations. The oscillation in each from is a common waveform with the same instantaneous frequency and phase. With each new frame there is a phase-resetting that is revealed by the HT. Tutorial in spatial EEG analysisWhat is at issue here is the finding that neural activity in the entire cerebral hemisphere of rabbits, cats and humans is globally organized into frames, and that each frame has a unique spatial pattern of amplitude and phase modulation of the beta-gamma oscillations. The oscillation in each from is a common waveform with the same instantaneous frequency and phase. With each new frame there is a phase-resetting that is revealed by the HT. Tutorial in spatial EEG analysisA simulated 20 Hz cosine has phase slip introduced by re-setting the phase at 70 ms.

Tutorial in spatial EEG analysisThe transformation to polar coordinates is demonstrated. Tutorial in spatial EEG analysisThe derivation of the analytic phase is demonstrated. The Hilbert trnsform in itself is not very interesting. Its importance is as a tool for the decomposition of an EEG trace into the analytic amplitude and analytic phase. Note that the Fourier phase and analytic phase converge to the same values, but only on average over a sufficiently long time interval to define the frequency from the average rate of change in the analytic phase. Tutorial in spatial EEG analysisThe most valuable property of the phase is given by its rate of change. This is often referred to as its instantaneous frequency. The term is an oxymoron equivalent to describing the length of a point or the thickness of a plane. Tutorial in spatial EEG analysisWhat is at issue here is the finding that neural activity in the entire cerebral hemisphere of rabbits, cats and humans is globally organized into frames, and that each frame has a unique spatial pattern of amplitude and phase modulation of the beta-gamma oscillations. The oscillation in each from is a common waveform with the same instantaneous frequency and phase. With each new frame there is a phase-resetting that is revealed by the HT. Here is a sample of the raw EEG from an 8x8 array 5.6x5.6 mm on the pial surface of a trained rabbit.Tutorial in spatial EEG analysisThe PSDt shows a peak in the theta range conforming to the frame rate, and a peak in the gamma range corresponding to the carrier frequencies of the contents of frames. Tutorial in spatial EEG analysisThe EEG has been band pass filtered (FIR) 20-80 Hz. The Hilbert transform is a line integral that gives the negative of the rate of change in amplitude at each point in terms of the Cauchy Principle Value,or simply the quadrature - 90 phase delay. Tutorial in spatial EEG analysisThe polar plot has time implicit in the rotation of a vector in the counterclockwise direction. Tutorial in spatial EEG analysisThe analytic amplitude resembles the envelope of the band pass filtered EEG, but it differs by inclusion of both the real and the imaginary parts. Owing to the negative feedback by which the oscillations are generated, the real part conforms to the output of excitatory neurons in neocortex (inhibitory neurons in the olfactory bulb), whereas the imaginary part conforms to the output of the inhibitory neurons in neocortex (excitatory cells in the bulb). Hence the analytic amplitude manifests the activity of both populations. The square of the amplitude gives an index of the power generated by the neurons in forcing electric current across the fixed extracellular resistance of the cortex. Therefore, the square of the analytic amplitude, A2(t), is an optimal measure of the rate of free energy dissipation of cortex, and should be used for correlation with fMRI and other measures of metabolic activity of cortex. Tutorial in spatial EEG analysisPhase slip can increase or decrease the instantaneous frequency, even to the point of negative frequencies. Methods that remove phase slip such as the Huang-Hilbert are detrimental to EEG analysis. Tutorial in spatial EEG analysisPhase slip tends to occur simultaneously or nearly so over wide areas of cortex. Tutorial in spatial EEG analysisThere is a strong negative correlation between analytic amplitude and analytic phase. Tutorial in spatial EEG analysisThe HT provides a powerful method for detecting phase transitions in cortical generators of the EEG. Tutorial in spatial EEG analysisDemonstration of the cinematographic hypothesis will require spatial analysis of the behavioral correlates of EEG. Tutorial in spatial EEG analysisWolfgang Khler introduced field theory into neuropsychology, much as Donald Hebb introduced network theory. Tutorial in spatial EEG analysisKohler made a serious error.Tutorial in spatial EEG analysisSperrys disproof of Khlers hypothesis led to decline of interesting EEG phenomena, accompanied by diminished attention to field properties of brain dynamics and neglect of the spatial properties of neocortical activity.