(Time )Frequency Analysis of EEG Waveforms - Timely Cost Busch... · (Time{)Frequency Analysis of...

(Time–)Frequency Analysis of EEG Waveforms

Niko Busch

Charite University Medicine Berlin; Berlin School of Mind and Brain

niko.busch@charite.de

niko.busch@charite.de 1 / 23

From ERP waveforms to waves

ERP analysis:

time domain analysis: when do things (amplitudes) happen?treats peaks and troughs as single events.

Frequency domain (spectral) analysis (Fourier analysis):

magnitudes and frequencies of waves — no time information.peaks and troughs are not treated as separate entities.

Time–frequency analysis (wavelet analysis):

when do which frequencies occur?

niko.busch@charite.de 2 / 23

From ERP waveforms to waves

ERP analysis:

time domain analysis: when do things (amplitudes) happen?treats peaks and troughs as single events.

Frequency domain (spectral) analysis (Fourier analysis):

magnitudes and frequencies of waves — no time information.peaks and troughs are not treated as separate entities.

Time–frequency analysis (wavelet analysis):

when do which frequencies occur?

niko.busch@charite.de 2 / 23

From ERP waveforms to waves

ERP analysis:

time domain analysis: when do things (amplitudes) happen?treats peaks and troughs as single events.

Frequency domain (spectral) analysis (Fourier analysis):

magnitudes and frequencies of waves — no time information.peaks and troughs are not treated as separate entities.

Time–frequency analysis (wavelet analysis):

when do which frequencies occur?

niko.busch@charite.de 2 / 23

Why bother?

(Time–)Frequency analysis complements signal analysis:

neurons are oscillating.

analysis of signals with trial-to-trial jitter.

analysis of longer time periods.

analysis of pre-stimulus and spontaneous signals.

necessary for sophisticated methods (coherence, coupling, causality, etc.).

niko.busch@charite.de 3 / 23

Parameters of waves

Oscillations regular repetition of some measure over several cycles.

Wavelength length of a single cycle (a.k.a. period).

Frequency 1wavelength — the speed of change.

Phase current state of the oscillation — angle on the unit circle. Runsfrom 0◦ (-π) — 360◦ (π)

Magnitude (permanent) strength of the oscillation.

niko.busch@charite.de 4 / 23

How to disentangle oscillations

Jean Joseph Fourier (1768—1830):”An arbitrary function, continuous or with

discontinuities, defined in a finite interval by an arbitrarily capricious graph canalways be expressed as a sum of sinusoids“.

niko.busch@charite.de 5 / 23

The discrete Fourier transform

The DFT transforms the signal from the time domain into the frequencydomain.

Requires that the signal be stationary.

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niko.busch@charite.de 6 / 23

The discrete Fourier transform

The DFT transforms the signal from the time domain into the frequencydomain.

Requires that the signal be stationary.

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niko.busch@charite.de 6 / 23

The discrete Fourier transform

The DFT transforms the signal from the time domain into the frequencydomain.

Requires that the signal be stationary.

Non-stationary signals:

When does the 10 Hz oscillation occur?DFT does not give time information.Time information is not necessary for stationary signalsFrequency contents do not change — all frequency components exist all thetime.How to investigate event–related spectral changes in brain signals?

niko.busch@charite.de 6 / 23

Event–related synchronisation / desynchronisation

Cut the signal in two time windows and assume stationarity in each half.

ERD/ERS1 = poststimulus power−baseline powerbaseline power ∗ 100

But why not use even smaller windows?

⇒ Windowed FFT / Short term Fourier transform.

1Pfurtscheller & Lopes da Silva (1999). Clin Neurophysiolniko.busch@charite.de 7 / 23

The short term Fourier transform (STFT) I

Assume that some portion of a non–stationary signal is stationary.

Important parameters:

window function (Hamming, Hanning, Rectangular, etc.)window overlapwindow length: width should correspond to the segment of the signal where itsstationarity is valid.

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Hanning windowHamming windowBoxcar window

niko.busch@charite.de 8 / 23

The short term Fourier transform (STFT) I

Assume that some portion of a non–stationary signal is stationary.Important parameters:

window function (Hamming, Hanning, Rectangular, etc.)window overlapwindow length: width should correspond to the segment of the signal where itsstationarity is valid.

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niko.busch@charite.de 8 / 23

The short term Fourier transform (STFT) I

Assume that some portion of a non–stationary signal is stationary.Important parameters:

window function (Hamming, Hanning, Rectangular, etc.)window overlapwindow length: width should correspond to the segment of the signal where itsstationarity is valid.

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niko.busch@charite.de 8 / 23

The short term Fourier transform (STFT) II

Window length affects resolution in time and frequencyshort window: good time resolution, poor frequency resolution.long window: good frequency resolution, poor time resolution.

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niko.busch@charite.de 9 / 23

Uncertainty principle

Werner Heisenberg (1901—1976):

Energy and location of a particle cannot be both known with infinite precision.a result of the wave properties of particles (not the measurement).

Applies also to time–frequency analysis:

We cannot know what spectral component exists at any given time instant.What spectral components exist at any given interval of time?

Spectral/temporal resolution trade off cannot be avoided — but it can beoptimised.

Good frequency resolution at low frequencies.Good time resolution at high frequencies.

niko.busch@charite.de 10 / 23

From STFT to wavelets

STFT: fixed temporal & spectral resolution

Analysis of high frequencies ⇒ insufficient temporal resolution.Analysis of low frequencies ⇒ insufficient spectral resolution.

Wavelet analysis:

Analysis of high frequencies ⇒ narrow time window for better time resolution.Analysis of low frequencies ⇒ wide time window for better spectral resolution.

niko.busch@charite.de 11 / 23

What is a wavelet?

Motherwavelet

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Zero mean amplitude.

Finite duration.

Mother wavelet: prototype function (f = sampling frequency).

Wavelets can be scaled (compressed) and translated.

niko.busch@charite.de 12 / 23

What is a wavelet?

Motherwavelet

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Zero mean amplitude.

Finite duration.

Mother wavelet: prototype function (f = sampling frequency).

Wavelets can be scaled (compressed) and translated.

niko.busch@charite.de 12 / 23

Wavelet transform of ERPs

ERP

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Wavelet transformed ERP

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Bandpass–filtered ERP

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... but be careful:

any signal can be represented as oscillations w. time-frequency analysisbut it does not imply that the signal is oscillatory!

niko.busch@charite.de 13 / 23

Important parameters of a wavelet

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Length how many cycles does a wavelet have?

e.g. 40 Hz wavelet (25 ms/cycle), 12 cycles ⇒ 250 ms length

σt standard deviation in time domain: σt = m2π∗f0

σf standard deviation in frequency domain: σf = 12π∗σt

time resolution increases with frequency, whereas frequencyresolution decreases with frequency.

niko.busch@charite.de 14 / 23

Evoked and induced oscillations I

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Evoked time-frequency representation of the average of all trials (ERP).

Induced average of time-frequency transforms of single trials.

niko.busch@charite.de 15 / 23

Evoked and induced oscillations II

Wavelet analysis of single trials reveals non–phase–locked activity.

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Wavelet transformed trials-

Evoked time-frequency representation of the average of all trials (ERP).

Induced average of time-frequency transforms of single trials.

niko.busch@charite.de 16 / 23

Phase–locking factor (PLF)

a.k.a. intertrial coherence (ITC) or phase–locking–value (PLV).

measures phase consistency of a frequency at a particular time across trials.

PLF = 1: perfect phase alignment.

PLF = 0: random phase distribution.

niko.busch@charite.de 17 / 23

The EEG state space

Frequency x phase locking x amplitude changes2

Evoked and induced activity are extremes on a continuum.

ERPs cover only small part of the EEG space.2Makeig, Debener, Onton, Delorme (2004). TICS

niko.busch@charite.de 18 / 23

Examples 1: spontaneous EEG

Stimulus pairs are presented at different phases of the alpha rhythm —sequential or simultaneous?3

If the stimulus pair falls within the same alpha cycle⇒ perceived simultaneity.

Does the visual system take snapshots at a rate of 10 Hz?

Simultaneity and the alpha rhythm

Figure from VanRullen & Koch (2003): Is perception discrete of continuous? TICS

3Varela et al. (1981): Perceptual framing and cortical alpha rhythm.Neuropsychologia

niko.busch@charite.de 19 / 23

Examples 2: pre-stimulus EEG power

Spatial attention to left or right.

Stronger alpha power over ipsilateral hemisphere.4

Attention: ipsi- vs. contra-lateral

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4Busch & VanRullen (2010): Spontaneous EEG oscillations reveal periodic samplingof visual attention. PNAS.

niko.busch@charite.de 20 / 23

Examples 3: analysis of long time intervals

Sternberg memory task with different set sizes.Alpha power increases linearly with set size.5

Effect of set size

5Jensen et al. (2002): Oscillations in the alpha band (9–12 Hz) increase withmemory load during retention in a short-term memory task. Cereb Cortex.

niko.busch@charite.de 21 / 23

Recommended reading

WWW:

EEGLAB’s time-frequency functions explained: http://bishoptechbits.blogspot.com/

FFT explained: http://blinkdagger.com/matlab/matlab-introductory-fft-tutorial/

Wavelet tutorial http://users.rowan.edu/ polikar/WAVELETS/WTtutorial.html

Books:

Barbara Burke Hubbard: The World According to Wavelets.

Steven Smith: The Scientist & Engineer’s Guide to Digital Signal Processing(http://www.dspguide.com/).

Herrmann, Grigutsch & Busch: EEG oscillations and wavelet analysis. In:Event-related Potentials: A Methods Handbook.

Papers:

Tallon-Baudry & Bertrand (1999) Oscillatory gamma activity in humans and its rolein object representation. TICS.

Samar, Bopardikar, Rao & Swartz (1999) Wavelet analysis of neuroelectricwaveforms: a conceptual tutorial. Brain Lang.

niko.busch@charite.de 22 / 23

Thank you...

... for your interest !

Please ask questions!!!

niko.busch@charite.de 23 / 23