(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

[email protected]

[email protected] 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?


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.).


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.


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“.


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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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?


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 [email protected] 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



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


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shifted Hanning window

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


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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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SPECTROGRAM, width = 1024

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102030


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.


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.


What is a wavelet?

Motherwavelet

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Gaussian

e−t2/2

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Cosine

|ejω0t|

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Wavelet

|Ψ(t)|

Zero mean amplitude.

Finite duration.

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

Wavelets can be scaled (compressed) and translated.


What is a wavelet?

Motherwavelet

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Frequency ( Hz )S

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ctr

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Zero mean amplitude.

Finite duration.

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

Wavelets can be scaled (compressed) and translated.


Wavelet transform of ERPs

ERP

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µV

Wavelet transformed ERP

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µVOz

Bandpass–filtered ERP

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Gamma–Band: ca. 30 – 80 Hz

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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!


Important parameters of a wavelet

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Am

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A Wavelet - time domain

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Frequency (Hz)

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plit

ude

B Wavelet - frequency domain

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.


Evoked and induced oscillations I

Average

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evoked/ phase-locked induced/ non-phase-locked

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

Induced average of time-frequency transforms of single trials.


Evoked and induced oscillations II

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

Average

ms

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Evoked/ phase-locked Induced/ non/phase-locked

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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.


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.


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


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


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.


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.


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.


http://www.dspguide.com/

Thank you...

... for your interest !

Please ask questions!!!


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