(Time )Frequency Analysis of EEG Waveforms

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1 (Time )Frequency Analysis of EEG Waveforms Niko Busch Charité University Medicine Berlin; Berlin School of Mind and Brain 1 / 23

2 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? 2 / 23

3 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? 2 / 23

4 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? 2 / 23

5 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.). 3 / 23

6 Parameters of waves Oscillations regular repetition of some measure over several cycles. Wavelength length of a single cycle (a.k.a. period). 1 Frequency wavelength the speed of change. Phase current state of the oscillation angle on the unit circle. Runs from (-π) 36 (π) Magnitude (permanent) strength of the oscillation. niko.busch@charite.de 4 / 23

How to disentangle oscillations Jean Joseph Fourier (1768 183): An arbitrary function, continuous or with discontinuities, defined

7 How to disentangle oscillations Jean Joseph Fourier ( ): An arbitrary function, continuous or with discontinuities, defined in a finite interval by an arbitrarily capricious graph can always be expressed as a sum of sinusoids. niko.busch@charite.de 5 / 23

8 The discrete Fourier transform The DFT transforms the signal from the time domain into the frequency domain. Requires that the signal be stationary. 5 Hz Time 5 Hz Time 1 Hz Time Sum of Time 4 2 FFT Spectrum 5 1 Hz FFT Spectrum Hz FFT Spectrum Hz FFT Spectrum Hz niko.busch@charite.de 6 / 23

9 The discrete Fourier transform The DFT transforms the signal from the time domain into the frequency domain. Requires that the signal be stationary. 1 EEG raw data 1 reverted raw data Time [sec] FFT Spectrum Hz Time [sec] FFT Spectrum Hz niko.busch@charite.de 6 / 23

10 The discrete Fourier transform The DFT transforms the signal from the time domain into the frequency domain. Requires that the signal be stationary. Non-stationary signals: When does the 1 Hz oscillation occur? DFT does not give time information. Time information is not necessary for stationary signals Frequency contents do not change all frequency components exist all the time. How to investigate event related spectral changes in brain signals? niko.busch@charite.de 6 / 23

11 Event related synchronisation / desynchronisation Cut the signal in two time windows and assume stationarity in each half. ERD/ERS 1 poststimulus power baseline power = baseline power 1 But why not use even smaller windows? Windowed FFT / Short term Fourier transform. 1 Pfurtscheller & Lopes da Silva (1999). Clin Neurophysiol niko.busch@charite.de 7 / 23

12 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 overlap window length: width should correspond to the segment of the signal where its stationarity is valid. 1.5 Hanning window Hamming window Boxcar window niko.busch@charite.de 8 / 23

13 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 overlap window length: width should correspond to the segment of the signal where its stationarity is valid. 1 EEG raw data shifted Hanning window windowed EEG signal niko.busch@charite.de 8 / 23

14 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 overlap window length: width should correspond to the segment of the signal where its stationarity is valid. 1 EEG raw data Time [sec] SPECTROGRAM, R = frequency time niko.busch@charite.de 8 / 23

15 The short term Fourier transform (STFT) II Window length affects resolution in time and frequency short window: good time resolution, poor frequency resolution. long window: good frequency resolution, poor time resolution. 1 EEG raw data Time [sec] SPECTROGRAM, width = 124 frequency SPECTROGRAM, width = 64 frequency time niko.busch@charite.de 9 / 23

Uncertainty principle Werner Heisenberg (191 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).

16 Uncertainty principle Werner Heisenberg ( ): 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 be optimised. Good frequency resolution at low frequencies. Good time resolution at high frequencies. niko.busch@charite.de 1 / 23

17 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. 11 / 23

18 What is a wavelet? Motherwavelet 2 e jω t 2 e t2 /2.15 Ψ(t) Cosine.5.5 Gaussian.5.5 Wavelet 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

19 What is a wavelet? Motherwavelet Hz 2 2 Hz Spectral Density Hz Time (s) Frequency ( Hz ) 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

Oz... but be careful: any signal can be represented as oscillations w. time-frequency analysis but it does not imply that the signal is oscillatory! niko.busch@charite.

20 Oz... but be careful: any signal can be represented as oscillations w. time-frequency analysis but it does not imply that the signal is oscillatory! 13 / 23 Oz Wavelet transform of ERPs ERP Bandpass filtered ERP µv 6. µv 2. s s Wavelet transformed ERP Gamma Band: ca. 3 8 Hz µv [uv].6 7. s Frequency [Hz] Time [s].

21 Important parameters of a wavelet Amplitude,8,6,4,2, -,2 -,4 -,6 A Wavelet - time domain -,15 -,1 -,5,,5,1,15 Time[s] Amplitude,8,6,4,2 B Wavelet - frequency domain, Frequency (Hz) Length how many cycles does a wavelet have? e.g. 4 Hz wavelet (25 ms/cycle), 12 cycles 25 ms length σ t standard deviation in time domain: σ t = m 2π f σ f standard deviation in frequency domain: σ f = 1 2π σ t time resolution increases with frequency, whereas frequency resolution decreases with frequency. niko.busch@charite.de 14 / 23

22 Evoked and induced oscillations I evoked/ phase-locked induced/ non-phase-locked Average ms 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

23 Evoked and induced oscillations II Wavelet analysis of single trials reveals non phase locked activity. Evoked/ phase-locked Induced/ non/phase-locked Wavelet-transformed trials Average ms 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

24 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 = : random phase distribution. niko.busch@charite.de 17 / 23

25 The EEG state space Frequency x phase locking x amplitude changes 2 Evoked and induced activity are extremes on a continuum. ERPs cover only small part of the EEG space. 2 Makeig, Debener, Onton, Delorme (24). TICS niko.busch@charite.de 18 / 23

26 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 1 Hz? Simultaneity and the alpha rhythm Figure from VanRullen & Koch (23): Is perception discrete of continuous? TICS 3 Varela et al. (1981): Perceptual framing and cortical alpha rhythm. Neuropsychologia niko.busch@charite.de 19 / 23

27 Examples 2: pre-stimulus EEG power Spatial attention to left or right. Stronger alpha power over ipsilateral hemisphere. 4 Attention: ipsi- vs. contra-lateral 5 4 frequency [Hz] time [s] log1[p] 8 15 Hz -.4 s.5 db Busch & VanRullen (21): Spontaneous EEG oscillations reveal periodic sampling of visual attention. PNAS. niko.busch@charite.de 2 / 23

5 Jensen et al. (22): Oscillations in the alpha band (9 12 Hz) increase with memory load during retention in a short-term memory task. Cereb Cortex. niko.busch@charite.

28 5 Jensen et al. (22): Oscillations in the alpha band (9 12 Hz) increase with memory load during retention in a short-term memory task. Cereb Cortex. niko.busch@charite.de 21 / 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

29 Recommended reading WWW: Books: Papers: EEGLAB s time-frequency functions explained: FFT explained: Wavelet tutorial polikar/wavelets/wttutorial.html Barbara Burke Hubbard: The World According to Wavelets. Steven Smith: The Scientist & Engineer s Guide to Digital Signal Processing ( Herrmann, Grigutsch & Busch: EEG oscillations and wavelet analysis. In: Event-related Potentials: A Methods Handbook. Tallon-Baudry & Bertrand (1999) Oscillatory gamma activity in humans and its role in object representation. TICS. Samar, Bopardikar, Rao & Swartz (1999) Wavelet analysis of neuroelectric waveforms: a conceptual tutorial. Brain Lang. niko.busch@charite.de 22 / 23

30 Thank you for your interest! Please ask questions!!! 23 / 23

Fourier and Wavelets

Fourier and Wavelets Why do we need a Transform? Fourier Transform and the short term Fourier (STFT) Heisenberg Uncertainty Principle The continues Wavelet Transform Discrete Wavelet Transform Wavelets