Time-Frequency analysis of biophysical time series. Arnaud Delorme CERCO, CNRS, France & SCCN, UCSD, La Jolla, USA

Transcription

1 Time-Frequency analysis of biophysical time series Arnaud Delorme CERCO, CNRS, France & SCCN, UCSD, La Jolla, USA

2 Frequency analysis synchronicity of cell excitation determines amplitude and rhythm of the EEG signal Hz Gamma Hz Beta 9-11 Hz Alpha 4-7 Hz Theta Hz Delta 1 second

3 Spectral phase and amplitude Imaginary Real Imag. Real F k (f,t)

4 Spectral phase and amplitude Imaginary Real Imag. Real F k (f,t)

5 function [a,b] = dft (y) % DFT - The Discrete Fourier Transform % [a, b] = DFT (y) % a, b are the cosine and sine components n = length (y); t = 2*pi*(0:n-1)/n; f = 2.0 / n; for j = 0:n2 cs = cos (j * t); ss = sin (j * t); a(j+1) = f * (cs * y); b(j+1) = f * (ss * y); end % boundaries n2 = floor (n / 2); a(1) = 0.5 * a(1); a(n2+1) = 0.5 * a(n2+1); b(1) = 0.0; b(n2+1) = 0.0; Loop on frequency Cosine component Sine component Multiply with signal From

6 Average of squared absolute values

7 Spectral power 0 Hz 10 Hz 20 Hz 30 Hz 40 Hz 50 Hz Average of squared amplitude Power (db) Frequency (Hz)

8 Overlap 50% Average

9 padding

10 Plot data spectrum using EEGLAB winsize, 256 nfft, 256 overlap, 128 (change FFT window length) (change FFT padding) (change window overlap)

11 Overlap 50% Padding Average Exercice - Read dataset eeglab_data_epoch_ica.set - Plot spectral decomposition 1- default (15% of data) 2- with 100% of data 3- with 100% and 50% overlap ( overlap ) 4- reduce window size ( winsize ) 5- reduce fft length ( nfft )

12 Spectrogram or ERSP 5 Hz 0 ms 10 ms 20 ms 30 ms 40 ms 50 ms 60 ms 10 Hz 20 Hz 30 Hz

13 Spectrogram or ERSP 5 Hz 0 ms 10 ms 20 ms 30 ms 40 ms 50 ms 60 ms 10 Hz 20 Hz Average of squared values 30 Hz 5 Hz 10 Hz 20 Hz 30 Hz 0 ms 10 ms 20 ms 30 ms 40 ms 50 ms 60 ms

14 Power spectrum and event-related spectral perturbation ERSP( f, t) = 1 n n k = 1 F k ( f, t) 2 Complex number Scaled to db 10Log 10 (ERSP) From Delorme and Makeig, J Neuro Methods, 2004

15 Absolute versus relative power Absolute = ERS Relative = ERSP (db or %)

16 Difference between FFT and wavelets FFT Wavelet Frequency

17 Wavelets factor Wavelet (0)= FFT Wavelet (1) 1Hz 2Hz 4Hz 6Hz 8Hz 10Hz

18 FFT In between Pure wavelet

19 Modified wavelets Wavelet (0.8) Wavelet (0.5) Wavelet (0.2)

20 Inter trial coherence same time, different trials Trial 1 Trial 2 Trial 3 amplitude 0.5 phase 0 amplitude 1 phase 90 amplitude 0.25 phase 180 POWER = mean(amplitudes 2 ) 0.44 or 8.3 db COHERENCE = mean(phase vector) Norm 0.33 A. Delorme, 2001

21 Phase ITC ITPC( f, t) = 1 n F k ( f, t) n k= F ( f, t) 1 k Normalized (no amplitude information) From Delorme and Makeig, J Neuro Methods, 2004

22 Power and inter trial coherence 5 Time-frequency power Condition 1 Condition 2 db ITC: trials synchronization

23 Channel time-frequency

24 Component time-frequency

25 Cross-coherence amplitude and phase 2 components, comparison on the same trials Trial 1 Coherence amplitude 1 Phase coherence 0 Trial 2 Coherence amplitude 1 Phase coherence 90 Trial 3 Coherence amplitude 1 Phase coherence 180 COHERENCE = mean(phase vector) Norm 0.33 Phase 90 degree

26 = = n k b k a k b k a k b a t f F t f F t f F t f F n t f ERPCOH 1 *, ), ( ), ( ), ( ), ( 1 ), ( Only phase information component a Only phase information component b Phase coherence (default)

27 Other spectral measures ( )( ) ( ) ( ) = = = = n k b b k n k a a k n k b b k a a k b a t f F t f F n t f F t f F n n t f F t f F t f F t f F t f corr , ), ( ), ( 1 ), ( ), ( 1 ), ( ), ( ), ( ), ( ), ( Covariance Standard deviation for a Standard deviation for b Amplitude correlation

28 Cross-coherence amplitude and phase 5 6 Condition 1 Condition 2 Phase (degree) Amplitude (0-1)

29 Component phase coherence

30 The Uncertainty Principle A signal cannot be localized arbitrarily well both in time/position and in frequency/momentum. There exists a lower bound to the Heisenberg s product: Δt Δf 1/(4π) Δf = 1Hz, Δt = 80 msec or Δf = 2Hz, Δt = 40 msec

31 Niquist frequency: Aliasing Signal (100 Hz) Sampling (120 Hz) 1 cycle e.g. 100 Hz sampled at 120 Hz From Karu, Signals and Systems Made Ridiculously Simple, Zizi Press

32 compare between conditions Dataset Face3.set - Extract epochs on objects - Extract epochs on faces - run time-frequency decomposition to compare between conditions Component 10 for objects (left) and faces (right) Number of data points >> newtimef({ ALLEEG(2).icaact(10,:) ALLEEG(3).icaact(10,:) }, EEG.pnts, [EEG.xmin EEG.xmax]*1000, ALLEEG(2).srate, 0, 'padratio', 1); Number of data points Sampling rate Cycles (0=FFT) padding

33 Compute coherence difference between components 4 and 9 >> newcrossf({ ALLEEG(2).icaact(10,:) ALLEEG(3).icaact(10,:) }, { ALLEEG(2).icaact(4,:) ALLEEG(3).icaact(4,:) }, EEG.pnts, [EEG.xmin EEG.xmax]*1000, ALLEEG(2).srate, 0, 'padratio', 4);

34 Statistics Parametric statistics: use mean and standard deviation (t-test, ANOVA, ) Non parametric statistics: do not depend on the shape of the distribution. Surrogate data: shuffle data and recompute measure of interest. Use the tail of the distribution to asses significance. Semi-parametric statistics.

35 Parametric statistics Assume gaussian distribution of data T-test: Compare paired/unpaired Samples for continuous data. In EEGLAB, used for grand-average ERPs. Paired Mean _ difference t = N 1 Standard_deviation Unpaired t = N Mean A B ( SD ) ( SD ) A Mean 2 2 B ANOVA: compare several groups (can test interaction between two factors for the repeated measure ANOVA) Variance intergroup N 1 Group F = Variance WithinGroup N NGroup

36 Non-parametric statistics Do not assume a distribution for the data χ2 is used to compare 2 or more unpaired samples χ = ( Observed expected ) / expected 2 2 i, j i, j i, j i, j Observed A Signal A binning expected time Observed B Signal B binning time

37 Randomization approach Original data a a a a a a a analyze difference X org b b b b b b analyze Courtesy of R. Oostenveld

38 Randomization approach Bootstrap or permutation 1 a a a b a b b analyze difference X 1 a b a b a b analyze

39 Randomization approach Bootstrap or permutation 2 b b a a a b a analyze difference X 2 a b a b a b analyze

40 Distribution can take any shape 5% 95% 5% 95% 5% 95%

41 Bootstrap for ERPs and time-frequency Channel/comp. 1 Significantly non-0 N trials Average Baseline period Bootstrap 1 average 0 Bootstrap 2 average 2.5% 97.5% Bootstrap 3 average Bootstrap k..

42 Permutation for comparing between conditions Help mask cross-coherence difference

43 Correcting for multiple comparisons Bonferoni correction: divide by the number of comparisons (Bonferroni CE. Sulle medie multiple di potenze. Bollettino dell'unione Matematica Italiana, 5 third series, 1950; ) Holms correction: sort all p values. Test the first one against α/n, the second one against α/(n-1) Clusters correction in Fieldtrip software MAX procedure