Analyses of Rhythmic Data - Mathematics & Statistics...

Analyses of Rhythmic Data

Society for NeuroscienceShort Course #2: Rhythms of the Neocortex

Mark KramerDepartment of Mathematics and Statistics

Boston University

Outline• Rhythms in health, sickness, in vivo, and in vitro.

• Motivates the study of rhythms, particularly . . .

• Quantification of rhythms in data

• Quantification of rhythmic interactions

• An introduction . . .

• Hands on MATLAB examples

• Discussion: future issues & questions

Get the data• Download example data:

http://makramer.info/sfn



Motivating questions

• How can we quantify rhythms in data?

• Power spectrum

• How can we quantify coupling between rhythms?

• Coherence

Load data & visualize

• Rhythmic

• It’s complicated

• How can we simplify?

>> load data.mat>> plot(t1,v1)

dt = 1ms

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−2

0

2

4

Time [s]

Volta

ge [m

V]

Download example data: http://makramer.info/sfn



0 5 10 15 20 25 30 35 40 45 50−50

−40

−30

−20

−10

0

Frequency [Hz]

Pow

er [d

B]Power spectrum

• Axes: Power [dB] vs Frequency [Hz]

• A simpler representation in frequency domain.Sum of four sinusoids at {7, 10, 23, 35} Hz

• How do we compute it?

Formula

• Idea: Write v[t] as sum of sines and cosines oscillating at different frequencies.

• Nice properties (we’ll skip)[MATLAB for Neuroscientists, Numerical Recipes in C]

• In MATLAB . . .

1

V [f ] =! !

"!v[t]e"2!iftdt (1)

P [f ] =|V [f ]|2

n(2)

df =1T

(3)

fNQ =f0

2(4)

C12[f ] =E{V1[f ]V #2 [f ]}

E{|V1[f ]|2}E{|V2[f ]|2} (5)

Fourier transform

Power (per unit time)

1

V [f ] =

!!

"!

v[t]e2!iftdt (1)

P [f ] =|V [f ]|2

n(2)

n = Length of v[t]

Data Sinusoids

MATLAB code

Incomplete: Must label the x-axis?

0 100 200 300 400 500 600 700 800 900 1000−50

−40

−30

−20

−10

0

Indices []

Pow

er [d

B]

>> pow = (abs(fft(v1)).^2)/length(v1);>> pow = 10*log10(pow/max(pow));>> plot(pow)


Clue?

1000 data pts

Matcheslength of v1



0 100 200 300 400 500 600 700 800 900 1000−100

−80

−60

−40

−20

0

Indices []

Pow

er [d

B]

Power spectrum x-axis• Indices & frequencies related in a funny way . . .

Examine vector pow:

df2

2df3

fNQ - df500

fNQ

501-(fNQ - df)

502-3df998

-2df9991

f > 0 f < 0

• Because data is real, f < 0 is redundant.

... ...

Frequency resolution

Nyquistfrequency

Index

Freq1000

01000-df

What remains?

Find df and fNQ

• What is df?

1

V [f ] =

!!

"!

v[t]e2!iftdt (1)

P [f ] =|V [f ]|2

n(2)

df =1

T(3)where T = Total time of recording.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−2

0

2

4

Time [s]

Volta

ge [m

V]

Ex:T = 1 sdf = 1 Hz

Q: How do we improve frequency resolution?A: Increase T or record for longer time.

Power spectrum x-axis

Examples• Demand 0.2 Hz frequency resolution.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5!4

!2

0

2

4

6

Time [s]

Volta

ge []

But, data may change during longer recordings . . .

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5!4

!2

0

2

4

Time [s]

Volta

ge []

Balance resolution requirements with consistency in data.

df = 1/5s = 0.2 Hz

Different spectra in 1st and 2nd half of data . . .

True signal: fs

• What is fNQ?

1

V [f ] =

!!

"!

v[t]e2!iftdt (1)

P [f ] =|V [f ]|2

n(2)

df =1

T(3)

fNQ =f0

2(4)The Nyquist frequency

where f0 = sampling frequency.

f0 >> 2 fs

The highest frequency we can observe.

f0 = 2 fs

f0 < 2 fs

Enough to reconstruct signal,

but just barely.

High frequency (in data) mapped to low frequency

(aliased).

Accurate reconstruction


Too expensive!

Sample:

2 samples/cycleMax freq we can observe at this sample rate!

All hope lost! Indistinguishable from true low frequency signals.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−2

0

2

4

Time [s]

Volta

ge [m

V]

Ex:f0 = 1000 HzfNQ = 500 Hz

Q: How do we increase the Nyquist frequency?A: Increase the sampling rate f0. [Hardware]

Moral: Sample fast enough to capture the highest frequency “true” signal.


dtSampling interval: Sampling frequency:= 1 msf0 = 1/dt

>> pow = (abs(fft(v1)).^2)/length(v1);

>> pow = 10*log10(pow/max(pow));

>> pow = pow(1:length(v1)/2+1);

0 5 10 15 20 25 30 35 40 45 50−50

−40

−30

−20

−10

0

Frequency [Hz]

Pow

er [d

B]

MATLAB code http://makramer.info/sfn

>> df = 1/max(t1); fNQ = 1/dt/2;

First half of data

>> faxis = (0:df:fNQ);Define df & fNQ

Frequency axis

>> plot(faxis, pow); xlim([0 50]);



Summary

• For finer frequency resolution: record more data.

• To observe higher frequencies: increase sampling rate.

>> pow = (abs(fft(v1)).^2)/length(v1);

• Many subtleties . . .

1

V [f ] =

!!

"!

v[t]e2!iftdt (1)

P [f ] =|V [f ]|2

n(2)

df =1

T(3)Frequency

resolution

1

V [f ] =

!!

"!

v[t]e2!iftdt (1)

P [f ] =|V [f ]|2

n(2)

df =1

T(3)

fNQ =f0

2(4)Nyquist

frequency

• Built-in routines: >> periodogram(...)Requires Signal Processing Toolbox

Tapers• Doing nothing, we make an implicit taper choice . . .

What we’re observing:

. . . Data goes on forever . . .

We we get:

0 0

But we observe it in a small window

0

1

Signal Rectangle taper

*

0 5 10 15 20 25 30 35 40

−30

−20

−10

0

Frequency [Hz]

Pow

er [d

B]

Spectrum we want

Tapers• The rectangle taper blurs the power spectrum.

0 5 10 15 20 25 30 35 40−40

−30

−20

−10

0

Frequency [Hz]

Pow

er [d

B]

What we get

Puresinusoid

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

Time [s]

near 20 Hz 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

Time [s]

Hann taper• Idea: smooth the sharp edges of rectangle taper.

>> st = s .* hann(length(s))’ ;

Taper reduces the “sidelobes”.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

−0.5

0

0.5

1

Time [s]

Hann Taper0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

−0.5

0

0.5

1

Time [s]

Data

Taper

0 5 10 15 20 25 30 35 40−40

−30

−20

−10

0

Frequency [Hz]

Pow

er [d

B]

Rect

0 5 10 15 20 25 30 35 40−40

−30

−20

−10

0

Frequency [Hz]

Pow

er [d

B]

RectHann

• Compute power spectrum of tapered data.Requires Signal Processing Toolbox

0 5 10 15 20 25 30 35 40 45 50−50−40−30−20−10

0

Frequency [Hz]

Pow

er [d

B]

Rect

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

0

2

Time [s]

v1:

Ex: Hann taper

• Good: Deeper baseline• Bad: Broader peaks & lose data at edges.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

Time [s]

Lost Lost

0 5 10 15 20 25 30 35 40 45 50−50−40−30−20−10

0

Frequency [Hz]

Pow

er [d

B]

Rect

Hann

Multi-taper Method• Idea: Apply lots of different tapers

• Chronux (www.chronux.org)>> mtmspectrumc(v1,params);

0 5 10 15 20 25 30 35 40 45 50−60

−40

−20

0

Frequency [Hz]

Pow

er [d

B]

Reduce sidelobes

Keep data edges

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time [s]

http://www.chronux.org


Spectrogram

• Visual inspection• Data characteristics change in time.

• What if signal characteristics change in time?

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

Time [s]

>> load data.mat>> plot(t2,v2) http://makramer.info/sfn



Spectrogram• Compute the spectrum (Hann taper) of all data

Q: Is this a good representation of the data?

0 5 10 15 20 25 30 35 40 45 50−60

−40

−20

0

Frequency [Hz]

Pow

er [d

B]

A: No, changing characteristics of the signal lost.

Time [s]

Freq

[Hz]

1 2 3 4 5 6 7 8 90

5

10

15

20

−40

−30

−20

−10

0

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

Time [s]

Spectrogram• Idea: Split up the data into windows &

Compute spectrum in each.

df =

0 10 20 30 40 50−30

−20

−10

0

Frequency [Hz]

Pow

er [d

B]

0 10 20 30 40 50−30

−20

−10

0

Frequency [Hz]Po

wer

[dB]

Different spectra at beginning and end of signal.

1Hz

Repeat for many overlapping windows . . .

>> [S,F,T]=spectrogram(v2,1s,0.5s,1s,1kHz)

A better representation of the data?

Time [s]

Freq

[Hz]

1 2 3 4 5 6 7 8 90

5

10

15

20

−40

−30

−20

−10

0

MATLAB code http://makramer.info/sfnRequires Signal Processing Toolbox

Window

Overlap

Padding

f0

>> S = abs(S);

>> imagesc(T,F,10*log10(S/max(S(:))));

Plot power [color] vs frequency and time

Can compute multi-taper spectrogram! (Chronux)



Conclusions & Refs

• We focused on power spectrum (not wavelets).

• Defined df and fNQ.

• Explored tapers and spectrograms.

MATLAB for Neuroscientists, Numerical Recipes in C

Chronux.org and Neuroinformatics Summer Course

EEGLab

References

Coherence• Idea: examine phase relationship between signals.

• Requires two signals.

v3a v3b• Requires multiple trials for each signal.

10.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

0

2

Time [s]0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

0

2

4

Time [s]

20.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−4

−2

0

2

Time [s]0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−4

−2

0

2

Time [s]

30.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−4

−2

0

2

4

Time [s]0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

0

2

Time [s]

>> load data.mat

Two variables: v3a & v3b

100Trials

1234

100

1000 indices1 ms 1000 ms

v3a0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−4

−2

0

2

Time [s]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−2

0

2

Time [s]

.

.

.

MATLAB code http://makramer.info/sfn



Power spectrum• Compute the power spectrum for each trial,

Then average over all trials.

Power at 10 Hz and 18 Hz.

Q: Are these rhythms coherent between v3a and v3b?

0 10 20 30 40 50−35

−30

−25

−20

−15

−10

−5

0

Frequency [Hz]

Pow

er [d

B]

v3a

0 10 20 30 40 50−35

−30

−25

−20

−15

−10

−5

0

Frequency [Hz]

Pow

er [d

B]

v3b

Coherence• For each trial, compute the phase at each frequency.

Re

Im

r

θ

= ( Re{V[f]}, Im{V[f]} )complex

Examine this complex plane for our data . . .

1

V [f ] =! !

"!v[t]e"2!iftdt (1)

P [f ] =|V [f ]|2

n(2)

df =1T

(3)

fNQ =f0

2(4)

C12[f ] =E{V1[f ]V #2 [f ]}

E{|V1[f ]|2}E{|V2[f ]|2} (5)

Fourier transform

Re{V[f]}

Im{V[f]} V[f]

θ = phase of V at frequency f.

2000

4000

6000

8000

30

210

60

240

90

270

120

300

150

330

180 0

v3a-v3b

Complex Plane at f =10 Hz

50

100

150

30

210

60

240

90

270

120

300

150

330

180 0

v3a


Plot complex plane at 10 Hz• For each trial, compute FT(data) & plot . . .

Coherent at 10 Hz?

Phase concentration = Nonzero mean vector = coherent

Draw the vector to each complex difference.Compute the mean vector

Examine their difference (trial by trial).Summarize:

v3a[f]Trial 1

v3a[f]Trial 2v3a[f]

Trial 3

v3a[f]Trial 4

v3a[f]-v3b[f]Trial 1

v3a[f]-v3b[f]Trial 2

Get trail #1 data.Compute FTEvaluate at f=10 HzPlot it . . .

50

100

150

30

210

60

240

90

270

120

300

150

330

180 0

v3b


100000

200000

300000

30

210

60

240

90

270

120

300

150

330

180 0

v3a-v3b


Coherent? Plot the complex difference

Phase dispersion = Zero mean vector = No coherence

500

1000

30

210

60

240

90

270

120

300

150

330

180 0

v3a


500

1000

30

210

60

240

90

270

120

300

150

330

180 0

v3b


Draw the vector to each complex difference.Compute the mean vector

Summarize:

Plot complex plane at 18 Hz• For each trial, compute FT(data) & plot . . .

Coherence• Idea: Examine the “angular concentration” of

vector differences in the complex plane.

Strong power does not imply coherence.

0 10 20 30 40 50−35

−30

−25

−20

−15

−10

−5

0

Frequency [Hz]

Pow

er [d

B]

v3a

0 10 20 30 40 50−35

−30

−25

−20

−15

−10

−5

0

Frequency [Hz]

Pow

er [d

B]

v3b

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Cohe

renc

e []

Coherentat 10 Hz

Not coherentat 18 Hz

Can compute multi-taper coherence! (Chronux)

18 Hz dominates 18 Hz dominates

Conclusions & Ref

• More about coherence:[Nunez et al., Electroenceph Clin Neurophys, 1997][Bruns, J Neurosci Methods, 2004]

• There are many coupling measuresCross correlation, phase consistency, Granger causality, cross-frequency coupling, . . .

[Pereda et al., Progress in Neurobiology (2005)]

• SfN AbstractsOnline:137 results for “coherence”.

Conclusions• We examined techniques to quantify

rhythms and their interactions in data.

• Many different techniques exist.

• Tutorial slides & MATLAB code availablehttp://makramer.info/sfn

• Contact me at SfN to talk more:Mark --- [email protected]

Thanks!



Extra Material

>> periodogram(v1,[],length(v1),1000Hz);

Signal Processing Toolbox

0 5 10 15 20 25 30 35 40 45 50−50

−40

−30

−20

−10

0

Frequency (Hz)

Pow

er/fr

eque

ncy

(dB/

Hz)

Periodogram Power Spectral Density Estimate

Taper Zero padding f0

•Use built-in MATLAB routine:

Correct axis!

Multi-taper spectrogram

Time [s]

Freq

[Hz]

1 2 3 4 5 6 7 8 90

5

10

15

20

−70

−60

−50

−40

−30

−20

−10

0

>> [S,T,F]=mtspectgramc(v2,[1s,0.5s],params)

>> imagesc(T,F,10*log10(S/max(S(:)))');

• Chronux (www.chronux.org)



Multi-taper coherency

>> [C,phi,S12,S1,S2,f]=coherencyc(v3a',v3b',params)

>> plot(f,C);

• Chronux (www.chronux.org)

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency [Hz]

Cohe

renc

e []



Coherence formalism

1

V [f ] =! !

"!v[t]e2!iftdt (1)

P [f ] =|V [f ]|2

n(2)

df =1T

(3)

fNQ =f0

2(4)

C12[f ] =E{V1[f ]V #2 [f ]}

E{|V1[f ]|2}E{|V2[f ]|2} (5)E: Sum over trials

cross spectrum

power spectrum

Note: could taper here!

sxy = zeros(ntrials, ttrials);sxx = zeros(ntrials, ttrials);syy = zeros(ntrials, ttrials);

for k=1:ntrials sxy(k,:) = fft(v3a(k,:)).*conj(fft(v3b(k,:))); sxx(k,:) = fft(v3a(k,:)).*conj(fft(v3a(k,:))); syy(k,:) = fft(v3b(k,:)).*conj(fft(v3b(k,:)));end coh = (abs(sum(sxy,1)).^2) ./ (sum(sxx,1) .* sum(syy,1));plot(faxis, coh(1:length(coh)/2+1))xlim([0 50]); ylim([0 1])xlabel('Frequency [Hz]')ylabel('Coherence')


Define cross and power spectra

For each trail,compute spectra.

Sum over trials.



Analyses of Rhythmic Data - Mathematics & Statistics...

Documents

Transcript of Analyses of Rhythmic Data - Mathematics & Statistics...