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analysis:nsb2016:week12

## Time-frequency analysis II: cross-frequency coupling

Goals:

• Use the spectrogram as a basis to reveal correlations in power across frequencies
• Understand how to compute and interpret autocorrelation and crosscorrelation functions for continuous data
• Apply the Hilbert transform to obtain signal phase over time for a given frequency
• Relate phase of one frequency to power at another by using histc()
• Quantify the strength and statistical significance of phase-amplitude coupling

Resources:

### Introduction

In the previous module we saw that a spectrogram can determine the frequency content of a signal over time. Often, spectrograms resulting from brain signals reveal rich patterns of interacting frequency bands. A simple example of this is the previously mentioned panel C of the first figure in the previous module: this shows that in the rat ventral striatum, gamma-50 and gamma-80 power tend to be slightly anticorrelated. More generally, we might ask, what structure is there in the spectrogram? What relationships exist between different frequency bands? Can we predict power based on the history of the signal? Beyond the power (amplitude) components of the spectrogram, an additional dimension is phase, which may enter in systematic relationships with power. For instance, in the hippocampus, gamma power tends to occur at certain phases of the slower theta (~8Hz) rhythm (Colgin et al. 2009). Such “second-order” relationships (built on the “first-order” spectrogram) are the focus of this module.

### Reminder: correlation

A simple yet powerful tool for characterizing dependencies between variables – such as power in two different frequency bands – is Pearson's product-moment correlation coefficient (“correlation coefficient” in short). For a finite sample, the correlation coefficient r can be computed as follows:

where x and y are the signals to be correlated, with means of x-bar and y-bar respectively. In other words, r is the covariance between X and Y, divided by the product of their standard deviations. It is 0 for independent (uncorrelated) signals, -1 for perfectly anticorrelated signals, and 1 for perfectly correlated signals.

Note that the correlation coefficient only captures linear relationships between variables, as can be seen from these scatterplots:

(source: wikipedia)

The MATLAB function corrcoef() computes the sample correlation coefficient, and in addition provides the probability of obtaining the reported r value under the null hypothesis that the signals are independent (“p-value”):

x = randn(100,4);  % uncorrelated data - 4 signals of 100 points each
x(:,4) = sum(x,2);   % 4th signal is now correlated with first 3
[r,p] = corrcoef(x)  % compute sample correlation and p-values; notice signal 4 is correlated with the others
imagesc(r) % plot the correlation matrix -- note symmetry, diagonal, and r values of ~0.5 for signal 4

corrcoef() computes the correlation coefficients and p-values for every pair of signals in a matrix – in this case, for 4 signals. We can see from its output that as expected, signal 4 is correlated with every other signal. Note also that, of course, the correlation of any signal with itself (the values along the diagonal) is 1.

Armed with this tool, we can compute the correlation matrix of a ventral striatal spectrogram:

% remember to cd to data folder first
cfg = [];
cfg.fc = {'R016-2012-10-03-CSC04a.Ncs'};

csc = restrict(csc,csc.cfg.ExpKeys.TimeOnTrack(2),csc.cfg.ExpKeys.TimeOffTrack(2)); % "risk" session only
Fs = csc.cfg.hdr{1}.SamplingFrequency;

[S,F,T,P] = spectrogram(csc.data,hanning(512),384,1:0.25:200,Fs); % spectrogram -- will take a while to compute!

[r,p] = corrcoef(10*log10(P')); % correlation matrix (across frequencies) of spectrogram

% plot
imagesc(F,F,r);
caxis([-0.1 0.5]); axis xy; colorbar; grid on;
set(gca,'XLim',[0 150],'YLim',[0 150],'FontSize',14,'XTick',0:10:150,'YTick',0:10:150);

(Note, this takes about a minute to run on my laptop.)

You should get:

As we would expect, nearby frequencies tend to be correlated. The blob around 80-100Hz indicates that these frequencies tend to co-occur. In contrast, there is a dark patch centered around (60,95) indicating that power in those frequencies is slightly anti-correlated. Note also the positive correlation around (20,60) suggesting that beta and low-gamma power co-occur. This pattern in the cross-frequency correlation matrix is characteristic for ventral striatal LFPs ( van der Meer and Redish, 2009).

☛ As a second-order method, how this plot looks is sensitive to the parameters of the (first-order) spectrogram. What in this plot changes if you vary the width of the window used to compute the spectrogram?

### Autocorrelation for continuous signals

The above analysis focused on the co-occurrence of power across different frequencies: it asks, “when power at some frequency is high, what does that imply about the power at other frequencies at that same time”?

However, often we are also interested in relationships that are not limited to co-occurrence – for instance, it is possible that power at a certain frequency predicts that some time later a power increase in some other band will occur. To characterize such relationships, which involve a time lag, we first need the concept of autocorrelation.

#### Autocorrelation of some synthetic signals

The correlation of a signal with itself is 1 by definition. However, what about the correlation of a signal with the same signal, one sample later? For white noise, where each sample is generated independently, this should be zero:

wnoise = randn(1000,1);
[r,p] = corrcoef(wnoise(1:end-1),wnoise(2:end)) % same signal, offset by one sample

Note that when passing two arguments to corrcoef() the output is a 2×2 matrix, where the diagonals are 1 and the off-diagonal elements are of interest.

We can generalize this example to computing correlations not just for a lag of one sample, as above, but for any lag. To do this, MATLAB provides the xcorr() function:

[acf,lags] = xcorr(wnoise,100,'coeff'); % compute correlation coefficients for lags up to 100
plot(lags,acf,'LineWidth',2); grid on;
set(gca,'FontSize',18); xlabel('time lag (samples)'); ylabel('correlation ({\itr})');

You should get

As expected, the autocorrelation of white noise is essentially zero for any lag (except 0). Note the use of LaTeX markup to italicize the r in the ylabel.

What if we try a signal that is not just white noise, but has a clear periodicity?

Fs = 500; dt = 1./Fs;
tvec = 0:dt:2-dt;

wnoise = randn(size(tvec));
sgnl = sin(2*pi*10*tvec) + wnoise; % 10 Hz sine wave plus Gaussian white noise

subplot(211);
plot(tvec,sgnl); grid on;
set(gca,'FontSize',18); title('signal');

subplot(212);
[acf,lags] = xcorr(sgnl,100,'coeff');
lags = lags.*dt; % convert samples to time
plot(lags,acf); grid on;
set(gca,'FontSize',18); xlabel('time lag (s)'); ylabel('correlation ({\itr})');
title('autocorrelation');

Now we get

The autocorrelation (acorr) has peaks at 0.1s, corresponding to the 10Hz component in the original signal, an idea formalized in the Wiener-Khinchin theorem we encountered in Module 11. An intuitive interpretation of this plot is: given a peak in the signal at time t = 0, what on average are the expected values at other times? For this signal, we can see that if there is a peak at 0, then we expect peaks at 0.1, 0.2, etc. and troughs at 0.5, 0.15… .

☛ What would you expect the autocorrelation to look like if the above code were modified to use a rectified sine wave (abs(sin(..)))? Verify your prediction.

You should notice that the autocorrelation now has twice the number of peaks compared to the original sine wave. In addition, this periodicity is perhaps not obvious in the original signal, but it is quite clear in the acorr. Autocorrelation functions average out noise that may be obscuring an underlying periodic signal; similarly, autocorrelations of grid cell rate maps often look much more impressive than the original (noisy) rate map!

A final difference is that there is now a baseline offset in the acorr. This occurs because after rectification, the original signal no longer has mean zero, and unlike corrcoef(), xcorr() does not actually subtract the mean from the signal(s).

☛ Subtract the mean from the rectified signal and plot the acorr again. Verify that the value at lag 1 matches the output of corrcoef() for lag 1.

#### Application to real data

Is there any periodicity in ventral striatal gamma power? That is, given high power at t = 0, what values do we expect at other times? If gamma power varies randomly, then we expect the acorr to look like that for white noise. If, on the other hand, high gamma power at time 0 tells us something about what powers to expect at other times, we will see something different.

First, let's get a finer-resolution version of the spectrogram:

% spectrogram with better time resolution (slow!)
Fs = 500;
csc.data = decimate(csc.data,4);
csc.tvec = downsample(csc.tvec,4);
[S,F,T,P] = spectrogram(csc.data,hanning(125),115,1:200,Fs);

What we need is a time series of gamma power, so that we can compute its autocorrelation. We will get this from the spectrogram, but it is worth noting that this is not the only way to obtain such a signal. Recall, for instance, that in Module 5 we used a filter to compute instantaneous power in the ripple band. An advantage of using the spectrogram is that it provides estimates of power over time across all frequencies, so it is easy to explore. However, more precise estimates can be obtained with the filtering method (as well as with better spectrograms such as the event-averaged one we computed with FieldTrip in Module 7).

For now we make do with what we have:

F_idx = find(F > 70 & F < 100); % high-gamma power
F_idx2 = find(F > 50 & F < 65); % low-gamma power

pwr = mean(P(F_idx,:)); % average across frequencies
pwr2 = mean(P(F_idx2,:));

[ac,lags] = xcorr(pwr2-mean(pwr2),50,'coeff'); % remember to subtract the mean!
lags = lags.*mean(diff(T));
figure;
plot(lags,ac)
set(gca,'FontSize',18); xlabel('time lag (s)'); ylabel('correlation ({\itr})');
title('low-gamma autocorrelation');

You should get:

Some differences with our reference acorrs from the previous section (white noise, and the 10Hz sine wave) are apparent. First, the central peak around t = 0 now has a certain nonzero width: if we have a peak in low-gamma power at time t = 0, power tends to stay elevated for a certain amount of time. Judging from the inflection point at around 0.2s, it appears that if low gamma power is high, it takes about 0.2s to return to a kind of baseline.

☛ What do you expect would happen to the width of this central peak if you were to change the width of the time window used to compute the spectrogram, say to twice the original width?

Second, a small dip in the baseline is apparent around 0.6s. This suggests that if we have high low-gamma power at t = 0, we are less likely than average to see high gamma power 0.6s later. This is an example of a “second-order” effect: we build on our “first-order” analysis (the spectrogram) to reveal further structure in how gamma power is organized over time.

### Cross-correlation for continuous signals

The autocorrelation computes the correlation of a signal with itself over some different time lags. The cross-correlation does the same but between two different signals. It asks: given a peak in signal 1 at time zero, what do we know about signal 2?

Let's apply this idea to ventral striatal gamma power by asking how high-gamma power varies given a peak in low-gamma power at t = 0:

F_idx = find(F > 70 & F < 100); % high-gamma power
F_idx2 = find(F > 50 & F < 65); % low-gamma power

pwr = mean(P(F_idx,:)); % average across frequencies
pwr2 = mean(P(F_idx2,:));

[ac,lags] = xcorr(pwr-mean(pwr),pwr2-mean(pwr2),50,'coeff'); % note, two different input signals now
lags = lags.*mean(diff(T));
figure;
plot(lags,ac)
set(gca,'FontSize',18); xlabel('time lag (s)'); ylabel('correlation ({\itr})');
title('xcorr (high-gamma relative to low-gamma at t = 0)');

You should see: