Having time-frequency results for individual channels is useful, however we hardly learn anything about functional relationships between different sources. Even if two channels have a spectral peak at say 100Hz, we don’t know if these signals are actually connected. Syncopy offers various distinct methods to elucidate such putative connections via the
connectivityanalysis() meta-function: coherence, cross-correlation and Granger-Geweke causality.
To have a synthetic albeit meaningful dataset to illustrate the different methodologies we start by simulating three autoregressive processes of order 2:
import numpy as np import syncopy as spy from syncopy import synthdata cfg = spy.StructDict() cfg.nTrials = 50 cfg.nSamples = 2000 cfg.samplerate = 250 # 3x3 Adjacency matrix to define coupling AdjMat = np.zeros((3, 3)) # only coupling 0 -> 1 AdjMat[0, 1] = 0.2 data = synthdata.ar2_network(AdjMat, cfg=cfg, seed=42) # add some red noise as 1/f surrogate data = data + 2 * synthdata.red_noise(cfg, alpha=0.95, nChannels=3, seed=42) spec = spy.freqanalysis(data, tapsmofrq=3, keeptrials=False)
We also right away calculated the respective power spectra
We can quickly have a look at a snippet of the generated signals:
data.singlepanelplot(trials=0, latency=[0, 1.])
All channels show visible oscillations as is confirmed by looking at the power spectra:
As expected for the stochastic AR(2) model, we have a fairly broad spectral peak at around 50Hz plus the 1/f like background.
One way to check for relationships between different oscillating channels is to calculate the pairwise coherence measure. It can be roughly understood as a frequency dependent correlation. Let’s do this for our coupled AR(2) signals:
coherence = spy.connectivityanalysis(data, method='coh', tapsmofrq=3)
The result is of type
CrossSpectralData, the standard datatype for all connectivity measures. It contains the results for all
nChannels x nChannels possible combinations. Let’s pick a few available channel combinations and plot the results:
coherence.singlepanelplot(channel_i='channel1', channel_j='channel2') coherence.singlepanelplot(channel_i='channel2', channel_j='channel1') coherence.singlepanelplot(channel_i='channel3', channel_j='channel1') coherence.singlepanelplot(channel_i='channel3', channel_j='channel2')
As coherence is a symmetric measure, we obtain exactly the same graph for both
channel1-channel2 combinations, showing high coherence around 50Hz. However as channel3 is completely uncoupled, there is no coherence with either
The plotting for
CrossSpectralData objects works a bit differently, as the user here has to provide one channel combination for each plot with the keywords
Coherence is a spectral measure for correlation, the corresponding time-domain measure is the well known cross-correlation. In Syncopy we can get the cross-correlation between all channel pairs with:
corr = spy.connectivityanalysis(data, method='corr', keeptrials=True)
As this also is a symmetric measure, we just look at only one channel combination for each channel pair:
corr.singlepanelplot(channel_i=0, channel_j=1, trials=1) corr.singlepanelplot(channel_i=0, channel_j=2, trials=1) corr.singlepanelplot(channel_i=1, channel_j=2, trials=1)
As a time domain measure, the cross-correlation is confounded by the 1/f background present in all 3 channels.
We can however use a bandpass filter around 50Hz first and then trial average the cross-correlations to unmask some short-lived (~0.1s) correlations between channel1 and channel2:
bp_filtered = spy.preprocessing(data, filter_type='bp', freq=[45, 55]) bp_corr = spy.connectivityanalysis(bp_filtered, method='corr', keeptrials=False) # look only at lags of max. 0.2 seconds bp_corr.singlepanelplot(channel_i=0, channel_j=1, latency=[0, 0.2]) bp_corr.singlepanelplot(channel_i=1, channel_j=2, latency=[0, 0.2])
Note that we can also look at the auto-correlation:
fig, ax = bp_corr.singlepanelplot(channel_i=0, channel_j=0, latency=[0, 0.2]) ax.set_title('Auto- and cross-correlation')
Have a look at the Preprocessing section to learn more about pre-processing data with Syncopy.
To reveal directionality, or causality, between different channels Syncopy offers the Granger-Geweke algorithm for non-parametric Granger causality in the spectral domain:
granger = spy.connectivityanalysis(data, method='granger', tapsmofrq=2)
Now we want to see differential causality, so we plot more channel combinations:
granger.singlepanelplot(channel_i=0, channel_j=1) granger.singlepanelplot(channel_i=1, channel_j=0) granger.singlepanelplot(channel_i=0, channel_j=2) fig, ax = granger.singlepanelplot(channel_i=2, channel_j=0) ax.set_ylim((-.05, 0.6))
This reveals the coupling structure we put into this synthetic data set:
channel2, but in the other direction there is no interaction. The oscillations in
channel3 are completely uncoupled.
As a spectral method, we did not need to filter out any 1/f component to uncover the coupling topology.
keeptrials keyword is only valid for cross-correlations, as both Granger causality and coherence critically rely on trial averaging.