# Chapter 11: Time/Frequency decomposition

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## Time/frequency decomposition

Time/frequency analysis characterizes changes or perturbations in the spectral content of the data considered as a sum of windowed sinusoidal functions (i.e. sinusoidal wavelets). There is a long history and much recent development of methods for time/frequency decomposition. The methods used in the basic EEGLAB functions are straightforward. Their mathematical details are given in a reference paper (Delorme and Makeig, 2004).

#### Decomposing channel data

KEY STEP 10: ERSP and ITC

To detect transient event-related spectral perturbation, or ERSP, (Makeig, 1993) (event-related shifts in the power spectrum) and inter-trial coherence (ITC) (event-related phase-locking) events in epoched EEGLAB datasets, select Plot > Time frequency transforms > Channel time-frequency (calling pop_timef.m. Below, we enter 14 (Cz) for the Channel number, .01 for the Bootstrap significance level, and set the optional parameter padratio to 16 as below (a very high over-sampling factor for frequencies, giving a smooth looking plot at quite some unnecessary computational cost). We let the other defaults remain.

Note the default "Wavelet cycles" entry of 3 0.5. As explained in the help message for the newtimef() function, this means that the wavelet used to measure the amount and phase of the data in each successive, overlapping time window will begin with a 3-cycle wavelet (with a Hanning-tapered window applied). The '0.5' here means that the number of cycles in the wavelets used for higher frequencies will continue to expand slowly, reaching half (0.5) the number of cycles in the equivalent FFT window at its highest frequency. This controls the shapes of the individual time/frequency windows measured by the function, and their shapes in the resulting time/frequency panes. Note: This information does not set the lowest frequency to be analyzed. By current default, the lowest frequency window is about 0.5 seconds long. Three cycles in 0.5 sec sets the lowest frequency analyzed to about 6 Hz. To make this lowest frequency near 3 Hz, we would need to add the Optional newtimef() argument 'winlen', xxx where xxx is the sampling rate of the data (EEG.srate, also shown on the blue EEGLAB menu window). This would specify that the window length ('winlen') at the lowest frequency should be xxx samples long (i.e., 1 sec long), meaning the lowest analysis frequency would be 3 cycles n 1 sec, i.e. 1 Hz. We press OK. The timef.m window below appears. The top image shows mean event-related changes in spectral power (from pre-stimulus baseline) at each time during the epoch and at each frequency (< 50 Hz). To inspect these changes more closely, click on the color image. A new window will pop up. Enabling Matlab zoom allows zooming in to any desired time/frequency window. The ERSP image shows a brief but significant decrease in power at about 370 ms at 8 Hz (click on the image to zoom in and determine the exact frequency), a power increase centered at 13.5 Hz and starting at 300 ms. More spectral changes occur at 20-28 Hz. Note that the method used to asses significance is based on random data shuffling, so the exact significance limits of these features may appear slightly different.

The upper left panel shows the baseline mean power spectrum, and the lower part of the upper panel, the ERSP envelope (low and high mean dB values, relative to baseline, at each time in the epoch).

The lower image shows is the Inter-Trial coherence (ITC) at all frequencies. We previously encountered the ITC when we explained the ERP_image_plotting.m function. A significant ITC indicates that the EEG activity at a given time and frequency in single trials becomes phase-locked (not phase-random with respect to the time-locking experimental event). Here, a significant ITC appears briefly at about 10 Hz (at about the same time as the power increase in the upper panel), but this effect might well become insignificant using a more rigorous significance threshold. Note that the time/frequency points showing significant ITC and ERSP are not necessarily identical. In this plot, ITC hardly reaches significance and cannot be interpreted. The help message for the timef.m function contains information about all its parameters, the images and curve plotted, and their meanings. #### Computing component time/frequency transforms

It is more interesting to look at time-frequency decompositions of component activations than of separate channel activities, since independent components may directly index the activity of one brain EEG source, whereas channel activities sum potentials volume-conducted from different parts of the brain.

To plot a component time-frequency transform, we select Plot > Time/frequency transforms > Component time-frequency (calling pop_timef.m. Enter 10 for the Component number to plot, [-500 1000] for the "Epoch time range", (FFT) for Wavelet cycles, and .01 for the Bootstrap significance level. Note that Morlet wavelets are used by default although it is also possible to use sinusoidal wavelets. We change padratio to 16 and add the optional argument 'maxfreq', '30' to visualize only frequencies up to 30 Hz. Again, we press OK.

Note: pop_timef.m decompositions using FFTs allow computation of lower frequencies than wavelets, since they compute as low as one cycle per window, whereas the wavelet method uses a fixed number of cycles (default 3) for each frequency. The following timef.m window appears. The ITC image (lower panel) shows strong synchronization between the component activity and stimulus appearance, first near 15 Hz then near 4 Hz. The ERSP image (upper panel) shows that the 15-Hz phase-locking is followed by a 15-Hz power increase, and that the 4-Hz phase-locking event is accompanied by, but outlasts, a 4-Hz power increase. Note that the appearance of oscillatory activity in the ERP (trace under bottom ITC image) before and after the stimulus is not significant according to ITC. #### Computing component cross-coherences

To determine the degree of synchronization between the activations of two components, we may plot their event-related cross-coherence (a concept first demonstrated for EEG analysis by Rappelsberger). Even though independent components are (maximally) independent over the whole time range of the training data, they may become transiently (partially) synchronized in specific frequency bands. To plot component cross-coherence, select Plot > Time-frequency transforms > Component cross-coherence, which calls pop_crossf.m. Below, we enter components 4 and 9 (Use any components in your decomposition), Bootstrap significance level to 0.01, set padratio to 16. We again press OK. In the crossf.m window below, the two components become synchronized (top panel) around 11.5 Hz (click on the image to zoom in). The upper panel shows the coherence magnitude (between 0 and 1, 1 representing two perfectly synchronized signals). The lower panel indicates the phase difference between the two signals at time/frequency points where cross-coherence magnitude (in the top panel) is significant. In this example, the two components are synchronized with a phase offset of about -120 degrees (this phase difference can also be plotted as latency delay in ms, using the minimum-phase assumption. See crossf.m help for more details about the function parameters and the plotted variables). One can also use Plot > Time-frequency transforms > Channel cross-coherence to plot event-related cross-coherence between a pair of scalp channels, but here relatively distant electrode channels may appear synchronized only because a large EEG source projects to both of them. Other source confounds may also affect channel coherences in unintuitive ways. Computing cross-coherences on independent data components may index transient synchronization between particular cortical domains.

#### Plotting ERSP time course and topography

Recall that spectral perturbations at a single-analysis frequency and channel or component in the single epochs (sorted by some relevant value) can be imaged using erpimage.m or by selecting Plot > Component|Channel ERP image.

Called from the command line (see EEGLAB script writing), the timef.m and crossf.m routines can return the data for each part of their figures as a Matlab variable. Accumulating the ERSP and ITC images (plus the ERSP baseline and significance-level information) for all channels (e.g., via an EEGLAB script) allows use of another toolbox routine, tftopo.m (currently not available from the EEGLAB menu).

In the next tutorial, we show more about how to import data and events into EEGLAB datasets.