## 2 Materials and Methods
### TGC Computation
#### 2.7.3 Extraction of Oscillations for Phase and Amplitude  
Band-Pass Filters: Filter Type  
One way of extracting the oscillatory components is band-pass filtering followed by the Hilbert transform. The filters applied usually have a zero phase shift to preserve the temporal relationship between phase and amplitude. At least three ways of constructing such filters have been used in PAC literature. The first one is the forward and backward (two-way) application of a Butterworth infinite impulse response (IIR) filter [21; 26; 43]. The second option is a two-way finite impulse response (FIR) filter as implemented in the EEGLAB function eegfill [7; 23; 31; 33; 34; 36; 40; 44; 45]. Finally, the third approach is a forward application of a FIR filter followed by shifting the data by the filter's group delay, as implemented in the function pop_eegfillnew from the firtilt plugin for EEGLAB [28; 35; 46; 47]. The orders and transition bands of the filters can be chosen in different ways, but they are usually kept at default values in the mentioned functions. In some papers, the exact filter type is not mentioned [39], or a partial description is provided, such as'second-order zero-phase shift' in [37], with later papers [12; 38] referencing this one regarding the methods. The EEGLAB package switched its default filter from pop_eegfill (calling eegfill) to pop_eegfillnew, and its website states that \"the heuristic for automatically determining the filter length in the legacy basic FIR filter (pop_eegfill) was inappropriate (possibly causingsuboptimal filtering or unexpected filter effects)\". At the same time, the number of papers using eegfilt (including recent ones) cited above shows that it is probably still the most popular method. Thus, we used eegfilt (with default parameters) as a starting point (in the 'basic' method below), but also explored the Butterworth filter (MATLAB function butter with n = 2 and forward-backward application by function filttilt) and pop_eegfiltnew (with default parameters). It should be noted that our signals after preprocessing contain boundaries from artifactual time interval rejection. The function pop_eegfiltnew uses DC padding around any continuous time interval, which arguably creates smaller distortions at the edges than filtering across a boundary [48]. Thus, in the versions of the algorithm without buffer zones or without selecting epochs (described below), we used pop_eegfiltnew rather than the functions eegfilt or butter. In the versions with buffer zones, a boundary will be separated from the PAC calculation interval by at least the buffer zone length, and the distortions are unlikely to affect the PAC. Band-Pass Filters: Band Definition  
Another issue besides the filter type is the choice of the band. Some papers compute the coupling for two fixed frequency bands, such as the theta band of 4-7 Hz and gamma band of 30-50 Hz used in [12; 37; 38; 41]. Other definitions of the gamma band include 30-40 Hz to avoid contamination with line noise in [35], 36-45 Hz in [40], 50-80 Hz in [33] and 50-70 in [34]. It is interesting to note that the last two studies (sharing three of the authors) used two almost mutually exclusive definitions of the theta band: 5-8 Hz and 3-5 Hz, respectively. Rather than using fixed bands and obtaining one PAC value, an exploratory analysis can be performed, i.e., computing the coupling for a series of bands. A comodulogram can then be produced, which is a plot with band-center frequencies on the axes and the corresponding PAC measure depicted by color. The widths of the bands can be fixed, e.g., 4 Hz for phase and 20 Hz for amplitude [45], or, respectively, 1 and 4 Hz [7], 2 and 40 Hz [36], 4 and 10 Hz [23; 31], 2 and 5 Hz [47], and 2 and 4 Hz [44]. It has been pointed out that if the bandwidth for amplitude is smaller than twice the modulating (phase) frequency, then it does not include the side peaks produced by the modulation, and PAC cannot be detected even if it is present [24; 25].