Morlet complex wavelets.

A wavelet (small wave) is a mathematical function of time (t) and frequency (f), with the following two features. First, its amplitude approaches zero at positive and negative infinity and it differs from zero only around the origin of the cartesian axes. That is, differently from other mathematical functions such as sinusoidal or infinite waves, a wavelet has a defined temporal length. Second, the sum of the area of a wavelet above the x axis is equal to the sum of the area of the wavelet below the x axis, hence the total sum of the area of a wavelet is equal to zero. Therefore, differently from a filter, a wavelet will not change the power spectrum of a signal. WTools computes Morlet complex wavelets at a 1 Hz sampling frequency according to:

where , is the temporal duration of the wavelet expressed in cycles of the oscillation frequency , is a Gaussian bell curve, is a complex sinusoid. Wavelets are normalized depending on the time-frequency transformation measure (see next paragraph): the normalization factor is equal to if the amplitude is taken, and it is equal to if the power is taken [39]. Wavelet normalization is common practice in time-frequency analysis (e.g., in EEGLAB), which we strongly recommend as it allows direct quantitative comparisons of frequency content across frequency values. Nevertheless, we still provide the option to omit wavelet normalization for back-compatibility with previous (beta) versions of WTools. As a sanity check, there is a perfect match between the wavelets specified by WTools and EEGLAB once wavelet normalization is applied to the WTools wavelet (Fig 2).

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Fig 2. Comparison of complex Morlet wavelets.

WTools wavelets (on the left) and EEGLAB wavelets (on the right) for the frequency range used in the present work (10-90 Hz; see section “Data analysis”). Please note that the default WTools pipeline computes wavelets with 1 Hz sampling frequency; here we plot wavelets with 10 Hz sampling frequency for visualization purposes only. The red plots show non-normalized wavelets for WTools. The blue plots show normalized wavelets for WTools and EEGLAB, with a perfect correspondence between the two. X axis: Time (s); Y axis: Wavelet amplitude (a.u.).

https://doi.org/10.1371/journal.pone.0323179.g002

Note that WTools uses a default value of for back-compatibility with beta versions of the toolbox and with previous infant EEG research that used the very same approach [2,18–20,1,23]. More precisely, these previous publications report a value of 3.5 cycles, which translates to 7 in the current version of the toolbox to fully align with the EEGLAB’s wavelet formula and resulting cycle value, thus avoiding potential confusion. Consequently, the very small oscillations at the edges of the wavelets are also considered. However, there is no other theoretical reason to choose this value. WTools implements the possibility to choose the number of cycles one thinks is the most appropriate for their experimental purposes. Furthermore, the wavelet transformation GUI enables plotting the Full Width at Half Maximum (FWHM) versus the wavelet frequencies. This feature is particularly useful when determining the appropriate optional padding to apply to the signals before transformation, as it suggests the minimum acceptable duration for the padding, helping to avoid the introduction of distortion at the edges of the EEG segments.

The last term of the equation can be written as a sum of sin and cos waves:

In practice, a wavelet is a complex sinusoidal wave enveloped into a Gaussian bell curve. Because a wavelet is a function depending on a frequency f, at each f the corresponding wavelet works as a magnifier that can reveal how much that frequency is present into the EEG signal. At the same time, because a wavelet has a defined temporal length, it can preserve some time information after the transformation of the signal.

Wavelets are wide at lower frequencies, that is the temporal information is relatively imprecise, but the frequency information is preserved. Wavelets become progressively narrower at higher frequencies, that is the time information is more accurate, but the frequency information is less precise. This creates an important difference with the short-time Fourier transform (STFT) approach, which is also widely used to uncover oscillatory activity in the EEG signal. During the STFT computation, the time information is kept fixed for all frequencies of interest, even when it could be higher (i.e., at higher frequencies). Conversely, wavelets optimize the window length for each frequency by appropriately scaling a unique function, the “mother wavelet”, so that it achieves the best possible compromise between uncovering the EEG oscillatory activity and preserving the maximum possible time information for each frequency of interest. In sum, time and frequency information respond to Heisenberg’s indetermination principle: the more accurately we know one of them, the less accurately we know the other. Given this constraint, wavelet transformation is the most informative approach for time-frequency analysis, preserving the time information as much as possible while reliably uncovering oscillatory information.

Time-frequency transformation.

After computing all the wavelets for each frequency in a range of interest, WTools uses continuous wavelet transformation. By default, all trials are analysed. However, the user can specify a subset of trials or even a specific single trial, enabling single-trial frequency extraction. By convolving the signal s(t) (cleared of its DC component) with the wavelet w(t, f) and taking the modulus of the resulting complex coefficients, WTools obtains the amplitude coefficients (the mean square root) of the wavelet transformation at a specific frequency f. The resulting vector represents how well the signal fits the wavelet at a the frequency f, that is how much of the frequency f is in that signal [39]. WTools performs the convolution for each experimental condition and channel, of each EEG segment and subject, for all frequencies of interest. Transformed segments are then averaged across trials:

where is the time-frequency coefficient at channel c, frequency f, time t and segment n of the EEG signal given by . Following the ERPWAVELAB data structure [27], the resulting MATLAB variable “WT” is a 3D matrix (Channel × Frequency × Time) that contains the average of the transformed segments. Note that, since WTools by default takes the modulus of the complex coefficients, the data are expressed in amplitude (μV) rather than power (i.e. μV²). As a result, the measure is slightly more sensitive to small frequency changes. However, since other established pipelines [26,28,29] use the power of the wavelet transform as the time-frequency measure, WTools also provides the option to compute and analyse the power. Thus, WTools offers flexibility to work with either the amplitude or the power of the wavelet transform, and users can display the data in dB. Importantly, since wavelet transformation reveals phase information, WTools provides access to the raw complex data before removing the complex components, allowing users to extract phase information directly from the transformed data.

WTools allows computing both total-induced and evoked oscillations [40,41]. Total-induced oscillations are computed by wavelet transformation of each EEG epoch before averaging all of them together (i.e., the pipeline we just described above): in this way, the oscillatory activity non-phase-locked to the onset of the stimulus is preserved. Total-induced oscillations are richer measures compared to evoked oscillations. Evoked oscillations are computed by running the wavelet transformation after averaging all the EEG epochs together, that is the wavelet transformation is performed on the average (i.e., it is the wavelet transformation of the ERP): in this way, the oscillatory activity that is non-phase-locked to the stimulus onset is averaged out and lost before the wavelet transformation. Evoked oscillations are poorer measures compared to total-induced oscillations and are slightly more sensitive to low-level features of the stimulus.

Edges chopping.

The continuous wavelet transformation assumes signal continuity, yet EEG segments inherently lack this continuity: at the segment’s start, a shift from no signal to signal with amplitude occurs, mirrored at the segment’s ends, causing significant frequency changes. This discontinuity distorts coefficient vectors at the edges, particularly noticeable at lower frequencies where wavelets are broader (see Fig 2). To address the impact of this distortion on EEG signal analysis, it is advisable to pre-cut wider segments. Alternatively, WTools offers an edge padding feature to compensate for signal absence at segment edges. Based on empirical evaluation, the minimally necessary length of extra signal needed at each edge can be estimated with the current formula (Gergely Csibra, personal communication):

where f is the frequency range of interest. For instance, if one wants to look at frequencies between 5–15 Hz, then it is necessary to consider t = 400 ms of extra signal at both left and right edges of the segment.

Baseline correction.

Differently from other toolboxes that adopt a divisive baseline [42], WTools performs by default a subtractive baseline similarly to baseline correction for ERPs. The average amplitude in a given time window of the pre-stimulus interval is computed and subtracted from the whole time-varying signal. Notice that there is not a common baseline for all frequencies; instead, each frequency of interest has its own baseline value that is used for correction only for that frequency. This approach makes the resulting spectrograms easy to read and highly comparable to ERPs, which is particularly handy for developmental research. Furthermore, adopting a divisive baseline assumes a multiplicative model, in which oscillations and activity scale by the same factor relative to baseline for each frequency [43]. This assumption may be incorrect when signal and noise contribute independently to the power spectrum (i.e., additive model; Gyurkovics et al., 2021), a very common problem in developmental studies for which a subtractive baseline may be more suitable. Nevertheless, for wider usability of the toolbox, WTools also offers options for subtractive baseline correction with normalization and divisive baseline correction [42].

WTools analysis.

To import data into WTools, epochs were transformed from the native EGI format to the MATLAB format using the appropriate NetStation Waveform export tool by EGI, and then the correspondent EEGLAB import plugin (v. 2021.1). In general, WTools calls existing EEGLAB plugins to import data. Each plugin takes as input a specific data format. Users must save the data in a format that is compatible with the EEGLAB plugin of interest, which is called by the WTools import routine (see: eeglab.org/tutorials/04_Import/Import.html). Note that data to be imported into WTools must conform to an event-related within-subject design with one single trigger per segment/epoch indicating the onset of the key event. Epochs were re-referenced to average reference and then subjected to time-frequency analysis. Data were analysed using both the default WTools pipeline (whose code is released with this publication) and custom pipelines that allowed a direct comparison with the EEGLAB output, as outlined below.

In the following, we describe the default WTools pipeline. We computed complex Morlet wavelets for the frequencies 10–90 Hz in 1 Hz steps. We calculated total-induced oscillations by performing a continuous wavelet transformation of all the epochs using convolution with each normalized wavelet (number of cycles = 7) and taking the absolute value (i.e., the amplitude) of the results [2]. To remove the distortion introduced by the convolution, we chopped 300 ms at each edge of the epochs, resulting in 1400 ms long segments, including 200 ms before and 1200 ms after stimulus onset. We used the average amplitude of the 200 ms pre-stimulus window as baseline, subtracting it from the whole epoch at each frequency.

To compare the WTools output to the EEGLAB output, we carried out a custom analysis that deviated from the default WTools pipeline in two ways. First, we took the squared value (i.e., the power) of the complex wavelet coefficient, in line with the EEGLAB pipeline. Second, we skipped baseline correction to avoid a mismatch between the two toolboxes (while WTools performs a subtractive baseline, EEGLAB implements a divisive baseline).

EEGLAB analysis.

Epochs were transformed from the native EGI format to the MATLAB format using the NetStation export tool and then imported by using the correspondent EEGLAB import plugin (v. 2021.1). Epochs were re-referenced to average reference and then subjected to time-frequency analysis. Data were analysed using various EEGLAB pipelines that allowed a direct comparison with the WTools output. Specifically, we carried out a custom analysis that deviated from the standard EEGLAB pipeline in two ways. First, we did not log-transform the results of the wavelet transformation (typically done to bring the results to the standard dB scale). Second, we skipped baseline correction to avoid a mismatch between the two toolboxes. Alongside these custom analyses, we processed the data using the standard EEGLAB pipeline. Please see the Supporting information (S1 Fig) for a comprehensive review of the analysis pipelines. For all analyses, we computed Morlet complex wavelets for the frequencies 10–90 Hz in 1 Hz steps. We calculated total-induced oscillations by performing a continuous wavelet transformation of all the epochs using convolution with each normalized wavelet (number of cycles = 7) and taking the squared value (i.e., the power) of the complex coefficients. Data were automatically chopped by EEGLAB to remove the distortion introduced by the convolution, resulting in 1400 ms long segments, including 200 ms before and 1200 ms after stimulus onset. For baseline correction, we entered the 200 ms pre-stimulus window as baseline.