Lucas Benjamin

Ghislaine Dehaene-Lambertz

Ana Fló
email: ana.flo@cea.fr

Remarks on the analysis of steady-state responses: spurious artifacts introduced by overlapping epochs

Keywords: steady state, neural entrainment, EEG, MEG, speech, infant, adult, statistical learning

Periodic and stable sensory input can result in rhythmic and stable neural responses, a phenomenon commonly referred to as neural entrainment. Although the use of neural entrainment to investigate the regularities the brain tracks has increased in recent years, the methods used for its quantification are not well-defined in the literature. Here we argue that some strategies used in previous papers, are inadequate for the study of steady-state response, and lead to methodological artefacts. The aim of this commentary is to discuss these articles and to propose alternative measures of neural entrainment. Specifically, we applied four possible alternatives and two epoching approaches reported in the literature to quantify neural entrainment on simulated datasets. Our results demonstrate that overlapping epochs, as used in the original Batterink and colleagues articles, inevitably lead to a methodological artefact at the frequency corresponding to the overlap. We therefore strongly discourage this approach and encourage the re-analysis of data based on overlapping epochs. Additionally, we argue that the use of time-frequency decomposition to compute phase coherence at low frequencies to reveal neural entrainment is not optimal.

Introduction

Neural Entrainment

It is well established that a stable and rhythmic stimulation elicits a stable and rhythmic neural response (e.g. [START_REF] J O U R N A L P R E -P R O O F Picton | Human auditory steady-state responses: Respuestas auditivas de estado estable en humanos[END_REF][START_REF] Regan | Steady-state evoked potentials[END_REF], a phenomenon usually referred to as neural entrainment or steady-state responses. An enhanced cortical response, measurable with electro or magneto-encephalography (EEG or MEG), appears at very specific frequencies in function of the periodicity of the stimulation. Although it is still discussed whether it results from multiple evoked responses to the stimulus train [START_REF] Capilla | Steady-State Visual Evoked Potentials Can Be Explained by Temporal Superposition of Transient Event-Related Responses[END_REF] or from the alignment of intrinsic brain oscillation to the input [START_REF] Doelling | An oscillator model better predicts cortical entrainment to music[END_REF], this enhanced response indicates which event the brain is tracking in a train of stimuli. It can be low-level properties of the sensory input (the "on-effect" of an image for example), or more abstract regularities, such as a face presented every four images [START_REF] De Heering | EEGLAB: An open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF]. Therefore, this phenomenon receives more and more attention due to its theoretical implications [START_REF] Buzsáki | Rhythms of the Brain[END_REF][START_REF] Giraud | Cortical oscillations and speech processing: Emerging computational principles and operations[END_REF], and also to its promising uses in special populations such as infants. For example, it has been used to study categorization in infants [START_REF] De Heering | EEGLAB: An open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF][START_REF] Peykarjou | Rapid Categorization of Human and Ape Faces in 9-Month-Old Infants Revealed by Fast Periodic Visual Stimulation[END_REF], the tracking of linguistic structures in adults [START_REF] Ding | Cortical tracking of hierarchical linguistic structures in connected speech[END_REF][START_REF] Ding | Characterizing Neural Entrainment to Hierarchical Linguistic Units using Electroencephalography (EEG)[END_REF], and speech segmentation in adults [START_REF] Batterink | Online neural monitoring of statistical learning[END_REF], 2019;[START_REF] Buiatti | Investigating the neural correlates of continuous speech computation with frequency-tagged neuroelectric responses[END_REF] and infants [START_REF] Choi | Preverbal Infants Discover Statistical Word Patterns at Similar Rates as Adults: Evidence From Neural Entrainment[END_REF][START_REF] Kabdebon | Electrophysiological evidence of statistical learning of long-distance dependencies in 8-month-old preterm and full-term infants[END_REF].

To quantify neural entrainment, frequency-domain analyses are preferred to conventional time-domain analysis for two reasons. First, it is generally easy to predict the frequencies at which the increased response should occur [START_REF] Zhou | Interpretations of Frequency Domain Analyses of Neural Entrainment: Periodicity, Fundamental Frequency, and Harmonics[END_REF] reducing the number of statistical comparisons. Second, frequency-domain analysis provides a better signal-to-noise ratio (SNR) [START_REF] Norcia | The steady-state visual evoked potential in vision research: A review[END_REF]. However, different methods can be used and have been used in the literature, some J o u r n a l P r e -p r o o f presenting a risk of bias when overlapping data are used [START_REF] Batterink | Online neural monitoring of statistical learning[END_REF], 2019;[START_REF] Choi | Preverbal Infants Discover Statistical Word Patterns at Similar Rates as Adults: Evidence From Neural Entrainment[END_REF]. Thus, our goal is to demonstrate the risk of bias when using overlapping data and provide artifact free alternatives. Additionally, we discuss the pros and cons of the different methods and then provide evidence with simulated data.

Although our point can be extended to all studies using neural entrainment, we decided to simulate a speech segmentation study. Neural entrainment is an adequate and extensively used tool to recover speech perception units within natural sentences [START_REF] Ding | Cortical tracking of hierarchical linguistic structures in connected speech[END_REF][START_REF] Ding | Characterizing Neural Entrainment to Hierarchical Linguistic Units using Electroencephalography (EEG)[END_REF] or artificial streams [START_REF] Buiatti | Investigating the neural correlates of continuous speech computation with frequency-tagged neuroelectric responses[END_REF][START_REF] Kabdebon | Electrophysiological evidence of statistical learning of long-distance dependencies in 8-month-old preterm and full-term infants[END_REF]. Because speech is a continuous stream of phonemes, syllables, and words, an important question notably concerns how infants succeed to chunk this stream in candidate words. It has been proposed that infants can take advantage of the probabilities of transitions between successive syllables that are constant within a word and drop at its boundaries [START_REF] Saffran | Statistical learning by 8-month-old infants[END_REF]. Thus, if a stream is composed of four tri-syllabic words randomly concatenated without repetition, the transition probability is 1 within a word and 0.33 between words. Neural entrainment appears as a suitable approach to investigate this experimental paradigm. As the syllables have equal duration, the neural response is entrained at the syllable rate, as many segmentation experiments report [START_REF] Batterink | Online neural monitoring of statistical learning[END_REF], 2019;[START_REF] Buiatti | Investigating the neural correlates of continuous speech computation with frequency-tagged neuroelectric responses[END_REF][START_REF] Choi | Preverbal Infants Discover Statistical Word Patterns at Similar Rates as Adults: Evidence From Neural Entrainment[END_REF][START_REF] Kabdebon | Electrophysiological evidence of statistical learning of long-distance dependencies in 8-month-old preterm and full-term infants[END_REF]. If the participant discovers the word regularity, the neural response should also be entrained at the word rate as also reported in the same studies.

Methodological considerations

When using rapid periodic stimulation, a sustained oscillatory activity is expected to emerge at the stimulation frequencies. The most direct analytical approach would be to transform J o u r n a l P r e -p r o o f the entire dataset in the frequency domain by applying, for example, the Fast Fourier Transformation (FFT) algorithm. However, this procedure is not the most commonly used because the signal's stationary nature allows obtaining a better SNR by dividing the signal into shorter segments aligned with the stimulation. In brief, having many data segments with aligned activity enables separating phase-locked from non-phase-locked activity. Therefore, two constraints oppose achieving a correct analysis of neural entrainment: on the one hand, to have long segments and, on the other hand, to have many of them.

Each epoch should be long enough for the signal and noise to be well represented, and thus, to obtain an optimal description in the frequency domain. The epoch's length determines the frequency resolution (fres = 1/epoch length), and the frequency resolution has to be at least half the frequency of the slower steady-state response to be able to see it in the frequency domain. In other words, the slower response must repeat at least once to be detected in the frequency domain. Moreover, longer epochs also provide a better SNR because, with a higher frequency resolution, the noise is spread over more frequency bins, while the signal remains restricted to specific frequencies [START_REF] Norcia | The steady-state visual evoked potential in vision research: A review[END_REF].

J o u r n a l P r e -p r o o f Conversely, dividing the recording into sub-segments aligned with the stimulus onset allows focusing on the coherent activity and provides a better SNR. One way of doing so is by averaging the responses across epochs. The averaging process enhances the phase-locked activity and reduces non-phase locked activity, such as in classical event-related potentials (ERP). Subsequently, entrainment appears as an increase in power at specific frequencies in the average signal.

The amplification of phase-locked activity by averaging allows the detection of entrainment even J o u r n a l P r e -p r o o f when the steady-state responses are much weaker than the background noise. Another way is by directly computing the phase coherence across the epochs. If activity phase-locked to the stimuli onset exists, the phases' distribution will not be uniform at certain frequencies, but phases will be coherent across epochs. The measurement of this coherence is called Inter Trial Coherence (ITC).

Many trials are needed to estimate the phase distribution and to measure the ITC properly. Therefore, in both computations, the same opposite constraints exist. Long segments provide a better frequency resolution. Many epochs allow a better enhancement of phase-locked activity or to better estimate the phases' distribution.

While most studies use multiple epochs to estimate the power spectrum or the phase coherence across them, the computations are differently implemented in practice. We have isolated two epoching approaches and four different calculations to estimate neural entrainment (see Fig. 1). We applied these methods to simulated data to evaluate their performance and potential biases.

Methods used in the literature

As mentioned before, the definition of epochs plays a crucial role in these analyses. Epochs have to be long enough to properly represent signal and noise, and at the same time, the more epochs, the better to detect phase-locked activity. In most of the papers, the segment's length is calculated to include 10-25 repetitions of the slower response. Although the epoch's length may affect sensitivity, it does not lead to artifactual measures of entrainment. Most papers have used independent epochs (e.g., [START_REF] De Heering | EEGLAB: An open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF][START_REF] Ding | Cortical tracking of hierarchical linguistic structures in connected speech[END_REF][START_REF] Kabdebon | Electrophysiological evidence of statistical learning of long-distance dependencies in 8-month-old preterm and full-term infants[END_REF], but some have used overlapping epochs to increase the total number of epochs [START_REF] Batterink | Online neural monitoring of statistical learning[END_REF], p. 200, 2019;[START_REF] Buiatti | Investigating the neural correlates of continuous speech computation with frequency-tagged neuroelectric responses[END_REF]Choi et al., 2020) (Fig 1). We believe that using overlapping epochs entails a methodological artifact; so, we decided to test both approaches.

J o u r n a l P r e -p r o o f

The four methods described in the literature further differ in the measures to estimate neural entrainment (power spectrum or phase coherence) and the procedure used to compute those measures (order of operations or whole epochs vs. time-frequency analysis). In Method 1, the power spectrum is computed for single epochs and then averaged across epochs to obtain an averaged power spectrum. Buiatti and colleagues used this approach in a speech segmentation task in adults [START_REF] Buiatti | Investigating the neural correlates of continuous speech computation with frequency-tagged neuroelectric responses[END_REF]. In this case, the entrainment measure depends on the presence of induced and evoked activity that is strong enough to emerge from the noise level. In Method 2, the average across epochs is obtained first, and its power spectrum is computed afterward. Here, nonphase locked activity is first reduced by the averaging step, which increases sensitivity to phaselocked activity. Method 2 is the most common approach used in numerous papers (e.g., [START_REF] De Heering | EEGLAB: An open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF][START_REF] Ding | Cortical tracking of hierarchical linguistic structures in connected speech[END_REF]. Notice that averaging the signal before computing the power spectrum makes Method 2 phase-dependent and similar to methods directly quantifying the ITC across all epochs, as Method 3 does. However, because Method 3 is exclusively phase-based, it is less dependent on amplitude changes of the signal than power-based methods. Method 3 is also widely used in the literature (e.g., [START_REF] Kabdebon | Electrophysiological evidence of statistical learning of long-distance dependencies in 8-month-old preterm and full-term infants[END_REF]. Method 4 has been implemented by Batterink and colleagues in speech segmentation experiments [START_REF] Batterink | Online neural monitoring of statistical learning[END_REF], 2019;[START_REF] Choi | Preverbal Infants Discover Statistical Word Patterns at Similar Rates as Adults: Evidence From Neural Entrainment[END_REF]. It also measures ITC, but through time-frequency analysis. Time-frequency analysis can be performed using different algorithms, notably by wavelet decomposition or by applying the Fourier transform to the data in short sliding time windows. While here we specifically discuss the use of wavelet decomposition (as implemented by Batterink and colleagues), both algorithms give similar results and entail the same type of properties and limitations. We argue that while time-frequency analysis is in theory applicable, it is less suitable for this particular context.

J o u r n a l P r e -p r o o f

The Matlab scripts of those different methods can be downloaded at https://github.com/neurokidslab/Neural-Entrainment.

Methods

Creation of the stimulated data

To compare the different methods and their risks of bias, we calculated the results of the four methods described above with independent and overlapping epochs on simulated data. To have datasets with a realistic noise structure, we used real EEG data, to which we added (or not) simulated steady-state responses.

We used EEG data from 27 asleep 5-month-old infants without stimulation. The data were recorded using a 128-electrode net (Electrical Geodesics, Inc.) referred to the vertex with a sampling frequency of 500 Hz, then bandpass-filtered [0.2, 40] Hz. The segments containing motion artefacts were removed. Only data from one frontal electrode (Fz) was kept for the simulations.

For each infant, we had between 25 and 48 minutes of artefacts-free data. From these artefactsfree data, we recreated 40 recordings of 720 s each (12 mn).

First, we simulated a study in which infants listen to randomly concatenated syllables with a fixed duration of 300 ms; thus, the neural signal should be entrained at 3.33 Hz. We did so by adding a half-sinusoidal wave (frequency 2.5 Hz) every 300 ms with amplitude 1 (Figure 2) to each of the 40 recordings. Second, we simulated a study in which infants discover the three-syllabic words (900 ms) embedded in the stream; thus, the neural response is entrained by both the syllable regularity and the word regularity (i.e. expected neural entrainment at 3.33 and 1.11 Hz).

We created the simulated data by adding a half-sinusoidal wave (frequency 0.8 Hz) every 900 ms J o u r n a l P r e -p r o o f with amplitude 2 (Figure 2) to each of the 40 recordings. The signal to noise ratio was 1/40 for the "syllabic" steady-response (300 ms), and 1/20 for "word" steady-response (900 ms) (the signal to noise ratio was estimated as the signal amplitude divided by the standard deviation of the original EEG data). We chose to use half-sinusoidal waves, because this is the simplest way of simulating evoked ERPs [START_REF] Yeung | Detection of synchronized oscillations in the electroencephalogram: An evaluation of methods[END_REF]. Moreover, half-sinusoidal waves result in the presence of harmonics in the frequency domain [START_REF] Zhou | Interpretations of Frequency Domain Analyses of Neural Entrainment: Periodicity, Fundamental Frequency, and Harmonics[END_REF], which are usually observed in this type of experiments (e.g. [START_REF] Ding | Cortical tracking of hierarchical linguistic structures in connected speech[END_REF]. Thus, we analyzed three sets of data, each one with 40 recordings: (i) original data, (ii) Simulated steady-state response at 3.33 Hz and (iii) Simulated steady-state response at 3.33 Hz and 1.11 Hz. Note that the third set corresponds to Batterink and colleagues' studies [START_REF] Batterink | Online neural monitoring of statistical learning[END_REF], 2019;[START_REF] Choi | Preverbal Infants Discover Statistical Word Patterns at Similar Rates as Adults: Evidence From Neural Entrainment[END_REF]. In their studies, the syllabic rate was 3.33 Hz and the word rate 1.11 Hz.

J o u r n a l P r e -p r o o f

Figure 2. Simulated steady-state response to the syllables (one every 300 ms) (top), and to the syllables and the words (one every 900 ms) (bottom).

Data preprocessing

The three sets of data were segmented in epochs of 10.8 s that either did not overlap (66 epochs), or have an overlap of 11/12 of their length (788 epochs), as in Batterink and colleagues' studies [START_REF] Batterink | Online neural monitoring of statistical learning[END_REF], 2019;[START_REF] Choi | Preverbal Infants Discover Statistical Word Patterns at Similar Rates as Adults: Evidence From Neural Entrainment[END_REF]. Notice that 10.8 s corresponds to 12 times the slower expected steady state-response response (i.e. 900 ms).

The analysis for each method was as follows. In Method 1, the FFT was applied to the data, the power spectrum was calculated for single epochs, and finally, the average power spectrum was obtained. Afterwards, the power spectrum at each frequency bin was normalized by the six adjacent frequency bins. In Method 2, the FFT was applied to the data, the average across epochs was computed, and the power spectrum was computed. As in method 1, the power spectrum at each frequency bin was normalized by the six adjacent frequency bins. In Method 3, the FFT was applied to the epochs, and the ITC was computed. The ITC was normalized by computing at each frequency bin the difference with the ITC at the adjacent six. In Method 4, a Morlet wavelet transformation was used to estimate the time-frequency ITC across epochs, by using the newtimef function of EEGLAB (Delorme & Makeig, 2004). The time dimension was then averaged to obtain a single ITC value at each frequency bin. We used the same parameters as Batterink and colleagues [START_REF] Batterink | Online neural monitoring of statistical learning[END_REF], 2019;[START_REF] Choi | Preverbal Infants Discover Statistical Word Patterns at Similar Rates as Adults: Evidence From Neural Entrainment[END_REF] (0.1 Hz step, with 1 cycle at 0.2 Hz linearly increasing until 45 cycles at 20.2 Hz) to faithfully reproduce their method and characterize the risk of artifact associated with it.

Results

J o u r n a l P r e -p r o o f

Original dataset

In the original dataset, no indices of neural entrainment are expected. It is indeed the case when non-overlapping epochs are used whatever the method: the response is flat across frequency bins (blue lines in Figure 3 left column for methods 1 to 4). By contrast, using overlapping epochs (blue lines in Figure 3 right column), creates a pronounced peak at the frequency corresponding to the overlap, for methods 2, 3 and 4, demonstrating a methodological artefact of the approach. The origin of the methodological artefacts stems from the multiple uses of the same data. Averaging the epochs (method 2) creates an artifactual periodic signal at the overlap frequency because the same data appears periodically. Thus, the mean signal power spectrum shows a peak at the frequency 1/overlap and its harmonics. A similar effect occurs when the phase coherence is estimated by using the FFT over the whole epoch or time-frequency analysis by computing the wavelet transform. Comparing the phase between epochs that share a large amount of data only shifted by a fixed value will result in an artificially high phase coherence at the frequency (and harmonics) of this overlap shift value. Interestingly, the presence of artifact when the time-frequency approach is used depends on the wavelet width and the amount of overlapping epochs. If the width of the wavelet does not cover the epochs' overlap, the results will be artifact-free. In the particular case of method 1 there are no visible artefacts when using overlapping epochs, because power is estimated over single epochs; therefore, the phase information is lost before the data of different epochs are averaged.

Simulated entrainment

When simulated steady-state responses are added, neural entrainment is detected at the syllable and word frequencies by methods 2, 3, and 4 (green and red lines in Figure 3). The peaks J o u r n a l P r e -p r o o f clearly appear relatively to the no-stimulation condition (blue line) in the case of independent epochs. When using overlapping epochs, the entrainment is added to the intrinsic bias due to overlapping epochs.

Method 1 was not sensitive enough to detect neural entrainment, which can be understood if we consider that the power is computed on single epochs and then averaged. Thus, the steadystate response has to be strong enough relative to the noise to emerge at the single epoch level. In other words, the method does not take advantage of the phase coherence across trials of the steadystate response. The sensitivity is slightly increased when the number of epochs is increased by using overlapping epochs but this method remains sub-optimal.

J o u r n a l P r e -p r o o f The ITC computed by performing a time-frequency analysis (method 4) shows a particular behavior. Entrainment appears at the frequency of the overlap and its harmonics, similarly to ITC computed on full epochs (method 3), but the signal is smoother than with the other methods. The J o u r n a l P r e -p r o o f smoothness of the signal affects both the estimate of neural entrainment resulting from the simulated steady-state response and the artefact due to overlapping epochs. This phenomenon is a consequence of the uncertainty principle in signal processing. The time-frequency approach allows investigating modulations either in the intensity or in the synchronization of the activity across trials over different frequency bands and at different time points. However, the uncertainty principle exposes a trade-off between frequency and time resolution (dt*dw>=1/2), that results in a smoothing of the measure of entrainment. With the parameters used here, at 3.33 Hz the width of the wavelet is 2.36 s, and at 1.11 Hz its width is 2.71 s, in both cases much smaller than the duration of a whole epoch (10.8s). The gain in time resolution, which has no interest here, comes with a loss in frequency resolution explaining the signal's smoothing.

The time-varying information provided by time-frequency approaches like wavelet decomposition is valuable in many contexts, but not meaningful in most steady-state stimulation experiments. In a steady-state experiment, the stimulus appears repeatedly, and the data are divided into sub-segments containing multiple instances of each stimulus. Therefore, looking for a time dimension within epochs is meaningless when all epochs are analyzed together. The strength of steadystate stimulation is that the neural response is periodic and stable, justifying a frequency domain analysis. To obtain a measure of phase coherence at each frequency using time frequency method, the measure must be averaged across the time dimension [START_REF] Batterink | Online neural monitoring of statistical learning[END_REF], 2019;[START_REF] Choi | Preverbal Infants Discover Statistical Word Patterns at Similar Rates as Adults: Evidence From Neural Entrainment[END_REF]. This step directly shows the lack of meaning of the time dimension in this context. Additionally, the computational cost of the time-frequency analysis on our simulated data set relative to the whole epoch approach was of the order of 10 4 times higher.

Comparison of the different methods

J o u r n a l P r e -p r o o f

To compare the sensitivity of the different methods more directly, we used the results obtained from the real data (dataset 1) and the real data plus the simulated signal at the syllabic and word rate (dataset 3). We assessed sensitivity as S = (E3(f) -E1(f)) / std(E1), where E3 is the normalized Power/ITC obtained from the dataset 3 (signal at the syllable and word frequencies), E1 is the normalized Power/ITC obtained from dataset 1 (no signal added), f is the frequency of interest.

The measure was divided by the standard deviation of E1 in the frequency-range 0.5-5 Hz, to estimate the signal relative to the noise level. We limited the analysis to the use of non-overlapping epochs.

We run 1-way ANOVAs with method as an independent factor to compare the methods' sensitivity (Figure 4). At the syllabic rate we observed a main effect of method (F(1,157) = 8.758, p = 0.00356). Pairwise comparison using Bonferroni correction showed that the effect was due to a difference between method 1 and 2 (p = 0.0018), method 1 and 3 (p = 8.7x10 -8 ), and method 3 and 4 (p = 0.005). All the other comparisons were not significant (p > 0.05). Results at the word rate were similar. We observed a main effect of method (F(1,157) = 6.018, p = 0.0153). Also in this case, pairwise comparison using Bonferroni correction showed that the effect was due to a difference between method 1 and 2 (p = 0.013), method 1 and 3 (p = 1.5x10 -5 ), and method 3 and 4 (p = 0.038). All the other comparisons were not significant (p > 0.05).

These results explicitly demonstrate the lower sensitivity of method 1 for detecting phase locked activity. Using time-frequency analysis to estimate the ITC also resulted in lower sensitivity than using the whole epoch, probably due to the loss of frequency resolution. Measuring neural entrainment based on the power of the average response (method 2) or the ITC (method 3) show comparable sensitivity. This can be understood, because, as mentioned earlier, both approaches J o u r n a l P r e -p r o o f rely on the presence of phase locked activity and both are obtained through FFT of the whole epoch.

Conclusions

Our simulations show that overlapping epochs unavoidably lead to artifactual entrainment at the overlap frequency. Nevertheless, this does not mean that overlapping epochs prevent the measurement of brain oscillations. The simulated neural entrainment is added to the methodological artefact. Thus, the main results of previous works using this approach might be valid especially when there was a difference with a control condition exposed to the same bias. In principle, if an adequate control condition is used, the entrainment derived from brain activity could be quantified.

J o u r n a l P r e -p r o o f

This is the case of one adult segmentation experiment [START_REF] Batterink | Online neural monitoring of statistical learning[END_REF], where the entrainment at the word and syllabic rates during the presentation of a structured stream of syllables was compared with the entrainment during a random stream of syllables. In a second paper, the authors investigated the effect of top-down attention by comparing speech segmentation while manipulating the attention to a competitive task in two groups of subjects [START_REF] Batterink | Statistical learning of speech regularities can occur outside the focus of attention[END_REF]. While there was no within-subjects control condition, we could assume that the methodological artefact is comparable between groups; thus, the comparison across groups would remain valid. In an infant segmentation task experiment using the same methodological approach [START_REF] Choi | Preverbal Infants Discover Statistical Word Patterns at Similar Rates as Adults: Evidence From Neural Entrainment[END_REF], the authors report an increase in entrainment to the word rate relatively to the syllabic rate during the familiarization. While the experiment did not have a control condition, in principle, the increase cannot arise from the methodological artefact.

Based on the above considerations, we do not question the validity of the data sets of Batterink and colleagues and their main results [START_REF] Batterink | Online neural monitoring of statistical learning[END_REF], 2019;[START_REF] Choi | Preverbal Infants Discover Statistical Word Patterns at Similar Rates as Adults: Evidence From Neural Entrainment[END_REF].

However, we believe that overlapping epochs should be avoided because they introduce an artefact that compromises the interpretation of the results, while not it does not providing a clear advantage.

For data transparency, we encourage the re-analysis of the dataset. Additionally, the time-frequency approach has an enormous computational cost compared to whole epoch methods, and the useless time dimension comes with a loss of frequency resolution. Therefore, we see no theoretical advantage in using time-frequency decomposition for a classical analysis of steady-state experiments. For the analysis of classical steady-state experiments, we recommend using whole epoch approaches such as method 2 or 3 on non-overlapping epochs.

Figure 1 .

 1 Figure 1. Description of the two epoching approaches and the four methods used to estimate neural entrainment. Note that methods 3 and 4 use the same approach (ITC) but on the whole epoch (method3) or on the time average of time frequency analysis (method 4)

Figure 3 .

 3 Figure 3. Results of the analysis with the different methods and simulations. The expected peaks at the syllable (3.33 Hz) and word (1.11) frequencies are clearly distinguishable from the control situation (blue line) when epochs are independent (left column, line 2 to 4). When overlapping epochs are used, artifactual peaks are present (right column). The time frequency approach used in method 4 (last line) inadequately smooths the frequency profiles.

Figure 4 .

 4 Figure 4. Comparison of the different methods to measure neural entrainment at the syllabic and word rate. The measure was computed as the difference between the entrainment when syllabic and word responses were added with the noise level of entrainment . Values were normalized by the standard deviation of the entrainment across frequencies in the range 0.5-5 Hz.

Acknowledgements

J o u r n a l P r e -p r o o f

This research has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 695710). We would like to thank Maria Chait for comments on the las version of the manuscript.


Ibtissem Bennacer
email: ibtissem.bennacer@univ-lemans.frremi.venant@univ-lemans.fr

Rémi Venant

Sébastien Iksal
email: sebastien.iksal@univ-lemans.fr

Towards a self-assessment tool for teachers to improve LMS mastery based on Teaching Analytics

Keywords: Teaching analytics, Learning Management System, Selfassessment, Peer recommendation, Clustering analysis, Principal Component Analysis

While learning management systems have spread for the last decades, many teachers still struggle to fully operate an LMS within their teaching, beyond its role of a simple resources repository. To elicit these learning situations, we suggest a web environment based on teaching analytics to provide teachers with self and social awareness of their own practices on the LMS. This article focuses on the behavioral model we designed on the strength of (i) a qualitative analysis from interviews we had with several pedagogical engineers and (ii) a quantitative analysis we carried out on three years of teachers' activities on an LMS at the scale of the University. This model describes teachers' practices through six major explainable axes: evaluation, reflection, communication, resources, collaboration as well as interactivity and gamification. It can be used by the institution to detect particular teachers who may be in need of specific individual support or conversely, experts of a particular usage of the LMS who could bring constructive criticism for its improvement. While instrumented in our environment, this model enables supplying teachers with self-assessment, automatic feedback and peer recommendations in order to encourage them to improve their skills with the LMS.

Introduction

The trend of using Learning Management Systems (LMS) is now spreading quickly across all areas of education. Most universities offer LMSs as a "one size fits all" technology solution for all teachers of any discipline. However, many teachers face several difficulties to integrate these platforms into their practices. The main problems of teachers appear to be technical or organizational, due to the lack of support and the lack of time devoted to its learning [START_REF] Fung | Factors affecting students' and teachers' use of lms-towards a holistic framework[END_REF]. Furthermore, most universities are hiring pedagogical engineers (PE), especially to support and train teachers in order to ensure a proper use of their LMS and ensure their pedagogical fit. With few PE compared to teachers, the formers struggle to support every teacher. For instance, in France, these problems were one of the reasons that led the Ministry of Higher Education to launch the HyPE-13 project(Hybridizing and Sharing Teachings). It aims to accompany teachers and students towards success by promoting the hybridization of training.

On the other hand, the use of LMS allows the capture of large amounts of quantitative data concerning the behavior of users and designers, and thus paves the way for Learning and Teaching Analytics (LA, TA). In our University, the LMS has been in place for more than 10 years. However, the University is facing the same issues we identified previously (LMS use expectations are not met and only 5 PE have to deal with more than 600 teachers). Our main objective is then to provide teachers with personal and social awareness [START_REF] Gutwin | Support for workspace awareness in educational groupware[END_REF], in order for them to engage in learning situations that aim at improving their LMS skills.

To reach this objective, we propose the design and the instrumentalisation of a teachers' behavior model to support their self-assessment and leverage peerlearning through automatic recommendations. We address here two first research questions: (i) How to model the exploitation of an LMS a teacher does and could do in an intelligible way ? (ii) What TA indicators can be instrument from this model to support self-assessment and enable feedback and recommendations?

Related work

Some researchers have focused on TA to understand how teachers deliver their lessons. For instance, to support the teacher inquiry process, [START_REF] Sampson | Teaching and learning analytics to support teacher inquiry[END_REF] identified TA as a necessary component, exploited in synergy with Learning Analytics (LA). For this purpose, [START_REF] Prieto | Teaching analytics: towards automatic extraction of orchestration graphs using wearable sensors[END_REF] used TA to automatically extract teachers' actions. To get out of the dependence on the technological context, [START_REF] Coomey | Online learning: it is all about dialogue, involvement, support and control-according to the research[END_REF] proposed a theoretical referential of good e-learning practices (DISC), while [START_REF] Peraya | Apprendre en présence et à distance: une définition des dispositifs hybrides[END_REF] empirically built a model to describe hybrid learning systems. On the other hand, some studies have been conducted to analyze teachers' behavior for different purposes as [START_REF] Whitmer | Patterns in blackboard learn tool use: Five course design archetypes[END_REF] aimed to uncover course design archetypes across multiple institutions and identified 5 groups consider courses with: mainly content and low interactions, one-way communication, strong peer interactions, more evaluation activities and those with a balance between content, communication and evaluation. Or, [START_REF] Regueras | Clustering analysis for automatic certification of lms strategies in a university virtual campus[END_REF] that proposed a method to automatically certify teachers' skills from LMS data and they were able to identify 6 types of courses based on teachers' practices (non-active, submission, deposit, communicative, evaluative, balance). Overall, these different studies show the importance of using analytical tools on the actions of teachers themselves, but it appears that the use of these behaviors for self-assessment has not yet been explored. In addition, they depict current platform usage, with the rejection of unused variables and cannot adapt to expected future uses.

Methodology

In order to qualify the current and expected teachers' uses of the LMS, we applied a quantitatively driven mixed method. We started with a quantitative analysis to deduce statistically different profiles of LMS use, in order to find groups of teachers or profiles of interest, based on the LMS log data. We performed a Principal Component Analysis (PCA) and a clustering analysis . PCA analysis allows to highlight diversity of the dataset in a reduced set of variables (components) while the clustering one aims to regroup the different instances of the dataset regarding their similarity. Afterwards, we conducted semi-structured interviews (i.e. : qualitative interview) with 3 female engineers on the same day (each lasted 40 to 50 minutes). In a series of open-ended questions prepared in advance to guide our interview, we collected information to improve the quantitative study which was analyzed by 2 researchers. This qualitative method was chosen because we needed the interviewee to answer freely, express a specific point of view, and bring out potential new working hypotheses.

We then performed a second quantitative analysis using the same previous method to address the engineer's comments by adding or modifying some variables. In order to design a behavior model that can handle both present and future expected usages of the LMS, we merge both results we obtained from this latest analysis and those we obtained from the interviews. Particularly, some of the discussed LMS features are still not used enough to appear in the results of the quantitative analysis. Moreover, the choice of the model axes (i.e.: the structure and how variables are grouped by axis) is also based on the results of the last PCA analysis and those of the qualitative interviews.

Finally, this model allowed to design several TA metrics. We applied clustering methods (K-Means, Dbscan, Agglomerative clustering and Gaussian Mixture) to be able to provide a social awareness based and defined interpretable scores to offer a more detailed personal awareness. Based on these metrics, we eventually designed a tool mainly dedicated to teachers, that supports selfassessment and awareness, but also can provide automatic peer recommendations using our model and metrics.

Teachers' behavior Model

Model definition

Through the intersection of the qualitative and quantitative studies, we designed a teacher behavioral model. It describes along six axes the behavior of teachers in a comprehensive way, and includes features that can be used to represent the current or potential use of the platform. The first axis (a.1 -Evaluation) represents the different tools used by the teacher to assess his students (quiz, assignemnt, attendance, calendar, grade). The second axis (a.2 -Reflection) concerns the LMS features that can provide teachers with a way to get feedback from students on their teaching and the digital resources they use (survey and choice). The third axis (a.3 -Communication) is devoted to the different means of communication used by the teacher to facilitate the transfer of information to the students and to improve the sharing between them (forum and chat). The fourth axis (a.4 -Resources) refers to the diversity of resources the teacher provides to students, and include then the file, book, folder, page, glossary and url features. Whereas the fifth one (a.5-Collaboration) concerns the promotion of collaboration between students with different LMS features (workshop, wiki, via, choice and data), the last axis (a.6 -Interactivity and Gamification) gathers the interactive or playful activities used by teachers to animate their courses and make them more attractive (lesson, course format, img, gallery, game and lti).

Teaching Analytics indicators

Based on the teachers' behavior model derived from the quantitative and qualitative analysis, we designed two TA metrics for awareness and self-assessment. The first metric is the LMS usage trends, it provides teachers with a current view by axis of their position relative to their colleagues. It was calculated by testing several clustering algorithms, and the results allowed teachers to identify the axes on which they are active and those on which they are not. The second metric propose two complementary scores for self-awareness to measure how the teacher profits from the LMS, based on the complete model we designed : (a) The score of curiosity that indicates the teacher's degree of curiosity according to each axis, takes into account the number of non null variables over all the teacher's courses. It aims to encourage to discover other LMS features within the axis.(b) The score of regularity that considers how often teachers exploit the features related to an axis with respect to their courses. In other terms, it helps validating a skill based on the repetition of practice.

Model and metrics exploitation

We started the development of a tool to engage teachers into learning situations regarding the different axes of our model. The main dashboard is represented in Figure 1. Once logged, the teacher can have an overview of his situation. Each axis is detailed within a card, with a different background color and subtitle whether the teacher was clusterized as active or inactive, and including the two different scores of curiosity and regularity. In the bottom right corner of the figure, we provide a radar visualisation that sums up the two scores for the teacher to have a quick comparative view of the different axes. Moreover, according to the different metrics our system can provide several automatic recommendations to improve the teacher's skill by recommending an active teacher if there is one, otherwise the system uses a fallback and recommends the PE.

Conclusion and perspectives

In this paper, we designed a behavioral model of teachers based on a qualitative analysis from interviews we had with three PE and a quantitative analysis we performed on teachers' activities on the University's LMS. This model describes teachers' practices through six major axes of mastery: evaluation, reflection, communication, resource, collaboration as well as interactivity and gamification. From this model, we designed several TA indicators. We proposed clustering Fig. 1. Teacher dashboard for self-assessment and recommendations models to make out non active and active teachers in a particular axis, as a metric for social awareness. For self-awareness, we took into account the complete model, including variables that relate unused features so far, into two different scores also proposed by axis (score of curiosity and score of regularity). However, our study presents several limitations. We have integrated all teacher traces on the University's LMS to analyze their behavior, knowing that many teachers use other technologies to manage their teaching, whom we do not have access to. Moreover, our study does not take into account what happens in a class, outside the technological environment and considers all teachers in the same way regardless of context. We will continue in the short term to refine our model with the inclusion and analysis of new features that would consolidate our axes and also our TA indicators. Indeed, once the first version of the tool will be operational, we will experiment it at the scale of our University to evaluate its usability, the interest teachers will show in it, and then test whether it allows inducing learning situations and if recommendations are followed and relevant.