Andrew Chang

Nathalie Bedoin

Laure-Helene Canette

Sylvie Nozaradan

Dave Thompson

Alexandra Corneyllie

Barbara Tillmann
email: barbara.tillmann@cnrs.fr

Laurel J Trainor

Atypical beta power fluctuation while listening to an isochronous sequence in dyslexia

Keywords: Dyslexia, Electroencephalography (EEG), Beta oscillation, Auditory perception, Entrainment electroencephalography (EEG), event-related potential (ERP), independent component analysis (ICA), standard deviation (SD)

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Introduction

Developmental dyslexia is a common reading disorder with a prevalence rate of around 5 to 10% [START_REF] Siegel | Perspectives on dyslexia[END_REF], featuring impairments in phonological awareness, such as difficulties in identifying rhyming words or syllabic stress patterns in speech [START_REF] Goswami | Impaired perception of syllable stress in children with dyslexia: A longitudinal study[END_REF]. Beyond deficits in speech perception, this impairment is also associated with deficits of auditory processing of rhythmic temporal regularity [START_REF] Goswami | A temporal sampling framework for developmental dyslexia[END_REF][START_REF] Goswami | Sensory theories of developmental dyslexia: Three challenges for research[END_REF][START_REF] Goswami | A neural oscillations perspective on phonological development and phonological processing in developmental dyslexia[END_REF], including detecting amplitude envelope onset, perceiving and producing rhythm, and extracting auditory rhythmic regularity [START_REF] Flaugnacco | Rhythm perception and production predict reading abilities in developmental dyslexia[END_REF][START_REF] Huss | Music, rhythm, rise time perception and developmental dyslexia: Perception of musical meter predicts reading and phonology[END_REF][START_REF] Leong | Rise time perception and detection of syllable stress in adults with developmental dyslexia[END_REF][START_REF] Leong | Impaired extraction of speech rhythm from temporal modulation patterns in speech in developmental dyslexia[END_REF]). The association between these two domains of auditory processing are also observed among typically-developing children: rhythm perception and the ability to synchronize to a beat are associated with phonological awareness, verbal short-term memory, rapid naming, and morphosyntactic accuracy in speech production (e.g., Woodruff Carr et al., 2014). Despite this evidence, we know little about the neural processing that underlies the rhythmic deficits in dyslexia, the focus of the present paper.

Neural oscillations have been proposed as an essential mechanism for tracking rhythm [START_REF] Haegens | Rhythmic facilitation of sensory processing: A critical review[END_REF][START_REF] Trainor | Is auditory perceptual timing a core deficit of developmental coordination disorder?[END_REF]. The power fluctuations of highfrequency oscillations (beta band: 15-25 Hz) entrain to rhythmic regularity in sound streams.

When participants listen to an isochronous tone sequence, the beta band power from auditory cortex decreases immediately following the onset of a tone, and then rebounds anticipatorily according to the onset time of the upcoming tone [START_REF] Cirelli | Beat-induced fluctuations in auditory cortical beta-band activity: using EEG to measure age-related changes[END_REF][START_REF] Fujioka | Internalized timing of isochronous sounds is represented in neuromagnetic beta oscillations[END_REF], consistent with a mechanism for predicting the onset time of the next expected tone. This entrainment activity can be modulated by hierarchical rhythmic structure or disrupted by a random sequence [START_REF] Fujioka | Beta and gamma rhythms in human auditory cortex during musical beat processing[END_REF][START_REF] Goswami | Sensory theories of developmental dyslexia: Three challenges for research[END_REF][START_REF] Iversen | Top-down control of rhythm perception modulates early auditory responses[END_REF][START_REF] Snyder | Gamma-band activity reflects the metric structure of rhythmic tone sequences[END_REF], and it is associated with auditory prediction, attentional processing and perceptual performance [START_REF] Chang | Beta oscillatory power modulation reflects the predictability of pitch change[END_REF][START_REF] Chang | Rhythmicity facilitates pitch discrimination: Differential roles of low and high frequency neural oscillations[END_REF][START_REF] Morillon | Motor origin of temporal predictions in auditory attention[END_REF]. Atypical beta power entrainment activities have been observed in populations featuring deficits in rhythm perception and tracking, such as patients with Parkinson's disease [START_REF] Te Woerd | Impaired auditory-tomotor entrainment in Parkinson's disease[END_REF][START_REF] Te Woerd | Entrainment for attentional selection in Parkinson's disease[END_REF] and children with stuttering [START_REF] Etchell | Abnormal time course of low beta modulation in non-fluent preschool children: a magnetoencephalographic study of rhythm tracking[END_REF]. However, it is unknown whether atypical beta power entrainment to auditory rhythmicity is associated with dyslexia.

In the present study, we measured the electroencephalographic (EEG) activities of individuals with dyslexia while they passively listened to an isochronous tone sequence. We hypothesized that the beta power fluctuation generated from auditory cortex is different between individuals with dyslexia and typical controls.

Methods

Participants

Thirteen adults with developmental dyslexia (seven women, ten right-handed; mean age = 23.2 years, SD = 2.95 years) and 13 matched controls (seven women, seven right-handed, mean age = 22.5 years, SD = 2.07) participated in the current experiment. Although handedness was not perfectly matched in the current study, the higher proportion of non-right-handers appears to be a feature of dyslexia (e.g., [START_REF] Eglinton | Handedness and dyslexia: A meta-analysis[END_REF]. All participants reported French as their native language, with one dyslexic participant reporting bilingualism with English. None reported auditory deficits. Education levels did not differ between the Dyslexic group (mean = 14.92 years; SD = 1.55) and the Control group (mean = 15.31 years, SD = 0.95), p = .45. Musical background, as measured by years of instrumental instruction, also did not differ between the Dyslexic group (mean = 2.00 years; SD = 2.16) and the Control group (mean = 1.69 years, SD = 2.13), p = 0.72.

Participants with dyslexia were part of a larger research project investigating dyslexia at the university level [START_REF] Abadie | Les étudiants dyslexiques à l'Université : Quels déficits cognitifs et langagiers?[END_REF]see Canette et al., 2020, Fiveash et al., 2020, for more information). As detailed in Table S2, they completed a set of language and neuropsychological tests, confirming the persistence of dyslexia for each participant. All participants with dyslexia reported having seen a speech therapist for a dyslexia diagnosis and for training designed to reduce reading difficulties for at least two years (and maximum 5 years) during childhood, which in France is generally defined as between 8 and 13 years old. They mainly have phonological difficulties (i.e., phonological or mixed forms of dyslexia, which are the most frequently observed forms of this neurodevelopmental pathology). Note that all participants with dyslexia had scores in the normal range for nonverbal intelligence (as measured by Raven's Matrices) and they also performed in the normal range for reading comprehension (average z-score placed them above the mean = 1.91, SD = 0.60). None of the participants reported psychiatric or neurological diagnoses or attention disorder with/without hyperactivity.

On the other hand, none of the controls reported a history of spoken or written language disorders. In particular we confirmed that, while at school, they did not have any difficulties in language understanding or production or learning to read, and none had been followed up by a speech therapist.

Written informed consent based on the French ethics procedure approval Committee (CPP Sud-Est II, 2007-009-3) was obtained from all participants prior to participating in the experiment, which was conducted in accordance with the guidelines of the Declaration of Helsinki. Participants received a compensation of 12 Euros per hour for their participation in the study.

Stimulus

During the EEG recording, participants passively listened for 5 minutes to an isochronous sequence of piano tones with an inter-onset interval of 500 ms, which is in the tempo range that gives rise to a strong sense of the beat [START_REF] Drake | The development of rhythmic attending in auditory sequences: attunement, referent period, focal attending[END_REF][START_REF] Merchant | Finding the beat: a neural perspective across humans and non-human primates[END_REF]. The tone was C4 (262 Hz), from the University of Iowa Musical Instrument Samples, and the amplitude envelope of the piano tone was percussive with 10 ms rise times. Tones were truncated to be 200 ms in duration, and a linear decay to zero was applied over the entire excerpt to remove offset artifact (Figure S3). For the behavioural post-test synchronization task, the same tone was played with the same inter-onset interval for either 10 repetitions (training trial) or 40 repetitions (experimental trials).

Procedure

To keep participants awake and still, they watched a silent black and white movie with Charlie Chaplin while they listened passively to the 5-minute isochronous tone sequence (and two other rhythmic patterns, not presented here) while EEG was continuously recorded 1 . This EEG recording was part of a longer experimental session and followed a priming experiment with an active task (reported in [START_REF] Canette | Regular rhythmic primes boost P600 in grammatical error processing in dyslexic adults and matched controls[END_REF][START_REF] Fiveash | A stimulus-brain coupling analysis of regular and irregular rhythms in adults with dyslexia and controls[END_REF]. After the EEG recording session, participants came back to the laboratory for a second testing session and 1 It is a common approach to have participants watch a silent movie during a passive listening task to keep them awake and reduce eyeblinks or eye movements during EEG/MEG recordings. As the movie is not time-locked to the tone sequence and is not rhythmic, its influence on EEG analyses is negligible (e.g., [START_REF] Kong | Differential modulation of auditory responses to attended and unattended speech in different listening conditions[END_REF][START_REF] Luo | Cortical encoding of acoustic and linguistic rhythms in spoken narratives[END_REF]. completed a set of behavioural tests on syntax processing and rhythm perception and production based on the complex Beat Alignment Task (Einarson and Trainor, 2015;data presented in Canette et al., 2020) and two rhythmic patterns (except for one dyslexic participant who did not come back to this second session); these data were presented in [START_REF] Canette | Regular rhythmic primes boost P600 in grammatical error processing in dyslexic adults and matched controls[END_REF]. In addition, relevant for the present experiment, participants performed a production task where they were required to tap along to the excerpts of the isochronous sequence used in the EEG session (i.e., inter-tone-onsets of 500 ms). Participants drummed with a stick on a drum pad (Roland, V-Drums) to each tone. Both the passive listening and production parts were implemented with the software Presentation (Neurobehavioural Systems), and stimuli were presented over headphones (Pioneer, HDJ-500). During a training trial, they tapped to a sequence of 10 tones. Following this, they completed two trials of 40 tones (i.e., 20 s duration) each.

EEG recording and preprocessing

The EEG signal was recorded with 95 Ag/AgCl active electrodes (actiCAP 96Ch Standard-2, Brain Products GmbH). The signal was recorded with a BrainAmp amplifier at a resolution of 16 bits, a sampling rate of 500 Hz, and with an analog low pass filter of 1000 Hz and high pass filter of 0.016 Hz. The ground electrode was placed at position AFz, the reference electrode on the tip of the nose, and an eye-movement monitoring electrode under the right eye at position Iz. Electrode impedances were kept below 20 kΩ, which is a well acceptable impedance levels for EEG recordings (e.g., [START_REF] Ferree | Scalp electrode impedance, infection risk, and EEG data quality[END_REF][START_REF] Luck | An introduction to the event-related potential technique[END_REF]. Continuous EEG data were transformed to an average reference offline. The EEG data was further processed in MATLAB with the FieldTrip toolbox (RRID: SCR_004849) (Oostenveld et al., 2011).

Independent component analysis (ICA) was used to remove artifact signals (Jung et al., 2000). To speed up the ICA performance, the continuous recordings were downsampled to 250 Hz, filtered between 0.7 and 83 Hz and then segmented into time window from -3 to 303 s timelocked to the first tone of the sequence. This window covered the entire stimulus sequence length. We went back to the unfiltered data, projected it to ICA space and the ICs reflecting artifact (identified by visual inspection, range of 2-13 ICs), including eye blinking, eye movement, electrocardiogram, and powerline noise, were excluded, and then the data in the ICA space was projected back to 95-channel space for all subsequent analyses.2

Modeling dipole sources for auditory cortex

In the present study, we focused on the responses generated from auditory cortex, as prior studies have shown that auditory beta oscillations are related to temporal prediction and rhythmic tracking [START_REF] Chang | Beta oscillatory power modulation reflects the predictability of pitch change[END_REF][START_REF] Chang | Rhythmicity facilitates pitch discrimination: Differential roles of low and high frequency neural oscillations[END_REF][START_REF] Cirelli | Beat-induced fluctuations in auditory cortical beta-band activity: using EEG to measure age-related changes[END_REF][START_REF] Fujioka | Internalized timing of isochronous sounds is represented in neuromagnetic beta oscillations[END_REF][START_REF] Fujioka | Beta-band oscillations represent auditory beat and its metrical hierarchy in perception and imagery[END_REF]. To achieve this goal, we used a dipole model as a spatial filter to largely separate the activities of auditory cortex from other sources, following the studies cited above, which is preferred to analyzing data on selected surface electrodes, as each surface electrode reflects a mixture of activities from many brain sources. The P1 ERP component (~60-90 ms) was used to localize bilateral auditory cortices, with the dipole locations (but not orientation) constrained to be bilaterally symmetric. We used the multiple source probe scan algorithm and the four-shell ellipsoid model included in the Brain Electrical Source Analysis (BESA, RRID: SCR_009530) software package. We chose P1 because (1) it is primarily generated from primary auditory cortex (Godey et al., 2001), (2) it is the dominant peak at fast presentation rates while the N1 peak is strongly reduced at fast rates relevant to the current study [START_REF] Näätänen | The N1 wave of the human electric and magnetic response to sound: a review and an analysis of the component structure[END_REF], and (3) previous studies with similar experimental designs also used P1 for localizing auditory cortex (e.g., [START_REF] Fujioka | Internalized timing of isochronous sounds is represented in neuromagnetic beta oscillations[END_REF]. The mean locations (averaged across participants) of the fitted dipoles were [±44. 23, -1.95, 7.26] (Talairach coordinates) with approximate mean orientations [0.2, 0.5, 0.8] (left) and [-0.2, 0.6, 0.8] (right). These locations are close to bilateral primary auditory cortices with orientations toward the mid-frontal surface area, consistent with typical auditory evoked potentials. The mean residual variances of the source fittings were 7.9% (range 3.4 to 16.3%) for the Control group and 7.0% (range 2.4 to 15.4%) for the Dyslexia group, and the residual variances were not different between groups (t(24) = 0.67, p = 0.509). Finally, the unfiltered continuous 95-channel EEG was projected into source-space EEG via the dipole model for further time-frequency analyses.

Note that it is important that we perform the analyses at dipole locations, as the MEG study of [START_REF] Fujioka | Internalized timing of isochronous sounds is represented in neuromagnetic beta oscillations[END_REF] reported that the beta power generated in motor regions fluctuated at the opposite phase of the beta power in auditory cortex. As the spatial resolution of EEG is less precise than MEG, these two signals are likely to be mixed and potentially cancel each other in EEG analyzed from the surface channels.

Time-frequency analyses

The unfiltered continuous source-space EEG data was segmented to include nonoverlapping epochs of 20 tones each. Each epoch was thus 11 s long, including 0.5 s prior to the first tone and 0.5 s following the twentieth tone as buffer periods to avoid edge artifacts for the time-frequency analysis. The first epoch at the beginning of each recording was excluded from further analysis, as the rhythmic temporal regularity might not yet be perceived. Subsequently, epochs were excluded that were greater than 150 μV to ensure artifacts were eliminated that were not removed by ICA. The mean number of usable trials was 23.8 in the Control group and 26.7 in Dyslexia group; this was not significantly different between groups (t(24) = 1.20, p = 0.244).

We focused on the induced (non-phase-locked) power in beta band, following our previous studies (e.g., [START_REF] Fujioka | Internalized timing of isochronous sounds is represented in neuromagnetic beta oscillations[END_REF]Chang et al., 2016a[START_REF] Chang | Beta oscillatory power modulation reflects the predictability of pitch change[END_REF][START_REF] Chang | Rhythmicity facilitates pitch discrimination: Differential roles of low and high frequency neural oscillations[END_REF][START_REF] Cirelli | Beat-induced fluctuations in auditory cortical beta-band activity: using EEG to measure age-related changes[END_REF].

For each participant, the unfiltered mean ERP waveform (averaged across trials) was subtracted from each epoch to obtain induced waveforms [START_REF] Fujioka | Internalized timing of isochronous sounds is represented in neuromagnetic beta oscillations[END_REF][START_REF] Cohen | Analyzing neural time series data: theory and practice[END_REF]. We used a short-time Fourier transformation with overlapped time windows (moving successively by one sample) to perform the time-frequency transformation for 15-25 Hz (frequency bin size = 1 Hz).

The window size was fixed at 500 ms and a Hanning taper was applied. For each frequency bin, the power was baseline corrected (percentage change) to the mean power for the 0-10 s window, then the beta power was obtained by averaging the power across 15-25 Hz bins. To further analyze the fluctuation of the beta power time series, we performed another Fourier transformation with a Hanning taper for the beta power time series for each 10 s epoch (singletrial), excluding the 0.5 s buffer periods at both ends.

Although some studies have defined the range of beta band to be as wide as 13-30 Hz, we defined it as 15-25 Hz for several reasons. (1) It is consistent with our previous studies (e.g., [START_REF] Fujioka | Internalized timing of isochronous sounds is represented in neuromagnetic beta oscillations[END_REF][START_REF] Chang | Rhythmicity facilitates pitch discrimination: Differential roles of low and high frequency neural oscillations[END_REF]. ( 2) Some other studies have interpreted activities below 15 Hz as alpha band (e.g., [START_REF] Ahveninen | Suppression of irrelevant sounds during auditory working memory[END_REF] and above 25 Hz as low-gamma band (e.g., [START_REF] Giroud | Asymmetric sampling in human auditory cortex reveals spectral processing hierarchy[END_REF]. Therefore, it is debatable whether to include these frequencies as beta band.

(3) Considering the imprecision of time-frequency analysis in terms of frequency resolution, activities closer to 13 or 30 Hz will be more likely contaminated by activities below 13 or above 30 Hz, respectively, compared to frequencies between 15 and 25 Hz.

For completeness, we also analyzed event-related potentials (ERPs) (see Supplementary Materials).

Tapping production task

We used circular statistics [START_REF] Berens | CircStat: A MATLAB toolbox for circular statistics[END_REF]Dalla Bella and Sowinski, 2015) to analyze the production performance of the synchronization task for isochronous sequences, leading to two measures related to the resulting vector R: its angle, representing synchronization accuracy and its length, representing synchronization consistency (ranging from 0 to 1, with 1 representing perfect consistency).

Statistics

We used a nonparametric cluster-based permutation test to analyze the EEG measure difference between groups with the FieldTrip toolbox (Oostenveld et al., 2011). In short, this procedure reduces the number of multiple comparisons by testing the statistical difference at the level of time and/or frequency clusters instead of individual time and/or frequency points, and it has been widely used for analyzing EEG and MEG data (see Maris and Oostenveld, 2007 for more details). Specifically, first we performed a two-sample t-test between the two groups (Dyslexia, Control) for each time or frequency sample as the initial scan for clustering. Second, we grouped adjacent frequency (Figure 1B) or time (Figure S1A and S1C) samples reaching a threshold of p < 0.05 in the initial scan into single clusters and summed the t-values of all samples within each cluster as a cluster-level statistic. Third, we built a null distribution with 1000 random iterations. In each iteration, we pooled all 26 participants (13 in Dyslexia group and 13 in Control group), randomly split participants into 2 equal-size groups, and performed the same calculations as in the first two steps. The null distribution was composed by the summed tvalues of the largest suprathreshold cluster of each iteration, and the final p-value was obtained by comparing the observed clustered summed t-value relative to the null distribution. In sum, if there was no group difference, the cluster(s) of the observed data should resemble the randomsplit data of the null distribution. The final p-value was corrected to two-tailed.

The circular statistics were computed in the Circular Statistics Toolbox for MATLAB [START_REF] Berens | CircStat: A MATLAB toolbox for circular statistics[END_REF]. Participant group differences for R angle and length were assessed with twotailed independent tests. Note that the Watson-Williams test assumes the data to be distributed according to a von Mises distribution, i.e., with concentration parameter (κ) equal or above 1. Therefore, we examined whether our 2 Hz beta power fluctuation time series met this assumption. The κ parameter was similar between compared datasets (difference range: 0.05-0.14), with only κ parameters of the left and right auditory cortex of the Control group above 1. Although our data did not fully satisfy the assumptions, the Watson-Williams test is considered robust against deviations from these assumptions [START_REF] Berens | CircStat: A MATLAB toolbox for circular statistics[END_REF].

Results

Power spectrum of beta power time series

The beta power time series of left and right auditory cortex are visualized in Figure 1A.

The Control group showed typical beta power fluctuations in that the peaks occurred at the approximate times of tone onsets (with anticipatory increase in power prior to tone onsets) and troughs between tones. This pattern is similar to those observed in previous studies (e.g., [START_REF] Chang | Rhythmicity facilitates pitch discrimination: Differential roles of low and high frequency neural oscillations[END_REF][START_REF] Fujioka | Internalized timing of isochronous sounds is represented in neuromagnetic beta oscillations[END_REF][START_REF] Fujioka | Beta-band oscillations represent auditory beat and its metrical hierarchy in perception and imagery[END_REF]. In contrast, the beta power fluctuations appear atypical in the Dyslexia group in that troughs occurred at the approximate times of tone onsets and peaks occurred between tones.

===========

Figure 1 =========== To examine the frequency characteristics of the beta power fluctuations, we frequencytransformed the beta power time series into a power spectrum for each participant. The spectra of beta power fluctuations (note that this is not the spectra of raw EEG waveform) are shown in Figure 1B. The cluster-based permutation tests on left or right auditory cortex did not show any significant differences in the power spectrum of beta power fluctuations between the Dyslexia and Control groups (range of power fluctuations 1-5 Hz). There was also no interaction between left/right auditory cortex and groups.

The frequency spectrum of beta power fluctuation (Figure 1B) did not show an obvious peak at 2 Hz as expected, despite the appearance of a power fluctuation around the 2 Hz tone onset rate in the averaged time domain waveforms (Figure 1A). Nevertheless, we speculated that the beta power fluctuations likely differed in phase between the two groups, and we extracted the phase of the beta power time series at 2 Hz (the stimulus presentation rate) from the frequency transformation above (Figure 1C) for subsequent exploratory analyses. Note that the phase of beta power time series refers to the peak/trough position of the fluctuations in the measured beta power relative to the onsets of the stimuli. A Watson-Williams two-sample test showed that the phase of the beta power time series differed significantly at right auditory cortex (F(1,24) = 12.90, p = .002), but not at left auditory cortex, although there was a trend for a difference (F(1,24) = 3.51, p = .073). The interaction between groups and left/right auditory cortex (taking the circular difference between hemispheres within each participant, and then performing a between-subject Watson-Williams test) was not significant (F(1,24) = 1.57, p = .222).

To investigate whether this significant right auditory cortex phase difference of beta power fluctuation between groups was specific to 2 Hz, we further performed the same analyses at 1 and 3 Hz in the right auditory cortex. At 3 Hz, there was no significant difference between groups, F(1,24) = 0.88, p = .356. Further, the group difference was larger at 2 Hz than at 3 Hz (taking the circular difference between frequencies within each participant, and then performing a between-subject Watson-Williams test), F(1,24) = 9.47, p = .005. Thus, the phase differences between groups were markedly larger at 2 Hz than at 3 Hz. At 1 Hz, the group difference was not significant, but there was a trend, F(1,24) = 3.33, p = .081. There was also a trend for larger group difference at 2 Hz than at 1 Hz, F(1,24) = 3.37, p = .079. Together these results suggest that the phase difference of beta band power fluctuation is likely specific to 2 Hz, the rate of tone presentation in the stimulus.

Tapping production task.

The two participant groups differed in neither synchronization consistency (vector length; t(23) = -0.10, p =.93) nor synchronization accuracy (vector angle; Watson-Williams two-sample test: F(1,23) = 0.00, p =.99). Both control and dyslexic participants performed well on the tapping task, reaching synchronization consistency (vector length) close to one (controls: .95 ±.05; dyslexics: .95 ±.04) and synchronization accuracy (vector angle) suggesting weak anticipation of the next tone (controls: -11 ms ± 13; dyslexics: -15 ms ± 39).

Discussion

The results of the current study revealed that the phase of beta power fluctuation was different between adult university students with dyslexia and typical controls while listening to an isochronous tone sequence. To the best of our knowledge, this is the first study reporting atypical beta power entrainment in dyslexia.

The atypical beta power fluctuation in dyslexia might reflect deficits in tracking and/or perceiving auditory rhythms. Previous studies on typical healthy participants showed that the fluctuation of beta power reflects entrainment to the rhythmic regularity of an auditory sequence.

Specifically, the peaks of the beta power align to the onset of the tones, and are associated with rhythm tracking and auditory perception and prediction [START_REF] Chang | Beta oscillatory power modulation reflects the predictability of pitch change[END_REF][START_REF] Chang | Rhythmicity facilitates pitch discrimination: Differential roles of low and high frequency neural oscillations[END_REF][START_REF] Cirelli | Beat-induced fluctuations in auditory cortical beta-band activity: using EEG to measure age-related changes[END_REF][START_REF] Fujioka | Beta and gamma rhythms in human auditory cortex during musical beat processing[END_REF][START_REF] Fujioka | Internalized timing of isochronous sounds is represented in neuromagnetic beta oscillations[END_REF][START_REF] Fujioka | Beta-band oscillations represent auditory beat and its metrical hierarchy in perception and imagery[END_REF][START_REF] Snyder | Gamma-band activity reflects the metric structure of rhythmic tone sequences[END_REF][START_REF] Iversen | Top-down control of rhythm perception modulates early auditory responses[END_REF]. Our neural findings are consistent with previous studies showing that individuals with dyslexia have deficits in perceiving the rhythmic modulation of nonverbal auditory streams and speech [START_REF] Goswami | Amplitude envelope onsets and developmental dyslexia: A new hypothesis[END_REF][START_REF] Goswami | A temporal sampling framework for developmental dyslexia[END_REF][START_REF] Goswami | Sensory theories of developmental dyslexia: Three challenges for research[END_REF][START_REF] Goswami | A neural oscillations perspective on phonological development and phonological processing in developmental dyslexia[END_REF][START_REF] Megnin-Viggars | Audiovisual perception of noise vocoded speech in dyslexic and nondyslexic adults: The role of low-frequency visual modulations[END_REF]). Nevertheless, it is important to note that we cannot eliminate all explanations based on factors beyond entrainment, such as, for example, that the participants in the Dyslexic group were more distracted by the visual movie, leading to less attention to the auditory stimuli.

It was unexpected that the frequency spectrum of beta power fluctuation (Figure 1B) did not show an obvious peak the at stimulus presentation rate (2 Hz). To the best of our knowledge, our study is the first one reporting the spectrum of beta power fluctuation, while most previous studies only reported its time-domain features. The lack of an obvious spectral peak could simply be due to a smaller signal-to-noise ratio in the current study, as the number of trials was smaller and the EEG recording time was shorter than in previous studies (e.g., [START_REF] Fujioka | Internalized timing of isochronous sounds is represented in neuromagnetic beta oscillations[END_REF]. It is also possible that the power fluctuation was not perfectly periodic; the moment-to-moment fluctuation rate might have drifted around 2 Hz on individual trials, and thus the 2 Hz peak was not obvious in the averaged spectrum with traditional Fourier-based time-frequency transformation approaches, which assume the signals to be stable over time. Future studies using more advanced cycle-by-cycle analysis approaches (e.g., [START_REF] Cole | Cycle-by-cycle analysis of neural oscillations[END_REF] are needed to investigate the dynamic beta oscillatory activities.

There are a few studies reporting an association between beta oscillations and dyslexia. A previous study reported that auditory steady-state response synchronization is reduced in the beta frequency range in dyslexia, and that this response is associated with literacy skills [START_REF] Van Hirtum | Atypical neural processing of rise time by adults with dyslexia[END_REF]. Differences in overall beta power at occipital channels between individuals with and without dyslexia have also been found during listening to noise-vocoded speech [START_REF] Power | Neural encoding of the speech envelope by children with developmental dyslexia[END_REF]. An atypical hemispheric asymmetry of beta power in response to linguistic tasks has also been associated with reading skill or processing phonological information in dyslexia (e.g., [START_REF] Penolazzi | Brain plasticity in developmental dyslexia after phonological treatment: A beta EEG band study[END_REF][START_REF] Spironelli | Dysfunctional hemispheric asymmetry of theta and beta EEG activity during linguistic tasks in developmental dyslexia[END_REF]. The present study extends these studies by indicating that the beta oscillatory activity in response to a simple, rhythmic (isochronous) sound sequence is also atypical in dyslexia.

Atypical beta power fluctuation has also been observed in other disorders featuring deficits in perceiving or tracking auditory rhythm. For example, Parkinson's patients are known to have deficits in tracking auditory rhythms [START_REF] Grahn | Impairment of beat-based rhythm discrimination in Parkinson's disease[END_REF][START_REF] Grahn | Neural mechanisms of rhythm perception: current findings and future perspectives[END_REF], and the phase of their beta power fluctuation while perceiving auditory or visual rhythms was opposite to that of the controls [START_REF] Te Woerd | Impaired auditory-tomotor entrainment in Parkinson's disease[END_REF][START_REF] Te Woerd | Entrainment for attentional selection in Parkinson's disease[END_REF]. Also, children with stuttering have deficits in perceiving auditory rhythm [START_REF] Chang | Relation between functional connectivity and rhythm discrimination in children who do and do not stutter[END_REF][START_REF] Falk | Non-verbal sensorimotor timing deficits in children and adolescents who stutter[END_REF][START_REF] Wieland | Evidence for a rhythm perception deficit in children who stutter[END_REF], and the phase of their beta power fluctuation was also opposite to that of controls [START_REF] Etchell | Abnormal time course of low beta modulation in non-fluent preschool children: a magnetoencephalographic study of rhythm tracking[END_REF].

Although previous studies investigating children with dyslexia showed deficits in their ability to tap to a beat or perceive a beat [START_REF] Muneaux | Deficits in beat perception and dyslexia: Evidence from French[END_REF][START_REF] Thomson | Rhythmic processing in children with developmental dyslexia: Auditory and motor rhythms link to reading and spelling[END_REF], in the present study, performance on a simple metronome tapping task did not differ between adults with dyslexia and typical controls, and we failed to find any associations between atypical beta power fluctuation and tapping production performance to the same isochronous sequence (Figure S2). In addition to the relatively small sample size and thus potentially insufficient statistical power, it is possible that (1) our task was too easy and thus not sensitive enough to reflect a deficit in rhythm processing in dyslexia. Indeed, in another study, our participants with dyslexia were worse than controls [START_REF] Canette | Regular rhythmic primes boost P600 in grammatical error processing in dyslexic adults and matched controls[END_REF] on performing more complex tapping task (e.g., tapping to the beat of musical excerpts). Furthermore, (2) individuals with dyslexia might find other ways and use different neural mechanisms to compensate as they get older (cf. [START_REF] Shaywitz | Neural systems for compensation and persistence: young adult outcome of childhood reading disability[END_REF][START_REF] Law | Morphological Awareness and Its Role in Compensation in Adults with Dyslexia[END_REF], making it challenging to directly associate beta oscillation with rhythmic abilities. In particular, previous behavioral studies have revealed greater deficits in tapping tasks for dyslexic adolescents than for dyslexic university students, who performed as well as controls on some of the measures [START_REF] Wolff | Rate and timing precision of motor coordination in developmental dyslexia[END_REF]; see also [START_REF] Thomson | Auditory and motor rhythm awareness in adults with dyslexia[END_REF]. Furthermore, the observed variability of the dependent measures of the participants with dyslexia in our study suggests that individuals might use different neural mechanisms.

In additional analyses examining ERPs (presented in the Supplementary materials), another common approach to examining auditory neural signature of dyslexia, we found a group difference in an early ERP component at the P1 latency at Fpz, and a marginal correlation between this component and the phase of beta power fluctuation (Figure S1). A possible post hoc explanation is that the atypical beta power fluctuation in dyslexia is associated with deficits in early auditory processing, such as encoding rise time and duration [START_REF] Bamiou | Aetiology and clinical presentations of auditory processing disorders-a review[END_REF][START_REF] Hämäläinen | Basic auditory processing deficits in dyslexia: Systematic review of the behavioral and event-related potential/field evidence[END_REF][START_REF] Sharma | Comorbidity of auditory processing, language, and reading disorders[END_REF][START_REF] King | Comorbid auditory processing disorder in developmental dyslexia[END_REF]. However, our study was not designed to examine this possibility, so future studies are needed with a larger sample size to investigate this possibility.

The current study did not investigate delta (1-3 Hz) oscillations or their cross-frequency coupling with beta oscillations, despite previously reported associations between delta oscillations and dyslexia. Both animal electrophysiology and human neuroimaging studies have shown that the delta oscillation phase time-locks and entrains to external rhythmic sensory input (see [START_REF] Haegens | Rhythmic facilitation of sensory processing: A critical review[END_REF] for a review) and it has been reported that atypical delta phase entrainment is associated with deficits of rhythm perception and tracking among individuals with dyslexia [START_REF] Colling | Neural entrainment and sensorimotor synchronization to the beat in children with developmental dyslexia: An EEG study[END_REF][START_REF] Di Liberto | Atypical cortical entrainment to speech in the right hemisphere underpins phonemic deficits in dyslexia[END_REF][START_REF] Molinaro | Out-of-synchrony speech entrainment in developmental dyslexia[END_REF][START_REF] Power | Neural entrainment to rhythmic speech in children with developmental dyslexia[END_REF][START_REF] Power | Neural encoding of the speech envelope by children with developmental dyslexia[END_REF]. However, recent studies have shown that it is challenging to reliably quantify delta phase entrainment because the frequency-domain signature of ERP activity typically overlaps with low-frequency oscillations in Fourier-based analysis [START_REF] Doelling | An oscillator model better predicts cortical entrainment to music[END_REF][START_REF] Haegens | Rhythmic facilitation of sensory processing: A critical review[END_REF]. For this reason, we have left the question of how delta and beta oscillations are related in dyslexia to future studies with materials and designs more appropriate for addressing this question. Note that the induced beta oscillations measured in the current study are unlikely to be confounded with ERPs because (1) the mean spectrum of the ERP showed that the power above 15 Hz was neglectable (Figure S4), and (2) we removed the phase-locked evoked activity from the induced activity in the beta band prior to conducting our analyses.

Understanding atypical beta power entrainment could potentially benefit individuals with dyslexia. Studies have shown that rhythmic auditory or musical training can improve the phonological awareness and reading skills in children with dyslexia, including randomized control trials (e.g., [START_REF] Cogo-Moreira | Effectiveness of music education for the improvement of reading skills and academic achievement in young poor readers: a pragmatic cluster-randomized, controlled clinical trial[END_REF][START_REF] Flaugnacco | Music training increases phonological awareness and reading skills in developmental dyslexia: A randomized control trial[END_REF][START_REF] Habib | Music and dyslexia: A new musical training method to improve reading and related disorders[END_REF][START_REF] Thomson | Auditory processing interventions and developmental dyslexia: A comparison of phonemic and rhythmic approaches[END_REF]. The efficacy of longer-term musical rhythm training in language remediation is likely related to the importance of rhythm for decoding the speech stream [START_REF] Flaugnacco | Music training increases phonological awareness and reading skills in developmental dyslexia: A randomized control trial[END_REF][START_REF] Kotz | Cortical speech processing unplugged: A timely subcortico-cortical framework[END_REF][START_REF] Overy | Dyslexia, temporal processing and music: The potential of music as an early learning aid for dyslexic children[END_REF][START_REF] Schön | Short-and long-term rhythmic interventions: Perspectives for language rehabilitation[END_REF].

In typically-developing children and in children with language disorders, presenting a rhythmic prime has been shown to immediately improve syntactic processing over the short term [START_REF] Bedoin | Temporally regular musical primes facilitate subsequent syntax processing in children with specific language impairment[END_REF][START_REF] Chern | New evidence of a rhythmic priming effect that enhances grammaticality judgments in children[END_REF][START_REF] Przybylski | Rhythmic auditory stimulation influences syntactic processing in children with developmental language disorders[END_REF]. Future studies are needed to investigate whether beta power entrainment is an underlying neural mechanism of these effects, and whether individual differences in beta power entrainment could predict the potential benefit of auditory rhythmic habilitation programs.

Figure 1. The beta (15-25 Hz) power at bilateral auditory cortices. (A) The beta power time series are presented as the mean ± standard error across typical (control) and dyslexic participants. (B) The spectra of beta power fluctuation (note that this is not the spectra of the raw EEG waveform). (C) The phases of beta power time series at 2 Hz. Each blue arrow represents the phase angle of beta power for a single participant; the phase angles and lengths of the red arrows represent the group-averaged angle and consistency of the angle distribution, respectively. The phase angles differed significantly between groups at the right auditory cortex. (p: p-value; a.u.: arbitrary unit)

Although the electromagnetic field of the headphones was not masked, which might lead to some contamination of EEG recordings from sound waveforms, any potential headphone artifact should have had a negligible influence on our beta band findings because: (1) the observed beta power fluctuation shape was similar to those of previous studies (see Power spectrum of beta power time series); (2) we observed group differences (while a potential artefact would have equally affected recordings of each participant); and (3) this potential artifact was most likely attenuated by the ICA procedure. A potential artifact might have had more influence on early ERP analyses (reported in the Supplementary Materials), which showed somewhat atypical scalp topography in comparison with previous studies.

Acknowledgements

We thank Raphaëlle Abadie for her help in the neuropsychological testing of the dyslexic participants. This research was supported by grants from ANR to BT and NB (Grant Agreement number ANR-16-CE28-0012), grants to LJT from the Canadian Institutes of Health Research (MOP 153130) and the Canadian Institute for Advanced Research, a visiting exchange fellowship to LJT from the Erasmus Mundus Auditory Cognitive Neuroscience network, grant to SN from the ERC (ERC Starting Grant 2018 number 801872 Rhythm and Brains), and a Vanier Canada Graduate Scholarship to AC. The team "Auditory cognition and psychoacoustics" is part of the LabEx CeLyA ("Centre Lyonnais d'Acoustique", ANR-10-LABX-60), and L-HC is supported by a post-doctoral grant of the LabEx CeLyA.

Conflict of Interest Statement

None


Catherine Goldstein
email: catherine.goldstein@imj-prg.fr

Diophantus redivivus: is Diophantus an early-modern classic?

Many early-modern mathematical books incorporated at least a part of Diophantus' Arithmetica, from Jacques de Billy's Diophanti Redivi Pars prior et posterior to John Kersey's Third and Fourth Books of the Elements of algebra or Jacques Ozanam's Recréations mathématiques. Diophantine questions regularly circulated among mathematicians of the time in the context of exchanges of problems or challenges [START_REF] Goldstein | Routine Controversies : Mathematical Challenges in Mersenne's Correspondence[END_REF]. It is thus tempting to consider Diophantus's opus magnum as a classic. However, I argued in my talk that, while Diophantus was indeed a classical author for early-modern mathematicians, his main work did not become a classical book.

The first point is easy to establish. According to the definition of the Dictionnaire de l'Académie [1, vol. I, p. 197 he who surpasses all the others by a long interval is their coryphaeus, and easily their prince.

Still, several historiographical issues are at stake. First of all, at least two authoritative versions of Diophantus' Arithmetica are referred to by most mathematicians, Franciscus Vieta's Zetetica [START_REF] Viète | Zeteticorum libri quinque[END_REF] and the heavily commented edition, with a Latin translation, published by Claude-Gaspard Bachet de Meziriac [START_REF] De Méziriac | Diophanti Alexandrini Arithmeticorum libri sex[END_REF]. These texts belong to different genres and provide their readers with different organizations and selections of Diophantine material, different symbolisms and textual marks, and even different ideas of what constitutes an adequate solution [START_REF] Morse | The Reception of Diophantus' Arithmetic in the Renaissance[END_REF] (see also Abram Kaplan's contribution to this workshop).

On the other hand, both, in various degrees, present Diophantus as the father of algebra, a clue followed by most early-modern mathematicians who followed them. Billy, mentioned above, for instance, goes on [4, Lectori Benevolo. . . ]: Diophantus remained within the limits of Arithmetic, not the vulgar kind that is taught to children and merchants, but another, more subtle one, that one calls Algebra and that is the science of unknown numbers starting from hypotheses.

Even this agreement poses problems: one of its consequences is that Diophantine questions are usually treated in textbooks on algebra, another that algebraic formulas are often presented as a natural and valid generalization of Diophantus's original, unique, solution in fractions. Some, like Fermat, advocated at the time against this trend, but in favor of a search for integer solutions only. However, none of these developments correspond to the current idea about Diophantine questions, with its emphasis on the description and computation of rational solutions [START_REF] Goldstein | Un Théorème de Fermat et ses lecteurs[END_REF][START_REF] Schappacher | Wer war Diophant[END_REF].

A last, intriguing, issue concerns the French scene particularly. In French literature, "classical" has been more and more associated with the new standards emphasized in the framework of, or parallel to, the courtly culture of Louis XIV's times; opposed to the heavy and contrived volumes of the schools [START_REF] Dhombres | Une mathématique baroque en Europe : réseaux, ambitions et acteurs[END_REF], classical texts, in this sense, were supposed to share a set of characteristics, such as elegance, simplicity, naturalness and correctness [START_REF] Viala | Qu'est-ce qu'un classique ?[END_REF][START_REF] Ribot | Qu'est-ce qu'un classique ?[END_REF]. A question is thus whether Diophantine problems found a home in mathematical books that can be qualified as classical, and how.

To study these issues, I analyzed in the talk several works linked to Diophantus's Arithmetica, focussing for the sake of time to French authors of the second half of the seventeenth century: Billy's Diophantus geometra [START_REF] Billy | Diophantus Geometra[END_REF] and Diophantus redivivus [START_REF] Billy | Diophanti redivivi Pars prior et Pars posterior[END_REF][START_REF] Cassinet | Problèmes Diophantiens en France à la fin du 17ème siècle[END_REF], Ozanam's manuscript Diophante reduit à la specieuse and Traité des simples, des doubles and des triples égalités pour la solution des problèmes en nombres [START_REF] Cassinet | Le Traité des simples, des doubles et des triples égalités, de Jacques Ozanam, dans le manuscrit autographe perdu puis retrouvé de la bibliothèque du chancelier d'Aguesseau[END_REF], as well as his Nouveaux elemens d'algebre [START_REF] Ozanam | Nouveaux elements d'algebre[END_REF], Jean Prestet's Elemens de mathematiques (both editions) [START_REF] Prestet | Elemens des mathematiques, ou principes generaux de toutes les sciences qui ont les grandeurs pour objet[END_REF][START_REF] Prestet | Nouveaux Elements des mathematiques[END_REF][START_REF] Asselah | Arithmétique et algèbre dans la 2e moitié du 17e siècle français : les Elemens et Nouveaux elements de mathématiques de Jean Prestet[END_REF], and finally Bernard Frenicle de Bessy's posthumous Traité des triangles rectangles en nombres [START_REF] Goldstein | Un Théorème de Fermat et ses lecteurs[END_REF]. For each of them, I surveyed the selection and organization of the material, as well as the individual treatment of some of the problems and their solutions. This shows that if Diophantus' Arithmetica remained an inspiration for new isolated mathematical problems, in algebra, number theory and even in Euclidean geometry, they entered into a large variety of partial reconstructions, within different genres. There was no agreement at the time on what was a satisfactory solution to Diophantine problems: even with algebraic formulas, for instance, and the claims of their authors to go further than Diophantus in providing infinitely many solutions instead of a single one, no proof of the infinity of the solutions was ever given, nor that they were all obtained. Attempts to restructure Diophantine problems and eventually integrate them in a text adapted to the new fashion (such as Ozanam's Traité des simples, des doubles and des triples égalités or Frenicle's Traité des triangles rectangles en nombres) either were not published in their original form or relied on a severe selection of a few problems and topics.

Contrary to other authors, thus, there was no Diophantus for the honnête homme. And early-modern Diophantus appears as classical author without a classical text.

  Ptolemy among the Astronomers, Euclid among the Geometers, is the very same as what Diophantus is among the Arithmeticians ;

	the Lawmakers,
	Statements establishing such a status for Diophantus abound. To give just one
	example among many, in his 1660 Diophantus geometra, Billy claims [4, Lectori
	Benevolo. . . ]:
	Who does not know Diophantus does not deserve the name of math-
	ematician ; indeed, what Cicero is among the Orators, Virgil among
	the Poets, Aristotle among the Philosophers, Saint Thomas among
	the Theologists, Hippocrates among the Physicians, Justinian among

], "classical"

[is]

only in use in this sentence: a classical author. That is: an ancient and much approved Author who is an authority on the subject he deals with. Aristotle, Plato, Livy, &c. are classical Authors.