Gabriele Arnulfo

Sébastien Rimbert
email: sebastien.rimbert@inria.fr

Manuel Zaepffel

Pierre Riff

Perrine Adam

Laurent Bougrain

Hypnotic State Modulates Sensorimotor Beta Rhythms During Real Movement and Motor Imagery

Keywords: hypnosis, event-related synchronization, event-related desynchronization, motor imagery, stroke rehabilitation, sensorimotor beta rhythms

Hypnosis techniques are currently used in the medical field and directly influences the patient's state of relaxation, perception of the body, and its visual imagination. There is evidence to suggest that a hypnotic state may help patients to better achieve tasks of motor imagination, which is central in the rehabilitation protocols after a stroke. However, the hypnosis techniques could also alter activity in the motor cortex. To the best of our knowledge, the impact of hypnosis on the EEG signal during a movement or an imagined movement is poorly investigated. In particular, how event-related desynchronization (ERD) and event-related synchronization (ERS) patterns would be modulated for different motor tasks may provide a better understanding of the potential benefits of hypnosis for stroke rehabilitation. To investigate this purpose, we recorded EEG signals from 23 healthy volunteers who performed real movements and motor imageries in a closed eye condition. Our results suggest that the state of hypnosis changes the sensorimotor beta rhythm during the ERD phase but maintains the ERS phase in the mu and beta frequency band, suggesting a different activation of the motor cortex in a hypnotized state.

INTRODUCTION

Hypnosis has been the subject of many debates or misunderstandings and its definition has constantly changed over time with currents of ideas or trends [START_REF] Salem | Soigner par l'Hypnose[END_REF]. To define the state of hypnosis better, since the 1950s, researchers have been studying this phenomenon under experimental laboratory conditions and investigated the cognitive, social, and psychological impact of this practice [START_REF] Oakley | Hypnotic suggestion: opportunities for cognitive neuroscience[END_REF]. Based on its findings, hypnosis can be defined as an altered state of attention, receptivity, and concentration during which the hypnotized person is captured by a suggestion made by the hypnotist [START_REF] Erickson | Hypnosis in painful terminal illness[END_REF][START_REF] Malrewicz | Erickson: de l'Hypnose Clinique à la Psychothérapie Stratégique[END_REF]. In practice, hypnosis sessions usually begin with an induction procedure that leads participants into a state of mental absorption and an augmented response to the suggestion [START_REF] Elkins | Advancing research and practice: the revised apa division 30 definition of hypnosis[END_REF]. Induction and suggestion procedures allow subjects to imagine experiences and gestures as real events [START_REF] Konradt | Motor imagery in hypnosis: accuracy and duration of motor imagery in waking and hypnotic states[END_REF].

Based on these findings, hypnosis can modulate cognitive processes (i.e., perceptual, motor, emotional) by producing observable behavioral changes and subjective experiences [START_REF] Landry | Brain correlates of hypnosis: a systematic review and meta-analytic exploration[END_REF]. For example, hypnosis leads the hypnotized person to relaxation, a change in perceptions of the body and/or the environment, an increased imagination, and careful control [START_REF] Spiegel | Trance and Treatment: Clinical Uses of Hypnosis[END_REF][START_REF] Rainville | Hypnosis modulates activity in brain sutructures involved in the regulation of consciousness[END_REF][START_REF] Oakley | Hypnotic suggestion: opportunities for cognitive neuroscience[END_REF]. This procedure is already applied to reduce pain, manage stress, strengthen the immune system, and to manage emotional problems [START_REF] Bryant | The additive benefit of hypnosis and cognitive-bahvioral therapy in treating acute stress dirsorder[END_REF][START_REF] Kupers | The cognitive modulation of pain: hypnosis-and placebo-induced analgesia[END_REF][START_REF] Verleger | Valid identification of blink artefacts: are they larger than 50 v in EEG records?[END_REF][START_REF] Elahi | Estimation of hypnosis susceptibility based on electroencephalogram signal features[END_REF]. In this article, we will focus on Ericksonian therapeutic hypnosis, which is already used in many hospitals or clinical centers for its benefits [START_REF] Erickson | Hypnotic Realities: The Induction of Clinical Hypnosis and Forms of Indirect Suggestion[END_REF]. Indirect suggestion is a characteristic of the Ericksonian hypnosis school since the metaphor is used to transmit the suggestion indirectly, through the patient's imagination.

Several studies have also confirmed the relevance of this approach for motor rehabilitation [START_REF] Wright | Hypnosis and rehabilitation[END_REF][START_REF] Raginsky | Creativity in hypnosis[END_REF][START_REF] Crasilneck | The use of hypnosis in the rehabilitation of complicated vascular and post-traumatic neurological patients[END_REF]. More precisely, hypnosis is particularly interesting for anxious patients, negativists or patients with a relatively low motivation for recovery [START_REF] Appel | Clinical hypnosis in rehabilitation[END_REF]. Moreover, hypnotic suggestion allows multiple states to be induce (e.g., being relaxed while focusing on feelings usually perceived during a real movement), which appears to be particularly relevant when performing motor imagery (MI) [START_REF] Konradt | Motor imagery in hypnosis: accuracy and duration of motor imagery in waking and hypnotic states[END_REF][START_REF] Muller | The facilitating effect of clinical hypnosis on motor imagery: an fmri study[END_REF][START_REF] Simpkins | Mindfulness and hypnosis: the power of suggestion to transform experience by M. Yapko[END_REF], i. e., the ability to imagine performing a movement without executing it [START_REF] Neuper | Imagery of motor actions: differential effects of kinesthetic and visual-motor mode of imagery in single-trial {EEG}[END_REF][START_REF] Guillot | Brain activity during visual versus kinesthetic imagery: an FMRI study[END_REF]. Nowadays, the MI task is fundamental in rehabilitation protocols after a stroke [START_REF] Zimmermann-Schlatter | Efficiacy of motor imagery in post-stroke rehabilitation: a systematic review[END_REF][START_REF] García Carrasco | Effectiveness of motor imagery or mental practice in functional recovery after stroke: a systematic review[END_REF], because it activates the motor cortex, resulting in better synaptic plasticity, and therefore enhances rehabilitation during the subacute phase [START_REF] Sharma | Motor imagery: a backdoor to the motor system after stroke?[END_REF][START_REF] Cincotti | EEG-based brain-computer interface to support post-stroke motor rehabilitation of the upper limb[END_REF][START_REF] Ang | Brain-computer interface for neurorehabilitation of upper limb after stroke[END_REF].

Thus, hypnosis seems to be a good practice to improve motor rehabilitation for patients, particularly by helping the person to perform MIs. This issue however, has to be investigated, and understanding the underlying brain phenomenon remains a mystery. Indeed, several studies reported a correlation between patients' suggestibility and neuronal activation but very few of them are interested in how the motor cortex responds during the hypnosis process [START_REF] London | Eeg alpha rhythms and susceptibility to hypnosis[END_REF][START_REF] Roelofs | Motor imagery during hypnotic arm paralysis in high and low hypnotizable subjects[END_REF][START_REF] Elahi | Estimation of hypnosis susceptibility based on electroencephalogram signal features[END_REF]. Other studies showed the effect of a suggested paralysis on motor areas, but the hypnotic induction used was very specific and not adaptable for stroke rehabilitation [START_REF] Roelofs | Motor imagery during hypnotic arm paralysis in high and low hypnotizable subjects[END_REF][START_REF] Haggard | Anomalous control: when 'free-will' is not conscious[END_REF][START_REF] Cojan | The brain under self-control: modulation of inhibitory and monitoring cortical networks during hypnotic paralysis[END_REF][START_REF] Cojan | Timecourse of motor inhibition during hypnotic paralysis: EEG topographical and source analysis[END_REF]. Interestingly, Muller et al. showed differences between hypnotic and normal states during a MI and suggested that hypnosis enhanced the motor control circuit engaged in the motor task by modulating the gating function of the thalamus [START_REF] Muller | Activation of thalamus in motor imagery results from gating by hypnosis[END_REF]. Unfortunately, no studies have yet confirmed their findings and revealed the effect of hypnosis on the electroencephalographic signal (EEG) of the motor cortex during motor tasks. The hypnosis technique could also alter the activity of the motor cortex.

The beta rhythm is well-known to be involved in sensorimotor processes. Under normal conditions, a voluntary movement or a MI are characterized by three distinct and easily identifiable phases in the EEG signal. Initially, when compared to a resting state, the movement preparation phase shows a gradual decrease of power both in the alpha (7-13 Hz) and beta (15-30 Hz) bands [START_REF] Pfurtscheller | Motor imagery activates primary sensorimotor area in humans[END_REF]. This is referred to as an eventrelated desynchronization (ERD) [START_REF] Pfurtscheller | Evaluation of event-related desynchronization (ERD) preceding and following voluntary self-paced movement[END_REF][START_REF] Pfurtscheller | Event-related EEG/MEG synchronization and desynchronization: basic principles[END_REF]. During movement or the motor imagery, a minimal power level is maintained in both bands [START_REF] Pfurtscheller | Event-related EEG/MEG synchronization and desynchronization: basic principles[END_REF][START_REF] Cheyne | MEG studies of sensorimotor rhythms: a review[END_REF]. Finally, 300-500 ms after the end of the task, there is an increase of power referred to as an event-related-synchronization (ERS) in the beta band, also known as post-movement beta rebound, lasting ∼1 s [START_REF] Salenius | Modulation of human cortical rolandic rhythms during natural sensorimotor tasks[END_REF][START_REF] Cheyne | MEG studies of sensorimotor rhythms: a review[END_REF][START_REF] Kilavik | The ups and downs of beta oscillations in sensorimotor cortex[END_REF]. Concurrently, in the alpha band, the power returns to a baseline after several seconds. Typically, these patterns occur in the motor cortex, in the contralateral hemisphere [START_REF] Pfurtscheller | Event-related cortical desynchronization detected by power measurment of the scalp EEG[END_REF][START_REF] Pfurtscheller | Motor imagery and direct brain-computer communication[END_REF][START_REF] Bai | Asymmetric spatiotemporal patterns of event-related desynchronization preceding voluntary sequential finger movements: a high-resolution EEG study[END_REF] but can appear bilaterally [START_REF] Pfurtscheller | Post-movement beta synchronization. A correlate of an idling motor area?[END_REF][START_REF] Fok | An EEG-based brain computer interface for rehabilitation and restoration of hand control following stroke using ipsilateral cortical physiology[END_REF][START_REF] Formaggio | Modulation of event-related desynchronization in robot-assisted hand performance: brain oscillatory changes in active, passive and imagined movements[END_REF]. ERD and ERS patterns are also observed during passive movement [START_REF] Müller | Event-related beta EEG changes during wrist movements induced by functional electrical stimulation of forearm muscles in man[END_REF], observed movement [START_REF] Avanzini | The dynamics of sensorimotor cortical oscillations during the observation of hand movements: an EEG study[END_REF], kinesthetic illusion [START_REF] Keinrath | Post-movement beta synchronization after kinesthetic illusion, active and passive movements[END_REF], or a median nerve stimulation [START_REF] Salenius | Modulation of human cortical rolandic rhythms during natural sensorimotor tasks[END_REF], raising fundamental questions about their sensorimotor role. To our knowledge, the impact of hypnosis on the EEG signal during movement or imagined movement has not been investigated. In particular, how ERD and ERS patterns are modulated for motor tasks may provide a better understanding of the potential benefits of hypnosis for stroke rehabilitation.

The goal of our study is to evaluate how hypnosis affects EEG signals in the motor cortex during two motor tasks (i. e., real movement and motor imagery). For this purpose, we recorded EEG signals from 23 healthy volunteers who performed motor tasks under normal and hypnotic conditions. We analyzed and compared the EEG modulations of the mu and the beta bands over the motor cortex for both conditions. The results suggest that hypnosis changes the sensorimotor beta rhythm during the ERD phase but maintains the ERS phase in the mu and beta frequency band, suggesting a different activation of the motor cortex in the hypnotized state.

MATERIALS AND METHODS

Participants

Twenty-three right-handed healthy unpaid subjects (10 females; from 18 to 30 years-old with a mean and standard deviation age 26.1 years ± 2.8) were recruited for this experiment. This study is in accordance with the WMA declaration of Helsinki on ethical principles for medical research involving human subjects [START_REF] World Medical | World Medical Association Declaration of Helsinki: ethical principles for medical research involving human subjects[END_REF]. In addition, participants signed an informed consent form, which was approved by the ethical committee of Inria (the COERLE, approval number: 2016-011/01) as it satisfied the ethical rules and principles of the institute. The subjects had no medical history and no experience with hypnosis which could have influenced the task. All were normotensive (BP < 140/90 mmHgl) and none were taking any cardiovascular medications or reported any history of neurological disorders.

Experimental Tasks

The aim of this study was to evaluate the impact of a hypnotic state on psychological and neurophysiological parameters during two motor tasks: (1) real movement (RM) and (2) motor imagery (MI). For the RM, the goal was to perform a simple sustained closing right hand-movement [START_REF] Nakayashiki | Modulation of event-related desynchronization during kinematic and kinetic hand movements[END_REF]. For the MI, the goal was to imagine performing the same movement without executing it [START_REF] Avanzini | The dynamics of sensorimotor cortical oscillations during the observation of hand movements: an EEG study[END_REF][START_REF] Liang | Improving the discriination of hand motor imagery via virtual reality based visual guidance[END_REF].

Experimental Environment

The experiment took place in a quiet confined room and was conducted by two researchers. One researcher assumed the technological process of the experiment. The second operator was a professional hypnotherapist in charge of instructing the subjects under both normal and hypnotic conditions. During the experiment, the subjects sat comfortably on a chair, with their right arm leaning on an armrest, their hand resting on an ergonomic pillow, without any muscular tension.

Experimental Design

The protocol contained two sessions (normal condition and hypnotic condition) split in two different runs (real movement and motor imagery) that were completed by the subject on the same day. The sessions were performed in a counterbalanced order determined by computerized randomization. Sessions 1 and 2 correspond to the normal and hypnotic conditions, respectively (Figure 1B). For each session, one run of 8 min of real movements and one run of motor imageries were performed in a randomized order. During a run, the subject had to perform a real movement task or a motor imagery task for 4 s, approximately once every 10 s according to triggers. A low frequency beep indicated when the subject had to execute the task (RM or MI) (Figure 1C). A high frequency beep indicated the end of the task. At the beginning of each run, the subject remained relaxed for 10 s. Breaks of a few minutes were planned between runs and sessions to prevent fatigue of the subject. Before the experiment, both tasks (RM and MI) were previously described to the subject and the subject trained to master them. The subjects were instructed to avoid swallowing or other movements during the recording phases.

Hypnotic Procedure

The induction procedure was performed by a professional hypnotist. For each subject, a standard script written in advance and based on Ericksonian hypnosis was used. The hypnotic condition (Session 2) was subdivided into several stages: (1) the induction pre-process, (2) the hypnotic induction, (3) suggestion during the experiment, and (4) the awakening (Figure 1D). First, the induction treatment phase consisted of explaining hypnosis with simple words to make a connection with the subject. Second, the hypnotic induction was composed by three steps : (2a) the point fixation, (2b) the body awareness, and (2c) the catalepsy. The point fixation phase (2a) consisted of asking the subject to fix a point to locate his attention and improve his receptivity to the hypnotist's inductions. The awareness phase (2b) of the body helped to increase the subject's body awareness. Specifically, particular attention was paid to the breathing, the position of the body on the chair, the room temperature, and the intensity of the hypnotist's voice. The aim of the catalepsy step (2c) was to consolidate the hypnotic state with a "lever" technique in order to obtain a deep trance. Then, the experiment consisted of two runs (RM and MI). The subject began randomly with one of them to avoid bias. Between the two runs, the hypnotist used a new step of catalepsy to maintain the state of hypnosis. Finally, the awakening phase allowed the subject to return to the normal state.

Personality and Cognitive Profile Assessment Using Questionnaires

At the beginning of the experiment, participants completed a psychometric questionnaire to assess different aspects of their personality and cognitive profile. Then, all the subjects completed the French version of Movement Imagery Questionnaire -Revised Second Version (MIQ-RS) [START_REF] Butler | The movement imagery questionnaire-revised, second edition (MIQ-RS) is a reliable and valid tool for evaluating motor imagery in stroke populations[END_REF][START_REF] Loison | Translation and validation of the french movement imagery questionnaire-revised second version (MIQ-RS)[END_REF], which allowed for evaluation of the subject's ability to realize visual or kinesthetic motor imaginations. At the end of each session, participants completed the same questionnaire based on a Lickert scale to assess the possible impact of hypnosis on several criteria: estimated time, body perception, memory, detachment, stress, fatigue, and motivation.

Physiological Recordings

EEG signals were recorded through the OpenViBE platform with a commercial REFA amplifier developed by TMS International. The EEG cap was fitted with 32 electrodes re-referenced with respect to the common average references across all channels over the extended international 10-20 system positions. The selected electrodes were FC 5 , 1A). These sites are localized around the primary motor cortex, the motor cortex, the somatosensory cortex and the occipital cortex, which allowed us to observe the physiological changes due to the RM and MI tasks. For this study, an external electromyogram (EMG) surface electrode was positioned on the right forearm of subjects for two reasons. First, we wanted to confirm that no movement was performed during the motor imagination task. In fact, based only on the experimenter's observation, it is very difficult to ensure that no micromovement is performed during the MI task. EMG activity during MI confirmed that there was no muscle activity during this mental task. Second, we needed to be able to test whether the real movements performed were similar to the normal condition despite the hypnotic state. Indeed, it has previously been shown that hypnotic suggestions may influence a movement strength, while increasing corticospinal excitability [START_REF] Takarada | Hypnotic suggestion alters the state of the motor cortex[END_REF]. Respiratory rate and heart rate were measured with a Datascope Passport2 developed by Mindray. Impedance was kept below 5 k for all electrodes to ensure that background noise in the acquired signal was low. No additional filtering was used during the recording.

FC 3 , FC 1 ,FC z , FC 2 , FC 4 , FC 6 , C 5 ,C 3 , C 1 , C z , C 2 , C 4 , C 6 , CP 5 , CP 3 , CP 1 , CP z , CP 2 , CP 4 , CP 6 , P 3 , P 1 , P z , P 2 , P 4 , PO 3 , PO z , PO 4 , O 1 , O z , O 2 (Figure

Signal Pre-processing

All offline analyses were performed using the EEGLAB toolbox [START_REF] Delorme | EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF] and MATLAB 2016a (The MathWorks Inc. Natick, MA, USA). The data were processed in the General Data Format (GDF). Raw EEG data were resampled at 256 Hz, high-pass filtered at 0.5 Hz using a FIR filter and divided into 8 s epochs, corresponding to 1 s before and 7 s after the motor task for each run. Then, the EEG signal baseline was defined 2 s before each trial, therefore, a specific baseline was chosen for the two sessions. The results were also visualized by applying a Laplacian filter [START_REF] Perrin | Spherical splines for scalp potential and current density mapping[END_REF] and confirmed those described below using the CAR method (Figure S1). Finally, based on the EMG activity, we removed trials that did not perform properly in the MI condition due to the presence of a micromovement or a real movement (Figure S2). For this purpose, we used the EMG electrode present throughout the experiment. We also eliminated trials when the EEG activity significantly exceeded ±50 µV, suggesting potential artifacts [START_REF] Verleger | Valid identification of blink artefacts: are they larger than 50 v in EEG records?[END_REF][START_REF] Moretti | Computerized processing of eeg-eog-emg artifacts for multi-centric studies in EEG oscillations and event-related potentials[END_REF]. The number of trials removed for each motor task and subject is described in Figure S3.

EMG Processing

The data from the EMG signal were resampled at 256 Hz, bandpass filtered between 5 and 450 Hz at order 4 (with a butterworth) and divided in 4 s epochs, corresponding to the motor task periods that appears between the two beeps. Then the rootmean-square (RMS) envelope was calculated for each subject for a 125 ms sliding window without overlapping (described in Farfán et al., 2010 (Figure 3). A grand average over all subjects (n = 23) was computed using values of the normal condition as a reference to normalize the signal of the hypnotic condition. The RMS represents the square root of the average power of the EMG signal for a given period of time. The RMS calculation is considered to have the largest information on the amplitude of the EMG signal because it provides a measure of the signal strength, while producing a waveform that is easily analyzed [START_REF] Fukuda | Root mean square value of the electromyographic signal in the isometric torque of the quadriceps, hamstrings and brachial biceps muscles in female subjects[END_REF].

Time-Frequency Analysis

To analyze the differences between each session (normal condition vs. hypnotic condition), we performed an event-related spectral perturbation (ERSP) between 8 and 35 Hz. We computed the ERSPs using the gain model approach [START_REF] Grandchamp | Single-trial normalization for eventrelated spectral decomposition reduces sensitivity to noisy trials[END_REF] which is equivalent to the "band power method" [START_REF] Pfurtscheller | Event-related EEG/MEG synchronization and desynchronization: basic principles[END_REF]. In this model, the eventrelated spectral power at each time-frequency point is divided by the average spectral power in a 2 s pre-stimulus baseline period for each frequency band. ERSP visualizes event-related changes in the averaged power spectrum relative to a baseline interval taken 2 s before each trial [START_REF] Brunner | EEGLAB-an open source matlab toolbox for electrophysiological research[END_REF]. Then, a logtransformed ERSP measure was used to highlight our results (1). The mean ERSP for frequency f and time point t is defined as

ERSP log (f , t) = 10 log 10 (ERSP % (f , t)) (1)
We used a 256-point sliding fast Fourier transform (FFT) window with a padratio of 4 and we computed the mean ERSP 1 s before the task to 7 s after the task (Figure 4). The ERSPs were computed separately for all EEG channels and then observed in the alpha and beta frequency bands. Since ERD and ERS are difficult to observe we used the averaging technique to represent the modulation of power for both conditions [START_REF] Pfurtscheller | Induced oscillations in the alpha band: functional meaning[END_REF][START_REF] Quiroga | Single-trial event-related potentials with wavelet denoising[END_REF].

Topographies

Brain topography allows us to display the possible changes over different electrodes on the scalp in order to localize which part of the brain was involved when the subject performed the requested task. We computed the topographic ERSPs in the beta (15-30 Hz) band for the normal condition and hypnotic condition (Figure 5).

Statistical Analysis

For the EMG processing, we chose to apply a paired t-test (twosided) to show the significant difference between both conditions (p-value = 0.0011).

A surrogate permutation test (p < 0.05; 3,000 permutations) from the EEGLAB toolbox was used to validate differences in term of time-frequency ERSPs and localization of this ERSPs with a good alpha level (<5%) (Figures 456). In addition to this analysis, we applied a false discovery rate (FDR) correction test in order to clarify how the false discovery rate was controlled for multiple comparisons. This test performed the drawing of data samples without replacement [START_REF] Manly | The generation of random permutations[END_REF].

We chose to apply a paired t-test (two-sided) to show the significant difference about questionnaire variations (Figure 2; p-value < 0.01).

Then, considering the use of Likert scales in the questionnaires, we computed Spearman's correlation to establish a correlation between questionnaires and ERD/ERS variations between the normal and hypnotic state (Figure 6A; p-value < 0.05).

RESULTS

The findings of this research can be defined as both behavioral and electrophysiological. First, the findings are of a behavioral nature, with the analysis of both the pre-and post-experimentation questionnaires and the EMG signal. Second, the findings are also of a electrophysiological nature, with the analysis of ERD and ERS modulations.

Behavioral Results

Questionnaires

After each (normal or hypnotic) condition, subjects filled in a post-experimentation questionnaire. The studied items were modification of body perception, lightness/heaviness of the right arm, lightness/heaviness of the body, heat/fresh sensation, event memorization, feelings of detachment, stress, motivation, and tiredness. In addition to these variables, the participants had to estimate for how long they had performed the condition while the duration for all runs was the same. Most of the perception items show significant differences between the normal and hypnotic conditions (Figure 2A). The body perception, light weight of the right hand, heavy weight of the body, warmness and detachment sensation show the biggest differences (p-value < 0.001). In addition, subjects estimated a longer duration for the hypnotic state (p-value < 0.01). The respiratory rate and heart rate show small differences, except for the respiratory rate during the RM which displays a small but significant difference.

Movement Analysis

The root-mean-square (RMS) envelope of the EMG signal for real movements shows an average difference of 14.9% between normal and hypnosis conditions. A student t-test was performed on the 40 trials for the 23 subjects, and this difference is significant (p-value = 0.0011). For the MI task, no significant EMG activity was detected.

Electrophysiological Results

ERD and ERS Modulations During Normal Condition

For the RM task (Figure 4B), an ERD appears 250 ms after the cue both in the mu band and the beta band, last for 3,250 ms, and stops 500 ms before the second cue. Then, an ERS appears in the beta band at 4,000 ms and last until 6,500 ms.

For the MI task (Figure 4D), the ERD is present from 250 to 3,000 ms and is mostly present in the beta band. There is no ERS in the beta band after the second cue, but a small one is present in the mu band (from 4,000 to 6,250 ms).

Concerning the topographic maps for these two tasks (Figure 5), the RM task shows a bilateral distribution of the ERD, centered around electrodes C3 and C4. The ERS is present on the C3 and FC2 electrodes. For the MI task, the ERD is mostly contralateral and centered on the C3 electrodes, the ERS is also mostly contralateral.

ERD and ERS Modulations During Hypnosis Condition

During the hypnotic condition, for the RM task (Figure 4A) we observed a smaller ERD than in the normal condition. It started at the same time, 250 ms after the cue, but quickly disappeared around 750 ms, and was only present on the beta band during this time window. The ERD reappeared at 2,000 ms in the band and disappeared at 3,500 ms. The ERS was mostly present in the beta band from 4,000 to 6,500 ms. There was a small ERS in the mu band from 5,000 to 6,000 ms.

For the MI task (Figure 4C), a small ERD appeared on both frequency bands from 250 to 1,000 ms. It was then replaced by an ERS on the beta band from 1,000 to 2,500 ms. Another ERD then appeared at 2,500 ms in the beta band (mostly between 14 and 16 Hz) and quickly disappeared 250 ms later. An ERS then took over on the beta band at 3,000 ms and lasted up to 5,500 ms.

Concerning the topographic map (Figure 5), for the RM task the ERD is bilateral and centered around electrodes C3 and C4. The ERS is mostly left-sided around C3. During the MI task, the ERD are also centered around electrodes C3 and C4, and the ERS was mostly centered around telectrode C3 from 4,000 to 5,000 ms.

Statistical Differences Between the Hypnotic Condition and the Normal Condition

For the RM task the absence of ERD from 0 to 2,000 ms in the beta band is the main significant difference (Figure 4). For the MI task, the statistical differences are mostly present on the beta band during most of the task duration, the absence of ERD during the task on the hypnotic condition, and the presence of an ERS is reflected in the statistical analysis. We can also observed a significant difference at 4,000 ms in the alpha band, explained by the presence of an ERS in the hypnotic condition.

Effect of the Randomized Sequence Order on the ERD

We also checked that there was no bias resulting from the sequence in which the conditions (hypnotic vs. normal) were achieved for each subject. Indeed, participants who started with the hypnotic condition may have a long-term effect influencing the next normal condition. Figure 6 indicates that there is no significant difference (at p-value < 0.05) in the ERD phase depending on the condition or task the subject started with, although subjects who started with the normal condition and the MI task seem to have minimal ERD.

Correlation Between Questionnaires and EEG

No significant correlation was found between the postexperiment questionnaire results and ERD and ERS patterns. We tested the correlation by considering all the items contained in the questionnaire (Figure 6). Only heat, freshness and detachment could be correlated with ERD and ERS variations (at p-value < 0.05). Most of the significant correlations have a value between 0.5 and 0.75 which means that every shift in perception was positively correlated with the other, as expected if the hypnosis induction was correctly performed on the subject.

DISCUSSION

In this study, we used the EEG technique to investigate whether a hypnotic suggestion impacted the ERD and ERS modulations that occur during the execution of a voluntary actual or imagined movement. The hypnotic suggestion used was made specifically to enhance the sensory feelings perceived during the two motor tasks. Our results show that the hypnosis technique significantly altered both the subject's subjective perceptions during the experiment and the ERD and ERS modulations resulting from motor tasks in the EEG signal. In this section, we will first discuss how behavior was modified with hypnosis. Second, we will analyze ERD and ERS modulations during the normal and hypnotic conditions. Finally, we will discuss the mechanism of the motor cortex activity under hypnosis and the potential impact of these findings for motor rehabilitation and the brain-computer interface (BCI) domain.

Hypnosis Effect on Behavior

Subjective Perceptions During Hypnosis

Altogether, our results reveal that hypnosis significantly alters many subjective perceptions, providing relative evidence of the effectiveness of hypnotic suggestion (Figure 2). Indeed, compared to a normal state, the subjects reported a modified perception of their body, alterations of body temperature and forearm weight corresponding to an increase or a decrease depending on the subject. With the hypnosis technique, such feelings that appear contradictory are possible [START_REF] Simpkins | Mindfulness and hypnosis: the power of suggestion to transform experience by M. Yapko[END_REF]. Similar reports were also found in other studies [START_REF] Kihlstrom | The domain of hypnosis, revisited[END_REF][START_REF] Landry | Brain correlates of hypnosis: a systematic review and meta-analytic exploration[END_REF]. Some perceptions experienced in a hypnotic state may be considered contradictory (e.g., freshness and warmth of the body, heaviness and lightness of the body, and more specifically of the right arm) but during the post-experiment testimonies, participants confirmed that multiple sensations were felt and varied with the moment of the experience. In addition, with hypnosis, subjects became more disengaged from the experiment and presented an upward time distortion (Figure 2). Time distortion in hypnosis is quite common but some studies suggest that subjects underestimate time during hypnosis [START_REF] Naish | The production of hypnotic time-distortion: Determining the necessary conditions[END_REF][START_REF] Haggard | Anomalous control: when 'free-will' is not conscious[END_REF] while others overestimate it [START_REF] Mozenter | Prospective time estimation and hypnotizability in a simulator design[END_REF]. No change can be observed in terms of stress, motivation, fatigue, or the memorization of events. There is also no evidence of any effect on the respiratory rate and/or heart rate, although it has previously been shown that hypnosis may have an effect on both parameters [START_REF] Yuksel | The effects of hypnosis on heart rate variability[END_REF]. However, in this study, the suggestion was centered on the increased sensations during motor tasks and was not essentially focused on relaxation, which could explain why these two parameters remained unchanged during the hypnosis state.

EMG Activity During Hypnosis

Our results highlight that the movement performed under hypnosis was slightly different from the one performed in real movement (Figure 3). Indeed, muscle contraction was reduced (-14.9%), reflecting the impact of the hypnotic condition on behavior.

ERD and ERS Modulations Under Hypnosis

In this article, we aimed to study EEG motor activity of two motor tasks (i.e., real and imagined movements) in a normal condition and after a suggestion of hypnosis. For this purpose, we have chosen to analyze the modulations of ERD and ERS that are usually present at the different stages of a real or imagined movement.

Motor Responses Without Hypnosis

To understand how hypnosis impacts ERD and ERS modulations, results obtained under the normal condition need to be compared with existing studies. Our results are consistent with the literature, highlighting the reliability of the motor brain responses obtained for both motor tasks.

Real movement task

Indeed, during a real movement, in a normal condition, Kilavik et al. showed that the power of the beta rhythm was very reduced during the dynamic phases of movement [START_REF] Kilavik | The ups and downs of beta oscillations in sensorimotor cortex[END_REF]. In agreement with previous findings, we could identify two consecutive ERD peaks which correspond to muscle contraction and relaxation, respectively. Our results show clearly that this decrease is characterized by an ERD during all motor executions (Figures 4A,B). A first ERD peak starts 200 ms after the first sound beep (see Figures 4A,B). In this article, the cue used was a sound beep emitted every 8-12 s (with a random delay of 2 s centered around 10 s), which implies that the movement is not completely voluntary and not too prepared. As a consequence, no preparatory phase prior to motor performance can be identified, which has already been shown in previous articles [START_REF] Alegre | Alpha and beta changes in cortical oscillatory activity in a go/no go randomly-delayed-response choice reaction time paradigm[END_REF][START_REF] Rimbert | Electroencephalographic modulations during an openor closed-eyes motor task[END_REF][START_REF] Rimbert | Median nerve stimulation based BCI: a new approach to detect intraoperative awareness during general anesthesia[END_REF]. A second peak was observed at 2,500 ms, suggesting that the subjects anticipated the end of the real motor execution. This hypothesis is supported by the analysis of the EMG activity (Figure 3) and is probably explained by the difficulties of maintaining motor performance for 4 s throughout the experiment. Typically, the end of the movement is followed by a predominant ERS between 16 and 22 Hz, better known as beta, rebound (Pfurtscheller et al., 1997;[START_REF] Kilavik | The ups and downs of beta oscillations in sensorimotor cortex[END_REF].

Topographic analysis (Figure 5) shows that the ERD is bilateral during a real movement while the post-movement ERS phase is contralateral, as previously established [START_REF] Pfurtscheller | Evaluation of event-related desynchronization (ERD) preceding and following voluntary self-paced movement[END_REF][START_REF] Bai | Asymmetric spatiotemporal patterns of event-related desynchronization preceding voluntary sequential finger movements: a high-resolution EEG study[END_REF].

In our task, the muscular contraction has to be maintained until the second beep. According to previous studies [START_REF] Kilavik | The ups and downs of beta oscillations in sensorimotor cortex[END_REF], a sustained movement goes with an increase in beta power. Even if beta power seems to increase between the two ERD peaks which correspond to the dynamic phase of the movement, the ERD is rather low. It may be due to the short delay between the two ERD peaks and the existence of preparatory power decrease related to the preparation of movement relaxation in a way that no ERS related to the sustained contraction can arise.

Motor imagery task

In a normal condition, the ERD of a MI is relatively similar to the one obtained with a RM but with a smaller amplitude. This is a very well-known observation. The rebound is also attenuated, or even almost non-existent (Figures 4C,D). Several studies have already shown that the intention to move generates modulations of ERD and ERS of smaller amplitudes compared to real movement, due to the activation of similar neural networks at a lower scale [START_REF] Filgueiras | The neural basis of kinesthetic and visual imagery in sports: an ALE meta-analysis[END_REF]. Again, it would seem that the MI was not maintained until the second beep, which may be explained by the difficulty of maintaining a MI for 4 s. Moreover, EMG also showed a premature end of muscular activity (Figure 3) Finally, in the literature it is common for the MI to generate a contralateral ERD during the task followed by an ERS [START_REF] Pfurtscheller | Beta rebound after different types of motor imagery in man[END_REF][START_REF] Hashimoto | EEG-based classification of imaginary left and right foot movements using beta rebound[END_REF][START_REF] Clerc | Brain-Computer Interfaces[END_REF], which is apparent in our results (Figure 5). Altogether, for the non-hypnotic condition, our results are highly consistent with the literature for both RM and MI.

Motor Responses With Hypnosis

Real movement task

In the introduction we made the hypothesis that a hypnotic suggestion oriented toward the sensation and full consciousness of movement would enable a more powerful motor brain response, and therefore a more important ERD and ERS. In opposition to our hypothesis, the results show that hypnosis during real movement can significantly reduce ERD during motor performance (Figures 4A,B), while the rebound, both in terms of amplitude and delay, is unchanged after a hypnotic suggestion.

Motor imagery task

For the MI task, hypnosis produces more changes. Indeed, the ERD is even more attenuated than in real movement. The rebound is also higher in terms of amplitude while latency is lower (Figures 4C,D). Hypnosis would therefore have a common effect for the two motor tasks: an attenuation of the ERD phase with a differential effect for MI, with a very pronounced rebound in the contralateral direction (Figure 5B). These results cannot be explained by reduced EMG activity under hypnosis. First, because several studies have shown that the strength of the movement has no impact on the ERD phase [START_REF] Kilavik | The ups and downs of beta oscillations in sensorimotor cortex[END_REF], and second, because the same modulation, i.e., attenuation of ERD, is also present with a motor imagery.

Are ERD and ERS Changes Related to Differences in Subjective Perceptions?

Our results show that hypnosis has an effect on a subject's perception (e.g., time distortion, body feelings, detachment) but also on the EEG signal with a significant attenuation of the ERD phase. Then the following question was raised: can the differences in subjective perception experienced by the subjects during hypnosis explain the differences in modulations recorded during the ERD phase? In order to answer this question, the correlogram results indicate very low correlation between variations in ERD and ERS between normal and hypnotic conditions and the item values of the post-experiment questionnaire (Figure 6). Only heat, freshness, and detachment could be correlated with ERD and ERS variations.

However, the ERD variation is directly correlated (p-value < 0.05) to the ERS change, which, to our knowledge, has not been shown in previous studies.

Motor Cortex Mechanism Under Hypnosis

Does Hypnosis Deactivate the Motor Cortex?

The absence of ERD during hypnosis for both motor tasks raises questions about the particular motor cortex mechanism in this altered state.

This reduction seems to be more important between the two peaks of the ERD, i.e., for the phase of sustained muscle contraction, but in fact with no significant difference (Figure 4A). Although it would appear that hypnosis leads to a global increase of power in the beta frequency band after the movement, our results for RM demonstrate that the rebound phase is not affected (Figures 4A,B).

Many authors have interpreted the ERD phase during a motor task as related to motor cortex activation [START_REF] Pfurtscheller | Functional brain imaging based on ERD/ERS[END_REF][START_REF] Neuper | ERD/ERS patterns reflecting sensorimotor activation and deactivation[END_REF], though it reflects the conjunction of multiple factors associated with sensory and cognitive aspects of motor control rather than pure motor processes [START_REF] Kilavik | The ups and downs of beta oscillations in sensorimotor cortex[END_REF]. Thus, the significant attenuation of ERD for the two motor tasks under hypnosis would result from a weaker activation of the motor cortex. However, Muller et al. highlighted with fMRI that hypnosis could enhance the motor control circuit engaged in motor imagery by modulating the gating function of the thalamus [START_REF] Muller | Activation of thalamus in motor imagery results from gating by hypnosis[END_REF]. Furthermore, hypnosis may create a specific state that may enhance the efficiency of the motor system and also increase corticospinal excitability during movement [START_REF] Takarada | Hypnotic suggestion alters the state of the motor cortex[END_REF]. Nevertheless, caution is required when studying hypnosis due to the variability between experiments, in particular in terms of the hypnotic induction used, the suggestion made, or the cognitive tasks performed by the subjects.

Increased automatism during hypnosis could explain an attenuation of the ERD phase despite the execution of the task [START_REF] Déry | Suggestion overrides automatic audiovisual integration[END_REF]. Indeed, in hypnotic states, subjects demonstrate extraordinary obedience by producing a movement in response to a suggestion that is described as an ideomotor response [START_REF] Haggard | Anomalous control: when 'free-will' is not conscious[END_REF]. Finally, under hypnosis, the motor cortex may be less involved to the benefit of other more internal structures, such as the thalamus, which is considered the location of a high subjective experiences [START_REF] Raz | Hypnotic suggestion and the modulation of stroop interference[END_REF][START_REF] Muller | Activation of thalamus in motor imagery results from gating by hypnosis[END_REF].

Use of Hypnosis for Stroke Rehabilitation

Nowadays, the MI task is fundamental in rehabilitation protocols after a stroke [START_REF] Zimmermann-Schlatter | Efficiacy of motor imagery in post-stroke rehabilitation: a systematic review[END_REF][START_REF] García Carrasco | Effectiveness of motor imagery or mental practice in functional recovery after stroke: a systematic review[END_REF]. The MI task activates the motor cortex, stimulating the injured area and all the evidence suggests that there is an impact on synaptic plasticity, resulting in better rehabilitation during the subacute phase [START_REF] Liu | Mental imagery for promoting relearning for people after stroke: a randomized controlled trial[END_REF][START_REF] Cincotti | EEG-based brain-computer interface to support post-stroke motor rehabilitation of the upper limb[END_REF][START_REF] Pichiorri | Brain-computer interface boosts motor imagery practice during stroke recovery[END_REF]. Interestingly, hypnosis has an effect on relaxation, concentration, or increased perception but is also particularly relevant for imaginative procedures [START_REF] Muller | The facilitating effect of clinical hypnosis on motor imagery: an fmri study[END_REF]. Moreover, many studies have also confirmed the beneficial role it plays in motor rehabilitation [START_REF] Wright | Hypnosis and rehabilitation[END_REF][START_REF] Crasilneck | The use of hypnosis in the rehabilitation of complicated vascular and post-traumatic neurological patients[END_REF], so we expected an increase of the ERD during the motor tasks (Figures 4A,D). Consequently, our results prompt us to be cautious about using the hypnosis technique since the suggestion used could be unproductive. In this study, the suggestion used by the hypnotherapist was to feel as many feelings as possible during the motor tasks, and a majority of subjects reported an easier way of imagining movement compared to the normal condition. Despite this positive feedback, our results showed that hypnosis involves a vanishing of the ERD during the RM and MI tasks, which implies a lower activation of the motor cortex or an activation according to other modalities. Further research investigating the EEG connectivity index be interesting to measure possible changes in the beta sensorimotor rhythm during hypnosis correlated with modulations related to the Default Mode Network (DMN). Indeed, several neurophysiological studies indicate that the state of hypnosis is associated with changes in connectivity [START_REF] Fingelkurts | Cortex functional connectivity as a neurophysiological correlate of hypnosis: an EEG case study[END_REF][START_REF] Jiang | Brain activity and functional connectivity associated with hypnosis[END_REF], especially in the DMN network [START_REF] Lipari | Altered and asymmetric default mode network activity in a "hypnotic virtuoso": an fMRI and EEG study[END_REF][START_REF] Landry | Elucidating unconscious processing with instrumental hypnosis[END_REF]. Although, McGeown et al. have shown that the induction of hypnosis can reduce pre-rest DMN activity without increasing activity in other cortical regions [START_REF] Mcgeown | Hypnotic induction decreases anterior default mode activity[END_REF]. To the best of our knowledge, only a few studies on the connectivity during motor tasks under hypnosis have been conducted. A new study combining EEG and fMRI would explore this issue and provide better guidance for the use of hypnosis practice for stroke rehabilitation.

Hypnosis in the BCI Domain

Brain-computer interfaces allow end users to interact with a system using modulations of brain activities (Jonathan Wolpaw, 2012). Most of these BCIs are based on the modulation of sensorimotor rhythms which are partially observable in EEG signals during a MI task. In order to improve BCI performance and allow a better control of MI, it has been suggested that some altered states of consciousness, such as mind-body awareness training (MBAT) [START_REF] Tan | Effect of mindfulness meditation on brain-computer interface performance[END_REF] or hypnosis [START_REF] Alimardani | Development of a real-time brain-computer interface for interactive robot therapy: an exploration of eeg and emg features during hypnosis[END_REF]) can be used, especially during the training phase. Interestingly, the hypnotic suggestion allows multiple states to be induced (e.g., being relaxed while focusing on the feelings usually perceived during a real movement), which appears to be particularly relevant to the MI task [START_REF] Konradt | Motor imagery in hypnosis: accuracy and duration of motor imagery in waking and hypnotic states[END_REF][START_REF] Simpkins | Mindfulness and hypnosis: the power of suggestion to transform experience by M. Yapko[END_REF]. Recently, we investigated this question in a study which shows that the attenuation of ERD induced by hypnosis had a negative impact on BCI performance. This prompts us to not recommend using hypnosis for BCI applications (Rimbert et al., 2019a).

CONCLUSION

In this study, we investigated the impact of hypnosis on the sensorimotor beta rhythm for two motor tasks (i.e., real movement and motor imagery). We conducted an experiment on 23 subjects, using both motor tasks under two different conditions: hypnosis and normal state. The hypnotic induction was Ericksonian, essentially based on relaxation, and the suggestion was aimed at improving feeling and focus during movement and motor imagery.

This work showed that the state of hypnosis changes the sensorimotor beta rhythm during the ERD phase but maintains the ERS phase in the mu and beta frequency band suggesting a different activation of the motor cortex in the hypnotized state. In addition, the subjects perception and behavior was changed during the hypnotic condition. These findings raise questions about the use of this technique for rehabilitation after a stroke or in the BCI domain.

FIGURE 1 |

 1 FIGURE 1 | (A) Electrode position. (B) Paradigm scheme representing the two different sessions: normal condition and hypnotic condition. Each session is composed of two runs of 8 min and a randomization of the order was applied between runs and sessions. (C) Trial design: the subject had to perform a real movement task or a motor imagery task for 4 s, approximately once every 10 s according to triggers. A low frequency beep indicated when the subject had to execute the task (RM or MI). A high frequency beep indicated the end of the task. (D) Description of the hypnosis session. The hypnotic condition (Session 2) was subdivided into several stages: (1) the induction pre-process, (2) the hypnotic induction, (3) suggestion during the experiment, and (4) the awakening.

FIGURE 2 |

 2 FIGURE 2 | (A) Bar plots representing the responses to the questionnaires concerning the 12 items related to the subject's perceptions for hypnotic (in blue) and normal (in orange) conditions. (B) Bar plots representing the time estimation for hypnotic (in blue) and normal (in orange) conditions. (C) Bar plots representing the respiratory rate and heart rate for hypnotic (in blue) and normal (in orange) conditions. *Corresponds to a p-value <0.05, **corresponds to a p-value <0.01 and ***corresponds to a p-value <0.001.

FIGURE 3 |

 3 FIGURE 3 | The root-mean-square (RMS) envelope of the EMG signal for real movement during both hypnosis (in blue) and the normal condition (in red).

FIGURE 4 |

 4 FIGURE 4 | (A,B) Time-frequency grand average analysis corresponding to an event-related spectral perturbation (ERSP) for Session 1 (Normal condition) and Session 2 (Hypnotic condition) for a real movement for electrode C 3 . (C,D) Time-frequency grand average analysis (ERSP) for Session 1 (Normal condition) and Session 2 (Hypnotic condition) for a motor imagery for electrode C 3 . Red corresponds to a strong ERS and blue to a strong ERD. Significant differences (p < 0.05) with a False Discovery Rate (FDR) correction are shown in the right part of the figure.

FIGURE 5 |

 5 FIGURE 5 | Topographic maps of ERD/ERS% (grand average, n = 23) in the Beta band (15-30 Hz) for a real movement (A) and a motor imagery (B) during two conditions: normal and hypnotic. Red corresponds to a strong ERS and blue to a strong ERD. The first vertical black line indicates when the motor task started and the second one to when it stopped. This figure is an extrapolation of 32 electrodes. Red electrodes indicate a significant difference (p < 0.05) with an FDR correction.

FIGURE 6 |

 6 FIGURE 6 | (A) Correlogram representing Spearman's correlations among the 12 items contained in the questionnaires, and ERD and ERS variation differences between normal and hypnotic conditions. Correlations with a p-value > 0.05 are considered as not significant and are shown in blank. Positive correlations are displayed in blue and negative correlations are in orange. The intensity of the color and the size of the circles are proportional to the correlation coefficients. (B) Topographic maps representing differences in term of ERSPs for the sequence in which the conditions (hypnotic and normal) were achieved for each subject. Red corresponds to a strong ERS and blue to a strong ERD. Significant difference (p < 0.05) with a False Discovery Rate (FDR) correction are shown in the right part of the figure.

Figure S1 |

 S1 Figure S1 | Time-frequency grand average with Laplacian filter analysis corresponding to an event-related spectral perturbation (ERSP) for Session 1 (Normal condition) and Session 2 (Hypnotic condition) for a real movement and a motor imagery for electrode C3. Red corresponds to a strong ERS and blue to a strong ERD. Significant differences (p < 0.05) with a False Discovery Rate (FDR) correction are shown in the right part of the figure.

Figure S2 |

 S2 FigureS2| The root-mean-square (RMS) envelope of the EMG signal for real movement and motor imagery (dashed line) during both hypnosis (in blue) and the normal condition (in red).

Figure S3 |

 S3 Figure S3 | Number of trials removed for each subject for real movement and motor imagery during hypnosis and normal condition.

Frontiers in Psychology | www.frontiersin.org

October 2019 | Volume 10 | Article 2341

ACKNOWLEDGMENTS

This research was supported by the Inria Project Lab BCI Lift. We also wish to give our thanks to all the participants.

DATA AVAILABILITY STATEMENT

The datasets generated for this study are available on request to the corresponding author.

ETHICS STATEMENT

The studies involving human participants were reviewed and approved by Ethical Committee of Inria (the COERLE). The patients/participants provided their written informed consent to participate in this study.

AUTHOR CONTRIBUTIONS

SR, MZ, and PA conceived and designed the experiments, performed the experiments. SR, MZ, PR, PA, and LB analyzed the data, contributed reagents, materials, and analysis tools, prepared figures and tables, authored or reviewed drafts of the paper, approved the final draft.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fpsyg. 2019.02341/full#supplementary-material Conflict of Interest: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


E L Houcein

E L Abdalaoui

AN ELEMENTARY QUESTION ON TRIGONOMETRIC POLYNOMIALS WITH COEFFICIENTS ±1 WHICH IMPLIES RIEMANN HYPOTHESIS

Keywords: Mathematics Subject Classification. 37B05, 37B45, 37E99 flat polynomails, semi-flat polyomials, Liouville function, Möbius function, Riemann hypothesis

whether they are published or not. The documents may come

Introduction

The purpose of this article is to present an elementary question on the norms of some polynomials with coefficients ±1 on the circle which implies Riemann hypothesis. This question can be seen as a special case in the weak form of the Erdös-Newman problem on the existence of the so called flat polynomials.

The sequence of L 2 -normalized polynomials (P n ) (that is, P n = 1 is said to be flat if it converge to 1 in some sense. The sequence is said to be ultraflat if the convergence holds for the uniform norm.

We warn the reader that this notion should not be confused with the notion of flatness introduced by Littlewood in his 1966's paper [START_REF] Littlewood | On polynomials n ±z m , n e αmi z m , z = e θi[END_REF]. For this later notion (which we attribute to Littlewood), the sequence of L 2normalized polynomials (P n ) is flat in the sense of Littlewood if there is a constant c, C > 0 such that c ≤ |P n (z)| < C, for all z with modulus 1. Therein, Littlewood asked if there exist a sequence of flat polynomials with coefficients of modulus 1 or ±1 which are flat in his sense. He further stated three conjecture on flatness [START_REF] Littlewood | On polynomials n ±z m , n e αmi z m , z = e θi[END_REF] and [START_REF] Littlewood | Some Problems in Real and Complex Analysis[END_REF]Problem 19]. Very recently, using Rudin-Shapiro polynomials combined with Spencer's six deviations lemma, P. Balister and al. constructed a flat polynomials in the Littlewood sense [START_REF] Balister | Flat Littlewood polynomials exist[END_REF]. However, it is easy to see that those polynomials are not L α -flat, for any α ≥ 0 1 . We further notice that Littlewood addressed Erdös-Newman question in several papers [START_REF] Littlewood | On the mean values of certain trigonometric polynomials[END_REF], [START_REF] Littlewood | On the mean values of certain trigonometric polynomials[END_REF], [START_REF]On the real roots of real trigonometrical polynomials[END_REF] . Therein, he established Erdös's result which say that the cosines polynomials with coefficients ±1 are not ultraflat. In fact, Littlewood established that those polynomials are not L αflat, for any α ≥ 0. He also provided a condition on the coefficients of a real trigonometric polynomials to insure that those polynomials are not L α -flat. But, it is seems that Littlewood had conflicting feelings about the existence of ultraflat polynomials. He further stated in [12, p.334] that the sequence of trigonometric polynomials

P n (θ) = N n=1 cos(2 2 n θ),
is L α -flat for any α > 0. But, according the central limit theorem for the Hadamard lacunary trigonometric series due to Salem-Zygmund [18, Vol 2, p. 264] combined with the technics in [1, p.176], it can be seen that for any 1 ≤ α < 2,

P n α -----→ n→+∞ Γ α 2 + 1 .
For more recent results on Erdös-Newman conjecture, we refer to [START_REF] El Abdalaoui | On the Erdös flat polynomials problem, Chowla conjecture and Riemann Hypothesis[END_REF] 2 .

Here, we are interest in the semi-flatness of an analytic trigonometric polynomials with Liouville or Möbius coefficients. A sequence of L 2 -normalized polynomials (P n ) is semi-flat if P n (z) is bounded above by some absolutely constant in some sense. Precisely, we consider L α -semi-flat polynomials with α > 2, that is, a sequence of analytic trigonometric polynomials (P n ) with coefficients in {±1, 0} such that P n (z) α is bounded above by some absolutely constant which may depend on α.

The plan of the paper is as follows. In Section 2, we recall a basic notions and the results that we need in the sequel. In section 3, we state and prove our main result.

Set-up and tools

Let us denote by the circle S 1 = z ∈ C, |z| = 1 and for any α > 1, we define the L α -norm of any trigonometric polynomials P by

P α = P (z) α dz 1 α ,
where dz is the Lebesgue measure on the circle. For any N ∈ N * , we denote by (ξ N,j ) N -1 j=0 , the the N th roots of unity given by

ξ N,j = e 2iπj N , j = 0, • • • , N -1.
The sequence of analytic trigonometric polynomials (P n ) is said to be L αsemi-flat polynomials with α ≥ 1 if P n (z) α is bounded above by some 2 Notice that it is not know if there exist almost everywhere flat polynomials with coefficients ±1.

absolutely constant which may depend on α.

The analytic trigonometric polynomials with Liouville or Möbius coefficients are defined as follows

P n (z) = n j=1 λ(n)z n , |z| = 1, (2.1)
Q n (z) = n j=1 µ(n)z n , |z| = 1 (2.2)
where λ and µ are respectively the Liouville function and the Möbius function. The Liouvile function is given by

λ(n) = 1 if n = 1;
(-1) r if n is the product of r not necessarily distinct prime numbers;

The Liouville function is related to another famous functions in number theory called the Möbius function. Indeed, the Möbius function is defined for the positive integers n by Let us further notice that the Dirichlet inverse of the Liouville function is the absolute value of the Möbius function.

µ(n) = λ(n) if n is not divisible
For the reader's convenience, we briefly recall some useful well-known results on the Riemann ζ-function. The Riemann ζ-function is defined, for s ∈ C, Re(s) > 1 by

ζ(s) = +∞ n=1 1 n s ,
or by the Euler formula

ζ(s) = p prime 1 - 1 p s -1
.

It is easy to check that ζ is analytic for Re(s) > 1. Moreover, it is wellknown that ζ is regular for all values of s except s = 1, where there is a simple pole with residue 1. Thanks to the functional equation

ζ(s) = 2 s π s-1 sin πs 2 Γ(1 -s)ζ(1 -s),
where Γ is the gamma function given by Γ(z) = +∞ 0

x z-1 e -x dx, Re(z) > 0.

We notice that the gamma function never vanishes and it is analytic everywhere except at z = 0, -1, -2, ..., with the residue at z = -k is equal to (-1) k k! . We further have the following formula (useful in the proof of the functional equation) Γ(s) sin πs 2 = +∞ 0 y s-1 sin y dy.

For the proof of it we refer to [16, p.88]. Changing s to 1s, we obtain

ζ(1 -s) = 2 1-s π -s cos πs 2 Γ(s)ζ(s). Putting ξ(s) = s(s -1) 2 π -s 2 Γ s 2 ζ(s), and
E(s) = ξ 1 2 + is . It follows that ξ(s) = ξ(1 -s), and
E(z) = E(-z).
We further remind that we have

ζ(s)Γ(s) = +∞ 0 x s-1 e x -1 dx, Re(s) > 1.
Therefore, it is easy to check that ζ has no zeros for Re(s) > 1. It follows also from the functional equation that ζ has no zeros for Re(s) < 0 except for simple zeros at s = -2, -4, • • • . Indeed, ζ(1s) has no zeros for Re(s) < 0, sin sπ 2 has simple zeros at s = -2, -4, • • • . It is also a simple matter to see that ξ(s) has no zeros for Re(s) > 1 or Re(s) < 0. Hence its zeros which are also the zeros of ζ lie in the strip 0 ≤ Re(s) ≤ 1. Hardy point out that for Re(s) > 1, it is easily seen that

+∞ n=1 (-1) n-1 n s = +∞ n=1 1 n s -2 +∞ n=1 1 2 s n s = 1 -2 1-s ζ(s),
This formula allows us to continue ζ analytically to half-plan Re(s) > 0 with simple pole at s = 1. We further have ζ(s) = 0 for all s > 0 since

+∞ n=1 (-1) n-1 n s > 0.
We thus conclude that all zeros of ζ are complex. The functional equation allows us also to see that if z is a zero then 1z and 1z are also a zeros. Whence, the zeros of ζ lie on the vertical line Re(s) = 1 2 or occur in pairs symmetrical about this line.

Conjecture (Riemann hypothesis (RH)). All nontrivial zeros of ζ lie on the critical line Re(s) = 1 2 . Here, we need the following characterization of Riemann hypothesis due to Littlewood [START_REF] Littlewood | Quelques conséquences de lhypothése que la fonction ζ(s) n'a pas de zéros dans le demi-plan Re(s) > 1 2[END_REF].

Lemma 2.1. The Riemann hypothesis is equivalent to

x n=1 λ(n) = o x 1 2 +ε as x -→ +∞, ∀ε > 0 (2.3)
For the proof of Littlewood, we refer also to [17, p.371], [5, p.261]. Notice that by applying Bateman-Chowla trick, the same conclusion can be drawn for µ.

We will further need the following fundamental inequalities from the interpolation theory due to Marcinkiewz & Zygmund [18, Theorem 7.10, Chapter X, p.30].

Lemma 2.2. For α > 1, n ≥ 1, and any analytic trigonometric polynomial P of degree ≤ n,

A α n n-1 j=0 P (ξ n,j ) α ≤ S 1 P (z) α dz ≤ B α n n-1 j=0 P (ξ n,j ) α , (2.4)
where A α and B α are independent of n and P .

For the trigonometric polynomials, Marcinkiewz-Zygmund interpolation inequalities can be stated as follows [START_REF] Zygmund | Trigonometric series[END_REF]Theorem 7.5, Chapter X, p.28].

Lemma 2.3. For α > 1, n ≥ 1, and any trigonometric polynomial P of degree ≤ n,

A α 2n + 1 2n j=0 P (ξ 2n+1,j ) α ≤ S 1 P (z) α dz ≤ B α 2n + 1 2n j=0 P (ξ 2n+1,j ) α , (2.5)
where A α and B α are independent of n and P .

main result and its proof

In this section, we start by stating our main result.

Theorem 3.1. Let α > 2 and suppose that the sequence of analytic polynomials

(P n ) is L α -semi-flat, that is, for each N ∈ N * , 1 √ N N j=1 µ(j)z j α < C α ,
for some constant C α . Then the Riemann hypothesis holds.

We are not able to see that the converse of our main theorem holds. although, we have the following Theorem 3.2. Suppose the Riemann hypothesis holds and for α > 2, the sequence

1 N 1+ α 2 N -1 k=1 N -1 n=1 λ(n)ξ n N,k α N ≥1 is bounded. Then the sequence of analytic polynomials (P n ) is L α -semi-flat.
Proof. [of the main theorem.] Assume that the Riemann Hypothesis does not holds. Then, according to Littlewood criterion (Lemma 2.1), there exist c > 0 and ǫ > 0 such that for infinitely many positive integers N , we have |M (N )| ≥ c.N 1 2 +ǫ . Let α > 1 such that αǫ > 1. Then, by Marcinkiewicz-Zygmund inequalities, we have

1 √ N N j=1 λ(j)z j α α ≥ A α N j=1 λ(j) α N α 2 +1 ≥ C α . N α 2 +αε N α 2 +1 = N αε-1 , Letting N -→ +∞, we conclude that 1 √ N N j=1 λ(j)z j α -----→ N →+∞ +∞.
This accomplishes the proof of the theorem.

Let us point out that we can also give a direct proof. Indeed, assume that for any α ≥ 1, we have Since α is arbitrary, it follows, with the help of (2.3), that RH holds. Now, let us give the proof of Theoreom 3.2.

Proof. [of Theorem 3.2.] Under our assumptions combined with Marcinkiewicz-Zygmund Theorem, it is easy to see that the sequence of polynomials (P n ) n≥1 is L α -semiflat for any α ≥ 0.

At this point we ask the following Question 3.3. Does the RH implies the semi-flatness. In another word, is the converse of Theorem 3.1 true?

  by the square of any prime; 0 if not Those two functions are related to Riemann ζ-function via the formulae

  Marcinkiewicz-Zygmund inequalities, for any α > 1, there exist A α such that

Acknowledgments. The author would like to thanks Sukumar Das Adhikari, Ramachandran Balasubramanian, Igor Shparlinski, Jean-Paul Thouvenot for a stimulating conversations on the subject and their sustained interest and encouragement in this work. He would like also to thanks the Ramakrishna Mission Vivekananda Educational and Research Institute and the organizers of the "Workshop on Topological Dynamics, Number Theory and related areas for the invitation.


Mathieu Rodriguez

Philippe Huchon

Nicolas Chamot-Rooke

Marc Fournier

Matthias Delescluse

Structural reorganization of the India-Arabia strike-slip plate boundary (Owen Fracture Zone; NW Indian Ocean

Keywords:

come   L'archive ouverte pluridisciplinaire

-Introduction

Strike-slip plate boundaries display a large variety of geological structures along their strike, especially in areas where the layout of the displacement zone is discontinuous or curved [START_REF] Mann | Global catalogue, classification and tectonic origins of restraining-and releasing bends on active and ancient strike-slip fault systems[END_REF]. These step-over areas favor the formation of releasing or restraining bends, according to the configuration of adjacent strike-slip fault segments and the local stress field [START_REF] Sylvester | Strike-slip faults[END_REF]. Detailed tectonic and stratigraphic investigations have revealed that strike-slip boundaries experience dramatic episodes of structural reorganization during their lifetime, marked by the formation of new structures and the abandonment of older ones (ten [START_REF] Brink | The anatomy of a pull-apart basin: seismic reflection observations of the Dead Sea[END_REF][START_REF] Wakabayashi | Stepovers that migrate with respect to affected deposits: field characteristics and speculation on some details of their evolution[END_REF][START_REF] Brothers | Tectonic evolution of the Salton Sea inferred from seismic reflection data[END_REF][START_REF] Schattner | What triggered the early-to-mid Pleistocene tectonic transition across the entire easternMeditterranean? Earth Planet[END_REF][START_REF] Brink | Geometry and subsidence history of the Dead Sea basin: a case forfluid induced midcrustal shear zone?[END_REF][START_REF] Le Pichon | The active main Marmara fault[END_REF], 2013). Models of structural evolution have proposed that continental strike-slip boundaries initiate as diffuse, en-_ echelon fault systems and become narrower with increasing maturity [START_REF] Tchalenko | Similarities between shear zones of different magnitudes[END_REF][START_REF] Wesnousky | The San Andreas and Walker Lane fault systems, western North America: transpression, transtension, cumulative slip and the structural evolution of a major transform plate boundary[END_REF][START_REF] Dooley | Analogue modelling of intraplate strike-slip tectonics: a review and new experimental results[END_REF][START_REF] Le Pichon | Propagation of a strike-slip plate boundary within an extensional environment: the westward propagation of the North Anatolian fault[END_REF]. This pattern of structural evolution is modulated by the layered rheology of the continental lithosphere (Le [START_REF] Pourhiet | Links between long-term and short-term rheology of the lithosphere: insights from strike-slip fault modeling[END_REF].

The structural evolution of oceanic strike-slip faults (with seismicity identified along their entire length) has been investigated only for a few cases, including the MacQuarie Fault (Australia-Pacific boundary; [START_REF] Massel | Neotectonics of the Macquarie ridge complex, Australia-Pacific plate boundary[END_REF][START_REF] Meckel | Influence of cumulative convergence on lithospheric thrust fault development and topography along the Australian-Pacific plate boundary south of New Zealand[END_REF] or the Azores-Gibraltar transform fault (Nubia-Eurasia boundary; [START_REF] Zitellini | The quest for the Africa-Eurasia plate boundary west of the strait of Gibraltar[END_REF][START_REF] Rosas | Analog modeling of strike-slip fault (lateral) propagation from an elastic to a viscous medium: insights from trials experiments[END_REF][START_REF] Miranda | Distributed deformation close to the Ac¸ores triple 'point[END_REF]. Here we focus on the oceanic India-Arabia strike-slip plate boundary, which initiated ∼90 Ma when India separated from Madagascar [START_REF] Bernard | Le bassin des Mascareignes et le bassin de Laxmi (océan Indien occidental) se sont-ils formés à l'axe d'un même centre d'expansion ?[END_REF] and started its motion toward Eurasia. The India-Arabia plate boundary experienced several episodes of migrations in response to India-Eurasia or Arabia-Eurasia collision (Rodriguez et al., 2014a(Rodriguez et al., ,b, 2016)). Multibeam mapping of the current India-Arabia plate boundary (Fig. 1), referred to as the Owen fracture zone (OFZ) in the Arabian Sea, revealed dextral morphological offsets of the Owen Ridge on the order of 10 -12 km (Fournier et al., 2008a(Fournier et al., ,b, 2011))-the Owen Ridge being a series of bathymetric highs uplifted 8.7 Ma ago (Rodriguez et al., 2014a(Rodriguez et al., , b, 2018)). Considering steady the current right-lateral motion of 3±1 mm yr -1 [START_REF] Demets | Geologically current plate motions[END_REF], these offsets indicate a recent reorganization of the OFZ, younger than the Late Miocene.

Using multibeam and seismic data, tied to nearby ODP-DSDP sites, we performed detailed structural and stratigraphic studies to investigate the age of formation of each major structure (mainly pull-apart basins) observed along the OFZ in order to determine its age of formation. The results show that the present-day expression of the entire, 800-km-long OFZ formed at 2.4 Ma, from the Aden-Owen-Carlsberg triple junction to the Makran subduction zone (Fig. 1) and involved the opening of conspicuous pull-apart basins (from south to north: the Beautemps-Beaupré Basin, Fig. 2; the 20° N Basin, Fig. 3; the Dalrymple Trough, Fig. 4).

-THE SEDIMENTARY RECORD OF STRIKE-SLIP TECTONICS ALONG THE OWEN FRACTURE ZONE

The Indus Turbiditic Channels

The OFZ crosses the distal Indus turbidite fan, which is fed from the east by the Indus canyon cutting through the NW Indian margin [START_REF] Von Rad | Late quaternary sedimentation on the outer Indus shelf and slope (Pakistan): evidence from high-resolution seismic data and coring[END_REF][START_REF] Rodriguez | Neotectonics of the Owen fracture zone (NW Indian Ocean): structural evolution of an oceanic strike-slip plate boundary[END_REF][START_REF] Rodriguez | Mode of opening of an oceanic pullapart: the 20 °N Basin along the Owen fracture zone (NW Indian Ocean)[END_REF]Rodriguez et al., , 2014a, b;, b;[START_REF] Bourget | Late quaternary megaturbidites from the Indus fan: Origin and stratigraphic significance[END_REF]. Turbiditic channels are observed on both sides of the OFZ (Figs. 3 and4). In terms of relative chronology, the turbiditic channels observed west of the OFZ (in the Owen Basin) predate its formation and indicate a period of limited tectonic activity, or a period when sedimentation rates were too high to record tectonics. East of the OFZ, the turbiditic channels trapped or deviated by strike-slip structures postdate the formation of the OFZ. Dating the different turbiditic systems provides good age brackets for the formation of the OFZ. The age of a turbiditic channel is estimated from the age of the first pelagic layer that covers it.

Fault-Controlled Contourite Drifts

Bottom currents influence the geometry of deep-sea sedimentary deposits and build conspicuous sedimentary formations referred to as contourite drifts [START_REF] Rebesco | Contourites and associated sediments controlled by deep-water circulation processes: state of the art and future considerations[END_REF]. In the vicinity of the OFZ, several fault-controlled contourite drifts are observed within the pelagic blanket lying over the turbiditic channels (Figs. 3 and4). The opening of pull-apart basins along the OFZ induced local perturbations of bottom current. The base of a fault-controlled drift indicates the minimum age of formation of the fault.

Angular Unconformities

Vertical motion of the seafloor (uplift or subsidence) related to faults results in major angular unconformities within the Indus fan, sometimes outlined by conspicuous fanning configurations recording the growth of the structure. Within the fanning configurations, numerous unconformities reflect the control of sealevel variations at the 105 years time-scale over the Indus fan sedimentation [START_REF] Bourget | Late quaternary megaturbidites from the Indus fan: Origin and stratigraphic significance[END_REF].

-AGE OF STRUCTURES ALONG THE OWEN FRACTURE ZONE

3.1 The Beautemps-Beaupré Pull-Apart Basin

The Beautemps-Beaupré Basin is a 120-km-long, 50-km-wide rhomboidal pull-apart basin located at the southern termination of the OFZ (Fig. 2; Fournier et al., 2008a,b). The Beautemps-Beaupré Basin is almost entirely filled in by Indus turbidites. The basin is bounded to the south by the Beautemps-Beaupré Ridge [START_REF] Rodriguez | The geological evolution of the Aden-Owen-Carlsberg triple junction (NW Indian Ocean) since the Late Miocene[END_REF], which corresponds to a tilted section of Indus turbidites (Fig. 2). Numerous angular unconformities are identified at the edges of the basin (Fig. 2), most of them being related to interruptions of Indus sedimentation related to sea-level variations [START_REF] Bourget | Late quaternary megaturbidites from the Indus fan: Origin and stratigraphic significance[END_REF]. The onset of the uplift of the Beautemps-Beaupré Ridge is recorded by a conspicuous angular unconformity and the base of the fanning configuration of a sequence of Indus sediments (Fig. 2; [START_REF] Rodriguez | The geological evolution of the Aden-Owen-Carlsberg triple junction (NW Indian Ocean) since the Late Miocene[END_REF]. This unconformity can be tracked within the Beautemps-Beaupré Basin, where it coincides with the onset of lateral variations in thickness of turbidites, which marks the onset of seafloor subsidence there. Since its uplift above the level of turbidites deposition, the Beautemps-Beaupré Ridge is blanketed by pelagic sediments that can be correlated with pelagic sediments on top of the nearby Owen Ridge (Fig. 2), where ODP sites provide stratigraphic constraints (Discoaster pentaradius; Shipboard Scientific Party, 1989). This unconformity is dated at 2.4 Ma (Rodriguez et al., 2014b[START_REF] Rodriguez | The geological evolution of the Aden-Owen-Carlsberg triple junction (NW Indian Ocean) since the Late Miocene[END_REF].

The 20°N Pull-Apart Basin

The 20° N Basin, named after its latitude, is an asymmetric, 90-km-long, 12-km-wide pullapart basin (Fig. 3; [START_REF] Fournier | Owen fracture zone: the Arabia-India plate boundary unveiled[END_REF]. The OFZ constitutes the western flank of the 20° N Basin, while imbricated systems of arcuate normal faults dissect its eastern flank (Fig. 3). The 20° N Basin is divided into three subbasins by the transverse faults (Fig. 3). Fossil turbidite channels are identified west of the basin, whereas the currently active channel is captured on the eastern side of the basin [START_REF] Rodriguez | Neotectonics of the Owen fracture zone (NW Indian Ocean): structural evolution of an oceanic strike-slip plate boundary[END_REF][START_REF] Bourget | Late quaternary megaturbidites from the Indus fan: Origin and stratigraphic significance[END_REF]. The most recent fossil turbidite channel to the west is dated at 3.4 ± 1.2 Ma [START_REF] Rodriguez | Mode of opening of an oceanic pullapart: the 20 °N Basin along the Owen fracture zone (NW Indian Ocean)[END_REF], which gives the maximal age of the opening of the basin (Fig. 3). A fault-controlled contourite drift, with a typical sigmoid geometry, is also identified on the top of the master fault (Fig. 3). The opening of the basin may have disturbed the course of the bottom current and triggered the building of the drift. The reflector marking the base of the drift can be correlated within the pelagic cover as far as the ODP sites located at the top of the Owen Ridge. The age of this reflector is 2.4 Ma [START_REF] Rodriguez | Mode of opening of an oceanic pullapart: the 20 °N Basin along the Owen fracture zone (NW Indian Ocean)[END_REF](Rodriguez et al., , 2014b)).

The Dalrymple Trough

The Dalrymple Trough marks the northern termination of the OFZ [START_REF] Edwards | Extension across the Indian-Arabian plate boundary: the Murray Ridge[END_REF][START_REF] Ellouz Zimmermann | Offshore Frontal Part of the Makran Accretionary Prism (Pakistan) the Chamak Survey[END_REF]. The southern segment of the Dalrymple Trough is a 150-km-long, 30-km-wide horsetail termination basin (Fig. 4), with numerous oblique splays connecting the OFZ [START_REF] Fournier | Owen fracture zone: the Arabia-India plate boundary unveiled[END_REF]Rodriguez et al., 2014b). The Dalrymple Trough is flanked to the east by the Murray Ridge. Indus turbidite channels dated at 3.7 ± 1 Ma to the west of the Dalrymple Trough predate its opening (Fig. 4). In contrast to the 20° N and Beautemps-Beaupré basins, the Dalrymple Trough has been isolated from the Indus infill subsequently to the uplift of the Murray Ridge. On transverse seismic profile (Fig. 4), the core of the Dalrymple Trough is expressed as a syncline. The last deformed layer can be fairly correlated with the Indus sequence at the border of the basin (Fig. 4). It coincides with the reflector marking a major angular unconformity in front of the Makran accretionary wedge (M-unconformity; [START_REF] Gaedicke | Seismic stratigraphy and correlation of major regional unconformities in the northern Arabia Sea[END_REF][START_REF] Ellouz Zimmermann | Offshore Frontal Part of the Makran Accretionary Prism (Pakistan) the Chamak Survey[END_REF]. Moreover, this reflector marks the base of a fault-controlled contourite drift close to the OFZ at the entrance of the trough (Rodriguez et al., 2014b). Here again, this reflector within well-bedded pelagic layers can be correlated from line to line to the location of the ODP sites. It is also dated at 2.4 Ma. However, the age of the northern segment of the Dalrymple Trough, connecting the Ornach Fault in Pakistan, remains to be constrained. Detailed tectono-stratigraphic studies indicate the present-day configuration of the entire 800-km-long OFZ formed at 2.4 Ma, expressed by the coeval opening of the Beautemps-Beaupré basin to the south, the 20° N basin, and the Dalrymple Trough to the north. Considering the 10 -12 km-morphological offsets were formed during the last 2.4 Ma implies a dextral rate of India-Arabia relative motion at ~ 4.2 ± 5 mm yr -1 . The India-Arabia boundary is located in this area since at least the Early Miocene and the first stages of seafloor accretion at the Sheba Ridge ago [START_REF] Fournier | Arabia-Somalia plate kinematics, evolution of the Aden-Owen-Carlsberg triple junction, and opening of the Gulf of Aden[END_REF]. The India-Arabia plate boundary has accommodated since then about 80-km of dextral relative motion [START_REF] Chamot-Rooke | Tracking Arabia-India motion from Miocene to Present[END_REF]. When the OFZ formed at 2.4 Ma, the India-Arabia boundary was therefore already a mature system. The timing of the formation of the OFZ at 2.4 Ma does not correspond to a clearly identified kinematic change [START_REF] Demets | Reconciling geodetic and geological estimates of recent plate motion across the Southwest Indian Ridge[END_REF], making the geodynamic driver of its formation unknown. Either the corresponding kinematic change has not been detected so far, or there is no kinematic change related to the onset of the OFZ. The formation of the OFZ may simply be the last step of a series of transient adjustments of the India-Arabia plate boundary since the last major kinematic change identified in the Indian Ocean between 6 and 8 Ma [START_REF] Demets | High-resolution estimates of Nubia-Somalia plate motion since 20 Ma from reconstructions of the Southwest Indian Ridge, Red Sea and Gulf of Aden[END_REF][START_REF] Demets | Reconciling geodetic and geological estimates of recent plate motion across the Southwest Indian Ridge[END_REF]. It is also possible that the 2.4 Ma episode of intensification of the Indian Monsoon [START_REF] An | Evolution of Asian monsoons and phased uplift of the Himalaya-Tibetan plateau since Late Miocene times[END_REF] might have played a role in the evolution of the strike-slip system through its effect on the Indus sedimentation rates (increase up to 500 m/Ma at 2.4 Ma; Shipboard Scientific Party, 1989).

Figure captions

Figure 1

 1 Figure 1 Multibeam bathymetric map of the Owen fracture zone. Inset shows the plate tectonic context in the northwestern Indian Ocean. (a) and (b) are enlargements of two areas where the lateral offset has been measured (respectively 10 and 12 km).

Figure 2 (

 2 Figure 2 (A) Multibeam bathymetric map of the Beautemps-Beaupré Basin (B 3 )[1], (B) North-South seismic profile across the Beautemps-Beaupré Basin, and (C) its interpretation (in blue, the post -2.4 Ma infill of the basin, in black the 8.8 Ma discordance that marks the uplift of the Owen ridge).

Figure 3 (

 3 Figure 3 (A) Multibeam bathymetric map of the 20°N Basin, OFZ: Owen Fracture Zone, SB1, SB2, SB3: subbasins, (B) West-East seismic profile across the sub-basin SB3 of the 20°N Basin, and (C) its interpretation (in blue, the post -2.4 Ma infill of the basin).

Figure 4 (

 4 Figure 4 (A) Multibeam bathymetric map of the Dalrymple Trough (inset is a bird's eye view from the southwest), (B) West-East seismic profile across the southern part of the Dalrymple Trough, and (C) its interpretation. The pink reflector dated at 2.4 Ma marks the beginning of the opening of the basin. The post -2.4 Ma infill of the basin is shown in blue.

Acknowledgments

This study is a synthesis of numerous works based on the multibeam and seismic dataset collected during the AOC, OWEN 1 and 2 surveys in the last decade. All scientists and cruise members (R/V Beautemps-Beaupré, SHOM) involved in these projects are warmly thanked for their help. We thank J. Duarte and T. Minshull for their detailed comments that helped us to improve this synthesis.