Raphaël Fargier
email: raphael.fargier@isc.cnrs.fr

Yves Paulignan

Véronique Boulenger

Padraic Monaghan

Anne Reboul

Tatjana A Nazir

Learning to associate novel words with motor actions: Language-induced motor activity following short training

Action words referring to face, arm or leg actions activate areas along the motor strip that also control the planning and execution of the actions specified by the words. This EEG study aimed to test the learning profile of this language-induced motor activity. Participants were trained to associate novel verbal stimuli to videos of object-oriented hand and arm movements or animated visual images on two consecutive days. Each training session was preceded and followed by a test-session with isolated videos and verbal stimuli.

We measured motor-related brain activity (reflected by a desynchronization in the µ frequency bands; 8-12 Hz range) localized at centro-parietal and fronto-central electrodes. We compared activity from viewing the videos to activity resulting from processing the language stimuli only. At centro-parietal electrodes, stable action-related µ suppression was observed during viewing of videos in each testsession of the two days. For processing of verbal stimuli associated with motor actions, a similar pattern of activity was evident only in the second test-session of day 1. Over the fronto-central regions, µ suppression was observed in the second test-session of day 2 for the videos and in the second testsession of day 1 for the verbal stimuli.

Whereas the centro-parietal µ suppression can be attributed to motor events actually experienced during training, the fronto-central µ suppression seems to serve as a convergence zone that mediates underspecified motor information. Consequently, sensory-motor reactivations through which concepts are comprehended seem to differ from those implicated in their acquisition.

Introduction

In their seminal work, [START_REF] Hauk | Somatotopic representation of action words in human motor and premotor cortex[END_REF] demonstrated that words referring to face, arm or leg actions differentially activated areas along the motor strip that are adjacent to or overlapping with areas activated by the movement described by the words. This result was taken to suggest that the referential meaning of action words has a correlate in the somatotopic activation of motor and premotor cortex. A number of imaging studies have since reported related findings [START_REF] Aziz-Zadeh | Congruent embodied representations for visually presented actions and linguistic phrases describing actions[END_REF][START_REF] Boulenger | Grasping ideas with the motor system: semantic somatotopy in idiom comprehension[END_REF][START_REF] Kemmerer | Neuroanatomical distribution of five semantic components of verbs: evidence from fMRI[END_REF][START_REF] Tettamanti | Listening to action-related sentences activates fronto-parietal motor circuits[END_REF] but disagreements about how to interpret these observations have also accumulated. Essentially, the debate focuses on the question as to whether language-induced cortical motor activity is a source or, rather, a consequence of language understanding (see [START_REF] Mahon | A critical look at the embodied cognition hypothesis and a new proposal for grounding conceptual content[END_REF] for a stimulating discussion). While advocates of "embodied theories" of semantics interpret the results as evidence that we use our motor system to understand language about actions (e.g., [START_REF] Allport | Distributed memory, modular subsystems and dysphasia[END_REF][START_REF] Fischer | Embodied language: a review of the role of the motor system in language comprehension[END_REF][START_REF] Gallese | The Brain's Concepts: The Role of the Sensory-Motor System in Reason and Language[END_REF][START_REF] Martin | Neural correlates of category-specific knowledge[END_REF]Pulvermüller, 1999), opponents of this view suggest that language-induced motor activity is an "after-effect" of language understanding that follows computations of abstract conceptual representations of action words performed elsewhere in the brain [START_REF] Damasio | Time-locked multiregional retroactivation: a systems-level proposal for the neural substrates of recall and recognition[END_REF][START_REF] Mahon | A critical look at the embodied cognition hypothesis and a new proposal for grounding conceptual content[END_REF]; see also [START_REF] Patterson | Where do you know what you know? The representation of semantic knowledge in the human brain[END_REF][START_REF] Simmons | The similarity in topography principle: Reconciling theories of conceptual deficits[END_REF] for related proposals). Given the difficulty of isolating the functional role of motor activity during language processing, this debate seems as yet irresolvable. However, studies that probe the trajectory of learning of motor-language associations may provide insight into the causal role of motor activity for language processing.

The arbitrary relationship between the phonological forms of words and their meaning necessarily entails that cortical motor activity observed during the processing of verbal action description is the result of learning. In fact, according to an idea first put forward by Pulvermüller (1999Pulvermüller ( , 2005)), links between brain regions involved in language processing and brain regions involved in action planning/execution could develop through Hebbian associative learning. Since "action words" (mostly verbs) are often acquired and experienced in the context of execution of the depicted actions [START_REF] Goldfield | Nouns before verbs in comprehension vs. production: the view from pragmatics[END_REF], and given Hebb's postulate that synchronous activity of neurons leads to the formation of neuronal assemblies [START_REF] Hebb | The Organization of Behavior[END_REF], Pulvermüller suggested that neural networks including perisylvian language areas and motor areas emerge with experience. By means of these shared circuits, perceiving an action word will then automatically trigger activity in motor regions of the brain (see also Del Giudice et al., 2008, andKeysers &Perret, 2004, for similar ideas regarding potential links between action execution and observation systems).

The assumption that association learning can rapidly generate such shared neural circuits is supported by studies showing that the selectivity of the brain's motor response to a given perceptual stimulus changes with learning. In a study by [START_REF] Catmur | Through the looking glass: counter-mirror activation following incompatible sensorimotor learning[END_REF], a group of participants were trained over three consecutive days to perform an action with the hand while observing a similar hand action.

Another group of participants were trained to perform the hand action while observing an action with the foot. After training, motor areas that showed greater responses to the observation of hand actions in the first group, responded more strongly to observation of foot actions in the second group (see also [START_REF] Catmur | Sensorimotor learning configures the human mirror system[END_REF]. Relatedly, [START_REF] Lahav | Action representation of sound: audiomotor recognition network while listening to newly acquired actions[END_REF] showed that shared circuits between auditory processes involved in music perception and motor processes involved in piano playing can emerge rapidly with training. In this study, non-musicians were trained to play a piano piece over a five day training period. When playing the piano, auditory feedback is naturally induced by each of the player's movements leading to a tight coupling between perception and action. Following training, simply listening to the newly acquired piece of music triggered activity in cortical motor regions. [START_REF] Mcnamara | Neural dynamics of learning sound-action associations[END_REF] reported similar results for sound-gesture associations. Finally, in a complex learning experiment, [START_REF] Kiefer | Experience-dependent plasticity of conceptual representations in human sensory-motor areas[END_REF] taught participants over a period of three weeks to categorize a large number of novel objects into different object categories by either pantomiming a functional action that could be associated with the object or by simply pointing to the objects. Following training, presentation of the objects triggered activity in visuo-motor and frontal motor regions when training involved pantomiming but not when it involved pointing (see also [START_REF] Weisberg | A neural system for learning about object function[END_REF] for similar results for novel tool use).

All these learning experiments clearly demonstrate that cortical motor responses to perceptual stimuli can rapidly change with experience. Consequently, it should not be surprising to find similar effects when we learn to associate novel word-forms with self-performed motor actions. Indeed, [START_REF] Revill | Neural correlates of partial lexical activation[END_REF], using fMRI, trained participants to associate novel verbal stimuli with motion changes of objects and demonstrated that following training, participants showed greater language-induced activation in cortical regions that had been linked to the processing of motion and to the processing of motion-related real words (e.g., left MT/V5 and its anterior part; [START_REF] Kable | Neural substrates of action event knowledge[END_REF][START_REF] Kable | Conceptual representations of action in the lateral temporal cortex[END_REF][START_REF] Martin | Discrete cortical regions associated with knowledge of color and knowledge of action[END_REF]. However, the low temporal resolution of fMRI means it is unclear exactly when this languageinduced motor activity becomes evident and how it evolves during training. To align the findings of [START_REF] Hauk | Somatotopic representation of action words in human motor and premotor cortex[END_REF] with research on the acquisition of word-referent relations, we therefore propose to monitor the temporal dynamics of learning word-action associations in adult participants, from the beginning of training to the point when associations are well established in behavior. Following the prediction by Pulvermüller (1999) and in line with previous learning experiments, we assume that associating novel verbal stimuli with a novel hand/arm movement (but not with a novel animated visual image) will lead to co-activation of perisylvian language regions and motor regions of the brain. Subsequent to training, the verbal stimuli alone should thus trigger activity in motor regions.

To study the dynamics of word-action associations, we analyzed oscillations of EEG (electroencephalogram) in the 8-12 Hz frequency range (i.e., "μ rhythms"; [START_REF] Gastaut | Electrocorticographic study of the reactivity of rolandic rhythm[END_REF] while participants listened to initially novel verbal stimuli, at four periods during training over two consecutive days. μ rhythms are typically present at rest but are suppressed, for instance, when performing a movement. This suppression is believed to reflect an event-related desynchronization of the EEG caused by an increase of neural activity in somato-sensory cortex [START_REF] Hari | Magnetoencephalographic cortical rhythms[END_REF]. In line with fMRI studies that indicated neural motor activity while listening to action sounds or observing actions [START_REF] Buccino | Neural circuits involved in the recognition of actions performed by nonconspecifics: an FMRI study[END_REF][START_REF] Kilner | Evidence of mirror neurons in human inferior frontal gyrus[END_REF][START_REF] Rizzolatti | Premotor cortex and the recognition of motor actions[END_REF], μ suppression has also been observed when participants hear the sound of actions [START_REF] Caetano | Actor's and observer's primary motor cortices stabilize similarly after seen or heard motor actions[END_REF] or when they observe a movement performed by others (see [START_REF] Pineda | Sensorimotor cortex as a critical component of an 'extended' mirror neuron system: Does it solve the development, correspondence, and control problems in mirroring?[END_REF], for a review). In the present study we therefore first identified EEG electrodes for which μ suppression was evident during movement observation, then we subsequently focused on these "electrodes of interest" for the analysis of μ suppression during processing of newly acquired action words.

Materials and method

Participants

Sixteen right-handed (scores 0.79 ± 0.2; [START_REF] Oldfield | The assessment and analysis of handedness: the Edinburgh inventory[END_REF] French native volunteers participated in the study (9 females; mean age 24.2 ± 4 years). None of the participants had a history of psychiatric or neurological disorders and all had normal or corrected-to-normal vision. Participants gave written consent to partake in the study in accordance to the Helsinki declaration (1968). They were paid for their participation.

Stimuli

The stimuli consisted of 20 different video clips (presented on a computer screen 80 cm from the participants) and 20 novel verbal stimuli. Half of the clips consisted in reaching-and-grasping movements towards a horizontally or vertically oriented cylinder. The object-directed movement was performed with the right hand. Each movement started from a rest position (the hand on a table in front of the actor's torso) and ended with the grasping of the object. Movements differed in the way the hand moved towards the object (e.g. a zigzag movement). The other half of the clips consisted of animated artificial images, which differed in shapes, colors and animations (see Figure 1). Ten pseudo-words and their temporally reversed counterparts (backward speech 1 ) served as verbal stimuli.

Pseudo-words were bisyllabic and were obtained by changing one or two letters from frequent French words (i.e. they respected the phonotactic rules of French). Average length of the verbal stimuli was 665 msec (± 123). Average length of video clips was 4600 msec (± 1170) for movement-clips and 3900 msec (± 1290) for image-clips. Verbal stimuli were presented binaurally through loudspeakers.

1 Backward speech was used to determine effects of stimulus familiarity. However, for these stimuli associations were not fully acquired even at the end of Day 2. Therefore, data for these stimuli are difficult to interpret and were not further considered.

Procedure

The experiment was conducted over two consecutive days. Each day included three sessions. Day 1 started with a test-session in which participants were instructed to listen attentively to the verbal stimuli and watch the video clips one by one. This first test-session was followed by a training-session in which participants learned to associate verbal stimuli and video clips. The training session was followed by a second test-session, which was identical to the first. Finally, a behavioral test assessed the effectiveness of learning. Day 2 started with a test-session followed by a training-session, a further test-session and the behavioral test. The exact procedure of each part of the experiment is described below. Note that EEG was recorded throughout all sessions. However, during training EEG recordings resulted in noisy signals and were thus not analyzed.

Day 1 2.3.1.1. First test session.

During the test-sessions the 20 verbal stimuli (pseudo-words and backward speech) and the 20 video clips (image or reaching/grasping movement) were presented in isolation in a pseudo-random order. For each trial, participants were requested to fixate on a crosshair at the center of the computer screen, which lasted for 500 msec, and to pay attention to the auditory or visual stimuli presented subsequently. At the end of each trial a grey screen appeared for 1sec, allowing participants to blink. Each stimulus was presented 5 times for a total of 200 trials.

Training session.

Each training trial started with a fixation cross, presented for 500 msec at the centre of the screen. The cross was followed by the presentation of a video clip (image or reaching/grasping movement). Participants were instructed to watch the video. A white screen of 500 msec marked the end of the clip. The same clip was then displayed again together with the verbal stimulus, which was presented shortly after the onset of the clip. When the clip depicted a movement (movement condition) the onset of the verbal stimulus coincided with the beginning of the movement and participants were requested to imitate the movement while listening to the verbal stimulus. When the clip showed an animated image, participants simply observed the image again while listening to the verbal stimulus (image condition). Participants were explicitly instructed to associate the verbal stimulus with the movement or the image depicted in the clip. To ensure that participants had finished imitating the movement before the beginning of the next trial, the experimenter triggered the trials manually. Each verbal stimulus/video clip pairing was displayed 15 times and the 300 trials in total were presented in pseudo-random order. A break was proposed to the participants after every 100 trials. The coupling of a given verbal stimulus with a given video clip was counter-balanced between participants such that the same verbal stimulus was associated with a movement for one participant and with an image for another. Identical to 2.3.1.1. 2.3.1.4. Behavioral task. In order to quantify learning performance, a behavioral test was performed as the final session on each day. For this, the 20 verbal stimuli were presented one by one and participants were requested to indicate if the stimulus was associated with an image or a movement in the previous training session. For verbal stimuli coupled with a movement, participants were asked to reproduce the movement. For verbal stimuli coupled with an image, they were asked to describe shapes, colors and animations.

Second test session.

Day 2

Day 2 started with a test-session, followed by a training-session, another test-session and the behavioral test. Except for a reduction in the number of repetitions in the training session (10 instead of 15) all experimental procedures were identical to those described for day 1.

Data acquisition

EEG data were recorded using BrainAmp amplifiers (Brain Vision recorder software, Brain Products GmbH, Munich, Germany). EEG was recorded from 32 scalp sites using the international 10-20 system (American Encephalographic Society, 1994), with a forehead ground. Impedance was 10 KΩ or less at the start of the recording. All scalp sites were referenced to the left mastoid. Horizontal and vertical eye movements were monitored using electro-oculograms (EOG) with a bipolar recording from electrodes placed on the outer canthi of the left eye. ERPs were sampled at 500 Hz.

Data analysis

EEG activity was analyzed using BrainVision Analyzer 2.0 software (Brain Vision analyzer software, Brain Products GmbH, Munich, Germany). First, a notch filter (>50 Hz) was applied. The EEG was then re-referenced to an average reference [START_REF] Bertrand | A theoretical justification of the average reference in topographic evoked potential studies[END_REF] excluding both hEOG and vEOG. A manual inspection of the raw data was performed before the application of an ocular correction ICA (512 ICA steps; convergence bound of 1.10 -7 ). Data were grouped according to the six conditions (movement video-clips; image video-clips; pseudo-words associated with an image; pseudo-words associated with a movement; backward speech associated with an image; backward speech associated with a movement) and epoched from -500 msec to 1500 msec relative to stimulus onset. Trials were baseline corrected (200 msec pre-stimulus baseline). Wavelet transformation in the 1-30 Hz domain (frequency steps of 20) was computed for each trial by convolving a Morlet wavelet with the signal with a width of 7 cycles (van Elk et al., 2010). Trials were averaged for each condition. Grandaverages were then calculated over all participants.

Statistical analysis

To test for statistical significance, the average amplitude of μ oscillations (8-12 Hz range) was computed for each condition and test-session over successive segments of 50 msec, starting from 0 msec to 1000 msec after stimulus onset (van Elk et al., 2010). Repeated-measures analyses of variance (ANOVAs) that contrasted μ oscillations in the "movement condition" to the "image condition" were then performed for all time-windows. Main factors included: day (first and second day of the experiment), test-session (first and second test-session of the day), condition (movement and image) and electrode (see Results section for the identification of "electrodes of interest"). Note that to reduce false positives, several consecutive intervals should show significant effects (p < .05) (see van Elk et al. 2010). Post-hoc (LSD-Fisher) analyses for test-sessions were therefore performed only when three or more consecutive time-bins showed a significant effect of the factor test-session and only within the time interval identified in the main analysis.

Results

Behavioral data

Participants had acquired all associations between verbal stimuli and depicted movements or images at the end of Day 1 and remained at ceiling at Day 2.

EEG data

Videos: Electrodes of interest

The onset of all videos triggered μ suppression over large parts of the head. To identify "electrodes of interest", we thus contrasted the topography of µ oscillations to videos that depicted the reaching-andgrasping movements to those that depicted the animated artificial images. Figure 2 shows spatial cartographies that present the difference in µ oscillations (in the 8-12 Hz frequency range) between the two conditions (movement-condition minus image-condition) within a time window of 250-1000 msec after video onset. Data are plotted separately for the four test-sessions. In line with previous studies showing reduced μ activity during preparation or observation of finger/hand movements over centroparietal sites contra-lateral to the performing hand (e.g. [START_REF] Pfurtscheller | Foot and hand area mu rhythms[END_REF][START_REF] Babiloni | Human cortical electroencephalography (EEG) rhythms during the observation of simple aimless movements: a high-resolution EEG study[END_REF], the contrast between movement-condition and image-conditions revealed a centro-parietal site over the left hemisphere at which stronger μ suppression was observed in the movement-condition. This pattern, which despite some variations could be restricted to electrodes CP1, CP3, P3 and Pz (Centro-Parietal cluster), was already present prior to training (i.e., during the first test-session of Day 1) and was observed in the remaining three test-sessions. Moreover, in agreement with data recently reported by van Elk et al. (2010), a cluster of electrodes over fronto-central regions showed stronger movement-than image-related μ suppression, especially at the second test-session of each day. This second region of interest involved electrodes F3, F4, Fz, FC1, FC3, FC2 and FC4 (Fronto-Central cluster). (movement-condition minus image-condition). Figure 3C plots µ activity (8-12Hz only) for each of the involved electrodes (black curves represent data for the movement-condition, red curves for the image-condition). These figures show that in the movement-condition as well as in the imagecondition µ suppression started to become evident between 200-400 msec after video onset. However, at centro-parietal electrodes, a tendency for stronger µ suppression during viewing of videos depicting the reaching-and-grasping movements was evident in all test-sessions. The ANOVA for this cluster of electrodes revealed no main effects, but a significant interaction between factors electrode x condition was observed for seven consecutive time-bins (from 450-850 msec post-stimulus onset; all p < .05).

The interaction between electrode x condition was due to variations of the effects at the different electrodes; see Suppl. material). This analysis thus indicates that at centro-parietal electrodes the effect of condition was present prior to training and was maintained throughout all test-sessions. At the fronto-central site, display of videos had little impact on µ activity prior to training. However, µ suppression intensified in the test-session that followed training, with a tendency for stronger µ suppression in the movement-condition. This pattern was observed on both days. The ANOVA confirmed the effect of test-session in sixteen consecutive time-bins from 200-1000 msec poststimulus onset (all p < .05). A significant interaction between electrode x condition was also seen during five consecutive time-bins from 600-850 msec after stimulus onset (all p < .05), confirming that µ suppression was stronger during viewing of videos depicting motor actions. Finally, a significant day x test-session x condition x electrode interaction in fourteen consecutive time bins from 300-1000 msec post-stimulus onset (all p < .05) was found, suggesting that the effect of condition was stronger on Day 2 compared to Day 1. In separate ANOVAs for each of the two days no significant effect of condition was evident on Day 1. However, on Day 2 a significant interaction between test-session x condition x electrode was observed for five consecutive time-bins (from 750-1000 msec post stimulus onset; all p < .05). Post-hoc analyses showed that the effect of condition was restricted to the second test-session of Day 2 (from 750-1000 msec; all p < .05).

Verbal Stimuli

Figure 4 plots the corresponding data for verbal stimuli. As expected, no µ suppression was observed prior to training. However, during the second test-session of Day 1 µ suppression became evident at centro-parietal electrodes in the movement-condition. No such effects were observed for stimuli associated with animated images. On Day 2 this language-induced µ suppression was no longer evident, neither before nor after further training. The ANOVA for the centro-parietal electrode cluster did not reveal any significant effects, but a t-test (pooled over electrodes) for the second test-session of Day 1 showed significant effects of condition for eleven consecutive time-bins (from 400-950 msec after stimulus onset; all p < .05). Comparable to what was observed for the videos, modulation of µ activity at fronto-central electrodes was also mainly observed in the second test-session of each day.

However, whereas on Day 1 stimuli associated with a motor movement resulted in µ suppression, on Day 2 a tendency for an increase of µ activity was observed. The ANOVA indicated a significant main effect of electrode and a significant interaction between day x condition for three consecutive timebins (from 300-450 msec post stimulus onset; all p < .05). The three-way interaction between day x condition x test-session, indicating a potential reversal of µ activity between the second test-session of Day 1 and Day 2, reached significance in one time bin only (from 300-350 msec after stimulus onset).

A separate ANOVA for Day 1 indicated a significant interaction between test-session x condition in three time-bins (from 300-450 msec post stimulus onset; all p < .05) and post-hoc analyses restricted the effect of condition to the second test-session (from 300-450 msec; all p < .05). For Day 2 the ANOVA (or t-test) did not reveal any significant effects.

Discussion

By analyzing action-related μ oscillations in the EEG of adult participants, the present study monitored the temporal dynamics of learning word-action associations from the beginning of training to the point when associations are well established in behavior. In agreement with previous associative-learning experiments (e.g., [START_REF] Catmur | Through the looking glass: counter-mirror activation following incompatible sensorimotor learning[END_REF][START_REF] Kiefer | Experience-dependent plasticity of conceptual representations in human sensory-motor areas[END_REF][START_REF] Lahav | Action representation of sound: audiomotor recognition network while listening to newly acquired actions[END_REF][START_REF] Mcnamara | Neural dynamics of learning sound-action associations[END_REF][START_REF] Revill | Neural correlates of partial lexical activation[END_REF][START_REF] Weisberg | A neural system for learning about object function[END_REF], our behavioral data revealed that links between verbal stimuli and motor actions or images are rapidly established with training. At the end of training on Day 1, all participants correctly described the actions or images the novel verbal stimuli referred to.

Performance stayed at ceiling on the following day. Moreover, during viewing of videos, strong and stable action-related μ suppression was seen over a centro-parietal region of the hemisphere contralateral to the performing arm during all four test-sessions. In line with results from previous studies (e.g., [START_REF] Pfurtscheller | Foot and hand area mu rhythms[END_REF][START_REF] Babiloni | Human cortical electroencephalography (EEG) rhythms during the observation of simple aimless movements: a high-resolution EEG study[END_REF] this pattern of μ suppression thus seemed a good indicator of motor activity triggered by the observation/execution of actions depicted in the videos.

Despite the fact that the topography of the identified region hinged on the specific contrast we performed (movement-condition minus image-condition), the pure association account by Pulvermüller (Pulvermüller, 1999;[START_REF] Gallese | The Brain's Concepts: The Role of the Sensory-Motor System in Reason and Language[END_REF] predicted that after training, simply listening to verbal stimuli should trigger the pattern of brain activity that was seen during viewing of videos. Consistent with this prediction, once associations with verbal stimuli were established, a similar but weaker pattern of µ suppression was observed over the centro-parietal region during listening to verbal stimuli. This language-induced μ suppression was observed on Day 1 but not on Day 2. Action-related µ suppression was also seen at electrodes over fronto-central regions of the two hemispheres. Yet, the pattern of μ oscillations at these electrodes was more complex and did not directly follow the prediction of the association account. Prior to training, activity at these electrodes was little affected by stimulus display. Moreover, action-related μ suppression for videos was not observed before the end of training on Day 2. Hence, it is unlikely that μ suppression at these electrodes was directly triggered by the observation/execution of actions depicted in the videos. This view is further supported by the fact that at fronto-central electrodes action-related μ suppression for verbal stimuli was seen prior to that of videos, that is, on Day 1. Then, on Day 2, training induced a tendency for an increase of action-related µ activity to verbal stimuli. Hence, while language-induced motor activity seen at centro-parietal electrodes could be attributed to motor events actually experienced during training, the same does not hold for fronto-central electrodes. Rather, the observed pattern of training-dependent and action-related μ oscillations to videos and verbal stimuli could imply that these electrodes capture activity from regions that serve as "convergence zones" [START_REF] Damasio | Time-locked multiregional retroactivation: a systems-level proposal for the neural substrates of recall and recognition[END_REF] between language and motor structures. According to [START_REF] Damasio | Time-locked multiregional retroactivation: a systems-level proposal for the neural substrates of recall and recognition[END_REF], convergence zones are local brain areas in association cortices that do not by themselves contain refined representations. Instead, these zones receive/send signals from/to different modalities and enable correlated activity to occur in different regions of the brain. By way of these convergence zones -which evolve through learning -the brain can bind information represented in distinct regions into coherent events (e.g. sound and action), and can enact formulas for the reconstitution (remembering) of representations of events in sensory and motor cortices. The fact that action-related μ suppression for verbal stimuli as well as for videos emerged concurrently with the acquisition of word-action associations, suggests that activity at fronto-central electrodes reflects the development of such convergence zones.

The possibility that language-induced sensory and motor activity originates from convergence zones in higher-level association cortices has also been suggested by [START_REF] Kiefer | The sound of concepts: four markers for a link between auditory and conceptual brain systems[END_REF]. In their study, processing words denoting objects for which acoustic features are relevant (e.g., telephone) activated the auditory association cortex without activating the primary and secondary auditory cortex.

According to [START_REF] Kiefer | The sound of concepts: four markers for a link between auditory and conceptual brain systems[END_REF], conceptual processing only involves a partial reinstatement of brain activity during perceptual experience because the modality-specific experience of sensory conceptual features (e.g., experience a "ringing" sound when reading the word "telephone") would interfere with action planning processes. Other researchers have developed similar arguments. [START_REF] Willems | Neural dissociations between action verb understanding and motor imagery[END_REF], for instance, compared brain activity during reading action words and during explicit motor imagery of the actions described by the words. Their results revealed task-specific, non-overlapping and uncorrelated activity in parts of the premotor cortex. Based on this observation, the authors concluded that in contrast to motor imagery, language-induced motor activity might reflect underspecified rather than explicit motor representations. The underspecified nature of this representation would assure rapid and flexible accommodation of incoming information during on-line sentence processing. Thus, an explicit action plan for a word like "grasp" would differ depending on whether the word refers to grasping a baseball or grasping a needle. In support of this assumption, van Elk et al. (2010) showed that the strength of µ suppression over fronto-central regions varies when the sentential context offers more explicit information about the designated action: µ suppression observed while reading the verb "jumped" differed depending on whether the verb indicated an action performed by an animal ("the deer jumped over the stream") or by a human ("the athlete jumped over the fence").

Our results are not directly comparable to results obtained in studies that investigated languageinduced sensory-motor activity for well-known words because our study captured dynamic processes that occur during word-referent acquisition and prior to memory consolidation. However, the basic features of our findings are compatible with the proposals made by Kiefer and colleagues. Hence, early during training the pattern of the language-induced μ suppression observed at centro-parietal electrodes resembled the pattern that was observed during perceptual experience. This motor-activity, which could be interpreted as reinstatement of brain activity during perceptual experience, reduced as training progressed. Concurrently, action-related μ suppression emerged at electrodes that seemed to capture activity from convergence zones (c.f. over fronto-central regions). At these electrodes motor-activity was not present prior to training but developed for both type of stimuli (verbal stimuli and videos) with the acquisition of word-action associations2 . Our training did not last long enough to capture a final, stable pattern. However, it is likely that over a more extensive training period, language-induced μ suppression would no longer be observed at centro-parietal electrodes, while at fronto-central electrodes solid patterns of action-related μ suppression would emerge for both verbal stimuli and videos.

Conclusion

According to embodied theories of semantics, cognitive processes are rooted in the body's interactions with the world. Within this theoretical frame, the present study probed the trajectory of learning motor-language associations to understand the functional role of neural motor activity observed during language processing (e.g. [START_REF] Hauk | Somatotopic representation of action words in human motor and premotor cortex[END_REF]. By associating novel verbal stimuli with motor actions or animated images, our results demonstrated that brain responses to verbal stimuli change rapidly with training (see also [START_REF] Shtyrov | Rapid cortical plasticity underlying novel word learning[END_REF] for related results) and form distinctions between the associated stimulus categories. Furthermore, verbal stimuli associated with motor actions trigger activity in motor regions of the brain. However, while this language-induced motor activity was initially very similar to motor activity observed during the perception of action, this pattern changed with training and suggested that overlapping activity between language and motor processes is confined to convergence zones between language and motor structures.

In line with suggestions made by other researchers (e.g., [START_REF] Kiefer | The sound of concepts: four markers for a link between auditory and conceptual brain systems[END_REF][START_REF] Sl | Neuroimaging studies of semantic memory: inferring "how" from "where[END_REF]van Elk et al., 2010;[START_REF] Willems | Neural dissociations between action verb understanding and motor imagery[END_REF], the present findings imply that language-induced motor activity observed for well-established words reflects underspecified rather than explicit motor representations. Such under-specification leads to a relative independence of the concepts in relation to its content that is identified in the similar but distinct well-described motor activity for action concepts or words. As a consequence, sensory-motor reactivations, through which concepts are comprehended, differ from those implicated in the acquisition of such concepts. Yet, even if these results might be interpreted as showing the first step of a gradual process of abstracting away from modality-specific representations (see [START_REF] Sl | Neuroimaging studies of semantic memory: inferring "how" from "where[END_REF], our findings nonetheless imply that, fundamentally, concepts are grounded in perception and action.

Figure 1

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Figures captionsFigure 1 :

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Figure 2 :

 2 Figure 2: Topography of µ oscillations (8-12 Hz range) averaged over a time window of 250-1000msec after stimulus onset while watching videos. Data shows the difference between the movement-and the image-condition for each of the four test sessions (pretest, test1, test2 and test3).

Figure 3 :

 3 Figure 3: Time-frequency analysis (5-30 Hz domain) performed over two clusters during presentation of videos: A) centro-parietal cluster (CP1, CP3, P3 and Pz) and B) fronto-central cluster (Fz, F3, F4, FC1, FC3, FC2 and FC4).

Figure 3 :

 3 Figure 3: C) Time-course of µ oscillations during presentation of videos at both centro-parietal (top row) and fronto-central (lower row) clusters for the four test sessions. Data are plotted from -200msec to 1000msec relative to video onset. Red curves represent data for the image condition, black curves for the movement condition. For both conditions, each curve represents one electrode.

Figure 4 :

 4 Figure 4: Time-frequency analysis (5-30 Hz domain) performed over two clusters during presentation of verbal stimuli: A) centro-parietal cluster (CP1, CP3, P3 and Pz) and B) fronto-central cluster (Fz, F3, F4, FC1, FC3, FC2 and FC4). Data are plotted from -200 msec to 1000 msec post-stimulus onset and for all test sessions. Top row: Movement condition. Middle row: Image condition. Lower row: "Movement minus Image" contrast. As indicated by the scale, a blue shaded area corresponds to a reduction of the amplitude of oscillations in a given frequency band while a red shaded area corresponds to an enhancement of the amplitude of oscillations.

Figure 4 :

 4 Figure 4: C) Time-course of µ oscillations during presentation of verbal stimuli at both centro-parietal (top row) and fronto-central (lower row) clusters for the four test sessions. Data are plotted from -200msec to 1000msec relative to video onset. Red curves represent data for the image condition, black curves for the movement condition. For both conditions, each curve represents one electrode.

  Figure 1 Click here to download high resolution image

Note that the delay in the emergence of action-related μ suppression for verbal stimuli and for videos could be related to our task. Given that our participants were requested to recall the action associated with the verbal stimuli and not the verbal stimuli associated with the action, action-related μ suppression in convergence zones might emerge first for verbal stimuli and then for videos.


Véronique Boulenger
email: veronique.boulenger@ish-lyon.cnrs.fr

Emmanuel Ferragne
email: emmanuel.ferragne@univ-paris-diderot.fr

Nathalie Bedoin

François Pellegrino

Derived

Derived contrasts in Scottish English: an EEG study

Keywords: Gradient Phonemicity Hypothesis, Derived contrasts, EEG, MMN, P3

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INTRODUCTION

Phonemicity and non phonemicity have often been assumed to be hard categories: if the occurrence of sound x instead of y is totally unpredictable, then x and y form a phonemic contrast, otherwise (i.e. if the distribution is predictable to some extent), x and y are allophones. However, empirical evidence suggesting that predictability is a matter of degrees has constituted a long-standing -though often overlooked -challenge to this binary view. Along with other authors [START_REF] Currie-Hall | A Probabilistic Model of Phonological Relationships from Contrast to Allophony[END_REF][START_REF] Labov | Principles of linguistic change[END_REF][START_REF] Scobbie | Quasi-phonemic contrast and the fuzzy inventory: Examples from Scottish English[END_REF], we contend that contrasts can be more or less phonemic, and we thus introduce the Gradient Phonemicity Hypothesis (GPH) and propose its assessment through perceptual experiments. In many varieties of Scottish English, duration is restricted to an allophonic role in simplex words. However, under the influence of the past morpheme <-ed>, a few duration-based derived contrasts emerge, yielding minimal pairs such as brood/brewed, need/kneed, side/sighed [START_REF] Scobbie | Standard English in Edinburgh and Glasgow: the Scottish Vowel Length Rule revealed[END_REF]. Such contrasts are atypical because they are conditioned by morphological factors, and their functional load is low. Therefore, following GPH, the perceptual correlates of their difference are expected to display intermediate values between those recorded for fullfledged phonemes and those elicited by allophones. In the present study, we investigated the potential differences in phonemicity between a clear phonemic contrast (relying on vowel quality) and a duration-based derived contrast through eventrelated potentials (ERPs) in an 'oddball' paradigm.

ELECTROPHYSIOLOGICAL CORRELATES OF SPEECH PERCEPTION

Over the last 20 years, an abundant literature has been produced on ERPs in speech perception, mostly using the 'oddball' paradigm (a 'deviant' stimulus is occasionally delivered within a sequence of 'standard' sounds). The deviant induces the automatic detection of unexpected information, followed by a switching of the attentional focus toward the new event. This chain of processes produces a cascade of cortical electrical signals, including the frontal mismatch negativity (MMN) peaking around 150-250 ms after deviance onset [START_REF] Naatanen | The perception of speech sounds by the human brain as reflected by the mismatch negativity (MMN) and its magnetic equivalent (MMNm)[END_REF], and the fronto-centrally distributed subcomponent of the P300 (i.e., P3a) peaking between 200 and 650 ms after deviance onset [START_REF] Polich | Updating P300: an integrative theory of P3a and P3b[END_REF]. The MMN is an automatic brain response induced by the detection of acoustic changes. However, it also reflects higher order perceptual processes, as its amplitude increases when the deviance relies on the activation of stored phonological knowledge [START_REF] Naatanen | The perception of speech sounds by the human brain as reflected by the mismatch negativity (MMN) and its magnetic equivalent (MMNm)[END_REF][START_REF] Pulvermuller | Language outside the focus of attention: the mismatch negativity as a tool for studying higher cognitive processes[END_REF]. It is consequently useful in the investigation of phonological processes. The P3a is assumed to index potentially non-conscious attention switches from the focus of voluntary attention to deviant sounds [START_REF] Polich | Updating P300: an integrative theory of P3a and P3b[END_REF]. The detection of regularity violations by low-level sensory processes triggers this brain response, which can be further modulated if the significant event activates pre-existing long-term memory traces. In our study, deviant events should therefore elicit an MMN and a P3a in both groups due to sensory oddity. However, since a change in vowel duration has potential phonological relevance only for Scottish listeners, higher amplitude is expected in Scottish but not in French participants.

METHODS

Participants and stimuli

Seven native Scottish and 19 native French speakers aged 18-26 years participated in the study. Scottish participants had been in France for less than 3 months at the time of testing. All participants were right-handed (mean score Edinburgh inventory = 92, SD 13, [START_REF] Oldfield | The assessment and analysis of handedness: the Edinburgh inventory[END_REF]) and had normal peripheral auditory thresholds (< 20 dB-SPL in 125-8000 Hz). They gave their written informed consent and were paid for their participation. The stimuli were taken from previous recordings of 15 Glaswegian speakers. The typical durations for the vowels in brood and brewed were approx. 70 and 175 ms respectively. One occurrence of the word brewed spoken by a male speaker was selected, vowel boundaries were identified as the limits of the steady-state portion of the formant pattern, and the vowel was time-compressed and resynthesized with the Praat program, yielding two words whose vocalic duration was 70 (brood) and 175 ms (brewed). One occurrence of bit and bet spoken by the same speaker was also selected. The four stimuli were peak-normalized, and their amplitude envelope was smoothed at stimulus onset and offset in order to avoid popping noises.

Procedure

Participants sat in an electrically and acoustically shielded chamber. They were instructed to watch a silent movie and to ignore the auditory stimuli that were delivered binaurally via headphones at a comfortable listening level. The stimuli were presented in an oddball sequence with 77% of standards and 23% of deviants with a stimulus onset asynchrony of 500 ms. Sequences of 3 to 6 standards separated consecutive deviants. The experiment was divided into four blocks of 642 stimuli each (522 standards and 120 deviants). In Block 1, brewed served as standard and brood as deviant while the reverse was true in Block 2. In Block 3, bit was used as standard and bet as deviant, and vice-versa in Block 4. The order of blocks was counterbalanced across participants.

EEG acquisition and analysis

EEG was acquired and analyzed according to standard procedure 1 . MMN waveforms were computed by subtracting standard-induced from deviant-induced ERP waveforms for each block and participant. Grand-averages were then calculated across all participants for each block. Based on visual inspection of the grand-averages at Fz, the MMN was defined as the maximal negativity peaking around 150-250 ms after deviance onset.

MMN amplitude was quantified for each participant and block within a time-window ranging between the zero-crossings on either side of the peak. MMN latency was determined as the point of maximal amplitude within the same time-window. P3a latency and amplitude were calculated similarly in the 250-400 ms range after deviance onset. For both MMN and P3a, the largest responses were found at electrode Fz; statistical analysis was thus restricted to this electrode. For the sake of clarity, blocks were averaged according to the type of contrast, namely a duration-based contrast (brewed/brood; Blocks 1 and 2) and a contrast relying on vowel quality (bet/bit; Blocks 3 and4). Amplitudes and latencies of the MMN and P3a were subjected to two-way repeated-measures ANOVAs including Native Language (Scottish English vs. French) as between-subject factor and Contrast (duration vs. vowel quality) as within-subject factor. When interactions were significant, multiple comparisons were performed to evaluate differences between conditions 2 .

RESULTS

MMN

Figure 1 displays the grand-averaged difference waves for each group and type of contrast. The difference waves revealed large negative deflections, identified as MMNs, peaking on average 187 ms after deviance onset, distributed over fronto-central sites and showing polarity inversion at mastoids. The ANOVA on MMN amplitude showed no significant effects of Native Language or Contrast and no significant interaction between the two factors. In the following, only significant results will be reported. The ANOVA on MMN latency revealed a significant main effect of Native Language (F(1, 21) = 5.64, p = .027), with shorter latency in Scottish (179 ms, SD 17) than in French participants (196 ms, SD 19).

P3a

As illustrated in Figure 1, a large positive wave, identified as P3a, was elicited at Fz for each type of contrast in each group. P3a peaked on average 298 ms after deviance onset and was maximal over fronto-central electrodes. The main effect of Native Language (F(1, 21) = 7.87, p = .01) showed that P3a latency was significantly shorter in Scottish (291 ms, SD 28) than in French participants (316 ms, SD 28). A main effect of Contrast was also (F(1, 21) = 5.86, p = .02), however we chose not to interpret this effect as latency measures were timelocked to deviance onset, which was difficult to estimate exactly for the two types of contrasts. The main effect of Native Language on P3a amplitude was significant (F(1, 21) = 6.20, p = .02), the positivity being larger in Scottish (4.31 µV, SD 0.8) than in French speakers (2.96 µV, SD 1.54). A Native Language × Contrast interaction also emerged (F(1, 21) = 5.47, p = .029; Figure 2): P3a amplitude was significantly smaller for the durationbased contrast (2.49 µV, SD 1.57) than for the contrast based on vowel quality (3.42 µV, SD 1.41) in French (p = .02) but not in Scottish participants (duration = 4.65 µV, SD 0.64; vowel quality = 3.97 µV, SD 0.85; p > .05). In addition, P3a amplitude for the duration-based contrast was larger in Scottish than in French participants (p = .004) whereas amplitude for the quality-based contrast did not differ between groups (p > .05). Note that the difference between Scottish and French speakers for the duration-based contrast was significant.

DISCUSSION

The present study tested the GPH by investigating whether a duration-based derived contrast in Scottish English can be considered as phonemic as a contrast based on vowel quality. In an oddball paradigm, we examined the MMN and P3a elicited by the perception of such contrasts in native Scottish and French speakers. Our results revealed that MMN latency was shorter in Scottish than in French participants, irrespective of the type of contrast. The MMN is thought to reflect memory traces for phonemes of the native language [START_REF] Naatanen | The perception of speech sounds by the human brain as reflected by the mismatch negativity (MMN) and its magnetic equivalent (MMNm)[END_REF][START_REF] Pulvermuller | Language outside the focus of attention: the mismatch negativity as a tool for studying higher cognitive processes[END_REF] and in addition, MMN peak latency has been shown to decrease as easiness of deviance detection increases [START_REF] Naatanen | Do event-related potentials reveal the mechanism of the auditory sensory memory in the human brain?[END_REF]. In our study, since Scottish English minimal pairs were used, deviance may have been detected more easily and more rapidly by native Scottish speakers as they have specific memory traces for these stimuli compared to French speakers who were not familiar with the stimuli. Regarding MMN amplitude, previous studies have shown enhanced MMNs for duration-based contrasts in native speakers of Finnish, where the long/short opposition is linguistically relevant compared to non-native speakers [START_REF] Nenonen | Linguistic relevance of duration within the native language determines the accuracy of speech-sound duration processing[END_REF]. This MMN enhancement has been proposed to indicate high accuracy in the processing of familiar, early learned, language-specific features. Our results did not show such a pattern, but further analysis with a larger number of participants is required. Regarding P3a, our study showed shorter latency and larger amplitude in Scottish participants. P3a is thought to index involuntary shifting of attention [START_REF] Polich | Updating P300: an integrative theory of P3a and P3b[END_REF] and P3a latency is known to reduce when the inherent salience of the stimulus increases [START_REF] Combs | P3a from auditory white noise stimuli[END_REF]. Here, Scottish English stimuli may have triggered faster and deeper processes in Scottish than in French speakers, hence the earlier and increased P3a. Interestingly, P3a amplitude was differentially affected by the type of contrast depending on native language. While P3a amplitude was comparable for the two types of contrasts in Scottish speakers, it was larger for the quality-based contrast in French speakers. P3a amplitude was further enhanced for the duration-based contrast in Scottish compared to French participants; no difference was observed between groups for the contrast based on vowel quality. Our results can be interpreted in the frame of the "significance hypothesis" [START_REF] Horvath | Do N1/MMN, P3a, and RON form a strongly coupled chain reflecting the three stages of auditory distraction?[END_REF] suggesting that P3a indexes processes that are activated by significant events such as stimuli carrying new information or stimuli that are stored in long-term memory. Vowel quality is relevant for phoneme discrimination and categorization in both Scottish English and French; speech-specific information may thus be stored in the long-term memory of Scottish and French speakers, hence the similar P3a amplitude for the contrast based on vowel quality in the two groups. In contrast, phonological length is relevant for Scottish English derived contrasts only, suggesting that information about vowel duration is stored in long-term memory only in native Scottish speakers. This may account for the larger P3a amplitude for the duration-based contrast in Scottish than in French participants. The small sample size allows drawing only tentative conclusions. Comparable MMN and P3a amplitudes for the contrasts based on duration and on vowel quality in Scottish speakers may provide first evidence that these two features are equally relevant for speech processing in Scottish English. In this view, derived contrasts may therefore be regarded as phonemic contrasts. Further analysis with a larger number of native Scottish speakers will allow precisely investigating this issue as well as potential ERPs differences between short and long deviants which may be subserved by partly distinct neural generators [START_REF] Takegata | Parameter-specific modulation of the mismatch negativity to duration decrement and increment: evidence for asymmetric processes[END_REF]. 1 EEG was continuously recorded from 32 scalp electrodes (international 10-20 system) using the Biosemi EEG system with a sampling rate of 512 Hz, filtered online between 0.1 and 100 Hz. Eye movements were monitored by recording horizontal and vertical electro-oculograms (EOG) with a bipolar montage of two electrode pairs. Data were analyzed with BESA software. Raw EEG recordings were segmented in 800 ms epochs starting 100 ms before deviance onset to 700 ms after onset. Epochs in which the EEG/EOG signal exceeded ±150 µV were discarded from analysis. ERPs were separately averaged for deviants and standards for each participant, block and electrode. Averages were baselinecorrected using the 100 ms period preceding deviance onset and re-referenced to linked mastoids. 2 One French participant was excluded from data analysis due to very noisy signal. Two other participants, one Scottish and one French, were identified as outliers when plotting ERPs amplitudes and were thus excluded from statistical analysis. Results are reported for 6 Scottish and 17 French participants.

Figure 1 :

 1 Figure 1: Grand-averaged difference waveforms at Fz in (a) Scottish and (b) French participants for the two types of contrast (duration: black, vowel quality: grey).

Figure 2 :

 2 Figure 2: Language × Contrast interaction on P3a amplitude. P3a amplitude is represented as a function of Language (Scottish English vs. French) and Contrast (duration (solid line) vs. vowel quality (dotted line)). (*) indicates a significant difference between conditions.Note that the difference between Scottish and French speakers for the duration-based contrast was significant.

ACKNOWLEDGEMENTS

Research supported by grants from Fondation Fyssen, the CNRS (PEPS) and the ANR (PI: E. Ferragne). We would like to thank Jennifer Krzonowski and Marie Garnier for running the EEG experiment.


Emmanuel Ferragne
email: emmanuel.ferragne@univ-paris-diderot.fr

Nathalie Bedoin

Véronique Boulenger

François Pellegrino

The perception of a derived contrast in Scottish English

Keywords: Gradient Phonemicity Hypothesis, perception, derived contrasts

This paper reports on ongoing experiments related to the perception of atypical phonological contrasts. It was hypothesized that the quasi-phonemic status of a derived contrast in Scottish English would induce atypical responses, yielding behavioural correlates with intermediate values between phonemicity and allophony. The results show that some of the Scottish participants produce identification slopes that are quite similar to the French allophonic control group while other Scottish listeners show more phonemic responses.

INTRODUCTION

Many phoneticians and phonologists take it for granted that a difference between two speech sounds can only reflect either a phonemic contrast (i.e. two categories) or allophonic variation (within one category). However, now and then, some authors have cast doubt on the strict categorical view [START_REF] Currie-Hall | A Probabilistic Model of Phonological Relationships from Contrast to Allophony[END_REF][START_REF] Labov | Principles of linguistic change[END_REF][START_REF] Scobbie | Quasi-phonemic contrast and the fuzzy inventory: Examples from Scottish English[END_REF]. Evidence from many languages in the world blatantly shows that among the entities analysed as phonemes, some constitute better candidates to phonemicity than others. The predictability criterion, according to which if the occurrence of a sound can be predicted from the phonological context, then this sound is an allophone, and if it is not predictable at all, then the sound is a phoneme, is commonly applied in a binary fashion. However, [START_REF] Currie-Hall | A Probabilistic Model of Phonological Relationships from Contrast to Allophony[END_REF] convincingly argued that the occurrence of a sound can be more or less predictable. Therefore, if predictability determines phonemicity, and if predictability is gradient, then it is not too farfetched to hypothesize that phonemicity could be gradient too. We introduce the Gradient Phonemicity Hypothesis (GPH), according to which the nature of the difference between two sounds in a language cannot be satisfactorily accounted for by the strict binarism underlying the phoneme concept. We contend that the relationship between two sounds is better described as belonging to the fuzzy set of phonemic contrasts with a certain degree of membership. We believe that this degree can be computed from various behavioural correlates collected during perceptual experiments. Scottish English provides a perfectly-suited phenomenon to begin testing our hypothesis experimentally. Contrary to most accents of English, the vowel system of Scottish English does not make use of phonological length [START_REF] Ferragne | Formant frequencies of vowels in 13 accents of the British Isles[END_REF][START_REF] Scobbie | Standard English in Edinburgh and Glasgow: the Scottish Vowel Length Rule revealed[END_REF]. However, closer attention reveals that some duration-based contrasts do emerge due to morphological conditioning. For example, the past morpheme <ed> causes the vowel in the word brewed to be appreciably longer than its counterpart in brood. Similarly, kneed/need and sighed/side (with additional vowel quality variation in the latter) are known to be distinct in many speakers of Scottish English. Derived contrasts constitute a puzzle for linguists because, as [START_REF] Harris | Derived phonological contrasts[END_REF] remarks, structuralists would count them as phonemes while generativists tend to regard them as mere allophones. This paper is a preliminary report on an ongoing series of perceptual experiments whose aim is to provide empirical information on the cognitive status of atypical contrasts such as the derived contrasts attested in Scottish English mentioned earlier. Because such contrasts are morphologically conditioned, and given their low functional load, our intuition is that their behavioural correlates could exhibit values that are intermediate between those of fully-fledged phonemic oppositions and those elicited by allophones.

EXPERIMENT

The experiment reported here is a labeling task involving stimuli varying in duration on a quasicontinuum between brood and brewed. Native speakers of French and Scottish English performed the task. French being a language without phonological length, the inclusion of French participants was meant to provide 'baseline' data on the perception of the duration of speech sounds.

Participants

The participants were 21 students from the University of Glasgow (ECO) and 19 students from the University of Lyon (FRA). The Scottish participants were paid, and most of the French received course credit for their participation.

Stimuli

From previous recordings involving 15 speakers from Glasgow, we had determined that typical duration values for the vowels in brood and brewed were approximately 70 and 175 ms respectively. One occurrence of the word brewed spoken by a male speaker was selected, vowel boundaries were identified as the limits of the steady-state portion of the formant pattern, and the vowel was timecompressed or expanded, and resynthesized with the Praat program, which yielded 12 stimuli whose vowel duration ranged from 40 to 205 ms in 15-ms steps. All stimuli were presented in the same carrier sentence (He said the word…) since speech rate is known to alter both the location of category boundaries and their internal structures. Each trial started with a 150-ms beep (300-Hz sine-wave) which was meant to focus participant's attention; then, the sentence was played after a 500-ms silent chunk.

Procedure

The Scottish part of the experiment took place in the recording studio at the Glasgow University Laboratory of Phonetics. The French experiment was carried out in a sound-attenuated booth at Laboratoire Dynamique Du Langage, Lyon. For both locations, the experiment was run with the same equipment: two laptop computers with relative output level equalized by means of an oscilloscope, and high-quality headset with TDH-39 earphones and Peltor noise-attenuating cups. The experiment was implemented as a Matlab standalone executable. At the beginning of the experiment, a synthesized /a/ with varying intensity was played and the method of adjustment was used to determine individual thresholds of hearing. Then all stimuli for the rest of the experiment were automatically adjusted so that their RMS intensity was 50 dB above threshold. This procedure was meant to provide a comfortable output level for all participants and to detect potential slight hearing deficiencies.

The French experiment started with a training phase during which participants were told to listen to brood (70 ms) and brewed (175 ms) in order to become familiar with what we called a 'short word' and a 'long word'. Once they felt confident that they had learned the difference between short and long, they were asked to move on to the experiment itself. They were prompted to decide as quickly as possible whether the word at the end of the carrier sentence was short or long. Then, they were asked to rate the goodness of the stimulus on a 1-7 scale. The Scottish experiment started with the recording of a short text designed to elicit derived contrasts. This procedure was chosen in order to estimate, for each participant, whether this contrast was present or absent of her production system. Then, the listening part was similar to that of the French experiment except that Scottish participants were asked to decide whether the word was brood or brewed. The screen prompts were written in the participants' native language. The 12 stimulus levels were presented 40 times in random order varying from a participant to the next, yielding a total of 480 trials.

Analysis

Logistic functionsrelating the proportion of stimuli identified as long (FRA)/brewed (ECO)were fitted to individual datasets with the Palamedes toolbox [START_REF] Prins | Palamedes: MATLAB routines for analyzing psychophysical data[END_REF] in order to determine the slope and the threshold of the identification function. The threshold corresponds to the point where the identification reaches 50%. A visual inspection of the results led to regard participants ECO22 and FRAFF as outliers in terms of slope for the former and threshold for the latter. As it turned out, their values lied outside the mean +/-two standard deviations interval computed from their group for the parameter in question. They were therefore removed from the rest of the analysis. A one-way ANOVA with Native Language as factor was conducted on Slope, and another one with Threshold as the dependent variable. Then, given the substantive variation in the Scottish data in terms of goodness ratings, identification functions, and speech production, individual data were closely inspected in an attempt to group the participants according to specific patterns of results. Goodness ratings, which were min-max normalized, would be expected to display a typical V-shaped curve if stimulus duration were plotted horizontally against goodness score. That is minimal goodness scores are expected for intermediate durations, and the further away from this intermediate point, the closer to the prototype of brood or brewed, and the higher the goodness. On the basis of the shape of goodness ratings, 3 groups were identified: Scottish-2 ('2 prototypes'; n = 9), with maximal (100%) or near maximal goodness scores given to both ends of the duration continuum, Scottish-1 ('1 prototype'; n = 7), with maximal goodness assigned to just one end of the continuum while the maximum score at the other end barely reached 60%, Scottish-3 (n = 4), with two participants showing a sawtooth pattern, and two participants whose scores increased monotonically as a function of duration. Scottish-3 were discarded, and we concentrated on whether the apparent existence of 1 vs. 2 prototypes could predict the steepness of the identification curve. A one-way ANOVA with factor Group (French, Scottish-1, Scottish-2) was computed on slope.

RESULTS

The statistical analyses showed a significant effect of Language on Slope F(1,36) = 10.28, p = .0028, but not on Threshold. As can be seen from Figure 1, the slope of the identification function is steeper for Scottish than French participants. With Group (Scottish-1, Scottish-2, French) as the factor, and Slope as the dependent variable, the ANOVA showed a significant main effect of Group: F(1,31) = 14.25, p < .0001. Post-hoc tests revealed significant differences between Scottish-2 and the other two groups, but not between Scottish-1 and French. The results are displayed in Figures 2 and3.

DISCUSSION

This study was a preliminary test of the Gradient Phonemicity Hypothesis. Our goal was to investigate whether the 'intermediate' linguistic status of a derived contrast in Scottish English triggered intermediate behavioural correlates. Identification curves were steeper in Scottish participants, which entails that the precision of the boundary between brood and brewed was, as expected, higher in the Scottish group. However, after inspecting individual goodness ratings, we were able to split the Scottish group into two subgroups: Scottish-1, with one prototype, and Scottish-2, with two prototypes. The analysis showed that the slopes in Scottish-1 participants were probably not different from those observed in French subjects. Scottish-2 had steeper slopes than French and Scottish-1 participants. We therefore tentatively suggest that the perception of the brood/brewed contrast in the Scottish-1 group was allophonic while it was more phonemic in Scottish-2 participants. Note that preliminary inspection of the production data does not seem to indicate that the likely presence of the contrast in production predicts group membership. The slope of the identification curve was implicitly accepted as a correlate of phonemicity: the steeper the slope, the more phonemic the contrast. The rationale for this claim stems from developmental studies involving labeling tasks, where the steepness of the response curve has often been equated with the presence of a reliable phonemic contrast in participants [START_REF] Bogliotti | Discrimination of speech sounds by children with dyslexia: comparisons with chronological age and reading level controls[END_REF][START_REF] Simon | Cross-language study of speech-pattern learning[END_REF]: the steeper the slope, the more reliable or fully-developed the contrast. Our experimental paradigm has often been used in the literature on categorical perception (CP) [START_REF] Repp | Categories and context in the perception of isolated steady-state vowels[END_REF][START_REF] Simon | Cross-language study of speech-pattern learning[END_REF]. However, the term 'categorical perception' has been avoided here because it is highly connoted and often refers to the extent to which identification predicts discrimination. This, of course, does not apply to our study. A question related to this is how come our response curves are so steep considering that vowels have long been known to be perceived almost non-categorically? Firstly, our task is by no means similar to a standard CP task with continua of isolated vowels, often varying on the spectral dimension. Vowels in word context are known to be perceived more categorically [START_REF] Repp | Categories and context in the perception of isolated steady-state vowels[END_REF]. Secondly, the nature of the task itself (labeling words, rather than vowels for Scottish participants) may have favoured steeper slopes, as the type of task can modulate correlates of CP [START_REF] Gerrits | Categorical perception depends on the discrimination task[END_REF]. Another objection could be that the steepness of the slope is a by-product of the durational step between successive stimuli. This is unlikely: our step size -15 msis smaller than the oneapprox. 20 msused successfully in a study on phonological quantity in Finnish [START_REF] Ylinen | The perception of phonological quantity based on durational cues by native speakers, secondlanguage users and nonspeakers of Finnish[END_REF], and it is smaller than the just-noticeable difference established by [START_REF] Henry | Discrimination of the duration of a sound[END_REF] . As pointed out earlier, this study is only preliminary. A more conclusive version will include the perception of a linguistically well-established phonemic contrast based on duration, such as the TRAP-START opposition in some accents in the north of England [START_REF] Ferragne | Formant frequencies of vowels in 13 accents of the British Isles[END_REF]. This will allow a more thorough test of the Gradient Phonemicity Hypothesis with 3 types of duration-based differences: allophonic for French listeners (brood/brewed, in the current article), phonemic for northern English listeners (TRAP-START), and quasi-phonemic (brood/brewed in the current article) with Scottish participants.

Figure 1 :

 1 Figure 1: Identification function for Scottish (thick line) and French (thin line) participants.

Figure 2 :

 2 Figure 2: Identification function for Scottish-1 (dashed line), Scottish-2 (thick line) and French (thin line) participants.

Figure 3 :

 3 Figure 3: Comparison of mean slopes between Scottish-1, Scottish-2, and French participants (ns: nonsignificant; *: significant .05 level).

ACKNOWLEDGEMENTS

Research supported by grants from Fondation Fyssen, the CNRS (PEPS), and the ANR (P.I.: E. Ferragne). Many thanks to Francesco Li Santi, Jane Stuart-Smith and the people at the University of Glasgow.


Alexis Benichoux
email: alexis.benichoux@irisa.fr

Emmanuel Vincent
email: emmanuel.vincent@inria.fr

Rémi Gribonval
email: remi.gribonval@inria.fr

A COMPRESSED SENSING APPROACH TO THE SIMULTANEOUS RECORDING OF MULTIPLE ROOM IMPULSE RESPONSES

Keywords: Room impulse response recording, convex optimization, compressed sensing

We consider the estimation of multiple room impulse responses from the simultaneous recording of several known sources. Existing techniques are restricted to the case where the number of sources is most equal to the number of sensors. We relax this assumption in the case where the sources are known. To this aim, we propose statistical models of the filters associated with convex log-likelihoods, and we propose a convex optimization algorithm to solve the inverse problem with the resulting penalties. We provide a comparison between penalties via a set of experiments which shows that our method allows to speed up the recording process with a controlled quality tradeoff.

INTRODUCTION

We focus on the recording of multiple room impulse responses. Up to now this is typically achieved by activating each loudspeaker or source in turn, with a silent interval equal to the expected duration of the impulse response in between [START_REF] Farina | Simultaneous measurement of impulse response and distortion with a swept-sine technique[END_REF]. The total recording duration is then N (D + K -1) where N is the number of sources, D the chirp duration and K the impulse response length in samples. An improvement [START_REF] Majdak | Multiple exponential sweep method for fast measurement of head-related transfer functions[END_REF] is to use time-overlapping but time-frequency disjoint chirps, which reduces the recording duration down to N K + D -1 when the system is linear. These techremain time-consuming e.g. in the context of the calibration of high-end 3D audio systems or the collection of binaural room impulse responses involving hundreds of loudspeakers. We investigate here possible improvements using state-of-the-art system inversion tools. This problem is equivalent to the estimation of the mixing filters in the context of convolutive source separation [START_REF] Sudhakar | Double sparsity: Towards blind estimation of multiple channels[END_REF].

The techniques in [START_REF] Aissa-El-Bey | Blind separation of underdetermined convolutive mixtures using their time-frequency representation[END_REF] and [START_REF] Sudhakar | Double sparsity: Towards blind estimation of multiple channels[END_REF] for mixing filter estimation assume each source to be active alone in a certain time interval. Once this time interval has been localized, the corresponding filters are estimated using a subspace method [START_REF] Aissa-El-Bey | Blind separation of underdetermined convolutive mixtures using their time-frequency representation[END_REF], or convex optimization [START_REF] Sudhakar | Double sparsity: Towards blind estimation of multiple channels[END_REF]. Alternative Convolutive Independent Component Analysis techniques [START_REF] Makino | Blind speech separation[END_REF] assume the number of sources to be at most equal to the number of sensors. Our work is to our knowledge the first to get rid of these two assumptions. We propose to take advantage of the a priori temporal structure of the filters to improve the iterative inversion of the linear system. In addition to the sparse prior introduced in [START_REF] Lin | Blind channel identification for speech dereverberation using ℓ1-norm sparse learning[END_REF] for single-source blind channel identification, we propose four new priors and a new multi-source inversion algorithm. Our approach is an example of compressed sensing [START_REF] Donoho | Compressed sensing[END_REF] [START_REF] Candès | Compressive sampling[END_REF], that is an emerging general approach to the recovery of structured signals from a smaller number of measurements. We show theoretically that white noise sources provide the most convenient system for inversion.

The structure of the paper is as follows. The formalization of the problem is described section 2. Section 3 corresponds to the study of the a priori structure of the filters. The implementation of the algorithm is detailed Section 4. The results shown in Section 5 show the potential of the proposed method.

APPROACH

The problem is formalized as follows : we represent the N sources of length T by the matrix S ∈ R N ×T , the filters of length K by the three dimensional array A ∈ R M ×N ×K and the M observations by X ∈ R M ×(T +K-1) . Assuming that the loudspeaker are linear, the convolutive matrix product ⋆ allows us to write

X = A ⋆ S = 0 @ X n≤N Amn * Sn 1 A m≤M . (1)
Earlier work [START_REF] Kowalski | Beyond the narrowband approximation: Wideband convex methods for under-determined reverberant audio source separation[END_REF] used convex optimization tools to recover S when A is known, using a sparsity prior on the sources.

Here we adapt the method in [START_REF] Kowalski | Beyond the narrowband approximation: Wideband convex methods for under-determined reverberant audio source separation[END_REF] to estimate A when S is known, by estimating lim λ→0 A λ where

A λ = argmin A  1 2 X -A ⋆ S 2 2 + λP(A) ff . ( 2
)
This limit is the solution of the constrained minimization problem

min A P(A) s.t. X -A ⋆ S 2 2 = 0. (3)
We choose for P the negative log-likelihood of a distribution suggested by the statistical analysis of a large family of filters.

STATISTICAL ANALYSIS OF A FAMILY OF FILTERS

The statistical theory of room acoustics [START_REF] Kuttruff | Room Acoustics[END_REF] treats each filter as a random i.i.d. signal whose amplitude envelope ρ(t) decays exponentially according to

ρ(t) = σ 10 -3t/t R , ( 4
)
where tR is the room reverberation time in samples, and σ a scaling factor. This theory assumes that a filter A ∈ R K follows a Gaussian distribution. In other work [START_REF] Lin | Blind channel identification for speech dereverberation using ℓ1-norm sparse learning[END_REF], A(t) is instead assumed to have a constant amplitude envelope and to be sparse, as it is formed by echoes at distinct instants. In order to evaluate the respective impact of both the envelope model and the sparsity model, we consider the following distributions : Laplacian with decaying envelope

P1(t) = 1 2ρ(t) e -|A(t)|/ρ(t) , (5)
Gaussian with decaying envelope

P2(t) = 1 √ 2πρ(t) e -A 2 (t)/2ρ 2 (t) , (6)
Laplacian with constant envelope

P3(t) = 1 2σ e -|A(t)|/σ , (7)
Gaussian with constant envelope

P4(t) = 1 √ 2πσ e -A 2 (t)/2σ 2 . ( 8
)
Figure 1 compares the average negative log-likelihoods of these four models over a set of 10 000 filters simulated by the image method [START_REF] Allen | Image method for efficiently simulating small-room acoustics[END_REF] for one source and one microphone at random positions spaced by 1 m, in a rectangular room of dimensions × 8 × 4 m with tR = 250 ms. For each model, the scaling factor σ is set in the maximum likelihood sense. Envelope modeling appears to be crucial : the likelihood of models P3 and P4 is much larger than that of P1 and P2 for large t. Sparsity has a weaker impact : the likelihood of P1 (and to a lesser extent that of P2) is larger than that of P2 for t ≤ 60 ms, but becomes similar for t > 60 ms. These observations lead us to propose a fifth hybrid model

P5(t) = ( P1(t) if t ≤ 60 ms P2(t) i t > 60 ms. ( 9
)
Assuming Gaussian white additive noise, maximum a posteriori estimation of the filters is equivalent to (2) with Pi = -log Pi.

ALGORITHM

solve (2), we use the FISTA (Fast Iterative Shrinkage-Thresholding) algorithm [START_REF] Beck | A fast iterative shrinkagethresholding algorithm for linear inverse problems[END_REF], which exploits the differentiability of the data fidelity term

L : A → X -A ⋆ S 2 2 , (10)
and the convexity and semicontinuity of Pi. So-called proximity operators are employed to overcome the non-differentiability of Pi.

Definition 1 For P : E → R semicontinuous and convex the proximity operator associated with P is the function

prox P : x ∈ E → argmin y∈E  P(y) + 1 2 x -y 2 2 ff
The general steps of FISTA are described in Algorithm 1. It relies on computation of the gradient of L, its Lipschitz constant L, and the proximity operator of the scaled penalty αP.

Algorithm 1 FISTA 1: A 0 ∈ R M N K , τ 0 = 1 2: for k ≤ kmax do Ãk = prox λ L P " A k-1 -∇L(A k-1 ) L " τ k = 1+ √ 1+4(τ k-1 ) 2 2 A k = Ãk + τ k-1 -1 τ k ( Ãk -Ãk-1 ) 3: end for
The computation of the gradient of L requires the introduction of the adjoint of the linear operator A → A ⋆ S. Denoting by Sn ∈ R T the time reversal of Sn, i.e. for t ≤ T , Sn(t) = Sn(T -t + 1), the adjoint operator is expressed using S * := " S1, . . . , SN " as

X → X ⋆ S * := " ( Sn * Xm)(t) " m≤M,n≤N,1≤t≤K . (11)
One may then write the gradient as

∇L(A) = (X -A ⋆ S) ⋆ S * . ( 12
)
The Lipschitz constant of ∇L is the greatest eigenvalue of the operator A → A⋆S⋆S * . We obtain this value using the power iteration algorithm as in [9, Algorithm 5].

The log-likelihood of the distributions introduced previously correspond to the ℓ1 and ℓ2 norms, whose proximity operators are well-known [START_REF] Kowalski | Beyond the narrowband approximation: Wideband convex methods for under-determined reverberant audio source separation[END_REF]. Denoting x + := max(x, 0)for x ∈ R, we obtain

prox αP 1 (A)m,n,t = ρ(t)Amn(t) |ρ(t)Amn(t)| " |Amn(t)| - α ρ(t) « + (13) prox αP 2 (A)m,n,t = Amn(t) 1 + α/ρ 2 (14) prox αP 3 (A)m,n,t = Amn(t) |Amn(t)| (|Amn(t)| -α) + (15)
prox αP 4 (A)m,n,t = Amn(t)

1 + α . ( 16
)
Concerning the hybrid model ( 9), we use (13) or ( 14) depending on the value of t.

We estimate the minima A λ for λ ∈ {1, 10 -1 , . . . , 10 -14 }, initializing each FISTA step at the minimum obtained for the previous value. We keep the last minimum obtained for λ = 10 -14 , and consider it as an estimate of the limit lim λ→0 A λ , i.e. the solution of (3). Note that the ℓ2 penalization P4 corresponds to the definition of the Moore-Penrose pseudo inversion.

EXPERIMENTAL RESULTS

The Matlab code allowing to reproduce the following experiments is available at the following address [13].

Role of the condition number

First we wish to study the contribution of the penalty depending of the invertibility of the problem. The system is composed of M (T +K -1) equations for M N K variables, therefore it is underdetermined if and only if the recording duration in samples satisfies

T + K -1 < Tc := N K. ( 17
)
Note that Tc is smaller than N K + D -1 which is the length of the recording required in [START_REF] Majdak | Multiple exponential sweep method for fast measurement of head-related transfer functions[END_REF]. Performance does not depend on the number M of microphones, in fact each microphone brings an independent problem.

The M N filters are a solution of the linear system, for m ≤ M Xm = S1 ⋆ Am1 + S2 ⋆ Am2 + . . . + SN ⋆ AmN .

In order to guide the choice of the source signals, we first show theoretically that the system is well conditioned if the sources are uncorrelated. To this aim we compute the condition number of the system i.e. the ratio of highest to lowest singular value. Note that this is only defined for a full rank system, hence for T ≥ Tc. In the under-determined setting, similar estimates can easily be obtained by exploiting the sparsity of the filters A.

The relation between the correlation of the sources and the condition number of the system is detailed in the following lemma proved in the Appendix. Using the usual cross-correlation function for n, n ′ ≤ N :

r nn ′ (k) = T -k X t=1 Sn(t)S n ′ (t + k), 0 ≤ k ≤ K -1. (19)
we introduce a measure of the maximum correlation between the sources r := max Lemma 1 For small r, the condition number of (18) obeys

n =n ′ or k =0 |r nn ′ (k)|, (20)
1 ≤ c ≤ maxn rnn(0) + r(N K -1) minn rnn(0) -r(N K -1) . (21)
Note that for white noises, r is small, and c is close to 1, leading to well conditioned system.

Performance as a function of the recording duration

In previous work [14, fig.2], we used human voice recordings, and we observed a transitory regime for T > Tc where the penalties still had an impact due to a large condition number. In this paper we want to choose the sources such that the system is well-conditioned, therefore we used white Gaussian noise, motivated by the above theoretical guarantees. We observed experimentally (results not shown here) that this choice remains experimentally valid for underdetermined systems. As a measurement of the error between the estimated filters A λ and the true filters A, we define the following ratio in decibels

SNR out A (A λ ) = 10 log 10 A 2 2 A λ -A 2 2 . ( 22
)
the solution is not unique, we expect to observe better results with the proposed regularizations : we then run the algorithm for several values of T . results shown in Figure 2 correspond to the case of N = 5 sources, M = 2 sensors, with filters of length K = 2753 (250 ms sampled at Hz) synthesized as in Section 2. For readability we express all the durations in ms the following. We obtain the critical value Tc = 1250 ms beyond which the system is overdetermined. We vary the length of the sources from T = 45 ms to T = 1500 ms.

observe in Figure 2 a clear jump around the critical value Tc after which the inversion made by all regularized algorithms yields the same solution, up to machine precision.

For T < Tc we observe the clear impact of all regularizations compared to the pseudo-inverse P4. The new penalties P1 and P2, corresponding to the Laplacian and Gaussian distributions with decaying envelope give the best results. For a speed-up of 50% in recording duration compared to [START_REF] Majdak | Multiple exponential sweep method for fast measurement of head-related transfer functions[END_REF], we achieve a recovery SNR out A = 25 dB.

Robustness to noise

We now add Gaussian white additive noise to the mixtures,

X = A ⋆ S + W. (23)
For each penalty Pi and each duration T + K -1 of the recordings, six experiments were made for a signal-to-noise ratio of 30, 40, 50, 60, 70 and 80 dB. We observe in Figure 3 that the noise decreases the overall performance, but has a smaller impact on the ℓ1 re-scaled penalty P1 than on the common pseudo-inversion P4. Not surprisingly, the two penalties lead to the same result once the sources are long enough for the solution to be unique. This experiment confirms the possibility to speedup the recordings even in the presence of noise. With an input SNR of 50 dB, the estimation fidelity SNR out

A is still 25 dB.

CONCLUSION

For the considered problem, the various a priori introduced as convex penalties provide better estimation of the filters than simple deconvolution using Moore-Penrose pseudo-inverse. The best results are achieved with the new proposed penalties based on a decaying envelope model. This method can speed up the recording, with a reasonable quality trade-off for noisy measurements. For large numbers of sources N in reverberant rooms, the expected speedup can be significant. Further experiments are needed to confirm the validity of the approach in such scenarii, by taking into account the nonlinearity of the loudspeakers, as well as other performance measures. Besides, we know that source separation informed by the filters provides better results [START_REF] Kowalski | Beyond the narrowband approximation: Wideband convex methods for under-determined reverberant audio source separation[END_REF] : this opens the way to the alternate estimation of both the sources and the filters with suitable penalties on the filters as opposed to [START_REF] Barchiesi | Dictionary learning of convolved signals[END_REF].

1 P 2 P 3 P 4 Figure 1 :

 12341 Figure 1: Comparison of the statistical models (5) to (8) over a set of generated filters for a reverberation time of 250 ms.

5 Figure 2 :

 52 Figure 2: Performance of the estimation of A with N = 5 white noise sources, depending on the duration of the signal.

1 P 4 Figure 3 :

 143 Figure 3: Estimation of A for six different additive noise levels, depending on the recording duration with N = 5 sources

Appendix

We derive from (18) the block matrix notation

and the correlation of the sources (19) appears since we have

Now c is the condition number of the N K × N K block matrix R = (Σ T n Σ n ′ ) n,n ′ ≤N , and the key is to choose the sources so that This work was supported by the EU FET-Open project FP7-ICT-225913-SMALL and the ANR project 08-EMER-006 ECHANGE. this matrix is highly diagonally dominant. Using Gerschogorin's disc theorem [START_REF] Serre | Matrices: Theory and applications[END_REF], we control its eigenvalues. For λ ∈ Sp(R), there exist n ≤ N such that

Then if r(N K -1) < minn rnn(0) we can conclude that c = |λmax| |λmin| ≤ maxn rnn(0) + r(N K -1) minn rnn(0) -r(N K -1) .

(28)