Céline Elie-Caille

Fedor Severin

Jonne Helenius

Jonathon Howard

Daniel J Muller
email: djmatmueller@biotec.tu-dresden.de

A A Hyman
email: aahathyman@mpi-cbg.de

Straight GDP-Tubulin Protofilaments Form in the Presence of Taxol

Keywords: AFM, atomic force microscopy; GMPCPP, guanylyl

come

SUMMARY

Microtubules exist in dynamic equilibrium, growing and shrinking by addition or loss of tubulin dimers from the ends of protofilaments. Hydrolysis of GTP in -tubulin destabilizes the microtubule lattice by increasing the curvature of protofilaments in the microtubule and putting strain on the lattice. The observations that protofilament curvature depends on GTP hydrolysis suggest that microtubule destabilizers and stabilizers work by modulating the curvature of the microtubule lattice itself. Indeed, the microtubule destabilizer MCAK has been shown to increase the curvature of protofilaments during depolymerization. Here we show that atomic force microscopy (AFM) of individual tubulin protofilaments provides sufficient resolution to allow imaging of single protofilaments in their native environment. Using this assay we confirm previous results for the effects of GTP hydrolysis and MCAK on the conformation of protofilaments. We go on to show that taxol stabilizes microtubules by straightening the GDP protofilament, and slowing down the transition of protofilaments from straight to a curved configuration.

RESULTS AND DISCUSSION.

The curvature of individual protofilaments can be assayed by AFM.

To look at the structure of individual protofilaments by AFM [START_REF] Drake | Imaging crystals, polymers, and processes in water with the atomic force microscope[END_REF][START_REF] Engel | Observing single biomolecules at work with the atomic force microscope[END_REF] , we used calcium to depolymerize microtubules polymerized with GMPCPP [START_REF] Müller-Reichert | Structural changes at microtubule ends accompanying GTP hydrolysis: information from a slowly hydrolyzable analogue of GTP, guanylyl (alpha,beta)methylenediphosphonate[END_REF] . The depolymerization products of GMPCPP microtubules were adsorbed to mica and imaged using tapping mode AFM in buffer (Fig. 1A). Numerous curled single protofilaments were observed with a curvature of 31  5 nm, close to the curvature of 36  10 nm measured previously using vitreous ice [START_REF] Müller-Reichert | Structural changes at microtubule ends accompanying GTP hydrolysis: information from a slowly hydrolyzable analogue of GTP, guanylyl (alpha,beta)methylenediphosphonate[END_REF] . These previous experiments showed that GMPCPP is not hydrolyzed under these conditions 3 ; they also showed that depolymerization of microtubules either by cooling or by calcium addition generated protofilaments with the same curvature. Therefore GMPCPP protofilaments depolymerized with calcium are thought to represent the structure of GTP protofilaments [START_REF] Müller-Reichert | Structural changes at microtubule ends accompanying GTP hydrolysis: information from a slowly hydrolyzable analogue of GTP, guanylyl (alpha,beta)methylenediphosphonate[END_REF][START_REF] Wang | Nucleotide-dependent bending flexibility of tubulin regulates microtubule assembly[END_REF] . We compared the curvature of GMPCPP protofilaments to those of GDP protofilaments (Fig. 1B). Protofilaments present during normal polymerization have radii of 21  4 nm (n=45) and protofilaments from calcium induced depolymerization of 21  2 nm (n=56) (Table 1), again confirming the results of cryo-EM (21.1  4.3 nm) [START_REF] Müller-Reichert | Structural changes at microtubule ends accompanying GTP hydrolysis: information from a slowly hydrolyzable analogue of GTP, guanylyl (alpha,beta)methylenediphosphonate[END_REF] . Therefore we conclude that AFM allows the analysis of the conformation of protofilaments under native conditions.

Taxol slows the straight-to-curved transition of GDP protofilaments.

We next examined the effect of taxol on the configuration of protofilaments. Taxol does not inhibit the hydrolysis of microtubule-bound GTP [START_REF] Melki | Continuous monitoring of Pi release following nucleotide hydrolysis in actin or tubulin assembly using 2-amino-6-mercapto-7-methylpurine ribonucleoside and purine-nucleoside phosphorylase as an enzymelinked assay[END_REF] , but slows down microtubule depolymerization [START_REF] Schiff | Promotion of microtubule assembly in vitro by taxol[END_REF] . Cryo-EM observations show that as with GMPCPP, taxol increases the length of tubulin monomers in microtubules from 4.01 to 4.17 nm [START_REF] Arnal | How does taxol stabilize microtubules?[END_REF] . This suggests that taxol could stabilize microtubules by preventing the protofilaments from curving and putting mechanical strain on the tubulin lattice. The small change in the size of the subunit would correspond to the decreased length associated with bending of the subunit.

Using AFM, we showed that microtubules polymerized with taxol contained a significant number of individual protofilaments (Fig. 1C). To exclude the possibility that the supporting surface induced formation of protofilaments, we showed that AFM topographs of glutaraldehyde fixed microtubules contained similar amounts and appearance of protofilaments compared to unfixed ones (data not shown). Thus our data are consistent with previous observations that individual protofilaments coexist with microtubules in freshly prepared taxol-stabilized microtubule samples [START_REF] Zhu | In vitro Studies of Microtubule Structures Using the MAC Mode TM AFM[END_REF] .

Taxol-stabilized protofilaments were mainly straight or slightly curved, but occasionally formed rings (Fig. 1C). The radius of curvature of slightly curved protofilaments was 245  105 nm (n=51), (compared to 31  5 nm for GMPCPP protofilaments (Table 1)).

The radius of closed rings (cf Fig. 1D) was 21 ± 2 nm (n =48), similar to the radius of GDP protofilament rings. To test whether the slightly curved taxol protofilaments represented transition states between straight and protofilaments rings, we examined the structure of protofilaments with time after polymerization. We prepared taxol-stabilized microtubules and imaged them by AFM 1, 4, 6, 7, 24 and 50 hours after polymerization (Fig. 2). These topographs show that straight protofilaments are very stable-up to four hours, most of taxol protofilaments are straight. The most likely explanation for these data is that taxol initially stabilizes protofilaments in a straight conformation. Because GTP is hydrolysed in taxol-stabilized microtubules [START_REF] Xiao | Insights into the mechanism of microtubule stabilization by Taxol[END_REF] , these data suggest that taxol actively straightens protofilaments. The majority of the protofilaments are in a straight configuration up to six hours after preparation. Given the time scale of microtubule life time during polymerization (minutes) it is likely that the protofilaments remain straight when taxol stabilizes microtubules. However at longer time scales, the protofilaments slowly convert to the curved conformation, and by 24 hours they eventually close up into rings, reminiscent of the structure of GDP protofilaments. How does taxol slow down the straight to curled transition of GDP protofilaments? One likely possibility is that taxol slows down the intrinsic structural transition from straight to curved that takes place in GDP protofilaments. However we cannot rule out that taxol itself is slowly inactivated in an aqueous environment.

Van Buren et al (2005) [START_REF] Vanburen | Mechanochemical model of microtubule structure and self-assembly kinetics[END_REF] proposed that taxol might stabilize microtubules by making the protofilaments less stiff rather than making them less curved: in both cases the mechanical strain energy penalty for tubulin in the microtubule lattice is reduced. We have no evidence indicating that taxol does not soften protofilaments because our AFM images provide no information about protofilament rigidity; however, we note that it is not established that taxol even has a softening effect on microtubules -see e.g. Mickey, B. and Howard, J. 11 who found that taxol actually increased slightly the flexural rigidity of GDP microtubules.

GTP hydrolysis and kink between tubulin monomers.

Previous experiments suggest that protofilament curvature increases after GTP hydrolysis because tubulin subunits in the lattice shorten, changing the repeat length of the tubulin dimers from 4.20 to 4.05 nm [START_REF] Hyman | Structural changes accompanying GTP hydrolysis in microtubules: information from a slowly hydrolyzable analogue guanylyl-(alpha,beta)-methylene-diphosphonate[END_REF] . This lead to the conclusion that a conformational change within tubulin subunits induced by GTP hydrolysis induces protofilament curvature. Recent structural analysis of the two alternative tubulin structures, both straight and curved protofilaments at atomic resolution have shown that indeed, GTP hydrolysis induces a 12° kink between the tubulin subunits [START_REF] Nogales | Structural intermediates in microtubule assembly and disassembly : how and why ?[END_REF] . The kink angle was obtained by reconstructing cryo-EM images recorded from double-layered tubes of GDP bound tubulin [START_REF] Wang | Nucleotide-dependent bending flexibility of tubulin regulates microtubule assembly[END_REF] . Thus, it was not possible to directly compare these tubulin polymer conformations with those of protofilaments induced by various stabilizing and destabilizing factors and conditions. Using high-resolution AFM topographs we were able to distinguish individual tubulin subunits in the protofilaments (Fig. 3). By averaging the tubulin distances contoured in the height profiles of 10 straight taxol-stabilized protofilaments such as shown in Fig. 3A we obtained a periodicity of 4.2 ± 1.6 nm (n=104) (Fig. 3B), corresponding well to the known size of monomeric tubulin, 4 nm [START_REF] Amos | Arrangement of subunits in flagellar microtubules[END_REF] . Second, combining tubulin periodicity and protofilament radius of curvature allowed us to calculate the kink angle established between the tubulin subunits (see materials and methods).

We calculated the kink in GMPCPP-stabilized protofilaments and GDP protofilaments, and compared this to the published data from other methods. GMPCPP protofilaments form curled protofilaments with an average radius of curvature of 31 ± 5 nm (n=35) in presence of 40 mM Ca 2+ . This gives a calculated value of 7.8 ± 1.2° (n=63, 40 mM Ca 2+ ) for the kink angle between dimers. This differs from the 5° kink found between subunits in tubulin sheets [START_REF] Nogales | Structural intermediates in microtubule assembly and disassembly : how and why ?[END_REF] . This difference may be because the kink of subunits within tubulin sheets is constrained [START_REF] Krebs | Structural rearrangements in tubulin following microtubule formation[END_REF] . The radius of GDP protofilament rings was 21 ± 2 nm (n=56) independently of whether 40 mM Ca 2+ or MCAK was present in the sample (Fig. 3F). With 30 subunits to a ring (Fig. 3D,E) the tubulin subunits within these protofilament had a kink of 12°, exactly that found between tubulin monomers of a GDP tubulin dimer in the relaxed, curved conformation [START_REF] Nogales | Structural intermediates in microtubule assembly and disassembly : how and why ?[END_REF] . Thus these data confirm that a change in the kink between tubulin subunits can account for the change in radius of curvature of the protofilaments. Next, we calculated the time-dependent changes in the kink angle between the subunits in taxol-stabilized protofilaments (Fig. 3F). 1 hour after preparation of taxol-stabilized protofilaments, tubulin monomers of protofilaments showed an average kink close to 0°, after 7 hours this kink increased slightly enhanced to ~1°. However, after 24 hours all taxol-stabilized protofilaments were in rings, with a calculated kink angle of 12°.

These results support previous analysis by Arnal and Wade (1995) [START_REF] Arnal | How does taxol stabilize microtubules?[END_REF] who looked at the periodicity of tubulin subunits in the lattice and showed that the most likely mechanism by which taxol stabilize microtubules is by preventing the GDP subunit adopting a kinked configuration after GTP hydrolysis.

MCAK increases the curvature of GTP protofilaments and kink between tubulin subunits.

Previous EM results suggested that MCAK induces curvature in GTP-like protofilaments [START_REF] Desai | Kin I kinesins are microtubule-destabilizing enzymes[END_REF] . Based on these and other observations [START_REF] Moores | A mechanism for microtubule depolymerization by KinI kinesins[END_REF]18 , it has been suggested that the microtubule destabilizing factors, such as KLP-7, bend tubulin dimers thus destabilizes microtubules [START_REF] Moores | A mechanism for microtubule depolymerization by KinI kinesins[END_REF]18 . To further test this idea, MCAK was added to microtubules grown both in the presence of GMPCPP or GTP with taxol. AMPPNP was added to inhibit MCAK and to tightly bind it to microtubules [START_REF] Moores | A mechanism for microtubule depolymerization by KinI kinesins[END_REF]

. After 5 min

Conformational transitions of protofilaments -8incubation the resulting structures were imaged by AFM. The AFM topographs showed some "intact" microtubule ends and numerous ring-like protofilaments (Fig. 1F,G). In contrast to the effects observed using only the MCAK motor core [START_REF] Moores | A mechanism for microtubule depolymerization by KinI kinesins[END_REF] , protofilaments obtained in the presence of the complete MCAK protein were predominantly single rings with a 19  2 nm (n=37) radius of curvature while no double or triple concentric rings were observed. The protofilament radius from taxol-stabilized microtubules 19  2 nm (Fig. 1G, Table 1), was similar to previously published EM data [START_REF] Moores | A mechanism for microtubule depolymerization by KinI kinesins[END_REF] . The radius of ring-like protofilaments from GMPCPP microtubules was 27  4 nm (n=43) (Fig. 1E, Table 1). From these data we calculated that MCAK increased the kink between tubulins from 7.8 ± 1.2° (n=63, 40 mM Ca 2+ ) to 8.9 ± 1.2° (n=43) (Table 1, Fig. 3F).

Thus, in agreement with Desai (1999) [START_REF] Desai | Kin I kinesins are microtubule-destabilizing enzymes[END_REF] and Moores 17 (2002), the AFM data indicate that MCAK bends tubulin protofilaments made of GTP tubulin. Interestingly, addition of MCAK to the sample of taxol-stabilized microtubules in the presence of ATP instead of AMPPNP did not induce formation of protofilament rings. We speculate that ATP hydrolysis and MCAK release from protofilaments occur more rapidly than protofilament bending. An alternative possibility is that the complete ATPase cycling allows disassembly of the protofilaments into tubulin dimers, suggesting that MCAK may induce a greater kink between tubulin monomers, increasing protofilament curvature.

The idea that microtubule stabilizers and destabilizers work through the curvature of protofilaments is a fascinating link between the global structural change of a microtubule and the small chemical change from GTP to GDP, within its tubulin dimers. By straightening the GDP protofilaments, taxol apparently prevents GTP hydrolysis from putting strain on the microtubule lattice. In this context, it is interesting that taxol-stabilized protofilaments had radii either greater than 200 nm or around 20 nm, and both populations were found in the same samples. This indicates that kinking occurs simultaneously in the 20-30 dimers of a protofilament, suggesting a longdistance cooperative interaction between dimers of a protofilament. Cooperative kinking may act as a growth-to-shrinkage switch and explain why microtubules undergo catastrophic disassembly. It will be fascinating to see how other modulators of microtubule stability alter the conformation of the protofilaments.

Taxol is used as an anticancer agent [START_REF] Sorger | Coupling cell division and cell death to microtubule dynamics[END_REF] . Its action on microtubules suggests a mechanism by which arrest in mitosis is due to changes in microtubule stability that induce apoptosis of cancerous cells. Our data on the effect of taxol on the microtubule lattice suggest a potentially more complex explanation for the efficiency of taxol. By straightening the protofilaments, taxol induces a more GTP-like configuration in the protofilament. Since proteins can distinguish the GTP from the GDP configuration microtubule lattice [START_REF] Severin | Kinetochores distinguish GTP from GDP forms of the microtubule lattice[END_REF][START_REF] Westermann | Formation of a dynamic kinetochore-microtubule interface through assembly of the Dam1 ring complex[END_REF] , perhaps putting the protofilaments into a GTP-like configuration triggers physiological changes in the apoptosis machinery that are different from those induced by simple microtubule depolymerization.

MATERIALS AND METHODS

Preparation of taxol-stabilized microtubule sample

Taxol-stabilized microtubules were prepared incubating tubulin (30 µM final concentration) for 20 min in BRB80 buffer, containing 1 mM GTP at 37 0 C followed by the addition of 170 µl of 10 µM taxol in BRB80. Then microtubules were pelleted by ultracentrifugation at 109,000 g (TLA 100 rotor, OptimaTM MAX Ultracentrifuge, Beckman) for 5 min, and resuspended in 50 µl BRB80 containing 10 µM taxol (BRB80-taxol). Taxol-stabilized microtubules are always diluted in a taxol-containing buffer.

Preparation of GMPCPP microtubule sample

GMPCPP microtubules were polymerized as described [START_REF] Howard | Preparation of marked microtubules for the assay of the polarity of microtubule-based motors by fluorescence microscopy[END_REF] at a tubulin concentration of 10 µM instead of 4 µM.

Calcium and MCAK/AMPPNP induced microtubule depolymerization

Microtubule depolymerization was induced by addition of calcium (40 mM final concentration) or MCAK/AMPPNP (0.2 M MCAK, 0.5 mM AMPPNP) to the microtubule sample in BRB80 as previously described [START_REF] Müller-Reichert | Structural changes at microtubule ends accompanying GTP hydrolysis: information from a slowly hydrolyzable analogue of GTP, guanylyl (alpha,beta)methylenediphosphonate[END_REF] . Recombinant MCAK was obtained as described [START_REF] Hunter | The kinesin-related protein MCAK is a microtubule depolymerase that forms an ATP-hydrolyzing complex at microtubule ends[END_REF] . The mixture was deposited on freshly cleaved mica immediately, and left at least 1 minute on the surface before extensive washing with BRB80 buffer. For each condition imaged by AFM we analyzed about 100 single protofilaments to determine their conformational statistics. Cartoons at the bottom relate

Determining protofilament radius, tubulin distance and kink between tubulin monomers

Figure 1 .

 1 Figure 1. Conformations of individual tubulin protofilaments imaged by AFM. (A) Products of calcium (40 mM) induced GMPCPP microtubules depolymerization. (B) Unstabilized GDP protofilaments imaged quickly after deposition on mica. (C) Individual protofilaments are present in the sample of taxol-stabilized microtubules and present either in straight, curved, and closed or open ring-like conformations. (D) After 2 days of aging, the taxol-stabilized protofilaments present a ring-like conformation, either closed (1), open (2) or even double ring (3). Addition of MCAK/AMPPNP induced ring-like protofilaments from GMPCPP-stabilized microtubules (E) and taxolstabilized microtubules (F,G). Nevertheless ends of taxol-stabilized microtubules maintain a straight (blunt) conformation (F). (G) Ring-like protofilaments obtained after MCAK addition to taxol-stabilized microtubules (inset shows higher resolution image). (H) MCAK/ATP, unlike MCAK/AMPPNP, does not induce accumulation of ring-like protofilaments. The gray levels of the AFM topographs correspond to vertical scales of 6 nm (A), 10 nm (B,D), 22 nm (C), 20 nm (E,H), 8 nm (G), and 30 nm (F). Scale bars of insets correspond to 20 nm.

Figure 2 .

 2 Figure 2. Straight-to-ring transition of tubulin protofilaments. (A) Gallery presenting the different protofilaments conformations observed from taxol-stabilized microtubules. The gray levels of all AFM topographs correspond to vertical scales of 22 nm. (B) Occurrence of the different structural intermediates depends on sample aging.For each condition imaged by AFM we analyzed about 100 single

  Figure 1 Click here to download high resolution image

Table 1 . Radius of curvature of tubulin protofilaments.

 1

	TABLES
	Sample			Protofilament radius of curvature
				(nm  SD)
	GDP protofilaments		21  4 (n=45)
	GDP protofilaments + 40 mM Ca 2+		21  2 (n=56)
	GMPCPP protofilaments + 40 mM Ca 2+ 31  5 (n=35)
	GMPCPP	protofilaments	+	27  4 (n=43)
	MCAK/AMPPNP
	Taxol-stabilized protofilaments (Fresh)		245  105 (n=51)
	Taxol-stabilized protofilaments + 40 mM Ca 2+	21  2 (n=57)
	Taxol-stabilized protofilaments (2 days	21  2 (n=48)
	old)
	Taxol-stabilized	protofilaments	+	19  2 (n=37)
	MCAK/AMPPNP

ACKNOWLEDGEMENTS

We thank E. Mandelkow for critical reviewing this manuscript. This work was supported by the Max-Planck-Society and the Deutsche Forschungsgemeinschaft.

We used "Image J" software to fit a circle of a given radius onto bent or ring-like protofilaments. Then we measured the periodicity of the height profile along the protofilament revealing the average distance between tubulin monomers. Both protofilament radius, r, and average tubulin distance, l, were used to calculate the kink according to kink(°)=2πr/(360•l). By determining the average radius of protofilaments at a certain time after preparing the sample we could then directly relate this time to the progression of the kink.

Sample adsorption to mica and AFM imaging

Samples were allowed to adsorb to freshly cleaved mica for at least 1 minute followed by rinsing (at least 3 times) with buffer (50 mM HEPES pH 7.4 adjusted with KOH, 200 mM KCl 2 mM MgCl 2 or BRB80). Mixtures of microtubules+40 mM Ca 2+ or microtubules+MCAK were deposited on freshly cleaved mica immediately after preparation, and left 1 minute for adsorption before extensively washing with BRB80 or HEPES buffer. AFM (Nanoscope III, Veeco) imaging was performed in buffer solution using tapping mode. The AFM cantilevers (NPS-oxide sharpened silicon nitride probes (Veeco) exhibiting spring constants of 0.32 or 0.58 N/m) were activated at their resonance frequencies ranging from 8.5 to 9.5 kHz. For the feedback controls, typical values of setpoint for imaging were between 0.5 to 1.5 V, depending on the drive amplitude. The oscillation amplitude was maintained at 5-10 nm for protofilament imaging. (F) Kink between tubulin monomers calculated from the radius of curvature of protofilaments observed under different conditions. Red line represents kink over protofilament curvature radius calculated for a subunit size of 4.2 nm. Experimental averages of radii are given for native and for taxol-and GMPCPP-stabilized protofilaments, in absence and in presence of calcium or MCAK. No time given in legend of (F) indicates that the structures were observed roughly 1h after protofilament preparation and immediately after MCAK or calcium addition to protofilaments. The gray levels of all AFM topographs correspond to a vertical scale of 22 nm.


Konstantinos Trochidis
email: konstantinos.trochidis@u-bourgogne.fr

Charles Delbé
email: charles.delbe@u-bourgogne.fr

Emmanuel Bigand
email: emmanuel.bigand@u-bourgogne.fr

INVESTIGATION OF THE RELATIONSHIPS BETWEEN AUDIO FEATURES AND INDUCED EMOTIONS IN CONTEMPORARY WESTERN MUSIC

This paper focuses on emotion recognition and understanding in Contemporary Western music. The study seeks to investigate the relationship between perceived emotion and musical features in the fore-mentioned musical genre. A set of 27 Contemporary music excerpts is used as stimuli to gather responses from both musicians and non-musicians which are then mapped on an emotional plane in terms of arousal and valence dimensions. Audio signal analysis techniques are applied to the corpus and a base feature set is obtained. The feature set contains characteristics ranging from low-level spectral and temporal acoustic features to high-level contextual features. The feature extraction process is discussed with particular emphasis on the interaction between acoustical and structural parameters. Statistical relations between audio features and emotional ratings from psychological experiments are systematically investigated. Finally, a linear model is created using the best features and the mean ratings and its prediction efficiency is evaluated and discussed.

INTRODUCTION

The expressive aspects of music are the most difficult to analyze structurally, inducing a large variety of emotional responses in humans. The richness of these responses is what motivates an engagement with music [START_REF] Bigand | Multidimensional scaling of emotional responses to music: The effect of musical expertise and of the duration of the excerpts[END_REF]. Many studies indicate the important distinction between one's perception of the emotion(s) expressed by music and the emotion(s) induced by music. Studies of the distinctions between perception and induction of emotion have demonstrated that both can be subjected to not only the social context of the listening experience, but also to personal motivation [START_REF] Juslin | Expression, perception, and induction of musical emotions: A review and questionnaire study of every day listening[END_REF].

Modeling the perception of expressive musical content is highly useful in MIR applications such as emotion based classification and recommendation systems, radio and TV broadcasting programs, and music therapy defining appropriate musical repertories for research in patients suffering from Alzheimer or Bibromilagic. Due to the highly conceptual elusiveness of emotions and the limitation of computational methods mainly based on low level features, modeling and prediction of emotion in music remains a particularly difficult task. Research on music and emotion has always focused in music genres such as Western Popular or Classical music.

To the best of our knowledge limited research has been conducted in the field of Contemporary Western music. The term "Contemporary art music" is used for Western art-tradition music written since 1945. Characteristic structures in Western music like systematic variation of tempo, mode and timbre, which are identified in Classical or popular modern music, do not necessarily exist in Contemporary art music. This raise questions such as: 1) Can an emotional response be triggered when the mentioned features are not present? 2) Which other features contribute to emotional reaction? and 3) Are the same features contribute to emotion generation in both musicians and non musicians?

The present paper deals with the above issues. Section 2 provides background material on previous research on music studies, while Section 3 presents the ground truth and the experiments carried out with musicians and nonmusicians volunteers. Section 4 describes the audio feature extraction and representation while, Section 5 discusses the statistical selection of features and modeling of music emotional regression. Discussion and conclusions are drawn in Section 6.

RELATED WORK

Many psychological models have been used in studies concerning music and emotion. The main approaches existing in the literature are the discrete and the dimensional models [START_REF] Juslin | Music and emotion: Theory and research[END_REF]. A comparison of the discrete and dimensional models of emotions in music can be found in [START_REF] Eerola | A comparison of the discrete and dimensional models of emotion in music[END_REF]. According to the categorical approach, emotions are conceptualized as discrete unique entities and contain a certain basic number of emotion categories from which all the emotional states are derived. There is an agreement towards researchers representing this approach as to five basic emotions: happiness, sadness, anger, fear and disgust [START_REF] Plutchik | The psychology and biology of emotion[END_REF], [START_REF] Kemper | How many emotions are there? Wedding the social and the autonomic components[END_REF].

In the dimensional approach, emotions are expressed on a Cartesian coordinate system according to two axes those of valence and arousal. The model depicted in figure 1 shows Russell's [START_REF] Russell | A circumplex model of affect[END_REF] circumplex model of affect, where the axes measure activation and pleasure. Happiness and Anger are located at the top of the vertical (arousal) axis, Serenity and Sadness are located at the bottom. On the horizontal axis (pleasure), Happiness and Serenity are more positive emotions than Anger and Sadness, and so these pairs are located on the right and left side respectively of this axis of Russel's space. In the late 1990s, Thayer [START_REF] Thayer | The biopsychology of mood and arousal[END_REF] proposed a two-dimensional mood model that uses individual adjectives which collectively form a mood pattern. This dimensional approach adopts the theory that mood is entailed from two factors: Stress (happy/anxious) and Energy (calm/ energetic), and divides music mood into four clusters: Contentment, Depression, Exuberance and Anxious/Frantic. Music emotion representation in an affective space gained lots of interest among researchers. Hevner's study [START_REF] Hevner | Expression in music: a discussion of experimental studies and theories[END_REF] arranges music emotion in a circle of eight clusters. Each cluster represents a certain emotion and contains seven to eleven adjectives grouped together. Shubert [START_REF] Schubert | Measuring emotion continuously. Validity and reliability of the two-dimensional emotion-space[END_REF] argues that emotional reaction depends on the interaction between different factors such as musical mode, tempo, loudness and pitch. In [START_REF] Bigand | The time course of emotional responses to music[END_REF] the time course of emotional responses to music is investigated in both untrained and trained musicians using musical excerpts of increasing duration ranging from 250 ms to 20 seconds. Findings show that less than 1s of music is enough to instill elaborated emotional responses in listeners

The four main emotion classes of Thayer's model were used as the emotion model in [START_REF] Lu | Automatic mood detection and tracking of music audio signals[END_REF]. Three different feature sets were adopted for music representation, namely intensity, timbre and rhythm. Gaussian mixture models were used to model each of the four classes. An interesting contribution of this work was a hierarchical classification process, which first classifies a song into high/low energy (vertical axis of Thayer's model), and then into one of the two high/low stress Classes.

Emotion recognition is modelled as a regression task in [START_REF] Yang | A regression approach to music emotion recognition[END_REF]. Volunteers rated a training collection of songs in terms of arousal and valence in an ordinal scale of values from -1 to 1 with a 0.2 step. The authors then trained regression models using a variety of algorithms and a variety of extracted features.

A model predicting perceived emotions based on a set of features extracted from soundtrack music is given in [START_REF] Eurola | Prediction of Multidimensional Emotional ratings in Music from Audio Using Multivariate Regression Models[END_REF]. Three separate data reduction techniques, namely stepwise regression, principal component analysis, and partial least squares are compared. The effectiveness of current music understanding processes and music intelligent systems is mainly hampered by the so called semantic gap between human perception and cognition and on the other hand by the low level music features which are mostly statistics of spectral and temporal characteristics in the signal.

Therefore, many researchers [START_REF] Laurier | Music mood representation from social tags[END_REF], [START_REF] Hu | Lyric text mining in music mood classification[END_REF], [START_REF] Kim | Music emotion recognition: A state of the art approach[END_REF] try to bridge the semantic gap between the low level features and high level semantics, which humans perceive and understand, by merging different modalities such as low level acoustical features and social data including lyrics, tags and web logs.

EXPERIMENTAL SETUP

We decided to analyze and explore Western Contemporary art music because several of its features are shared differently by other musical genres. The main concept that best describes contemporary music is confusion in listening. Harmony does not necessarily play an important role. Thus, there is a difficulty of extracting and interpreting harmonic information because of the lack of tonal reference system. Composition contains clustered sounds and disharmonic intervals very different compared to the ones found in Western Classical music or modern popular music. Instrumentation is very complex and musicians use their instruments for producing sounds very different from those encountered in the Classical repertoire. Mixed Electro acoustic and traditional instruments raises the problem of which information of inharmonic sounds related to timbre can be captured and represented.

Method

The data used in this paper are based on a previous study [START_REF] Bigand | The time course of emotional responses to music[END_REF]. Twenty participants without musical training (referred to as non musicians) and 20 with an average of 10 years of musical training and instrumental practice participated in this experiment. A set of 27 musical excerpts of Contemporary music is selected by music theorists and psychologists according to several constraints. All excerpts were expected to convey a strong emotional experience. They were chosen to illustrate a large variety of emotions, and to be representative of key musical periods of Contemporary Western music. The excerpts showed a great variation in musical structure including harmony, rhythm, tempo, timbre and instrumentation. The participants were asked first to listen to all excerpts and then focus their attention on their private emotional experience. Next, they were asked to look for excerpts that induced similar emotional experience based on arousal and valence dimensions (whatever that may be) and to drag the corresponding icons in order to group these excerpts. They corresponded either to the beginning of a musical movement, or to the beginning of a musical theme or idea. An average duration of 30s sounded appropriate. We adopted a dimensional approach for emotional labelling because it avoids a strict classification and accounts for similarity and dissimilarity of emotions. Linguistic labels remain problematic and may simplify the emotional reaction and further disregard the difference between induced and perceived emotions.

The groupings of participants were then converted into a 27x27 matrix of co-occurrence. Each cell of the matrix indicated the average number of times that two excerpts were grouped together. The subtraction of the average matrix of occurrence from 1 resulted in a matrix of dissimilarity. The matrices obtained were highly correlated for musicians and non musicians, (r= .65, p < .001). The resulting matrices were analysed with MDS and cluster analysis methods. The locations of the 27 excerpts along the two principal dimensions are presented in Figures 2 and3. The vertical axis represents musical excerpts that varied obviously by their arousal level (with low arousal pieces at the bottom, and high arousal pieces at the top). The horizontal axis represents presumably musical excerpts that differ by their emotional valence (with positive valence on the right and negative valence on the left).

AUDIO FEATURE EXTRACTION

Low-level acoustical features

A theoretical selection of musical features was made based on music characteristics such as timbre, harmony, rhythm and dynamics. A total of 324 features where extracted from the music excerpts representing information related to the above concepts. The MIR Toolbox for MATLAB was used to compute the various low and high level descriptors [START_REF] Lartillot | MIR in Matlab (II): A Toolbox for Musical Feature Extraction From Audio[END_REF].

Rhythmic features

A rhythmic analysis of the music signals was performed. Descriptors such as the fluctuation (the rhythmic periodicity along auditory frequency channels), the estimation of notes onset times and the number of onsets per second were computed. Finally, the tempo was estimated.

Timbre features

Mel Frequency Cepstral Coefficients (MFCCs) are used for speech recognition and music modeling. To derive the MFCCs, the signal was divided into frames and the amlit ude spectrum for each frame was calculated. Next, its logarithm was taken and converted to Mel scale. Finally, the discrete cosine transform was implemented. We selected the first 13 MFCCs. Another set of 4 features related to timbre textures were extracted from the Short-Term Fourier Transform: spectral centroid, spectral rolloff, spectral flux and flatness which indicate whether the spectrum distribution is smooth or spiky. The size of the frames used to compute the timbre descriptors was 0.05 sec half-overlapped.

Tonal features

The signals were also analyzed according to the tonality context. Descriptors such as the Chromagram (energy distribution of the signals wrapped in the 12 pitches), the key strength (i.e. the probability associated with each possible key candidate, through a cross correlation with the Chromagram and all possible key candidates), the tonal Centroid (a six dimensional vector derived from the Chromagram corresponding to the projection of the chords along circles of fifths or minor thirds) and the harmonic change detection function (flux of the tonal Centroid) were extracted.

Dynamic features

We computed information related to the dynamics of the music signals such as the RMS and the percentage of low energy frames to see if the energy is evenly distributed throughout the signals or certain frames are more contrasted than others For all features a series of statistical descriptors were computed such as the mean, the standard deviation and the linear slope of the trend along frames, i.e. the derivative. For all Mel Frequency Cepstral Coefficients the first and second derivative were computed. Also the maximal periodicity detected in the frame-by frame evolution of the values was estimated through the computation of the autocorrelation sequence and the amplitude of this periodicity.

High-level Contextual features

While low-level descriptors such as loudness (perception of sound intensity) or pitch (perception of fundamental partials) account for perceptual aspects of music, higherlevel ones, such as pulse, harmony or complexity account for contextual aspects, i.e. they refer to the cognitive perception and aspects of music. Many models using low level features successfully predicted the dimension of Arousal. The retrieval, however, of Valence has proved to be difficult to measure by using only low level information [START_REF] Fornari | Computer Music Modeling and Retrieval. Genesis of Meaning in Sound and Music[END_REF]. In order to tackle this problem, we used a set of five high level features in conjunction with the low level descriptors which are described above.

Pulse Clarity

This descriptor measures the sensation of pulse in music. Pulse can be described as a fluctuation of musical periodicity that is perceptible as "beatings" in a sub-tonal frequency band below 20Hz. The musical periodicity can be melodic, harmonic or rhythmic as long as it is perceived the listener as a fluctuation in time.

Articulation

Articulation usually refers to the way in which a melody is performed. If a pause is clearly noticeable in between each note in the melodic, the articulation of the melody is staccato, which means "detached". On the other hand, if there is no pause in between the notes of the melody then the melody is legato, meaning "linked". This feature attempts to estimate the articulation from musical audio signals by attributing to it an overall grade that ranges continuously from zero (staccato) to one (legato).

Mode

This feature refers to a computational model that detects between major and minor excerpts. It calculates an overall output that continuously ranges from zero (minor mode) to one (major mode).

Event density

This descriptor measures the overall amount of simultaneous events in a musical excerpt. This can be melodic, harmonic and rhythmic, as long as they can be perceived as independent entities by the human cognition.

Brightness

This descriptor measures the sensation of how bright of a music excerpt is felt to be Attack, articulation, or the unbalance or lacking of partials in other regions of the frequency spectrum can influence its perception.

FEATURE SELECTION

From the selected features, only those whose correlation with the ratings is sufficiently statistically significant (with a p-value lower than .05) are selected. The selected features are ordered from the most correlated to the least correlated ones. Features that are not sufficiently independent with respect to the better scoring ones (with a normalized cross correlation exceeding 0.6) were removed as well. In order to see how these acoustic features may account for the present data, a normalized stepwise regression of the coordinates of the pieces on the two axes was performed using the best features. Table 1 and 2 provide the outcome of the multiple linear regression analysis of the acoustic features over the coordinates of the pieces for musicians and non musicians. The resulting model provides a good account of the arousal for musicians (adjusted R² = 0.72, see Table 1), with the periodicity amplitude of flatness (β = 0.60) and the entropy of the magnitude of the highest peak in the chromagram (β = -0.43) contributing the most, followed by the flux of the tonal centroid and the mean derivative of the 3d mfcc band. On the other hand, the regression model provided a moderate account of valence with R² = 0.57, with the mean of pulse clarity (β = 0.74) (i.e. the perceived sensation of pulse) contributing the most, followed by the mean of articulation (β =0.68) and brightness (β =-0.40). The outcome of the multiple regression analysis of the acoustic features over the coordinates of the pieces for non musicians is presented in Table 2. One can see that the results are very similar to that of the musicians. The resulting model provides a good account R² = 0.67 of the arousal for the non musicians, with the periodicity amplitude of flatness (β = -0.58) and the entropy of the magnitude of the highest peak in the chromagram contributing the most (β = -0.46) followed by the flux of the tonal centroid. The regression model provided a moderate account of valence with R² = 0.62, with the mean of pulse clarity and the mean derivative of the 10 th mfcc band contributing the most, followed by the mean of brightness.

Valence

CONCLUSIONS

In the present paper the relationships between music features and emotion perception in the case of Contemporary Western music are investigated. A systematic analysis of the musical stimuli shows that low level spectral and temporal features such as flatness and chroma features are efficient in modeling the emotion perception of arousal dimension, while high-level contextual information such as articulation, pulse clarity, mode and brightness succeed to measure the more cognitive nature of valence. The results contradict the widespread opinion that understanding of contemporary western music is restricted to highly trained listeners. It is shown that the emotion processing mechanism is quite similar for musicians and non musicians with the same low level spectral, temporal features correlated with arousal and high level contextual features correlated with valence dimension. Contemporary Western music can serve successfully as stimulus for studying the emotional processing mechanism in music. An emotional response can be still triggered when characteristic structures and features of Western popular or Classical music are not present. Future work will explore the effectiveness of new features extracted from physiological signals such as EEGs to bridge the semantic gap between high level knowledge related to the cognitive aspects of emotion and low level acoustical features. Furthermore, a larger Contemporary music dataset will be constructed and new audio features will be designed and tested to allow for better statistical results.

Figure 1 .

 1 Figure 1. Russell's Circumplex model of emotion.

Figure 2 .

 2 Figure 2. Geometrical representation for the 27 Contemporary music excerpts in musicians, resulting from the MDS analysis.

Figure 3 .

 3 Figure 3. Geometrical representation for the 27 Contemporary music excerpts in non-musicians, resulting from the MDS analysis.

Table 1 .

 1 Outcome of the multiple linear regression analysis of the acoustic features over the coordinates for musicians.

		β	Arousal	β
	Pulse_clarity	0.74	FlatnessPeriodAmp	0.60
	Articulation	0.68	chromagramPeakstd	-0.43
	brightness	-0.40	Tonal_hdcf	0.33
	Event_density	-0.19	Dmfcc_mean_3	-0.18
	Mode_Mean	0.17	FlatnessPeriodFreq	-0.06

Table 2 .

 2 Outcome of the multiple linear regression analysis of the acoustic features over the coordinates for non musicians.

	Valence	β	Arousal	β
	Articulation	0.83	FlatnessPeriodAmp	0.58
	Pulse_clarity	0.68	chromagramPeakstd	-0.46
	brightness	-0.50	Tonal_hdcf	0.25
	Event_density	-0.19	FlatnessPeriodFreq	-0.19
	Mode_mean	0.05	RoughnessPeriodFreq -0.009


Caspar Addyman
email: addyman@bbk.ac.uk

Denis Mareschal
email: d.mareschal@bbk.ac.uk

Elizabeth Thomas
email: elizabeth.thomas@u-bourgogne.fr

Learning to perceive time: A connectionist, memory-decay model of the development of interval timing in infants

Keywords: Infancy, cognitive development, interval timing, embodied learning

 emerges naturally from this network and (3) that motor activity can synchronize independent timing mechanisms across different sensory modalities.

Introduction

Adults from all cultures and backgrounds can accurately anticipate short-term events, can catch objects thrown to them, and can perform the enormous range of human activities that require timing. This universality may seem to imply that the sense of time does not rely on learning or embodiment, a position implicitly taken by the well-known pacemaker-accumulator models of time perception (Gibbon, 1997;[START_REF] Church | Properties of the internal clock[END_REF][START_REF] Gibbon | Scalar timing in memory[END_REF] and multiple-oscillator/coincidence-detection models [START_REF] Miall | The storage of time intervals using oscillating neurons[END_REF][START_REF] Church | Alternative representations of time, number and rate[END_REF]. But there are a number of problems with these models, most important among them, the problem of resetting the system for every event for which a time judgment might later be required. Recently, the reset problem has been avoided by suggesting that time perception depends on memory-trace decay [START_REF] Staddon | Interval timing: memory, not a clock[END_REF]. Our central hypothesis is that adult timing capacities are learned in early infancy and require grounding in motor activity. For this reason, we developed a model of time-perception learning in infants.

As far as we know this is the first developmental model of interval timing. It is a memory-based connectionist model of how infants learn to perceive time and has two further novel features. First, it uses the uncertainty of a memory for an event as the measure of how long ago that event happened. This is in contrast to other memory models which use relative intensity (e.g [START_REF] Staddon | Multiple time scales in simple habituation[END_REF]. Secondly, it is a developmental model in which embodimentspecifically, infant motor activity -is crucial to the calibration of time-perception within and across sensory modalities. Learning and coordinated motor activity in infancy play a key role in synchronising different timing mechansism and permitting developmental predictions. In addition, the scalar property of interval timing [START_REF] Gibbon | Scalar expectancy theory and Weber's law in animal timing[END_REF] emerges naturally from the model. This paper is organized as follows. We begin by briefly discussing the various classes of models of time perception. We then present the theoretical justification and framework for our connectionist model of timing based on memorytrace decay. We focus, in particular, on the need to calibrate this model via repetitive motor activity. Finally, we present the details of the model, including how it is calibrated by replicable signals in the motor system, and demonstrate that the well-known scalar property of interval timing [START_REF] Gibbon | Scalar expectancy theory and Weber's law in animal timing[END_REF] is a natural by-product of its operation. We also show how this model could provide developmental predictions.

Background

Time perception is central to human cognition (e.g., [START_REF] Grondin | Psychology of time[END_REF][START_REF] Zakay | Temporal cognition[END_REF]. Not only does it allow us to organize and make sense of physical events, it also underlies micro and macro level social interactions. At the level of the individual, time perception is linked to executive control, delayed gratification, and decision making. It, therefore, comes as no surprise that Immanuel Kant described time as an inner sense that structures and makes possible cognition. Human time perception is generally divided into three categories: precision timing (less than 500 ms), interval timing (500 ms. to 5 mins.), and longer-term time perception, each of which appears to implicate by a different neurocomputational system [START_REF] Buhusi | What makes us tick? Functional and neural mechanisms fo interval timing[END_REF]. This paper will be concerned only with interval timing, which lies between cellularly-driven precision timing and longer-term sequence-based, schemadriven timing. This intermediate level is believed to share a common processing core with spatial and numerical magnitude estimation abilities [START_REF] Walsh | A theory of magnitude: common cortical metrics of time, space and quantity[END_REF][START_REF] Brannon | Temporal discrimination increases in precision over development and parallels the development of numerosity discrimination[END_REF].

There are currently three major explanatory paradigms for interval timing. The first relies on an internal pacemaker that emits regular, short pulses that are counted by an accumulator. The number of pulses stored in the accumulator gives the measure of the time that has passed [START_REF] Staddon | Multiple time scales in simple habituation[END_REF][START_REF] Church | Properties of the internal clock[END_REF][START_REF] Gibbon | Scalar timing in memory[END_REF]Meck, 1984, Droit-Volet and[START_REF] Droit-Volet | Temporal bisection in children[END_REF]. The second class of models relies on multiple neuronal oscillators with coincidence detectors associating particular patterns of firing with given time intervals [START_REF] Church | Properties of the internal clock[END_REF]Broadbent, 1984, Matell and[START_REF] Matell | Neuropsychological mechanisms of interval timing behavior[END_REF]. Process-decay models constitute the final class, where the estimation of the passage of time is derived from the decay of memory traces [START_REF] Staddon | Multiple time scales in simple habituation[END_REF][START_REF] Lewis | Remembering the time: A continuous clock[END_REF].

One of the most significant problems with both pacemaker/accumulator and oscillator/coincidence-detector models is the necessity of resetting the accumulators or oscillators for every event that could potentially be timed. Are there multiple accumulators, one for each possible event? What triggers their resetting? What, exactly, constitutes "an event?" Memory models of time-perception do not have this problem. The activation pattern for any event for which there is a memory trace will decay to a greater or lesser extent depending on how long ago the event took place. The memory-trace decay determines the perception of the amount of time that has passed.

However, models that rely on memory-trace decay need to be calibrated against the duration of events in the world. We suggest that infant motor activity is a plausible way in which this calibration could be achieved early in development. Numerous authors [START_REF] Lakoff | Women, Fire, and Dangerous Things: What Categories Reveal About the Mind[END_REF][START_REF] Glenberg | What Memory is For[END_REF][START_REF] Thelen | A dynamic systems approach to the development of cognition and action[END_REF][START_REF] Piaget | The Child's Construction of Reality[END_REF] have convincingly argued for the importance of motor activity in structuring early perceptual and cognitive development. We believe that infant time perception needs to be included in the range of cognitive phenomena structured by motor activity.

There is now ample evidence that infants as young as 4 months are able to keep track of short time intervals and can respond to the violation of an expectation based on a regular time interval (Colombo and Richman, 2002;[START_REF] Adler | Infants' visual expectations and the processing of time[END_REF][START_REF] Brannon | Temporal discrimination increases in precision over development and parallels the development of numerosity discrimination[END_REF][START_REF] Clifton | Heart-rate conditioning in the newborn infant[END_REF]. However, this ability is not entirely mature and continues to develop well into childhood (e.g., [START_REF] Friedman | Developmental perspectives on the psychology of time[END_REF][START_REF] Droit-Volet | Perception of the duration of auditory and visual stimuli in children and adults[END_REF][START_REF] Goldberg | Emergent forms: Origins and early development of human action and perception[END_REF], in concert with children's improved attentional control.

Theoretical Framework

We suggest that the development of interval time perception in infants is intimately linked to coordinated motor activity. Three main lines of argument support this hypothesis:

1. Repetitive motor activity is ubiquitous in early infancy (e.g., [START_REF] Piaget | The Child's Construction of Reality[END_REF]Droit-Volet, Clement and Fayol, 2008), even being present in newborns (Van der Meer, Van der Weel and Lee 1995; [START_REF] Lewandowsky | The rewards and hazards of computer simulations[END_REF]. This movement is perfectly correlated with the time that it takes to complete it. When a baby moves its hand to touch its mother's face, this is a motor activity, but it is also a temporal activity, because it takes a certain amount of time for the baby's hand to move from its starting position to its final position. This fact will be used to calibrate our activation-decay model of time perception.

2. Areas of the brain that are important in motor activity have recently been found to play a role in perceptual timing [START_REF] Ivry | The representation of temporal information in perception and motor control[END_REF]Rao, Meyer and Harrington, 2001). It is now known that the disruption of input from the motor cortex to the associative areas of the parietal cortex selectively interferes with perceptual timing [START_REF] Bueti | Sensory and association cortex in time perception[END_REF].

3. Memory-trace decay is known to exist in the brain and is a well-studied and neurophysiologically plausible phenomenon [START_REF] Staddon | Multiple time scales in simple habituation[END_REF][START_REF] Lewis | Remembering the time: A continuous clock[END_REF]. In contrast, after 25 years of research the neurological evidence for accumulator models remains equivocal [START_REF] Staddon | Interval timing: memory, not a clock[END_REF][START_REF] Buhusi | What makes us tick? Functional and neural mechanisms fo interval timing[END_REF]. In addition, if time perception is shown to develop from embodied sensorimotor origins in infancy into more abstract adult representations then the present theory could provide a framework to explain how adult temporal concepts appear embodied despite limited neural data in adults [START_REF] Kranjec | Are temporal concepts embodied? A challenge for cognitive neuroscience[END_REF].

The scalar property

The scalar property for interval-timing [START_REF] Gibbon | Scalar expectancy theory and Weber's law in animal timing[END_REF] is a widely replicated finding in both adult humans and rats (see [START_REF] Gibbon | Timing and time perception[END_REF] for a book length summary). It states that time-perception errors (E) increase as a scalar function of the length of the time interval (I) to be predicted. An interval twice as long (2I) produces errors that are twice the large (2E). This is surprising because the Central Limit Theorem predicts that error in cumulative processes grows more slowly -the expected error for an interval length 2I would be 2E . The mechanisms of the model that we propose in the present paper solves this problem, timeinterval prediction uncertainty in our model does, indeed, increase in a scalar fashion.

The need for calibration

Memory-trace models determine the passage of time in terms of how much a memory trace has decayed. But memory-trace decay cannot be directly decoded into temporal information, unless there is (or has been) some means of grounding that decay in meaningful, repeatable event sequences. The central hypothesis of this paper is our suggestion that infants ground memory-trace decay through self-generated, repetitive corporeal motions (such as arm movement). A simple example illustrates this idea.

Each time a baby moves its hand from a vertical position to a horizontal position (Figure 1), it takes approximately the same amount of time and this allows different amounts of memory-trace decay to be grounded in amounts of body or arm movement.

Imagine a baby playing with some toys. The baby picks up a colored block, waves it about a bit before bringing the object to its mouth for a closer inspection. The length of time between when the baby first noticed the toy and when it arrives at its mouth will be directly proportional to the amount of arm movement the baby has carried out. In another case, imagine that a baby is lying in a cot and hears a noise to his or her left. The baby may have to manouver its whole body in order to orient its head towards the sound. Again the amount of movement is proportional to the time since the original sound.

Moreover, similar movement sequences will take broadly similar lengths of time. So body and arm movement can serves as a rough temporal yardstick for visual and auditory memory-trace decay. In this way, over time an association between how long ago an event took place (as measured by activation decay of a memory trace) and limb movement is gradually learned. Infant body movement serves as a metronome for the timing of memory decay.

Visual and auditory events come in all shapes, sizes and intensities. The initial activation associated with an event can therefore vary considerably in amplitude. The fading memory trace for a high-intensity event will not correspond directly to the fading memory trace for a low-intensity event. That correspondence can be established via constantvelocity body movement. The "yardstick" of repeatable, predictable body movement in space is what we propose to align the time perception for the two events.

Visual and auditory are often, but not always, correlated Many visual events are not accompanied by a simultaneous production of sound and vice-versa. It is also important to note that the calibration of visual and auditory memorytrace decay rates does not have to happen simultaneously.

Architecture of the network

For this model we used a simple connectionist architecture (Figure 2) with a "visual" and an "auditory" pathway leading to the same set of "arm position nodes." Note that arm position here is chosen as an easily graspable instance of the more general notion of proprioceptive configuration. The input layer consisted of 41 units for each sensory modality which represented a memory as time evolving guassian. These inputs connected to two independent sets of 10 hidden nodes which both connected to the same 10 output nodes which encoded arm position as a binary vector. During training a particular arm position is associated with a particular time interval in one or the other modality and the network is trained using back-propogation of error. At test Figure 1. The two lines of graphs show hypothetical activation decay sequences (" fading gaussians" ) over visual columns (α=0.0045, β = 0.001, ϕ = 0.0105) and over auditory columns (α=0.0048, β = 0.001, ϕ = 0.0105) at 500 time-step intervals over 3000 time steps. Uncertainty (σ ) in both modalities grows linearly and leads to the scalar property of interval timing.

the gaussian corresponding to a particular time interval is presented in a single modality and the predicted arm position that is output serves as a proxy estimate of the amount of time since the original sensory event.

Arm position encoding

Here we use simplified topographic representations of limb position with binary encoding. Position 1 is given as [1 0 0 0… ] , position two as [1 1 0 0 … ] and so on. Although movement is continuous we do not code intermediate positions. Similarly, we are using a completely linear representation. This is an approximation. Further research with this model will attempt to direcly fit data from classic studies of infant timing abilities which used predictive reaching as diagnostic measure [START_REF] Hofsten | Predictive reaching for moving objects by human infants[END_REF] and infant electromyography (EMG) data collected as part of this research programme.

Memory decay using fading gaussians

We begin with a cluster of neuronal columns in visual cortex. The central column of the 41-column cluster is assumed to receive input from the sensory interface. The initial activation level of this column depends on the intensity of the input stimulus, which we assume to be a visual or auditory stimulus of short duration. Once the stimulus is no longer present, the peak activation value of the central column will decrease and activation will spread to neighboring columns. The activation of the i th column at time step t is designated by Ai(t), Activation at time t+1 is determined by the following equation:

A i (t+1)=α( A i-1 (t)+ A i+1 (t))+(1+β-ϕ) A i (t)
where α determines the amount of activation spreading between adjacent columns; β determines the level of self-excitation ϕ determines the amount of activation leakage.

The evolution of activation in this cluster of columns, which we refer to as "fading gaussians," is illustrated by the series of graphs at the top and bottom of Figure 1, indicating activation decay in the visual (top) and auditory (bottom) columns.

Results

We report three preliminary results from our simulations 1 . First, we show that the network does, indeed, gradually learn to associate the fading-gaussian input profiles with the various arm positions. The network gradually learns to associate the various stages of the fading visual or auditory input profiles with arm positions (Figure 3). Second, after training, we test it to determine the amount of error roduced for each time interval (Figure 4). 1 The MATLAB code for these simulations can be downloaded from http://www.cbcd.bbk.ac.uk/people/affiliated/caspar/time This testing is done by inputting to the network the activation gaussians corresponding to 500, 1000, 1500, etc time steps of decay (here denoted as intervals from 1 to 90 seconds). We note the average error produced from 20 seperate estimates at each time interval. Figure 4 shows that error increases approximately as a linear function of the time interval being measured, which is what Weber's Law for interval timing predicts [START_REF] Gibbon | Scalar expectancy theory and Weber's law in animal timing[END_REF]. Third, in a final simulation, we show how, via arm movement, the visual and auditory fading gaussians produce correlated outputs. In other words, whether a time interval is measured in the visual modality or in the auditory modality, the output (i.e., the estimate for the length of the time interval) will be the same.

Simulation 1: Development of interval timing

This first simulation looks at the performance of a network trained in one modality (Figure 3). A naive network is initialised with small random weights and is trained with a set of randomly presented gaussian activations. Each of these is associated with a particular amount of arm movement and the network is trained to predict these values. Figure 3 shows the average output of 20 networks during training across the full range of possible time intervals. The predicted arm position given by the network is translated to the corresponding time interval to plot the figure.

As can be seen the network learns to predict the intervals quite effectively in the middle of the range. It over estimates timing on short intervals and underestimates it on long intervals. In part this is due the lack of granularity of the binary encoding. However, this general pattern of responses has been found in children's estimates of time (Droit-Volet, 2003). This model could provide an explanation for this developmental effect. Simulation 2: The scalar property

The second simulation demonstrates how the scalar property of interval timing is a natural feature of this model. With a network trained to make predictons of arm location from memory decay, we a took a set of twenty responses for 50 different time points between 1 and 90 seconds. The average of these responses is plotted in Figure 4, together with error bars representing one standard deviation. The scalar property says that the size of the errors is proportional to the length of the interval. In other words error divided by interval should be a constant. We also plot this relative proportion in Figure 4. This proportion is constant which is what Weber's law predicts.

The bump appears to be due to the limited granularity of using a binary representation of arm position. It is important to note that the simplifying assumptions about bodily representaion (binary, linear) are likely to impair network performance. Using a non-binary representation would provide greater information. Whilst Fitts Law states that the motor system obeys a power law which suggests that motor representations also possess scalar properties. Similarly, in the current model, body position is used both to calibrate the sensory modalites and as a proxy representaion of time quantities. As an infant matures it is likely that its time representations will become more abstract, although it is beyond the scope of this paper to demonstrate this. In line with [START_REF] Walsh | A theory of magnitude: common cortical metrics of time, space and quantity[END_REF] ATOM model, we expect the brain to recruit regions of the cortex that represent quantity logarithmically. Future iterations of this model will investigate both these refinements.

Simulation 3: Calibration of independent modalities

The first two simulations looked at the performance of a network trained in one modality. Here we investigate how embodiment can work as a mechanism to synchronise and calibrate time interval estimates in different modalities. Droit-Volet (2003) showed that children's interval timing can differ across auditory and visual domains suggesting that there is some independence of these measures and that a development mechanism for calibration is required. In our model, each modality is trained separately but is tied to the same underlying bodily representation. To show how this leads to calibration and synchronisation of clocks in different modalities, we took representative networks trained in two modalities and looked at their outputs across the full range of possible intervals (1 to 90 seconds.) Correlating these outputs for the two independent networks showed how well calibrated and synchronised the networks were. In Figure 5, we plot the evolution of this correlation as the networks both gain greater experience.

Conclusion

We have presented a new model of interval timing in infants. It is the first developmental model of time perception and has two further novel features. It is based on memory uncertainty and it utilizes embodied learning to calibrate timing across different perceptual modalities. We have demonstrated that this model captures the scalar property of interval timing and certain developmental effects. One prediction of this model is that restricting a baby's movement would impair his or her time perception. This precise prediction is part of an ongoing research project involving babies.

Figure 2 :

 2 Figure 2: Schematic representation of network.

Figure 3 :

 3 Figure 3: Learning in a single modality network. Each line represents the output of partially trained networks. As learning progresses the lines tend to converge to the 'perfect prediction' diagonal. (Each line represents an averaged over 20 equivalently trained networks.)

Figure 4 :

 4 Figure 4: The scalar property in a trained network. The top line shows the average prediction at each interval and the standard deviations from 20 networks. The lower line shows this error as a relative proportion of the predicted interval.

Figure 5 :

 5 Figure 5: Correlation between network predictions in two independent modalities with training.

Acknowledgments

This research was funded by UK ESRC-ORA RES-360-25-0056 and French government ANR grant ORA-10-056.