Gaël Prado

Yacine Amarouchene

Hamid Kellay

Incompressible-compressible transition in falling granular jets

Keywords: 45, 70, -n -Granular systems

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Particles or grains flowing from hoppers and silos have attracted the attention of scientists since the seminal work of Hagen [1][2][3]. Contrary to molecular fluids, the flow rate is constant and does not depend on the height of the granular layer in sharp contrast with the Torricelli law describing fluids emptying from vessels. It is also known that the mass flow rate is mainly fixed by the ratio between the aperture at the exit D and the particles diameter d. Beverloo proposed an empirical expression (Q ∼ (Dkd) 5/2 with k ∼ 3-5 with a characteristic velocity scaling as √ gD; g being the gravity acceleration) that is widely used for large aspect ratios D/d. However, deviations are observed for smaller aspect ratios [START_REF] Andreotti | Les milieux granulaires[END_REF][START_REF] Beverloo | [END_REF][START_REF] Nedderman | Static and Kinematics of Granular Materials[END_REF]. It was anticipated that this deviation may be due to a lower density near the exit [START_REF] Van Zuilichem | [END_REF]8].

Recently, several studies focused on the properties of granular jets falling under the action gravity from hoppers or funnels, the main interest lying in describing the surface properties of these jets [9,[START_REF] Rowlinson | Molecular Theory of Capillarity[END_REF]. An analog of the surface tension has been characterized but the bulk properties remain poorly studied [START_REF] Schaflinger | [END_REF][12][13][14][15][16]. It is the purpose of the present paper to investigate the bulk properties of granular jets falling under gravity. These jets were observed to thin as they propagate downstream, just like liquid jets, with a roughly constant density suggesting incompressible-like behavior [14][15][16]. However, less attention has been paid to the understanding of the conditions that set the incompressible nature of granular jets. Through a systematic (a) E-mail: y.amarouchene@loma.u-bordeaux1.fr study of the effect of the particle size, ambient pressure, funnel diameter on the thinning properties of granular jets, a clear "Incompressible-Compressible" transition is observed in our experiments. The control parameter for this transition turns out to be the aspect ratio D/d. This transition occurs for low aspect ratios near 15 for which the departure from the Beverloo law occurs. The transition observed here can be rationalized using a continuum hydrodynamic model for granular media by taking into account the role of the boundaries.

In these experiments, we have used various funnels and granular materials. The funnels are all open topped and made of plastic or glass with an exit half-angle of the order of 2.5 degrees. In the following, the exit diameter of the funnel is noted D (varying from a few mm to 35 mm). The granular materials used are made of glass spheres (ρ B = 2500 kg • m -3 ) of mean diameter ranging from d ≈ 79.5 µm to d ≈ 1 mm (standard deviation δd ∼ 25 µm). Movies of the flowing granular material are made with a Phantom v640 camera with a typical acquisition rate of 4000 frames • s -1 and a typical spatial resolution of 20 µm • pixel -1 . Experiments conducted under vacuum are done using a 2 m long and 9 cm wide transparent PMMA cylinder sealed at its top and its bottom. Measurements of the jet diameter and the beads velocities are carried out using various image processing routines including PIV and PTV (Particle Imaging Velocimetry and Particle Tracking Velocimetry, respectively). Experimental details can be found in previously published papers [15,16]. Note that the granular jet thins for small grains (d < 337 µm or D/d > 15). However for larger grains the thinning does not occur (d > 512 µm or D/d < 10), and for even larger grains the jet spreads out and becomes less dense. If we keep the grain diameter d fixed, a similar behavior is observed while varying the funnel diameter D with thinning jets for large D and jets that spread out for small D. The jets for D/d < 10 are clearly less dense than for D/d > 10. Furthermore, they seem to become less dense as they propagate. The mean density or volume fraction φ 0 at the exit can be estimated by the simultaneous measurement of the mass flux Q and the mean velocity V 0 (which is close to the velocity of grains measured near the wall inside the funnels and is of the order of 10 cm/s) at that location. Using mass conservation, we have φ 0 = 4Q/(ρ B V 0 πD 2 ). The results obtained for φ 0 are shown in fig. 1(b) for different aspect ratios (D/d): for fixed D and variable d or fixed D and variable d. Our main observation is the decrease of the mean volume fraction below D/d ∼ 15. For higher aspect ratios (D/d > 20), a saturation at φ 0 ∼ 0.5 is reached. This is close to the value expected for random loose packings.

An analysis of the granular jets diameter vs. the distance from the exit z provides a means to characterize the transition (corresponding to pictures shown in fig. 1(a). The diameters H(z) are extracted using image analysis of the light transmitted through the jet. The radial intensity profile for each z is adjusted by a Gibbs law or a Hyperbolic tangent profile [15,16]. This analysis permits to measure the thickness or diameter of the jets H(z) vs. distance z as displayed in fig. 2(a) for a funnel diameter D = 5 mm. Note that the thinning is less pronounced as d increases. For the largest beads (D/d = 5; d = 1 mm), we observe expanding jets. These jets expand linearly vs. the distance z with a mean angle of the order of 5 • . For an intermediate range of aspect ratios 10 < D/d < 20, the jets thin at first, but beyond a characteristic distance z d the jet spreads out. z d decreases as the aspect ratio D/d decreases. For D/d < 15, z d is close to zero.

When a liquid (density ρ and surface tension σ) emerges from the exit of a funnel or from the opening at the bottom of a reservoir, it is accelerated downward by gravity: the velocity grows with z as V (z) = 2gz + V 0 2 . While the jet is stretched by gravity, it simultaneously thins. In the incompressible limit, mass conservation (V (z)H 2 (z) ∼ constant) implies that the average diameter H of the jet decays with z as

H(z) = H(0)(1 + 2 F r z H(0) + 1 W e (1 -H(0) H )) -1/4 with F r = V0 2 gH(0) and W e = ρ V0 2 H(0)
2σ being, respectively, the Froude number (ratio inertia/gravity) and the Weber number (ratio inertia/capillarity) [START_REF] Middleman | Modeling Axisymmetric Flows: Dynamics of Films, Jets, and Drops[END_REF]. In the limit W e 1 the dimater of the jet is well approximated by:

H(z) = H(0)(1 + 2 F r z H(0) ) -1/4
and is independent of both σ and ρ. For instance, this condition is fulfilled for water jets [START_REF] Middleman | Modeling Axisymmetric Flows: Dynamics of Films, Jets, and Drops[END_REF]. Therefore, it is difficult to measure a surface tension from the thinning of jets in the limit W e 1 even for a relatively high surface tension liquid like water (σ = 72 mN/m).

The granular jet diameters H(z) for D/d > 15 are well approximated by this law using an exponent of -1/4 and F r as a free parameter. This indicates that the granular jet thinning is mainly affected by inertia and that capillary forces are weak as has been found before [15,16] but most importantly this highlights that the packing density of the jet remains roughly constant during the free fall. As a consequence, for aspect ratios D/d > 15, once rescaled by F r and H(0), all the mean profiles of the jets collapse on a master curve as shown in fig. 2(b). Here, the Froude number F r contains only the mean velocity as a free parameter. The values for V 0 extracted from the Froude fitting of the normalized jet evolution along with the values measured at the surface of jet using either PIV or PTV are shown in fig. 2(c). The measurements using PIV or PTV are similar and follow the same trend as the values obtained from Froude fitting albeit smaller by a factor 2/3. This suggest the existence of a non flat velocity profile. Note that for all the data presented, H(0) is simply related to D by the linear relation H(0) = D -1.5d while for liquids H(0) = D. This highlights the discrete nature of granular fluids.

For aspect ratios D/d < 15, the thinning is less pronounced and the incompressibility condition previously described no longer holds. An alternative to describe them is to introduce a functional form

H(z) = H(0)(1 + 2 F r z H(0) ) -α
with a compressibility index α. The volume fraction evolution is then simply given by φ(z)

= φ 0 (1 + 2 F r z H(0) ) 2α-1
2 . For α = 1/4, we recover the incompressible liquid-like limit (φ = const), while for α < 1/4 the jets become less dense as they propagate downstream. The values of F r and α for all the jets corresponding to fig. 2(a) are presented in fig. 2(d). Indeed, the value of α is constant and close to 1/4 for D/d > 15 and monotonically decreases towards zero as the aspect ratio D/d decreases. The Froude number decrease as D/d increases in the incompressible regime. This indicates that the average exit velocity decreases with D/d for D/d 15 as shown in fig. 2(c). The mean velocity displays the same trend as the velocity measured at the surface using either PIV or PTV suggesting the existence of velocity profile inside the funnel. This is to be contrasted with the results obtained for granular flows through apertures in silos where the exit velocity is of the order of (gD) 1/2 for a fixed grain size (see [8]).

The compressibility index α is shown in fig. 3 for various experimental conditions (various D and d). This exponent depends only on the aspect ratio D/d. The surrounding medium (air) seems to have no influence on this diagram since the same results are observed under vacuum (13 mbar) both in the compressible and incompressible granular jet regimes. For D/d > 15, other measurements made at a much lower pressure (0.25 mbar) farther downstream corroborate this observation for the incompressible limit. Indeed the jets continue to thin as in the presence of air even at 0.25 mbar. This may suggest either a weak influence of the surface tension previously reported [15,16] in the limit W e 1 and D/d > 15 or that the incompressible nature of a granular jet is set by other mechanisms. For instance, in a recent numerical work, it was suggested that the collimating properties of granular jets arise from dissipative inelastic collisions [START_REF] Waitukaitis | [END_REF].

To gain insight into how and why the transition at D/d ∼ 15 occurs, let us briefly summarize our reasoning to rationalize the experimental results presented here. It is known that grains close to solid boundaries in granular flows are most of the time hotter in terms of granular temperature (see GDR MiDi experiments in [START_REF] Andreotti | Les milieux granulaires[END_REF]). Furthermore, this temperature drops away from the boundaries. Thus, it is natural to expect a heat flux directed from the walls where the particles are hotter to the center of the funnel where the temperature is smaller. The grains are colder at the center of the channel because this heat flux must be balanced by dissipation. The dissipation is due to inelasticity of the collisions either between the grains or between the grains and the walls. The dissipation may also have a frictional origin. One may, according to this scenario, anticipate a denser and colder region at the center of the funnel and a less dense and hot region near the walls. If the thickness of this region near the boundary becomes of the order of the funnel diameter, one may then expect less dense jets at the exit. In order to quantify this scenario, we model the granular medium inside the funnel using a continuum approach without mean flow for simplicity. We will assume that the behavior of the jet right after the exit is dictated by the properties of the granular medium inside the funnel. Figure 4 displays a schematic of the system of interest.

Continuum modeling of granular media relies on modified versions of the Navier-Stokes equations derived from the kinetic theory of colliding inelastic hard spheres. One of the simplest sets of equations that still capture the effect of the inelastic collisions has been introduced a few years ago by Grossman, Zhou and Ben-Naim to describe the steady-state properties of granular gases in two dimensions for a wide range of densities below the close-packing density ρ c = 2 √ 3d 2 [19]. Here, in the steady [20]). Following Janssen, the pressure P is kept constant and given by P = ρ gR 2µK (with Janssen coefficient K and friction coefficient µ) [START_REF] Andreotti | Les milieux granulaires[END_REF]. The order of magnitude of the dynamic Janssen pressure used for our simulation (µK = 1) is 2-3 times higher than the value usually assumed for the static case as reported in [21]. Two boundary conditions are needed to solve numerically the problem at hand. We choose to impose a zerotemperature gradient dT dx x=0 = 0 at the center of the funnel and a fixed temperature T at the boundary equal to the value we have measured experimentally as we will detail below.

The temperature near the inner wall of the funnel above the outlet is achieved by measuring the velocity of the grains using transparent glass funnels. Figure 5(a) and fig. 5(b) show the probability distribution functions of the transverse and longitudinal velocity at the wall rescaled by the standard deviation for different grain sizes and a funnel diameter D = 5.3 mm. All the data collapse for both components of the surface velocity. An almost Gaussian behavior for the most probable events and algebraic for large velocities is observed in agreement with previous reported results [22]. The standard deviation for both components of the surface velocity δU 2 1/2 and δV 2 1/2 used to collapse all the probability distribution functions shown in figs. 5(a) and (b) are displayed in fig. 5(c). The granular temperature T defined using T 1/2 ≡( δU 2 + δV 2 ) 1/2 is also displayed in fig. 5(c). The order of magnitude of T 1/2 is a few cm/s and increases monotonically vs. the grain size d. The increase is close to the (gd) 1/2 scaling expected from dimensional arguments for the standard deviation of the velocity. A deviation is however observed for large d (D/d < 15). The measured temperature at the wall (necessary to impose boundary conditions in the numerical simulation) is well approximated for all grain diameters by

T = T ∞ • (1 -exp(d/d c ))
with T ∞ = 0.0038 and d c = 0.592 mm as fitting parameters. For all the data presented here, an anisotropy of the order of 20% (defined as ( δV 2 1/2 -δU 2 1/2 )

δV 2 1/2
) is present independently of the bead size d as shown in the inset of fig. 5(c).

These measurements allow us to fix the value of the temperature at the wall. The numerical resolution using this fixed temperature T measured at the wall vs. the grain diameter d is shown in fig. 6. The density profiles vs. x, rescaled by the close-packing volume fraction, are shown in fig. 6(a). For all the results presented, the volume fraction at the wall is always smaller than the value at the center of the domain of calculation. A clear decrease of the volume fraction at the center of the funnel is observed for the lowest aspect ratios D/d, while high densities are observed for the high aspect ratios D/d. The temperature profiles displayed in fig. 6(b) show a pronounced exponential drop far from the boundaries in agreement with previously reported results in the literature [19]. The cooling is more significant for the highest aspect ratios. The characteristic length ξ of the exponential decrease of the temperature (vs. the distance to the heated wall) rescaled either by the grain diameter d or diameter D is shown in fig. 6(c). For large aspect ratios D/d, ξ is close to 5d as observed in several previous investigations [19]. For values of D/d ∼ 15, a sharp increase of ξ is observed. ξ may reach more than half the diameter D. Numerical simulations using the 2D axisymmetric equations given in [23] give similar results. The results of these numerical simulations can be compared to the experimental data for the mean volume fraction. The results are shown in fig. 7 as shown by Garzó and Dufty [24]. The decrease of the volume fraction observed close to D/d ∼ 15 corroborates previous attempts to explain deviations from the empirical law of Beverloo as suggested in the literature [START_REF] Van Zuilichem | [END_REF]8]. Furthermore, as displayed in fig. 7(a), there is a good agreement with the numerical simulation if we focus on the volume fraction φ min measured experimentally near the inner wall of the funnel. To estimate φ min experimentally, we counted the number of grains N wall near the wall and then divided N wall by the volume of a truncated wedge of thickness d corresponding to the measurement zone. This provides another test of the various hypotheses made for the temperature and the pressure fields in the model. We also attempted to tune the roughness at the wall boundaries in the funnel by randomly gluing grains of diameter d ∼ 337 µm to the inner walls of the funnel. In this case, lower densities (than for the smooth wall funnel) are observed for a fixed aspect ratio close to the transition as displayed in fig. 7(b). This observation highlights the crucial role of the roughness at the walls: the temperature at the wall is expected to be higher for rough walls but remains difficult to measure.

Several questions remain. Our main assumption is that the temperature of the grains near the wall is fixed. But what sets the temperature near the wall? This is a difficult problem and the proper understanding of the appropriate boundary conditions remains a major challenge for continuum hydrodynamic models of granular media. One needs to solve the complete set of coupled conservation equations (mass, energy temperature and momentum [25]). Even in that case, a closure relation for the pressure has to be specified. This relation may go beyond the simple one proposed here. Our simple model also neglects any rearrangements of the grains within the bulk of the medium in the presence of flow.

Rearrangements of the grains especially for small aspect ratios when the granular medium is flowing may also induce variations in density. In short, a full description of the transition observed here requires further theoretical work.

In conclusion, we have described a Compressible-Incompressible transition for flowing granular jets that depends solely on the aspect ratio D/d at the exit of the vessel. Surprisingly, the incompressible liquid-like behavior observed in the limit of high aspect ratios seems to find its origin in the inelasticity of collisions independently of the ambient fluid pressure or the grain sizes. A simple model using continuum hydrodynamic equations to describe the granular medium seems to capture the essential features of this transition.

Fig. 1 :

 1 Fig. 1: (Colour on-line) (a) Snapshots of granular jets near the exit of a funnel (diameter D = 5 mm) for different grain sizes d (from left to right 1000 µm ; 512 µm; 337 µm; 240 µm; 159 µm and 112 µm). Note that the granular jet thins for small grains, d < 337 µm. For larger grains, d ∼ 337 µm, the thinning does not occur. For even larger grains, d = 1000 µm, the jet spreads out and becomes less dense. (b) Mean volume fraction φ0 estimated by the simultaneous measurement of the mass flux Q and the mean velocity V0 at the exit of the funnel for different grain sizes d vs. the aspect ratio D/d. Note the decrease of φ0 for D/d below 15 while for higher aspect ratios (D/d > 20), a saturation at φ0 ∼ 0.5 is observed.

Figure 1 (

 1 Figure 1(a) shows snapshots of granular jets falling under the action of gravity for different grain sizes d while the funnel diameter D is kept constant (D = 5.3 mm).Note that the granular jet thins for small grains (d < 337 µm or D/d > 15). However for larger grains the thinning does not occur (d > 512 µm or D/d < 10), and for even larger grains the jet spreads out and becomes less dense. If we keep the grain diameter d fixed, a similar behavior is observed while varying the funnel diameter D with thinning jets for large D and jets that spread out for small D. The jets for D/d < 10 are clearly less dense than for D/d > 10. Furthermore, they seem to become less dense as they propagate. The mean density or volume fraction φ 0 at the exit can be estimated by the simultaneous measurement of the mass flux Q and the mean velocity V 0 (which is close to the velocity of grains measured near the wall inside the funnels and is of the order of 10 cm/s) at that location. Using mass conservation, we have φ 0 = 4Q/(ρ B V 0 πD 2 ). The results obtained for φ 0 are shown in fig.1(b) for different aspect ratios (D/d): for fixed D and variable d or fixed D and variable d. Our main observation is the decrease of the mean volume fraction below D/d ∼ 15. For higher aspect ratios (D/d > 20), a saturation at φ 0 ∼ 0.5 is reached. This is close to the value expected for random loose packings.An analysis of the granular jets diameter vs. the distance from the exit z provides a means to characterize the

Fig. 2 :

 2 Fig. 2: (Colour on-line) (a) Granular jet diameter vs. the axial distance z for D = 5 mm and various grain sizes d. (b) Granular jet diameter vs. the rescaled axial distance (F r/2) -1 z/H(0) for D = 5 mm and various grain sizes d. (c) V0 extracted from the thinning jet diameter ( V0 ) and from PIV and PTV measurements at the wall vs. aspect ratio D/d for D = 5 mm and various grain sizes d. (d) Froude number F r and compressibility index α vs. aspect ratio D/d for the data in (b).

Fig. 3 :

 3 Fig. 3: (Colour on-line) Compressibility index α vs. the aspect ratio D/d for various experimental conditions.

Fig. 4 :

 4 Fig. 4: (Colour on-line) Schematic of the system of interest for the numerical simulation. The length of the system is R = D/2. The heated wall is at x = D/2.

Fig. 5 :

 5 Fig. 5: (Colour on-line) Probability distribution functions of the velocity of the grains near the wall for various grain sizes and a funnel diameter D = 5 mm at a distance z = -20 mm above the outlet. (a) Rescaled transverse component U . (b) Rescaled longitudinal component V . The thick dotted line corresponds to a Gaussian fit while the thin dotted line corresponds to a power law. (c) Standard deviation of U , V and temperature T vs. d for D = 5 mm; inset: anisotropy of the velocity fluctuations vs. d under the same conditions.

Fig. 6 :

 6 Fig. 6: Simulation results for different aspect ratios D/d: (a) volume fraction profiles; (b) temperature profiles; (c) characteristic length of the exponential decrease of the temperature away from the wall (rescaled by the grain size d and the domain size of the computation D) vs. the aspect ratio D/d.

  (a) for different aspect ratios (D/d) while keeping fixed D or d. The results of the numerical simulation are superimposed to the experimental data by setting an asymptotic volume fraction ∼0.54 as observed experimentally for large D/d. The agreement is good considering the severe simplifications used in the model (2D vs. 3D and zero mean flow).In terms of density, the Compressible-Incompressible transition occurs for 0.3 < φ 0 < 0.4. This may correspond to the transition from a low-density kinetic regime to a high-density collisional regime that occurs for φ ∼ 0.2-0.3

Fig. 7 :

 7 Fig. 7: (Colour on-line) (a) Mean volume fraction φ0 at the exit of the funnel for different grain sizes d vs. the aspect ratio D/d. Black circles: results from the numerical simulation with the temperature at the wall fixed by the experimentally measured value. White crosses show the volume fraction φmin at the wall vs. D/d for D = 5 mm. White circles show φmin from the simulation. (b) For D = 13 mm, snapshots of granular jets without and with wall roughness (337 µm); first pair of images d = 337 µm; second pair of images d = 1000 µm.

We would like to thank J.-F. Boudet for discussions. This research is supported by CR Aquitaine Grants No. 2006111101035, No. 20091101004 and by ANR Grant No. ANR-09-JCJC-0092.


Jérémy Fix

Julien Vitay

Nicolas P Rougier

A distributed computational model of spatial memory anticipation during a visual search task

Some visual search tasks require the memorization of the location of stimuli that have been previously focused. Considerations about the eye movements raise the question of how we are able to maintain a coherent memory, despite the frequent drastic changes in the perception. In this article, we present a computational model that is able to anticipate the consequences of eye movements on visual perception in order to update a spatial working memory.

Introduction

In the most general framework of behavior, the notion of anticipation is intimately linked with the possibility to predict the consequences and the outcomes of a given action. If we consider that any action is goal-motivated, then an action is carried out in the first place because it is anticipated that this action will lead to a situation in which the goal can be reached more directly. In this framework, anticipation can be viewed as a prediction of the future and is tightly linked to the notion of goal-directed behavior. However, there also exists more structural reasons why anticipation is necessary.

For example, when dealing with both accurate and very fast movements like catching a ball or scanning a visual scene, brain representations should be updated very quickly (even in advance in some cases) in accordance with the task that is carried out. The problem in this context is that the time scale required for carrying out such tasks may be dramatically smaller than the time scale of a single neuron. Moreover, those neurons are also in interaction with other neurons in the network and the resulting dynamic may be even slower. One solution to cope with this problem is to use a forward predictive model that is able to anticipate the consequences and outcomes of a motor action. The resulting dynamic at the level of the model is then faster than the dynamic of its components.

Let us consider the ability to anticipate changes in the visual information resulting from an eye saccade. This anticipation is known to be largely based on unconscious mechanisms that provide us with a feeling of stability while the whole retina is submerged by different information at each saccade; producing a saccade results in a complete change in the visual perception of the outer world.

If a system is unable to anticipate its own saccadic movements, it cannot pretend to obtain a coherent view of the world, because each image would be totally uncorrelated from the others. One stimulus being at one retinal location before a saccade could not be easily identified as being the same stimulus at another retinal location after the saccade. Consequently, the saccadic eye movements should be anticipated in order to keep the coherence of the scene and to be able to track down interesting targets. A number of works have already addressed the specific problem of visual search of a target among a set of distractors. However, most of the resulting models do not deal with the problem of saccadic eye movements that produce drastic changes in the available visual information.

Using neural fields introduced by Amari [START_REF] Amari | Dynamical study of formation of cortical maps[END_REF] for the one dimensional case and later extended to higher dimensions by Taylor [START_REF] Taylor | Neural bubble dynamics in two dimensions[END_REF], we would like to address in this paper the specific problem of anticipation during visual search using a purely distributed and numerical neural substrate. After briefly reviewing literature related to visual search in the first section, we introduce a very simple visual experiment that helps to illustrate the underlying mechanisms of the model that is detailed in that same section.

Visual search

Visual search is a cognitive task that most generally involves an active scan of a visual scene to find one or several given targets among distractors. It is deeply anchored in most animal behaviors, from a predator looking for a prey in the environment, to the prey looking for a safe place to avoid being seen by the predator. Psychological experiments may be less ecological and may propose, for example, to find a given letter among an array of other letters, measuring the efficiency of the visual search in terms of reaction time (the average time to find the target given the experimental paradigm). In the early eighties, [START_REF] Treisman | A feature-integration theory of attention[END_REF] suggested that the brain actually extracts some basic features from the visual field in order to perform the search. Among these basic features, which have been recently reviewed by [START_REF] Wolfe | Visual search[END_REF], one can find features such as color, shape, motion, or curvature. Finding a target is then equivalent to finding the conjunction of features, which may be unique, that best describes the target. In this sense, [START_REF] Treisman | A feature-integration theory of attention[END_REF] distinguished two main paradigms (a more tempered point of view can be found in [START_REF] Duncan | Visual search and stimulus similarity[END_REF]).

Feature search refers to a search where the target differs from all distractors by exactly one feature. Conjunction search refers to a search where the target differs from distractors by at least one of two or more features.

What characterizes feature search best is a constant search time that does not depend on the number of distractors. The target is sufficiently different from the distractors to pop out. However, in the case of conjunction search, the mean time needed to find the target is roughly proportional to the number of distractors that share at least one feature with the target (cf. Figure 1). These observations lead to the question of how a visual stimulus could be represented in the brain. The explanation given by Treisman and Gelade [START_REF] Treisman | A feature-integration theory of attention[END_REF], the so-called Feature-Integration Theory, proposes that elementary features are processed in separated feature maps. Competition inside one map would lead to feature search, based on the idea that the item differing the most from its background would win the competition and be represented. For targets differing from distractors by more than two features, there cannot be any global competition. This would mean that finding the target requires successively scanning every potential candidate until the correct target is found. This explains the dependence of the search time on the number of similar distractors in conjunction search tasks.

The main prediction of this theory is that processing visual inputs is not a global feed-forward processing, but more an iterative and sequential process on sensory representations. We describe below the strategies used by the brain to achieve this sequential search, by putting emphasis on saccadic eye movements and visual attention. The scope of this article is therefore to model the cognitive structures involved in the sequential processing of visual objects, and not the visual processing of the features alone.

Fig. 1. Feature search can be performed very quickly as illustrated on the left part of the figure; the disc shape pops out from the scene. However, as illustrated in the right figure, if the stimuli share at least two features, the pop out effect is suppressed. Hence, finding the disc shape with the stripes going from up-left to down-right requires an active scan of the visual scene.

Saccadic eye movements

The eye movements may have different behavioral goals, leading to five different categories of movements: saccades, vestibulo-ocular reflex, optokinetic reflex, smooth-pursuit and vergence. However, in this article we will only focus on saccades (for a detailed study of eye movements, see [START_REF] Leigh | The Neurology of Eye Movements[END_REF], [START_REF] Carpenter | Movements of the Eyes[END_REF]).

Saccades are fast and frequent eye movements that move the eye from the current point of gaze to a new location in order to center a visual stimulus on the fovea, a small area on the retina where the resolution is at its highest. The velocity of the eyes depends on the amplitude of the movement and can reach up to 700 degrees per second at a frequency of 3 Hz. The question we would like to address is how the brain may give the illusion of a stable visual space while the visual perception is drastically modified every 300 ms.

While the debate to decide whether or not the brain is blind during a saccade has not been settled (see [START_REF] Li | Saccadic suppression of displacement : separate influences of saccadic size and of target retinal eccentricity[END_REF][START_REF] Burr | Eye movements, keeping vision stable[END_REF][START_REF] Kleiser | Neural correlates of saccadic suppression in humans[END_REF][START_REF] Ross | Changes in visual perception at the time of saccades[END_REF] for the notion of saccadic suppression and [START_REF] O'regan | A sensorimotor account of vision and visual consciouness[END_REF] for a discussion about the necessity of a saccadic suppression mechanism), the coherence between the perception before and after a saccade cannot be established accurately solely based on perception. One solution to consider is that the brain may use an efferent copy of the voluntary eye movement to remap the representation it has built of the visual world. Several studies shed light on presaccadic activities in areas such as V4 and LIP where the locations of relevant stimuli are supposed to be represented. In [START_REF] Moore | Visual representations during saccadic eye movements[END_REF], the authors suggest that "the pre-saccadic enhancement exhibited by V4 neurons [...] provides a mechanism by which a clear perception of the saccade goal can be maintained during the execution of the saccade, perhaps for the purpose of establishing continuity across eye movements." In [START_REF] Merriam | Active vision in parietal and extrastriate cortex[END_REF], the authors review evidence that LIP neurons, whose receptive field will land on a previously stimulated screen location after a saccade, are excited even if the stimulus disappears during the saccade. In a recent study, Sommer and Wurtz [START_REF] Sommer | Influence of the thalamus on spatial visual processing in frontal cortex[END_REF] showed neurons in FEF that receive projections from the superior colliculus that could explain the origin of a corollary discharge signal responsible for the pre-saccadic activity exhibited by these neurons.

Visual attention

Focusing on a given stimulus of the visual scene is a particular aspect of the more general concept of attention that has been defined as the capacity to concentrate cognitive resources on a restricted subset of sensory information ( [START_REF] James | The principles of psychology[END_REF]). In this context of visual attention, only a small subset of the retinal information is available at any given time to elaborate motor plans or cognitive reasoning (cf. change blindness experiments presented in [START_REF] O'regan | A sensorimotor account of vision and visual consciouness[END_REF], [START_REF] Simons | Current approaches to change blindness[END_REF]). A visual scene is not processed as a whole but rather processed by successively focusing on interesting parts of it, possibly involving eye movements, but this is not necessary. The selection of a target for an eye movement is then closely related to the notion of spatial attention ( [START_REF] Moore | Control of eye movements and spatial attention[END_REF]) that is classically divided into two types: overt attention, which involves a saccade to center a stimulus on the fovea, and covert attention, in which no eye movement is triggered. These two types of spatial attention were first supposed to be independent ( [START_REF] Posner | The attentional system of the human brain[END_REF]) but recent studies such as the premotor theory of attention proposed in [START_REF] Rizzolatti | Reorienting attention accross the horizontal and vertical meridians[END_REF] (see also [START_REF] Chelazzi | A neural basis for visual search in inferior temporal cortex[END_REF], [START_REF] Kowler | The role of attention in the programming of saccade[END_REF], [START_REF] Craighero | Action for perception : a motor-visual attentional effect[END_REF]) consider that covert and overt attention rely on the same neural structures but the movement is inhibited in covert attention. A more general discussion about the covert and overt stages of action can be found in [START_REF] Jeannerod | Neural simulation of action : A unifying mechanism for motor cognition[END_REF].

The deployment of attention on a specific part of the visual information can be the consequence of two phenomena. Firstly it can rely on the saliency of a stimulus, compared to its surrounding (for example a sudden strong flash light); this is known as bottom-up attention. Secondly, it can also depend on the task in which the subject is involved, which may need to enhance some parts of the perception (for example, imagine that you have to find an orange among apples and bananas, the color information could be a good criteria to find the target rapidly).

In [START_REF] Moran | Selective attention gates visual processing in the extrastriate cortex[END_REF], the authors shed light on the neural correlates of attention on the response of neurons in the visual and temporal cortices. If we consider a specific neuron tuned to a given orientation in its receptive field, one can distinguish several cases:

the response of the neuron is high when an oriented bar with the preferred orientation (called good stimulus) is presented in its receptive field the response of the neuron is low when an oriented bar with an orientation different from the preferred one (called bad stimulus) is presented in its receptive field the response is between the two preceding ones when both a good and bad stimulus are presented When a monkey is involved in a task that requires to select one of the two stimuli, for example the good one, the response of the neuron is enhanced. The study of this suppressive interaction phenomena was extended by further authors ( [START_REF] Luck | Neural mechanisms of spatial attention in areas v1, v2 and v4 of macaque visual cortex[END_REF], [START_REF] Reynolds | The role of neural mechanisms of attention in solving binding problem[END_REF], [START_REF] Treue | Attentional modulation of visual motion processing in cortical areas mt and mst[END_REF]).

As we will see in section 3.2, we do not deal with how the salience of the visual stimuli is computed, whether or not it is a bottom-up or top-down processing. The main points are that for each location in the visual space, we are able to compute its behavioral relevance, and that considering eye movements necessarily implies dealing with overt attention.

Computational models

Over the past few years, several attempts at modeling visual attention have been engaged ( [START_REF] Koch | Shifts in selective visual attention : Towards the underlying neural circuitry[END_REF], [START_REF] Tsotsos | Modeling visual attention via selective tuning[END_REF], [START_REF] Wolfe | Visual attention[END_REF], [START_REF] Itti | Computational modeling of visual attention[END_REF], [START_REF] Hamker | A dynamic model of how feature cues guide spatial attention[END_REF]). The basic idea behind most of these models is to find a way to select interesting locations in the visual space given their behavioral relevance and whether or not they have already been focused. The two central notions in this context have been proposed by [START_REF] Koch | Shifts in selective visual attention : Towards the underlying neural circuitry[END_REF] and [START_REF] Posner | Components of visual orienting[END_REF]:

saliency map inhibition of return (IOR).

The saliency map is a single spatial map, in retinotopic coordinates, where all the available visual information converge in order to obtain a unified representation of stimuli, according to their behavioral relevances. A winner-take-all algorithm can be easily used to find which stimulus is the most salient within the visual scene, and thus identify its location as the locus of attention. However, in order to be able to go to the next stimuli, it is important to bias the winner-take-all algorithm in such a way that it prevents going backward to an already focused stimulus. The goal of the inhibition of return mechanism is precisely to feed the saliency map with such a bias. The idea is to have another neural map that records focused stimuli and inhibits the corresponding locations in the saliency map. Since an already focused stimulus is actively inhibited by this map, it cannot pretend to win the winner-take-all competition, even if it is the most salient.

The existence of a single saliency map is still not proved. In [START_REF] Hamker | A dynamic model of how feature cues guide spatial attention[END_REF] the author proposes a more distributed representation of these relevances, making a clear anatomical distinction between the processing of the visual attributes of an object and its spatial position (according to the What and Where pathways hypothesized by [START_REF] Ungerleider | Two cortical visual systems[END_REF], see also [START_REF] Goodale | Seperate visual pathways for perception and action[END_REF]). In this model, spatial competition occurs in a motor map instead of a perceptive one. It exhibits good performances regarding visual search task in natural scene, but is restricted to covert attention. In most of the previously proposed models, the authors do not take into account eye movements and the visual scene is supposed to remain stable: scanning is done without any saccade. During the rest of this article, we will keep the saliency map hypothesis, even if controversial, in order to illustrate the anticipatory mechanism.

A model of visual search with overt attention

The goal of our model is to show the basic mechanisms necessary to achieve sequential search in a visual scene using both overt and covert attention. Using a saliency map, we need to compute the location of the most interesting stimulus that will be processed to achieve recognition. This focus of attention on a stimulus has to be displaced in two situations. First, in covert attention this focus has to be dynamically inhibited to represent another stimulus. There is therefore a need for an inhibition-of-return mechanism than can inhibit the current focus of attention. Moreover, we have to memorize the locations of previously attended stimuli, by the means of a dynamic spatial working memory.

The second situation to consider is when eye movements can center the stimulus that is being attended to. The spatial working memory has to be updated by the eye movement so that its state corresponds to the post-saccadic locations of memorized stimuli. This is where an anticipatory mechanism is mandatory.

To describe these mechanisms, we first present an experimental setup for which previous computational models would fail to achieve efficient sequential search. We then present the architecture of our model and report simulated results.

Experiment

In order to accurately evaluate the model, we setup a simple experimental framework in which some identical stimuli are drawn on a blackboard and are observed by a camera. The task is to successively focus (i.e. center) each one of the stimuli without focusing twice on any of them. We estimate the performance of the model in terms of how many times a stimulus has been focused. Hence, the point is not to analyze the strategy of deciding which stimulus has to be focused next (see [START_REF] Findlay | Eye scanning of multi-element displays: I. scanpath planning[END_REF][START_REF] Findlay | Eye scanning of multi-element displays: Ii. saccade planning[END_REF] for details on this matter). In the context of the proposed model, the strategy is simply to go from the most salient stimulus to the least salient one, and to randomly pick one stimulus if the remaining ones are equally salient.

Figure 2 illustrates an experiment composed of four identical stimuli where the visual scan path has been materialized. The effect of making a saccade from one stimulus to another is shown and underlines the difficulty (for a computational model) of identifying a stimulus before and after a saccade. Each one of the stimuli being identical to the others, it is impossible to perform an identification based solely on features. The only criteria that can be used is the spatial location of the stimuli. Fig. 2. When scanning a visual scene, going for example from stimulus 1 to stimulus 4, as illustrated in the left figure, the image received on the retina is radically changed when each stimulus is centered on the retina, as illustrated in the right figures. The difficulty in this situation is to be able to remember which stimuli have already been centered in order to center another one. The figures on the stimuli are shown only for explanation purpose and do not appear on the screen; all the stimuli are identical.

Model

The model is based on three distinct mechanisms (cf. Figure 3 for a schematic view of the model). The first one is a competition mechanism that involves potential targets represented in a saliency map that were previously computed according to visual input. Second, to be able to focus only once on each stimulus, the locations of the scanned targets are stored in a memory map using retinotopic coordinates. Finally, since we are considering overt attention, the model is required to produce a camera movement, centering the target on the fovea, used to update the working memory. This third mechanism works in conjunction with two inputs: current memory and parameters of the next saccade. This allows the model to compute quite accurately a prediction of the future state of the visual space, restricted to the targets that have already been memorized. Fig. 3. Schematic view of the architecture of the model. The image captured by the camera is filtered and represented in the saliency map. This information feeds two pathways: one to the memory and one to the focus map. A competition in the focus map leads to the most salient location that is the target for the next saccade. The anticipation circuit predicts the future state of the memory with its current content and the programmed saccade.

The model is based on the computational paradigm of two dimensional discrete neural fields (the mathematical basis of this paradigm can be found in [START_REF] Amari | Dynamical study of formation of cortical maps[END_REF] for the one dimensional case, extended to a two dimensional study in [START_REF] Taylor | Neural bubble dynamics in two dimensions[END_REF]). The model consists of five n×n maps of units, characterized by their position, denoted x ∈ [1..n] 2 and their activity as a function of their position and time, denoted u(x,t). The basic dynamical equation that follows the activity of a unit at position x, depends on its input I(x, t). Equation ( 1) is the equation proposed in [START_REF] Amari | Dynamical study of formation of cortical maps[END_REF], discretized in space.

τ. ∂u(x, t) ∂t = -u(x, t) + baseline + 1 α I(x, t) (1)
We distinguish two kinds of units. The first are sigma units that compute their input as a weighted sum of the activity of afferent neurons, where afferent neurons are defined as neurons in other maps. We also consider lateral connections that involve units in the same map. If we denote w af f the weighting function for the afferent connections and w lat the weighting function for the lateral connections, the input I(x, t) of a unit x at time t can be written:

I(x, t) = af f w af f u af f (t) + lat w lat u lat (t), (2)
where equations 3 and 4 define the lateral and afferent weighting functions as a Gaussian and difference of Gaussians, respectively.

w af f (x, y) = A.e x-y 2 a 2 with A, a ∈ IR * + , x, y ∈ [1..n] 2 (3)
w lat (x, y) = B.e x-y 2 b 2 -C.e x-y 2 c 2 with B, C, b, c ∈ IR * + , x, y ∈ [1..n] 2 (4)
The second kind of units we consider are sigma-pi units ( [START_REF] Rumelhart | A general framework for parallel distributed processing[END_REF]), which compute their input as a sum of the product of the activity of afferent neurons. We also consider the lateral connection term so that the input of a unit x at time t can be written:

I(x, t) = i∈I w af fi j∈Ei u af fj (t) + lat w lat u lat (t). ( 5
)
All the parameters of the previous equations used in the simulation are summarized in the appendix. We now describe how the different maps interact. Since the scope of this article is the anticipation mechanism, the description of the saliency map, the focus map and the working memory will not be accurate; a more detailed explanation, with the appropriate dynamical equations, can be found in [START_REF] Vitay | Using neural dynamics to switch attention[END_REF].

Saliency map

The saliency map, also referred to as input in the following, is computed by convolving the image captured with the camera of a robot used for the simulation with Gaussian filters. The stimuli we use are easily discriminable from the background on the basis of the color information. This computation leads to a representation of the visual stimuli with Gaussian patterns of activity in a single saliency map. We do not deal with how this saliency map is computed, whether or not it is due to bottom-up or top-down attention. We only consider that we are able to compute a spatial map, in retinotopic coordinates, that represents the behavioral relevance of each location in the visual space. We point out again that this is one of our working hypothesis, detailed in section 2.3.

Focus

The units in the focus map have direct excitatory feedforward inputs from the saliency map. The lateral connections are locally excitatory and widely inhibitory so that a competition between the units within the map leads to the emergence of only one stimulus in the focus map. This mechanism is not just a dynamical winner-take-all algorithm because the winning stimulus will still be represented in this map, even if the other stimuli in the visual scene become comparatively more salient through time, but it has to be explicitly inhibited. This focused stimulus is considered the next target to focus on and the movement to perform to center it on the fovea is decoded from this map. This map then codes the parameters of the next saccade to make.

Working memory Once a stimulus has appeared within the focus map and because it is also present in the saliency map at the same location, it emerges within the working memory. Both the excitations from the focus map and the saliency map (at a same location) are necessary for the emergence of a stimulus in the working memory area. If the focused stimulus changes, it will not be present anymore in the focus map such that an additional mechanism is needed to maintain it in the memory. It is not shown on the schematic illustration (3) but the memory consists of two maps, wm and thal wm, that share excitatory connections in two ways: the first map excites the second and the second excites the first, weighted so that the excitation is limited in space.

Memory anticipation The memory anticipation mechanism aims at predicting

what should be the state of the working memory after an eye movement centers another stimulus in the focus map before the movement is triggered. This map, filled with sigma-pi units, has two inputs: units of the focus map and units of the working memory. If we denote wm(x, t) the activity of unit x of the working memory at time t, and f (x, t) the activity of unit x of the focus map at time t, we define the input I(x, t) of unit x in the anticipation map as:

I(x, t) = w sigma-pi y∈IR 2 wm(y, t)f (y -x, t) + af f w af f u af f (t) (6)
The input of each unit in the anticipation map is computed as a convolution product of the working memory and the focus map, centered on its coordinates. To make (6) clearer, the condition of the sum is weaker than the one that should be used: since the input maps are discrete sets of units, the two vectors y and y-x mustn't exceed the size of the maps. The equation ( 6) should also take into account that the position eye centered is represented by a bell-shaped pattern of activity centered in the focus map, so that an offset should be included in the first sum when determining which unit of the focus map multiplies wm(y, t) From ( 1) and ( 6), the activity of the units in the anticipation map, without lateral connections, satisfies [START_REF] Findlay | Eye scanning of multi-element displays: I. scanpath planning[END_REF].

τ. ∂u(x, t) ∂t = -u(x, t) + baseline + w sigma-pi y∈IR 2 wm(y, t)f (y -x, t) (7)
Then, the shape of activity in the anticipation map converges to the convolution product of the working memory and the focus map. Since the activity in the focus map has a Gaussian shape and the working memory can be written as a sum of Gaussian functions, the convolution product of the working memory and the focus map leads to an activity profile that is the profile in the working memory translated by the vector represented in the focus map. This profile is the prediction of the future state of the working memory and is then used to slightly excite the working memory. After the eye movement, and when the saliency map is updated, the previously scanned stimuli emerge in the working memory as a result of the conjunction of the visual stimuli in the saliency map and the prediction of the working memory, that is, the prediction is combined with the new perception. This is exactly the same mechanism as the one used when a stimulus emerges in the working memory owing to the conjunction of the activity in the saliency map and the focus map.

Simulation and results

The visual environment consists of three distributed but identical stimuli that the robot is expected to scan successively exactly once. A stimulus is easily discriminable from the background, namely a green lime on a white table. A complete activation sequence of the different maps is illustrated on Figure 4. The saliency map is filled by convolving the image captured from the camera by a green filter in HSV coordinates such that it leads to three distinct stimuli 1 .

At the beginning of the simulation (Figure 4a), only one of the three stimuli emerges in the focus map, thanks to the strong lateral competition that occurs within this map. This stimulus, which present in both the focus map and the saliency map, emerges in the working memory. The activation within the anticipation map reflects what should be the state of the saliency map, restricted to the stimuli that are in the working memory after the movement that brings the next targeted stimulus into the center of the visual field. During the eye movement (Figure 4b), no visual information is available and the parameter τ in ( 1) and ( 7) is adjusted so that only the units in the anticipation map remain active, whereas the activity of the others approach zero. After the eye movement and as soon as the saliency map is fed with the new visual input, the working memory is updated thanks to the excitation from both saliency and anticipation map at a same location: the prediction of the state of the visual memory is compared with the current visual information. A new target can now be elicited in the focus map thanks to a switch mechanism similar to that described in [START_REF] Vitay | Using neural dynamics to switch attention[END_REF], but not detailed here. This mechanism acts like the inhibition of return presented in section 2.3; the memorized locations in the working memory are inhibited in the focus map, therefore biasing the competition in it, so that only a stimulus that was not already focused can be the next target to focus.

In order to illustrate more explicitly the role of the anticipatory signal, we now consider a second experiment. In this experiment, the visual scene consists of only two identical stimuli (Figure 5).

The task is the same as the previous one, namely, the robot must scan each stimulus only once, but the experimental conditions are slightly different: we enforce the robot to scan these targets in a predefined order. To bias the spatial attention toward one of the two targets, we first increase the intensity of the leftmost target. Then, when the saccade to center that target is performed, we refresh the display and increase the intensity of the rightmost target. In that way, the scenario is as follows:

1. Select the leftmost target. 2. Focus on that target. 3. After the saccade, when the display is refreshed, select the rightmost target. 4. Perform the saccade to center the rightmost target.

The visual bias we add makes us able to get the same experimental conditions over the trials. During a trial, we record the activity of the neurons whose receptive field covers one of the five positions, denoted x0, x1, x2, x3, and x4 in the figure, in the four maps: visual, focus, wm and anticipation. In a typical trial, we will have a target at x1 and x3, then, after the first saccade, the targets will be at x2 and x4, to finally occupy, after the last saccade, the positions x0 and x2 (Figure 6, top).

Moreover, two conditions are considered; in the first one (Figure 6), the anticipation is enabled, whereas in the second one (Figure 7), the anticipatory signal is disabled. At the beginning of the trial, the targets are at positions x1 and x3 so that the neurons in the visual map at these positions are excited (dashed line) whereas the neurons at the other positions remain silent. The two positions x1 and x3 compete for the spatial attention. Since we added a bias toward the target at position x1, the spatial attention is on target x1, rather than on target x3, so that the activity of the neuron at position x1 in the focus map (solid line) grows, whereas the activity of the neuron at position x3 in the same map decreases to zero. The attention on target x1 enables it to emerge in the working memory (dash-dot line). The task is now to produce an eye movement that will center that target. The anticipatory mechanism predicts that when that target is centered, it will occupy the position x2; the activity of the neuron at position x2 grows (dotted line). As soon as the saccade is performed, we refresh the display. The two targets now occupy the positions x2 and x4. The bias toward the rightmost target enforces that target to be attended. The activity of the neurons at position x4 in the focus map and the working memory grows. Whereas the target at position x4 emerges in the working by the conjunction of an activity in the visual input and the focus map, the target at position x2 emerges thanks to the visual input and the anticipatory signal. As we can see in Figure 7, in which the anticipatory signal was disabled, the position of the first attended target cannot be updated at position x2. Finally, a saccade to center the target at position x4 is performed. In the case that the anticipation is present, the new positions of the two targets are in the working memory at x0 and x2, whereas when there is no anticipation, only the last attended target is in the working memory. In this paper, we have presented a continuous attractor network model that is able to anticipate the consequences of its own movements by actually predicting the visual scene as it is supposed to be after the execution of an action. Furthermore, the model also illustrates how this information is used in the context of a serial search of a target among a set of distractors. Each already focused target is kept within a working memory area that is updated with regards to eye movements.

The model is of a completely distributed nature and does not require any central supervisor. All the units in the model satisfy a dynamical equation. When dealing with this kind of dynamic model, the integration time of the units is a critical factor as shown in [START_REF] Rougier | Emergence of attention within a neural population[END_REF], which shares some ideas with the present model. It means that in our case, even if we make the hypothesis that the perception is available during the saccade (ignoring also that the perception is smeared), the working memory could be updated dynamically with the perception only if the movement's speed doesn't exceed a critical limit. In the case of saccadic eye movements, it is then necessary to have an anticipatory mechanism. We are definitely speaking about anticipation since a prediction about the future perception is used to maintain a coherent memory which is mandatory to accomplish the task we designed. It is nonetheless not limited to that particular case since scanning several potential targets is one of the basic primitives we use when performing a visual search task.

The question of learning the underlying transformation of the anticipatory mechanism, namely the convolution product of the focus map and the working memory, remains open and is still under study. We did implement a learning mechanism, under restrictions and strong hypotheses that rely heavily on the difference between the pre-saccadic prediction and the post-saccadic actual perception. This self generated signal is able to measure to what extent the prediction is correct or not. Hence, it is quite easy to modify the weights accordingly. The main difficulty during learning remains the sampling distribution of examples within the input space, which is a well known problem in information and learning theory. Without any additional motivational system that could bias the examples according to a given task, it is quite unrealistic to rely on a regular distribution of examples.

Finally, the coherence of the visual world is solely based on an anticipatory mechanism that ultimately allows the identification of identical targets before and after a saccade, despite drastic changes in the visual perception. The prediction of the future state of the visual memory enriches the perception of the visual world in order, for example, to prevent focusing twice on the same stimulus. Of course, this model does not pretend to be complete nor accurate and does not tackle a number of problems that are directly related to visual perception. However, we think that the possibility to unconsciously anticipate our own actions using a dynamic working memory could be extended to other motor tasks involving other types of perception as well.

Fig. 4 .Fig. 5 .

 45 Fig. 4. A sequence of evolution of the model during an overt visual scan trial. a) One of the three stimuli emerges in the focus map and the anticipation's units predict the future state of the visual memory (the maps wm and thal wm). b) During the execution of the saccade, only the units in the anticipation map remain active. c) The focused stimulus emerge in the memory since it is both in the saliency map and the anticipation map at the same location. d) A new target to focus is elicited. e) The future state of the memory is anticipated. f) The saccade is executed and only the prediction remains. g) The two already focused stimuli emerge in the memory. h) The attentional focus lands on the last target.

Fig. 6 .

 6 Fig.6. Case with the anticipatory signal enabled. We record the activity of neurons whose receptive field covers one of the five positions x0, x1, x2, x3 and x4, in the four maps: visual, focus, wm and anticipation. During the trial, we add a bias toward one of the targets so that the attention directs to the biased target (that target is shown by the arrow). Each subplot represents the activity of the neurons in each map at a given position. The dashed line represents the activity of the neuron in the visual map, the solid line the activity of the neuron in the focus map, the dash-dot line the activity of the neuron in the working memory and the dotted line the activity of the neuron in the anticipation map. Please read the text for explanations on these curves.

Fig. 7 .

 7 Fig.7. The experiment is the same as in Figure6except that the anticipatory signal is disabled.

A video of the model is available at http://www.loria.fr/ fix/publications.php

Appendix

Dynamic of the Neurons

Each sigma neuron loc in a map computes a numerical differential equation given by equation [START_REF] Findlay | Eye scanning of multi-element displays: Ii. saccade planning[END_REF], which is a numerized version of that proposed in [START_REF] Amari | Dynamical study of formation of cortical maps[END_REF] and [START_REF] Taylor | Neural bubble dynamics in two dimensions[END_REF]:

Each sigma-pi neuron loc in a map computes a numerical differential equation given by equation ( 9

where σ(x) is a semi-linear function assuring that 0 ≤ σ(x) ≤ 1, τ is the time constant of the equation, α is a weighting factor for external influences, aff is a neuron from another map and lat is a neuron from the same map. To know how the set of afferent neurons E loc is determined in the case of a sigma-pi map, please refer to the section 3.

Connections intra-map and inter-map

The lateral weight from neuron lat to neuron loc is:

where dist(loc, lat) is the distance between lat and loc in terms of neuronal distance on the map (1 for the nearest neighbor). In the case of a "receptive field"-like connection between two maps, the afferent weight from neuron aff to neuron loc is:

In the case of the sigma-pi connections, all the weights are the same:

The connections in the model are described the following table:


F

luid flows come in a variety of states, from slow and perfectly predictable to fast and horribly complex. Turbulent flows represent this last category, which is actually the most common type of flow on the surface of the Earth, and on other planets for that matter. Such a complex and unpredictable state has been depicted in a variety of ways but the most common representation is that of swirls and eddies mingling with each other to give the impression of something uncontrollable. But this inherent complexity has not prevented scientists from all sides from trying to confront this problem. It is somehow a cliché now to describe turbulent flows as the last unsolved problem of classical mechanics.

Several attempts have been undertaken to identify the problem so as to be able to craft a solution. In this respect, the work of L. F. Richardson 1 comes to mind, as he noted that turbulent flows disperse and mix (pollutants, dust, smoke and so forth) fast and well. He then devised experiments to measure such dispersions by tracking the separation of weather balloons in the atmosphere. His work showed that 'turbulent dispersion' is much more efficient and much faster than, say, thermal diffusion, and allowed him to put numbers to the process now known as Richardson dispersion. This quantitative aspect has not prevented Richardson from using a qualitative representation as well, stating that "big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity…" in an attempt to represent 'turbulence' as an assembly of interacting swirls exchanging energy with each other.

Such an approach is still being used today to teach students of turbulence how eddies transport energy injected at some large scale in the system all the way to the smallest scales where viscosity will transform it into heat [START_REF] Pope | Turbulent flows[END_REF][START_REF] Frish | Turbulence: The legacy of A. N. Kolmogorov[END_REF] . At this point of the story another character enters the set: A. N. Komogorov 4 . His theory of turbulent flow is still around, despite its limitations, and brings a quantitative explanation of what Richardson measured with his floating balloons. The idea is still the same: big whirls break into smaller ones down to viscosity. In turbulence speak one says that energy cascades from the large to the small scale, at a constant injection rate (or dissipation rate) [START_REF] Pope | Turbulent flows[END_REF][START_REF] Frish | Turbulence: The legacy of A. N. Kolmogorov[END_REF] . The direct cascade of energy of three-dimensional (3D) turbulence was born along with an expression for it [START_REF] Pope | Turbulent flows[END_REF][START_REF] Frish | Turbulence: The legacy of A. N. Kolmogorov[END_REF][START_REF] Kolmogorov | [END_REF] : the -4/5 law.

I explicitly mention three dimensions because this picture of small eddies feeding on larger ones breaks down if the spatial dimension is changed from three to two [START_REF] Frish | Turbulence: The legacy of A. N. Kolmogorov[END_REF]5,6 (Fig. 1). This concept was put forth in the late 1960s 7 and early 1970s 8 where the cascade is said to be an inverse cascade of energy: big eddies or whirls now feed on smaller ones, up to infinity (so to speak, as the container size will cut such transfers). Two-dimensional turbulence -that is, turbulence in flat space -changes the character of energy transfers between scales for fundamental reasons. Eddies in three dimensions stretch but in flat space all they can do is deform and eventually merge. This picture of turbulence in flat land has been made quantitative through numerical simulations at first, and through different experiments that either use soap films (see Fig. 2) or very thin layers of fluid 5,6 . Upscale energy transfers have therefore been predicted and found. Such a picture favours the presence over long timescales of large vortices dominating the flow and has been used to explain the robustness and long lifetime of such objects as Jupiter's red spot 9 . Notice that the word thin comes back and for good reason: eddy or vortex stretching has to be suppressed to obtain such upscale energy transfers, and to suppress it, the thickness of the layer has to be very small. This limits the validity of such an upside-down turbulent picture to a small category: very thin layers.

But whether this picture is applicable in other situations such as atmospheric flows and at what scales remains a question with many answers. This is where the Letter by

TURBULENCE Thick puddle made thin

For turbulent flows, the energy transfer between large and small whirls depends on the dimension of the fluid. Imposing large-scale shear on a three-dimensional system can unexpectedly induce two-dimensional behaviour. Hua Xia and collaborators 10 comes into play because it shows that this picture can be obtained in even thicker layers, which extends the importance of such inverse transfers to a much wider category of flows of importance for geophysics and oceanography. How did they do this? Well the answer comes from the use of an extra-large-scale flow imposed on the thick layer. This layer, an electrolyte solution, was first driven by a so-called electromagnetic forcing scheme (small magnets below the cell cause deviations in the motion generated by an imposed current in the solution). The flow observed in this thick layer (several millimetres) shows much movement in the direction perpendicular to the plane of the layer, or in the third dimension.

Hamid Kellay

By imposing a large-scale vortex through the use of an additional layer of magnets placed near the upper surface of the layer, a large vortex was superimposed on the flow. The presence of this large vortex slowly suppressed the velocity variations in the third dimension and rendered the behaviour of the flow in the thick layer similar to that in very thin layers: all the flow features were confined to the plane of the layer. The consequences of such an operation is the recovery of features common to turbulence in very thin layers, upscale energy transfers, for a thick pool of fluid for which previous work ruled out any two-dimensional behaviour.

The authors argue that the shear engendered by the large-scale vortex is responsible for the suppression of 3D movements. This discovery is important for a variety of phenomena for which if the flow were rather 2D instead of 3D several curious features can be explained. The authors invoke two major examples: atmospheric turbulence in the range between 10 and 500 km and turbulence in the tachocline layer beneath the surface of the Sun. Strong shear between the outer convective layer of the Sun and its radiative interior make turbulence in the tachocline layer a candidate for a two-dimensional phenomenology. As many questions, such as its stability and its role in the Solar dynamo, are still open, an understanding of the nature of the turbulence in the tachocline layer is highly important. Also, the shear engendered by planetary-scale motions may render turbulence in the atmosphere two dimensional. This will help in modelling such systems, partially explaining features of their wave-number spectra, and reveal how energy is exchanged between the widely different scales that enter the problem. Although the argument about atmospheric turbulence or turbulence in the tachocline layer is certainly not closed by this study, an important step towards settling it has been achieved through this work.

Hamid Kellay is at Laboratoire Ondes et Matière d'Aquitaine, Université Bordeaux 1, 33405 Talence, France. e-mail: hamid.kellay@u-bordeaux1.fr

Figure 1 |

 1 Figure 1 | Downscale and upscale energy transfers. a, In three dimensions, big eddies break up into smaller and smaller eddies. b, In two dimensions the small eddies give rise to larger and larger eddies.

Figure 2 |

 2 Figure 2 | Image from a soap film showing upscale transfers as small vortices detach from the teeth of a comb and grow as they travel downstream from the grid.