Jean-Charles Quinton

Bernard Girau

Predictive neural fields for improved tracking and attentional properties

Predictive capabilities are added to the competition mechanism known as the Continuum Neural Field Theory, in order to improve and extend its attentional properties. In order to respect the distributed and bio-inspired nature of the model, the prediction is introduced as an internal stimulation, directly determined by the past field activity. Building on mathematical developments and optimization techniques, performance is ascertained on a 2D tracking application where the system must robustly focus on a target despite rapid movements, noise and distracters. In addition to a consistent gain on previously observed capabilities, the extended model can also track stimuli with full occlusions and static obstacles on the trajectory.

I. INTRODUCTION

The Continuum Neural Field Theory (CNFT) is a particular kind of dynamic neural field model that implements lateral competition within cortical maps [START_REF] Amari | Dynamics of pattern formation in lateral-inhibition type neural fields[END_REF], [START_REF] Wilson | A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue[END_REF]. These models rely on the hypothesis that the generic structure and topological organization of the cortex are relevant and maybe necessary to explain a wide range of cerebral phenomena. Following the classical results on the laminar structure of the cortex, the correlated activities found across cortical layers [START_REF] Mountcastle | Modality and topographic properties of single neurons of cat's somatic sensory cortex[END_REF] and the multiscale organization deduced from cytoarchitectural and functional differences [START_REF] Brodmann | Brodmann's 'Localisation in the Cerebral Cortex[END_REF], [START_REF] Hubel | Receptive fields, binocular interaction, and functional architecture in the cat's visual cortex[END_REF], several computational neuroscience models have been proposed in this context [START_REF] Burnod | An adaptive neural network: The cerebral cortex[END_REF].

The CNFT can be reduced to a single differential equation that describes the evolution of the membrane potential of neurons over entire cortical maps. The simplified version used in this paper is given by the following single-layer field equation of lateral inhibition type (∀x ∈ M ):

u t+dt (x) = f 1 - dt τ u t (x) + dt τ (c(x) + i(x)) (1)
where f is chosen such as to maintain the membrane potential u in [0, +∞). An input signal i is provided to the field and the competition term c is defined as a continuous convolution with a lateral connection kernel w by: c(x) =

x ∈M w(x -x )u t (x )dx w(dx) = Ag(dx, a) -Bg(dx, b) g(dx, σ) = e -dx 2 σ 2 [START_REF] Amari | Dynamics of pattern formation in lateral-inhibition type neural fields[END_REF] Jean-Charles Quinton is with the INRIA/Cortex project and Bernard Girau is with the UHP/LORIA/Cortex project, Campus Scientifique, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France (email: quintonj@loria.fr).

Under adequate conditions, so-called bubbles of activity emerge on the neural field in response to external stimulation. These bubbles can be defined as spatiotemporally coherent patches of activity that exhibit robust attentional properties, such as focusing on and tracking stimuli despite the presence of noise or distracters [START_REF] Rougier | Emergence of attention within a neural population[END_REF]. Additionally, the non-linear dynamics of the CNFT accounts for the rapid selection of a single target even when exposed to a set of equally active stimuli.

Competition is made possible by introducing a lateral connectivity pattern following a difference of Gaussians profile. Building on mathematical developments for the conditions of convergence toward bubbles of activity [START_REF] Taylor | Neural bubble dynamics in two dimensions: Foundations[END_REF], [START_REF] Alexandre | On practical neural field parameters adjustment[END_REF] and using genetic algorithms to select optimal parameters for the equation, we showed that there is a tradeoff between accuracy and robustness [START_REF] Quinton | Exploring and optimizing dynamic neural fields parameters using genetic algorithms[END_REF]. Indeed, correctly constraining the connectivity pattern to a Mexican hat profile with local excitation and global inhibition almost guarantees the emergence of an activity bubble, but the speed of the moving stimulus and the amount of noise present in the stimulation either result in a tracking lag or the system loosing its target. This limitation can be explained by the fact that the competition term and thus the solutions to the equation are fundamentally stationary, which is quite paradoxical when considering applications such as motion tracking or spatiotemporal pattern recognition. Albeit the focus dynamically emerges from a differential equation, it is not made for dynamic inputs, it just adapts to them. As a consequence, the focus bubble resists to movement and prefers noise or distracters reinforcing its activity over any moving stimulus, even when they provide a much weaker stimulation compared to the target. Put in the broader context of cognitive agents, this phenomenon pinpoints the limits of purely reactive bottom-up attentional systems when confronted to situations that offer multiple potentialities. Some kind of "internal model" is needed to discriminate between dynamics, yet it does not need to be very complex or abstract.

When considering applications to sensorimotor systems such as robots, the controller can be modeled by a distributed network of local contingencies, each predicting the outcome of specific actions [START_REF] Quinton | Multilevel anticipative interactions for goal oriented behaviors[END_REF]. Indeed, delays are intrinsic to interactive systems and fast interactions are often required (keeping balance with a relatively slow feedback loop is a paradigmatic example), which implies that the agent should anticipate the dynamics. In that case, the CNFT can be seen as a competition mechanism acting upon distributed anticipatory representations. Competition is then not only beneficial to perception by contributing to covert attentional mechanisms, but is also required for action selection as all anticipatory representations are simultaneously active. Furthermore, actions participate to overt attention (consider saccadic eye movements for instance), making competition a crucial and recurrent component of the sensorimotor loop [START_REF] Fix | A dynamic neural field approach to the covert and overt deployment of spatial attention[END_REF].

Whatever the perspective taken on the potential place of the CNFT within a cognitive architecture, preferences or predictions thus need to be integrated to reactions in a single coherent model, where they can influence each other. Even though iterating over the CNFT computations or switching to more complex integration schemes might sometimes help to better converge on the stimuli and reduce the tracking lag, this has a high computational cost incompatible with real-time constraints. In this paper, we therefore evaluate the consequences of adding a predictive input signal within the CNFT equation, as the basic equation cannot cope anyway with all the other problems mentioned above.

Despite the broad scope of application the combination of a predictive system and the CNFT model may have, we here only analyze the benefits and potential drawbacks of the introduction of a single predictive term for the CNFT attentional properties (focus, accuracy, robustness). In addition to mathematical developments and qualitative results in one dimension, the performance will be quantitatively assessed on the artificial 2D visual tracking application introduced by Rougier et al. [START_REF] Rougier | Emergence of attention within a neural population[END_REF].

II. PROPOSED MODEL

In this paper, we propose an extension of the original equation (Eq. 1). We simply decompose the input signal i as a weighted sum of a bottom-up stimulation s and a topdown prediction p:

i(x) = αp(x) + (1 -α)s(x) (3)
where α is chosen such as to guarantee the primacy of the external stimulation on the inner prediction (α ∈ [0, 0.5]).

The predictive term can then be considered as a corticocortical modulation to be contrasted with the thalamic input from the senses, a division that is also found in neural field models of multimodal integration [START_REF] Ménard | Model of multi-modal cortical processing: coherent learning in self-organizing modules[END_REF]. To remain as simple as possible relatively to the competition between the immediate external stimulation and future-oriented inner projections, α is fixed and a single predictor is considered (see Fig. 1 for a graphical representation). Under these conditions, we will determine in the following sections how the predictor should be defined to optimally improve the performance.

In more realistic settings, a variable number of predictors (associated with different potentialities, goals and values) as well as adaptive coefficients can be introduced. The corresponding weights should then reflect the adequation of the predictors with the current needs of the agent and dynamics of the environment. The focus here is on the flexibility of the model, and a simple change in Eq. 3 facilitates the introduction of an arbitrary number of predictors that can be evaluated in an online manner (details cannot be given in this short paper).

Keeping the convolution kernel untouched and thus symmetrical makes it furthermore easy to optimize the computations, either by using linear algebra techniques (such as truncated singular value decomposition for matrix approximation) or shifting to continuous approximations of the entire field activity (Gaussian mixture models for instance) [START_REF] Quinton | A sparse implementation of dynamic competition in continuous neural fields[END_REF]. Instead of considering a linear combination of predictors, it is however possible to directly learn asymmetrical kernels that bias the competition toward predicted states as in [START_REF] Cerda | Bio-inspired visual sequences classification[END_REF]. Although this latter approach may provide an increased expressive power and the system may assimilate faster dynamics, it is also harder to evaluate the kernels correctness online. Moreover, it needs the system to learn complete and well-separated trajectories whereas our predictive neural fields simply handle local predictions and are able to cope with close trajectories.

The predictor p can be defined as a transformation of u t in the geometric sense, using the past activity over the entire field to predict the movement of the bubble. As local linear movements can be combined to produce arbitrary tra-jectories (as Locally Weighted Regression can approximate any function [START_REF] Cleveland | Robust locally weighted regression and smoothing scatterplots[END_REF]), we will only consider linear predictors in presence of a single bubble. From now on, v represents the instantaneous speed vector of the tracked stimulus, so that a bubble centered on x at time t should move to location x + vdt at time t + dt. This does not restrict the generality of the approach, but allows us to directly express p(x) as:

p t (x) = u t (x -γ(v)dt) (4)
Taking γ equal to the identity seems natural as it means that the prediction, if correct, will be aligned with the stimulus. It thus introduces an asymmetry between the stimulus, that benefits from both the activity of s and p, and distracters that may be equally active in the external signal but do not receive a positive influence from p. This bias in the amplitude is demonstrated on Fig. 2.

Nevertheless, due to the inertia of the CNFT equation (as the inputs get integrated with a ratio of dt τ ), an infinite number of iterations would still be required to converge on the moving stimulus. Moreover, the stationarity and nonlinear dynamics of the CNFT results in oscillations of the bubble. Indeed, it tends to stabilize at the target location, with its amplitude increasing until the stimulus and competition inputs get compensated by the leaking term. As the target moves away from the bubble, the latter will then be destabilized as the stimulation enters the inhibitory part of the lateral kernel. The bubble amplitude will decrease, allowing it to move back to the target position (see Fig. 3).

As a consequence, taking the identity for γ leads to an unbreachable lag behind moving stimuli. To quantitatively evaluate the error made in the tracking of the target, we compute the distance between the center of the stimulus x s and the center of mass of the focus activity u. The resulting functional d, that depends on the γ function, is defined in Eq. 5. If convergence occurred and a single bubble emerged, it should be symmetrical and centered on the stimulus so that the distance should be null. As an aside, x s is only known to the designer for computer generated inputs, where input (left) Without prediction, a stimuli and a distracter of equal intensity are not differentiated by the CNFT. Any of them may thus act as an attractor in Eq. 1, even when the convergence already occurred but the distracter is located on the trajectory (x d ). (right) If the prediction matches the actual stimulus movement (xp = xs) and not the distracter, the symmetry is broken and the selection or tracking is biased. scenarios are carefully specified.

p x = s x s x 1 0 s = i d x d x x s ) ® + (1{ ®p = i 1 ® 1{ (without prediction)
t d (a) (b) w d 0 =0 v 5 : =0 v 1 : =0 v 5 : =0 v 1 : =0 v =0 v 0 0 d 0 d
d(γ) = x s -x∈M u(x)xdx x∈M u(x)dx (5)
Given the parameters of the CNFT equation and for a single iteration of the integration scheme, we therefore try to find γ that minimizes the tracking error whatever the speed of the stimulus to follow. The following section aims at deriving such an optimal γ function by analyzing the reciprocal interactions between the stimulus and both the focus and prediction bubbles of activity.

III. OPTIMAL PREDICTION

According to preliminary experiments, it has appeared that γ needs to overestimate the position of the bubble so as to compensate for the intrinsic latency of the equation. Nevertheless, these experiments have shown that the optimal γ function is not constant, nor easy to find. However, due to the slow variations of the optimal value relatively to the speed of the stimulus to follow, it is possible to improve the attentional properties of the CNFT using the same parameter value for a wide range of input trajectories. We therefore propose here to analytically find such value based on the whole set of parameters.

A. Analysis of interactions

To give a rough idea of the mathematical resolution of this problem, we can get inspiration from the Gaussian mixture model introduced in [START_REF] Quinton | A sparse implementation of dynamic competition in continuous neural fields[END_REF]. Let g s = (x s , S), g u = (x u , U ) and g p = (x p , P ) respectively be the Gaussian components approximating the stimulus, focus and predicted bubbles. Each couple corresponds to the center and amplitude of the associated bump of activity so that s(x) ≈ S × g(x -x s ) for instance. Starting from a state where the focus bubble was perfectly aligned with the stimulus at x = 0 implies that x u = 0, x s = vdt and x p = γ(v)dt. The prediction being reduced to a translation, the predicted and focus bubbles have the same amplitude P = U (see Fig. 4). By translating Eq. 1 and Eq. 3 into interactions between the various components as in [START_REF] Quinton | A sparse implementation of dynamic competition in continuous neural fields[END_REF], we can update their amplitude as follows:

g u = (0, U ) U = (1 + dt τ (-1 + w(0)))U g s = (vdt, S ) S = dt τ ((1 -α)S + w(vdt)U ) g p = (γ(v)dt, P ) P = dt τ (α + w(γ(v)dt))U (6)
The new bubble position can then be computed as the barycenter of the components, where the weights correspond to the amplitudes. The optimal γ function is then obtained for:

d(γ) = 0 ⇔ U x u + S x s + P x p U + S + P = x s (7)
As the components considerably overlap due to the small timestep required for the competition to work, this approximation only captures the essence of the interactions that determine the efficiency of the prediction on tracking performance. This component based model should therefore not be used for precisely choosing the optimal γ function. Yet, an accurate albeit more complex analytical resolution based on definite integrals is possible.

B. Analytical resolution

To find the optimal γ analytically, we only consider a manifold M = (-∞, +∞), take f (u) = max(u, 0) in Eq. 1 and provide a single Gaussian stimulation to the system:

s t (x) = Sg(x -vt, c) (8)
We also approximate the localized bubble of activity on which this equation converges as the following Gaussian function: 4. Graphical illustration of the input, prediction and focus in 1D (distances between bumps on the x-axis are not to scale). The focus at time t (gray) is translated from position 0 by γ(v)dt to produce the prediction (red). The stimulation moves at speed v, which makes its position go from 0 (previous) to vdt for this iteration (blue). By updating the CNFT equation, the focus activity u t+dt is produced (black). For the tracking to be optimal, γ must be chosen such as to minimize the error distance d(γ).

u t (x) = U g(x -x t , σ) (9)
p s t u dt ) v ( °= p x vdt = s x =0 u x activity x U S ) °( d dt + t u Fig.
where x t corresponds to the center of the bubble distribution at time t (parameters U and σ are dependent variables of the equation parameters and input stimulation). For simplification purpose, we suppose that the position of the bubble at time t is 0 and that the stimulus moved at position vdt (see Fig. 4). The instantaneous prediction p can then be defined by:

p t (x) = U g(x -γ(v)dt, σ) (10)
Starting from Eq. 5 when d(γ) = 0 and x s = vdt then transforming it to eliminate the divisor, we more generally study the following function: [START_REF] Mountcastle | Modality and topographic properties of single neurons of cat's somatic sensory cortex[END_REF] This function is ideally equal to 0 when the focus moves to the same position as the stimulus.

e I (γ) = x∈I u t+dt (x)xdx -vdt x∈I u t+dt (x)dx
If the function f was absent from the equation, the integration interval I could be taken equal to (-∞, +∞) and computations would be drastically reduced. However, this leads to a simple expression

(γ(v) = v α τ dt -1 + √ π(aA -bB) + 1)
that does not capture accurately the relationships between the parameters. The reason is that the integral of c(x) is negative and far greater in absolute value than all the other components. We thus need to restrict the integration interval to [vdt -θ, vdt + θ], with θ chosen as the minimal value to satisfy the relation u(vdt + θ) = 0. The interval is symmetric around vdt, which corresponds to the ideal value of the center of the bubble at t + dt, and this is equivalent to only taking into account the positive values of u(x). Of course θ depends on the width of the gaussian σ, that can be chosen to minimize the difference between the integral of the Gaussian and actual bubbles, which is x∈I (u(x) -g(x, σ))dx .

Having thus reduced the integration interval, a factorized expression of e I (γ) can be found. Mathematical details are given in the appendix. The final expression does not get really simpler by differentiating it, and series developments are required for a direct approximation of the function from which we can derive the optimal γ. For example with functional parameter values (A = 20, B = 15, a = 0.1, b = 1, dt = 0.1, τ = 0.3, α = 0.5) and v = 0.1, we obtain an optimal value γ(v) = 8.04 with σ = θ = 0.07, which is a good approximation of the value obtained experimentally (γ(v) 8).

C. Interpretation

This difference in the location of the anticipatory feedback (γ(v)dt) and the predicted location of both the stimulus and bubble (vdt) can be interpreted in terms of action and perception. The injected activity p is the action performed on the CNFT by the predictor, which is partially decoupled from the actual movement observed in u. This is very similar to inner actions observed between cortical areas (the prefrontal cortex being considered for instance as part The first snapshot always corresponds to t = 0, where the algorithm gets some time to converge on one of the stimuli (and there is only one but for scenario C ). The second snapshot is associated to a later time, chosen to be representative of the input dynamics. The position of the target, its trajectory as well as the occluded area for scenario G are superimposed on the fields. of the motor cortices [START_REF] Fuster | Cortex and Mind: Unifying Cognition[END_REF]). At this internal level again, perception-action loops are required to monitor the effect of modulations between neural maps.

A parallel can also be made with the correlations we learn about signals sent from the brain to muscles (motor commands) and the sensory feedback perceived in return (proprioception). Such relationships between the afferent and efferent signals in neural maps can be used to influence the dynamics of our body and bias the decisions (for instance by modulating central pattern generators) without having a direct control over the dynamics of the muscles.

IV. RESULTS

The discrete implementation described in [START_REF] Rougier | Emergence of attention within a neural population[END_REF] is used as a reference, and the same evaluation process is adopted. Choosing adequate parameters, global inhibition guarantees a single bubble will emerge, and we thus evaluate the performance of the model using the distance defined in Eq. 5.

To test the robustness and attentional properties of the model, we provide 2D artificial dynamic inputs to the system. On a toric manifold mapped on [-0.5, 0.5] 2 and approximated by a square matrix of size 50 × 50, bell-shaped stimuli of standard deviation 0.1 are used for both the target and distracters. The input dynamics are defined by the following scenarios (scenarios C, D and E being taken from [START_REF] Quinton | Exploring and optimizing dynamic neural fields parameters using genetic algorithms[END_REF]): C Competition: 2 distant stimuli are introduced at time t = 0. Their intensity are governed by S 1 = 0.5 -0.5 sin(π × (t/10)) and S 2 (t) = 0.5 + 0.5 sin(π × (t/10)).

D Distracters: 1 stimulus follows a circular trajectory of radius 0.2 around the point (0, 0) at 30 deg/s from t = 0. From t = 1, 30 distracters are added and take new random positions on the field every second. E Noise: 1 stimulus (same as above). At t = 1, Gaussian noise with a standard deviation of 0.5 is added and refreshed every second. F Fixed distracter: 1 stimulus (same as above). At t = 5, a fixed distracter is added at (0, -0.2), i.e. on the trajectory of the target. G Occlusion: 1 stimulus (same as above, except for a speed of 10 deg/s). At t = 30, a fixed distracter is added out of the trajectory, and the stimulus is then occluded on part of the trajectory for 0 < x < 0.5 and -0.1 < y < 0.1.

Except for scenario C , the amplitudes of the target and distracters in the input signal are identical and arbitrarily taken to 1.0. The initial difference in intensity or the delay introduced before distracters and noise are added guarantee that the model will converge on the chosen stimulus during the first few iterations. In order to prove the efficiency and usefulness of the prediction term, optimal parameters as defined in [START_REF] Quinton | Exploring and optimizing dynamic neural fields parameters using genetic algorithms[END_REF] have been independently selected for each scenario. This guarantees the standard equation is not simply outperformed because of a poor calibration of its dynamics.

The mean error over entire simulations and averaged over several trials for non deterministic scenarios (noise and random distracters) is given in Table I for the model in 3 different conditions: without prediction (reference) and with both a correct and incorrect prediction. In the table, gray cells highlight the best performance for each scenario, most often when the equation is extended with an adequate predictor. It might be argued that introducing a correct prediction into the equation should of course lead to performance improvement, as we bias the dynamical attractor landscape toward the right trajectory. Another interpretation is to say that prediction stands for a prior probability distribution in the selection and tracking process. However, the efficiency of such method on an already robust yet complex dynamics is far from obvious, not even mentioning the importance of the γ function. The detailed analysis of the dynamics on the different scenarios also gives precious insights into the potential of coupled competition-prediction systems. G Fig. 6. Error distance as a function of time for various scenarios with no prediction (blue), a correct prediction (red) or an incorrect one (black). The black lines represent critical events: equal stimuli intensity for scenario C , going over the static distracter for scenario F , or entering/exiting the occluder in G . When the error goes above 0.1, one can consider the target has been lost as it entered the inhibitory part of the lateral competition kernel.

Scenario C instantiates the competition property of the CNFT between distant stimuli, and since they are not moving, the ideal prediction corresponds to a speed v = 0. Initially, the system must choose a single target from two almost identical stimuli (S 1 S 2 0.5), and the slightest asymmetry in their intensities should lead to a non-linear bifurcation and the selection of the most salient target. Fig. 6C shows that the convergence occurs at the same rate with or without prediction, which is normal since the target is not moving and stationarity is the default in the original equation. However, in the case of an incorrect prediction constantly anticipating a movement, the bubble progressively drifts. Such an anticipation that constantly conflicts with the target dynamics should not remain active when competing with better predictors, but it is not the case here as the effect of a single predictor is considered. Also, it should be noted that the prediction increases the self-maintenance of the bubble activity relatively to the stimulation because of the reduced input weight it takes in the equation. This results in the increasing hysteresis phenomenon observed on the graph, the model keeping track of the initial target long after t = 10 (time at which the dominant stimulus changes). The delayed shifts visible on the figure reflect the increased attention and sustained focus activity with prediction. Such extension of the time spent focusing on a target is desired, as the system should track the target as long as it can be perceived. Whatever the difference in intensity required for the bifurcation to occur, the model is still able to relax and focus on a new stimulus when the stimulation totally disappears.

Scenario D for distracters and E for noise test the robustness of the model to random and non random perturbations in the input. Again, the parameters of the equation are chosen such as to maximize the model performance on these scenarios only, and performance is therefore higher when compared with results presented in [START_REF] Rougier | Emergence of attention within a neural population[END_REF], where tradeoffs are necessary to instantiate all attentional properties at once. When integrated into the lateral competition term c through a convolution with the kernel w, noise and distracters are equivalent to shifts of the input target location. Whereas the probability of a random noise to produce a large translation in any particular direction is quite low, distracters introduce clear oriented distortions in the local activity. As a consequence, an incorrect predictor simply acts as a distracter close to the target trajectory, which does not fundamentally hinder performance for scenario D as many distracters are already present on the field, in contrast to scenario E and as reflected in Table I. In addition to the results given in the table demonstrating a gain of 33% to 50% when a roughly correct prediction is used, More generally, testing the model with an invalid prediction shows that the system is able to fall back on a reactive tracking behavior. The effect of bad predictions on the dynamics is limited because of a conjunction of factors. First, the α coefficient in Eq. 3 gives primacy to the external stimulation. The non coherent stimulus and predicted bubble activity are spread out and their intensity is therefore reduced. As a consequence and because of the way the γ function pushes the prediction outside the excitatory range of the lateral competition kernel, predictions that are not confirmed by the external stimulation get strongly inhibited. Therefore, even without associating a variable weight to the predictor based on its level of assimilation of the dynamics, the influence of the prediction is highly constrained by the dynamics of the CNFT and stimulation.

Finally, scenario F and G correspond to tasks the standard model alone cannot perform correctly. Although the mean error values of Table I display a gain of only 50-60% with a correct prediction, this is mainly due to the experimental setup, and the model in the other conditions is in fact not able to succeed at tracking the target with a fixed distracter or an occluder on the trajectory. The inability for the standard model to discriminate between the target and distracter (Fig 6F ), as well as to maintain a moving bubble during several timesteps without stimulation (Fig 6G) is made flagrant by looking at the error curves as a function of time. Due to the toric nature of the field and the periodic trajectories of the stimuli, a large error (> 0.2) might sometimes go down progressively or oscillate, which does not always mean the algorithm was able to focus again on the correct target. It just means for instance that the focus shifted to a static distracter (the most stable attractor for the standard equation) while the target moves toward or away from it. This is the case for the no predictor and bad predictor conditions on scenario F at the end of the simulation, after they both lost the target when going over the distracter. This is also particularly striking for scenario G with the bad predictor, because the high integration constant τ required to increase the inertia of the standard model (selected by the genetic algorithms) leads to a self-maintaining bubble with its own dynamics.

V. CONCLUSIONS

We presented an extension of the inhibitory single layer CNFT model to integrate predictive signals. This new model displayed improved attentional properties and tracking capabilities over the standard version, with a fallback on the original dynamics would the prediction be incorrect. In particular, it is now possible for the stimulus to go over stationary or moving distracters without loosing track of the selected target. Whereas the excitatory projection from the prediction here amplifies the hysteresis phenomenon observed when switching between targets, inhibitory projections can be used to ease and fasten transitions to unexpected stimuli. We also showed that the relationship between the predictive modulation to introduce in the equation and its effect on the dynamics is non-trivial but coherent within the context of distributed interactive systems.

Since the paper focused on the method and benefits of combining dynamic competition and prediction within a single model, a number of key issues have been overlooked. 7. Mean error distance for different levels of noise with (red) or without prediction (blue). Due to the dynamics of stabilization through activity feedback acting against the mobility of a low amplitude bubble, a large level of noise might lead to a more reactive system. With a single spatiotemporally coherent stimulus to track on the field, this explains the otherwise strange performance boost observed when the standard deviation goes over 0.8, as the system cannot spontaneously focus on another target.

Most of these questions are addressed by ongoing work, and the potential solutions should at least be introduced here. First, where does the predictor come from? As the CNFT is able to reactively track the stimuli when the prediction is absent or invalid, the resulting bubble can be used to progressively learn the predictor and improve performance. Echoing the arguments made in the previous sections, any complex trajectory can be decomposed into a set of local linear predictions, the dynamic competition smoothing out the transitions. Such linear anticipations can be generated in an ad hoc manner, by converting the observed bubble movement into anticipations whenever the accuracy drops below a given threshold. A variation/selection mechanism is conceivable as well, thus removing the need to evaluate the γ function a priori. Indeed, the "action" and its consequences could be totally decoupled and only confirmed anticipations would be reinforced.

When the system encounters a wide range of dynamics, several predictors are needed to cover the space of potentialities, but only those best adapted to the current context should be active. For this reason, a variable weight should be associated to each predictor and to the external stimulation, even though the latter should keep primacy. In the same way as local anticipations can get reinforced during learning, the immediate and relative level of assimilation of the dynamics by the predictors can be used to discriminate between situations. The presented model, by building on the normativity of the anticipation, could then be applied to online spatiotemporal pattern recognition.

APPENDIX

Let the definite integral of a Gaussian be:

g [x1,x2] (µ, σ) = x2 x1 g(x -µ, σ)dx (12) = σ √ π 2 erf x 2 -µ σ -erf x 1 -µ σ
where erf is the standard Gaussian error function defined by:

erf (z) = 2 √ π z 0 e -x 2 dx (13)
For the computation of the centroid of the focus, we also need the definite integral of g(x -µ, σ)x:

µ m = µ 1 σ 2 2 + µ 2 σ 2 1 σ 2 1 + σ 2 2 σ 2 m = σ 2 1 σ 2 2 σ 2 1 + σ 2 2 ( 16
)
Proof: The initial expression can be written as e -ω , with w defined by:

ω = σ 2 2 (x -µ 1 ) 2 + σ 2 1 (x -µ 2 ) 2 σ 2 1 σ 2 2 = (x -µ m ) 2 σ 2 m + ε ε = 1 σ 2 m -µ 2 m + σ 2 2 µ 2 1 + σ 2 1 µ 2 2 σ 2 1 + σ 2 2 = (µ 1 -µ 2 ) 2 σ 2 1 + σ 2 2
Using Eq. 16, the convolution of a Gaussian bubble with the lateral kernel on a point is trivially obtained:

c(x) = +∞ -∞ w(x, x )u(x , t)dx (17) = σ √ π aA √ a 2 + σ 2 g x, a 2 + σ 2 - bB √ b 2 + σ 2 g x, b 2 + σ 2 U
We now focus on the relationship between U and S for the optimal γ, which is given by:

U = 1 -α λ S (18)
λ = τ dt + 1 - τ dt g(vdt, σ) -αg((γ(v) -v)dt, σ) -σ √ π aA √ a 2 + σ 2 g vdt, a 2 + σ 2 - bB √ b 2 + σ 2 g vdt, b 2 + σ 2
Proof: For the optimal γ, we are looking for a stable value of U between t and t + dt. Because the location vdt should be the center of the new bubble, so that u t+dt (vdt) = U > 0, we can write: which trivially leads to the result by using Eq. 17.

We now evaluate the simplified expression of e I = e [vdt-θ,vdt+θ] . Since (1 -α)Sgd I (vdt, c)) is null, as the stimulation is symmetric around vdt (this remains true for other inputs as long as they remain symmetric in the interval I), we can factorize the expression as follows: With U remaining a function of γ (Eq. 18), and η almost a constant (as γ still slightly constrains the values of θ and σ).

Fig. 1 .

 1 Fig. 1. Graphical illustration of the predictive CNFT dynamics. Although 2 bubbles are competing for the focus on the field u t , a single stimuli from the stimulation s confirms the rightward translation anticipated by the predictor p (predicted and actual movements are indicated by arrows). Combining the different fields (terms in Eq. 1) and applying the non linear function f gives the new focus activity u t+dt , the inhibitory field induced by the competition eliminating most non coherent activities. Convergence thus occurs on the stimulus that matches the prediction.

Fig. 2 .

 2 Fig. 2. Comparison of the input activity i with or without prediction.(left) Without prediction, a stimuli and a distracter of equal intensity are not differentiated by the CNFT. Any of them may thus act as an attractor in Eq. 1, even when the convergence already occurred but the distracter is located on the trajectory (x d ). (right) If the prediction matches the actual stimulus movement (xp = xs) and not the distracter, the symmetry is broken and the selection or tracking is biased.

Fig. 3 .

 3 Fig. 3. (a) Typical convergence and tracking error curves for 3 different speeds for γ(v) = v. Although produced by simulation, the results have been smoothed out for a qualitative analysis of the dynamics and axis scales are thus irrelevant. Starting from a distance d 0 between the target and bubble, the tracking accuracy deteriorates as the speed increases. Oscillations described in the main text can be observed, as the bubble moves closer and away from the target. (b) The range of the oscillations are reproduced, and superimposed on the integrated lateral kernel function w. Oscillations result from an alternation between positive (increasing the amplitude) and negative weights (facilitating the movement).

Fig. 5 .

 5 Fig.5. Illustration of scenarios C to G (please refer to the main text for their exact description). Each scenario is represented by 2 snapshots of the input stimulation. The first snapshot always corresponds to t = 0, where the algorithm gets some time to converge on one of the stimuli (and there is only one but for scenario C ). The second snapshot is associated to a later time, chosen to be representative of the input dynamics. The position of the target, its trajectory as well as the occluded area for scenario G are superimposed on the fields.

  Fig 7 also shows the mean error committed as a function of the noise level, i.e. the standard deviation of the Gaussian distribution with values in the interval [0, 1]. The bias in activity from the prediction results in a relative decrease of the noise level, similarly to what was illustrated on Fig 2.

  Fig.7. Mean error distance for different levels of noise with (red) or without prediction (blue). Due to the dynamics of stabilization through activity feedback acting against the mobility of a low amplitude bubble, a large level of noise might lead to a more reactive system. With a single spatiotemporally coherent stimulus to track on the field, this explains the otherwise strange performance boost observed when the standard deviation goes over 0.8, as the system cannot spontaneously focus on another target.

-

  gx [x1,x2] (µ, σ) = -σ 2 2 (g(x2 -µ, σ) -g(x1 -µ, σ)) + µg [x1,x2] (µ, σ)(14)To simplify the expressions in the following sections, we also introduce a function gd:gd [x1,x2] (µ, σ) = gx [x1,x2] (µ, σ) -vdt × g [x1,x2] (µ, σ) x2 -µ, σ) -g(x1 -µ, σ)) + (µ -vdt)g [x1,x2] (µ, σ)(15)The product of non-normal Gaussian distributions is as well needed for the convolution:g(x -µ 1 , σ 1 )g(x -µ 2 , σ 2 ) = e x -µ m , σ m )

U = 1

 1 vdt) + αp(vdt) + (1 -α)s(vdt))

σ 2 gd I 0, a 2 + σ 2 - bB √ b 2

 222 + σ 2 gd I 0, b 2 + σ 2


Roberto A Vazquez

Bernard Girau

Jean-Charles Quinton

Visual attention using spiking neural maps

Visual attention is a mechanism that biological systems have developed to reduce the large amount of visual information in order to efficiently perform tasks such as learning, recognition, tracking, etc. In this paper, we describe a simple spiking neural network model that is able to detect, focus on and track a stimulus even in the presence of noise or distracters. Instead of using a regular rate-coding neuron model based on the continuum neural field theory (CNFT), we propose to use a time-based code by means of a network composed of leaky integrate-and-fire (LIF) neurons. The proposal is experimentally compared against the usual CNFT-based model.

I. INTRODUCTION

Attention enables us to dynamically select and enhance the processing of the most relevant stimuli and events at each moment [START_REF] Busse | The spread of attention across modalities and space in a multisensory object[END_REF]. Visual attention is a powerful mechanism that enables perception to focus on a small subset ("where to look") of the information picked up by our eyes [START_REF] Maunsell | Feature-based attention in visual cortex[END_REF], based on both bottom-up and top-down cues [3].

The control of focal visual attention involves an intricate network of brain areas, spanning from the primary visual cortex to the prefrontal cortex. Selecting where to attend next is primarily controlled by the dorsal visual processing stream (or "where/how" stream) which comprises cortical areas in the posterior parietal cortex, whereas the ventral visual processing stream (or "what" stream), comprising cortical areas in the inferotemporal cortex, is primarily concerned with localized object recognition [START_REF] Ungerleider | Two Cortical Visual Systems[END_REF].

Several computational theories and models have been developed for how attention may be attracted towards a particular object in the scene rather than another [START_REF] Treisman | A feature-integration theory of attention[END_REF], [START_REF] Itti | Computational modelling of visual attention[END_REF], [START_REF] Koch | Shifts in selective visual attention: towards the underlying neural circuitry[END_REF]. Recently, the authors in [START_REF] Rougier | Emergence of attention within a neural population[END_REF] have demonstrated that bottomup (i.e. stimulus driven) attention may be seen as an emergent property of a neural population using the Continuum Neural Field Theory. From a pool of neurons spread over two maps, one input map feeding a focus map, a bubble of activity emerges within the focus map at the precise location of a stimulus presented within the input map. Furthermore, when noise or distracters are added, the bubble of activity stays focused on the chosen stimulus and then, between several simultaneously possible objects, the model is able to maintain visual attention onto the one stimulus it first focused.

The goal of this paper is to show that similar attentional properties (detection, focus and tracking of a stimulus) can Roberto Vazquez is with the Intelligent Systems Group, Faculty of Engineering, La Salle University, Benjamín Franklin 47, Col. Condesa, 06140, Mexico City, Mexico (email: ravem@lasallistas.org.mx) -Bernard Girau and Jean-Charles Quinton are with the LORIA/Cortex project -Campus Scientifique, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France (email: bernard.girau@loria.fr). This work was supported by the INRIA CorTexMex Research Associate Team. be obtained by a population of spiking neurons, with an improved robustness in the presence of noise or distracters. The visual attention model we propose is based on a simple spiking neural model, the so-called leaky integrate-and-fire (LIF) neurons. This model explicitly handles temporal events (spikes) that are usually "hidden" in the rate-coding neuron models used by the continuum neural field theory. This proposal is experimentally compared against the CNFT-based model corroborating its robustness to noise and distracters while achieving the main visual functionalities depicted in [START_REF] Rougier | Emergence of attention within a neural population[END_REF]: competitive behavior, tracking and target switching.

Section II describes the motivation for using spiking neurons within neural models of visual attention, while introducing both CNFT-based and LIF-based models. Our model is precisely defined in section III. Its properties are experimentally validated in section IV.

II. MOTIVATION AND RELATED WORK

Neural fields, and more specifically CNFT-based neural fields, have proved very powerful to build neural models able to perform more or less complex visual tasks such as visual attention [START_REF] Rougier | Emergence of attention within a neural population[END_REF], scene exploration [START_REF] Fix | Anticipatory Behavior in Adaptive Learning Systems: From Brains to Individual and Social Behavior[END_REF], overt attention [START_REF] Fix | A dynamic neural field approach to the covert and overt deployment of spatial attention[END_REF] or motion discrimination [START_REF] Cerda | Bio-inspired visual sequences classification[END_REF]. These models are massively distributed, so that we aim at using them in autonomous embedded systems thanks to their hardware parallel implementation. Preliminary attempts such as in [START_REF] Torres-Huitzil | On-chip visual perception of motion : A bio-inspired connectionist model on FPGA[END_REF] have shown that their dense local interactions are too demanding to define efficient parallel implementations of these models. On the other hand, the study in [START_REF] Girau | Massively distributed digital implementation of an integrate-and-fire LEGION network for visual scene segmentation[END_REF] has shown that simple spiking neurons may be efficiently assembled on a hardware device such as an FPGA (Field Programmable Gate Array) thanks to their reduced communication bandwidth: whatever the level of complexity of the internal computations of each neuron, simple 1-bit messages (spikes) are exchanged between neurons. This has motivated the will to define populations of simple spiking neurons that mimic the properties of the CNFT-based models for visual attention. This is not the first attempt to define spiking neural fields. A related approach may be found in [START_REF] Chevallier | Visual focus with spiking neurons[END_REF]. In order to precisely situate the contribution of our model with respect to [START_REF] Rougier | Emergence of attention within a neural population[END_REF] and [START_REF] Chevallier | Visual focus with spiking neurons[END_REF], we now give more details about these two visual attention models that are able to perform a tracking task of a target even in the presence of very strong noise or in the presence of a lot of distracters.

A. CNFT visual attention model

The Continuum Neural Field Theory is a kind of dynamic neural field model that implements lateral competition within cortical maps [START_REF] Amari | Dynamics of pattern formation in lateral-inhibition type neural fields[END_REF], [START_REF] Taylor | Neural bubble dynamics in two dimensions: Foundations[END_REF]. The CNFT can be reduced to a single differential equation that describes the evolution of the membrane potential of neurons over cortical maps. This is a continuous approximation of the evolution of large populations of neurons that interact through excitatory and inhibitory connections: close neurons tend to be reciprocally excited, distant neurons inhibit themselves.

In [START_REF] Rougier | Emergence of attention within a neural population[END_REF] the authors propose a visual attention model based on a discrete version of the CNFT. This model is highly robust and it is able to track a static or moving target in the presence of noise with high intensity or despite a lot of distracters possibly more salient than the target. The main hypothesis about the target stimulus is that it has a spatiotemporal continuity that should be observable by the model.

A neural position is labeled by a vector x, which represents a two-component quantity designing a position on a manifold M in bijection with [0, 1]

2 . The membrane potential of a neuron at position x and time t is denoted by u (x, t). The lateral connection weight function w (xx ′ ) is a difference of Gaussian function applied to distance |xx ′ |. An afferent connection weight s (x M ′ , x M ) applies to the local stimulus received at position x M in manifold M from position x M ′ in manifold M ′ . The membrane potential u (x, t) satisfies the following equation:

τ ∂u(x,t) ∂t = -u (x, t) +h + 1 α M w M (x -x ′ )f [u (x ′ , t)] dx ′ + 1 α M ′ s (x, y) I (y, t) dy (1)
where f represents the mean firing rate as some function of the membrane potential u of the relevant cell, I (y, t) is the external stimulus at position y and time t in M ′ , h is the neuron threshold and α is a scaling term.

Lateral connection weights are given by:

w M (x -x ′ ) = Ae |x-x ′ | 2 a 2 -Be |x-x ′ | 2 b 2 (2)
and afferent connection weights are given by:

s (x, y) = Ce |x-y| 2 c 2 (3)
Furthermore, the activity of a neuron is bounded between 0 and 1: if u (x, t) > 1, u (x, t) = 1, and if u (x, t) < 0, u (x, t) = 0.

The model consists of two maps, input and focus, each of them being of size n × n units. The input map corresponds to an entry that is feeding the focus map, whereas the focus map represents a cortical layer whose units possess localized receptive fields on the surface of the input. Each unit x ij of the focus map receives its input from the input map using Eq. 3 which corresponds to a localized receptive field. While the input map does not have any lateral interaction or feedback, the units in the focus map are laterally connected using a difference of Gaussians.

B. LIF visual attention model

In [START_REF] Chevallier | Visual focus with spiking neurons[END_REF] the authors propose a visual attention model based on Leaky Integrate and Fire neurons. This work can be seen as a direct transformation of the CNFT-based model of [START_REF] Rougier | Emergence of attention within a neural population[END_REF] by using spike-based computations. It may also be seen as the attentional part of a more general model of covert attention that includes an additional pre-attentional part [START_REF] Chevallier | Covert attention with a spiking neural network[END_REF]. This visual attention model is again able to focus and stay focused even when the stimulus moves. In addition, they show that noisy backgrounds and distracters only have a small influence on the behavior of the model. Indeed, the experimental results are close to those observed with a CNFT model, although the CNFT-based model seems to be more accurate than the LIF-based model. The authors also experimentally validate the fast computation time achieved by their model.

Again, this model consists of two maps, input and focus, each of them being of size n × n units. Depending on the parameters used to model a neuron, this set of neurons can either integrate the information over a predefined temporal window or act as a synchrony detector (emitting spikes when inputs are condensed in a small period of time). Neurons from the input map behave as integrators and neurons from the focus map as synchrony detectors.

The input map translates an input stimulus into spike trains. Each unit of the input map is connected to each unit of the focus map through a Gaussian mask. A mask is a static weight matrix and it defines a generic projection of a neural map onto another. The weight matrix values of the Gaussian mask can be viewed as the parameters of a Gaussian image filter. On the other hand, this projection could be related to the afferent connections given by Eq. 3.

The focus map is laterally connected using a difference of Gaussians as in [START_REF] Rougier | Emergence of attention within a neural population[END_REF], see Eq. 2, which mutually excites adjacent neighbors and inhibits distant ones. This lateral connectivity alone is not sufficient to maintain a self-sustained activity. For this purpose, it needs the spikes from the input map to have an ongoing activity.

III. PROPOSED MODEL

Instead of using rate-coded neurons as in [START_REF] Rougier | Emergence of attention within a neural population[END_REF], the model we propose uses leaky integrate-and-fire neurons as in [START_REF] Chevallier | Visual focus with spiking neurons[END_REF].

A. Definition

Our model mostly consists of a 2D neural map of size n×n LIF neurons that emit their spikes at discrete times. The membrane potential V i of neuron i is given by the following differential equation:

τ ∂V i ∂t = -g leak (V i (t) -E leak ) + γ • I i (t) (4)
where g leak and E leak are the conductance and the reversal potential of the voltage-independent leak current, and τ is the membrane time constant. I i (t) is the input stimulus given at time t, and γ is a scaling term. Each neuron is associated with a pixel, i.e. the pixel luminance determines the input term I i (t) of the corresponding neuron. This differential equation is approximated through a simple Euler scheme (see [START_REF] Cessac | A discrete time neural network model with spiking neurons. rigorous results on the spontaneous dynamics[END_REF] to assert that simple discrete-time generalized LIF neurons have the same expressive power as the underlying continuous models). Whenever the membrane potential V reaches the threshold θ, a spike is fired, and V is instantaneously reset to rest.

Finally, the membrane potential V i of neuron i takes into account the influence of incoming spikes by means of the following equation:

V i (t + dt) = V i (t) + τ ∂V i ∂t dt + I syn i (t) γ (5)
where I syn i (t) is the synaptic current due to the action of other neurons of the network describing the influence of incoming spikes on the membrane potential. It is given by the following equation:

I syn i (t) = j w ij S j (t) (6)
where w ij is a weight matrix given by a difference of Gaussians (see Eq. 2) centered at neuron i, which excites adjacent neighbors and inhibits distant ones. Finally, the instantaneous spike S j (t) is given by:

S j (t) = 1 V j (t) ≥ θ 0 V j (t) < θ (7)

B. Model positioning

Although this model uses the same kind of LIF neurons as in the model proposed in [START_REF] Chevallier | Visual focus with spiking neurons[END_REF], there are three main differences that impact the computational cost of the model and its behavior.

First of all, our proposal could be seen as a reduced version of the model in [START_REF] Chevallier | Visual focus with spiking neurons[END_REF], since the focus neural map is the only one composed by LIF neurons. The input map simply sends the stimulus information to the focus map. Therefore the input stimulus is not transformed into spike trains as in [START_REF] Chevallier | Visual focus with spiking neurons[END_REF], it directly feeds the focus map. This simplification not only induces a lower computational cost 1 , above all it has a strong influence on the future ability of the model to be mapped onto hardware parallel devices, since it avoids the complex connection topology required by two neural maps interconnected in a retinotopical way with multiple overlapping receptive fields.

Another difference is related to the use of filters. In [START_REF] Chevallier | Visual focus with spiking neurons[END_REF] the output of the input map is filtered by means of a Gaussian filter and then it is sent to the focus map. Our model does not apply any Gaussian filter, the receptive fields in the input map being directly received by the focus map without any type of weighted afferent connections. The lateral weights of the focus map appear as sufficient to spatially smooth out the activity of the neurons over small receptive fields.

A third difference is the γ scaling term that appears both in the evolution of the membrane potential with respect to the received stimulus and in the influence of the incoming spikes. This parameter enables us to easily balance both kinds of influences. The experimental results in section IV includes a detailed study of the impact of such a balance on both robustness and behavior of the model.

Finally, we not only propose a LIF-based model that exhibits similar properties as the initial CNFT-based model, we perform a large experimental study so as to quantitatively compare the degree of robustness of both models.

IV. EXPERIMENTAL RESULTS

Before studying the robustness of the model in the presence of perturbations, it is necessary to validate its ability to provide the attentional properties that are typical of the usual CNFT-based model.

A. Experimental setup

In order to test the proposed model, we use an experimental setup similar to [START_REF] Rougier | Emergence of attention within a neural population[END_REF] and [START_REF] Chevallier | Visual focus with spiking neurons[END_REF]: a stimulus follows a circular path on a n × n pixels input image with either noise or distracters in the background. The stimulus is a Gaussian patch whose center is localized at (x c , y c ) of width W and intensity I given by:

S (x, y, r, θ, W, I) = Ie -(x-xc) 2 W 2 e -(y-yc ) 2 W 2 (8)
where x c = r sin θ and y c = r cos θ.

On the one hand, distracters are exact copies of the Gaussian patch, but they lack spatio-temporal continuity. On the other hand, the added noise is assumed to be independently drawn from a Gaussian distribution with different variance levels σ and mean zero.

Parameters for generating the synaptic weight connections are defined as: α = n/2, A = 25/α, a = 5/n, B = 12.5/α and b = 75/n. It must be noted that these weights could be tuned to optimize each type of expected behavior as in [START_REF] Quinton | Exploring and optimizing dynamic neural fields parameters using genetic algorithms[END_REF], but in this work, we rather adopt the approach of [START_REF] Rougier | Emergence of attention within a neural population[END_REF] where the authors study the ability of the model to simultaneously exhibit all properties with the same set of weights that are experimentally chosen.

During the integration step dt = 0.1, τ = 1. Based on eq. 4, we choose to increase the influence of the input compared to the lateral connections by setting γ = 10. This gaining factor helps the neurons in the focus map to rapidly generate a spike. The exact influence of γ is analyzed in IV-C.

As in [START_REF] Rougier | Emergence of attention within a neural population[END_REF] and [START_REF] Chevallier | Visual focus with spiking neurons[END_REF], the input image only contains the stimulus target until the first spikes appear on the focus map. In order to ensure that the spikes emerge in the focus map, we wait ten computational steps (one second) before the Gaussian patch starts to move, or before adding some distracters or noise. After that, we generate a new image each 10 computational steps.

To validate the accuracy of the proposal, we compute the normalized euclidean distance between the stimulus center and the center of the activity in the focus map. Following the same definition as in [START_REF] Chevallier | Visual focus with spiking neurons[END_REF], the center of activity is defined as the center of all spikes emitted by the focus map during the integration steps of the image presentation. This center of activity is computed as in [START_REF] Rougier | Emergence of attention within a neural population[END_REF] by means of the following equation:

(x c , y c ) = i,j i n a (i, j) i,j a (i, j) , i,j j n a (i, j) i,j a (i, j) (9)
where a (i, j) is the value at position (i, j).

As explained in this section, our experiments use synthetic inputs, since we aim at comparing the models with the same criteria already used in similar studies ( [START_REF] Rougier | Emergence of attention within a neural population[END_REF], [START_REF] Chevallier | Visual focus with spiking neurons[END_REF], [START_REF] Quinton | Exploring and optimizing dynamic neural fields parameters using genetic algorithms[END_REF]) that clearly quantify the visual attention properties alone. The question of how the models perform on real video sequences is not addressed in this paper. Nevertheless, it must be noted that all these attentional models may be used for real video sequences, adding a pre-attentional processing part (as in [START_REF] Chevallier | Covert attention with a spiking neural network[END_REF] or [START_REF] Torres-Huitzil | On-chip visual perception of motion : A bio-inspired connectionist model on FPGA[END_REF]), but in this case the performance of the model strongly depends on the pre-attentional processing and its ability to properly extract moving objects.

B. Behavioral properties

Following the scenarios defined in [START_REF] Quinton | Exploring and optimizing dynamic neural fields parameters using genetic algorithms[END_REF] to characterize the main expected properties of visual attentional models, we may summarize these properties as competition (ability to focus on a specific target when multiple targets appear in the visual stimulus), tracking (ability to track a focused target that moves), switching (ability to focus on a new target when the previous one disappears), and tolerance to perturbations (noise and/or distracters). Since the latter is directly linked to the study of the robustness of the model in IV-C, we consider here the three first attentional properties.

1) Competitive behavior: The proposed model exhibits a competitive behavior with the same set of synaptic weights used throughout this paper. Furthermore, even when targets are too close, the model is able to focus on only one target after a small number of computational steps.

The only constraint that should be satisfied is that for a pair number of targets, they should not be placed regularly and equidistantly from the center of the focus map: this is a well-known side effect of pure synchronous evaluation when interactions are perfectly symmetrical, that results in all bubbles cyclically appearing and disappearing. Any level of noise of asynchronous evaluation avoids this side effect [START_REF] Rougier | Synchronous and asynchronous evaluation of dynamic neural fields[END_REF].

2) Tracking: The model is able to track a focused target that moves in the visual scene. Nonetheless, we found that the velocity at which the targets update their position should be at least two computational steps, with the chosen setup. Again, this is a well-known property of the CNFT-based model: when the target moves too fast, it rapidly goes out of the excitatory range of the emerged bubble, and thus becomes unable to attract it towards the new position of the target. As expected, we also observed that faster tracking can be achieved with an appropriate set of synaptic weights.

3) Switching: Again, the qualitative behavior is preserved: when the initial target disappears, a self-sustained bubble of activity remains a few computational steps at the last location of the bubble, but its amplitude decreases and the system finally focuses on another target. In addition, we observed that the reactivity of the model is strongly linked to the spiking threshold of the neurons: a higher threshold reduces the ability of the bubble to have a long self-sustained activity.

C. Robustness

Three types of experiments were performed to evaluate the robustness of the proposal. The first set of experiments evaluates the robustness of the proposal in the presence of noise. The second set of experiments verifies the robustness of the proposal in the presence of distracters. Finally, in the third set of experiments, we study the effect of changing the gaining factor γ in the accuracy and reactivity of the proposed model. In all experiments, we consider perturbations added to a bell-shaped input such as depicted in figure 1.

In [START_REF] Rougier | Emergence of attention within a neural population[END_REF] and [START_REF] Chevallier | Visual focus with spiking neurons[END_REF], the authors assume that noise or distracters are added each computational step. It is important to notice that in those experiments the time that noise or distracters remain in the same position might be too short to significantly alter the behavior of the model. In this paper we study the behavior of the model when the noise or distracters are updated at the same rate as the target as well as when distracters and noise are updated at a slower or faster rate than the target, so as to assert that some results are not obtained only thanks to an unfair instability of the perturbations, as it might sometimes be the case in [START_REF] Rougier | Emergence of attention within a neural population[END_REF] and [START_REF] Chevallier | Visual focus with spiking neurons[END_REF].

1) Robustness of the model in the presence of noise:

During the first set of experiments we could observe that the proposed model was able to track the stimulus even when the intensity of the noise was greater than the input stimulus (see Figure 2(a) and Figure 2(b)). The obtained results show that the proposed model is highly robust to the presence of noise (see Figure 4(a)). Despite the high intensity of the noise added to the input, the proposed model stays focused on the input target (see Figure 3).

As was demonstrated in [START_REF] Rougier | Emergence of attention within a neural population[END_REF], the CNFT visual attention model also provides highly acceptable results when the target is perturbed by some noise (see Figures 4(b)). However, the accuracy of the CNFT model starts to decrease when the intensity of the noise added to the background exceeds the intensity of the target (variance greater than 0.5).

As can be observed from Figures 4(a) and 4(b), the time that noise remains without change modifies the accuracy of the model. If noise changes rapidly, for example every computational step (0.1 second), the accuracy that the model provides is higher than the accuracy achieved when the noise changes slowly. This phenomenon may be easily explained by the inertia of the bubble of activity that is less disturbed by transient perturbations. Nevertheless, the accuracy obtained with the proposed method is still highly acceptable when noise is updated every two seconds. On the contrary, under the same conditions, the accuracy achieved by the CNFT- Fig. 3. Trajectories followed by the bubble of activity in the focus map when the target changes its position each ten computational steps along its circular trajectory. Each column presents the trajectory followed by the bubble when the input is perturbed by different levels of Gaussian noise (0.2, 0.4, 0.6 and 0.9). Each row presents the trajectory followed by the bubble when the noise is updated at different rates (1, 5, 10 and 20 computational steps).

based model decreases more rapidly when noise shows some inertia.

The LIF visual attention model of [START_REF] Chevallier | Visual focus with spiking neurons[END_REF] also provides highly acceptable results when the target is perturbed by noise, though the influence of the noise inertia is not studied. Anyway, our model provides comparable results even when the background is highly noisy, despite its simplicity (one map of LIF neurons).

2) Robustness of the model in the presence of distracters: During the second set of experiments, we observed that the proposed model was also robust in the presence of several distracters (even more salient than the target), see Figure 5(a) and Figure 5(b). The obtained results show that the proposed model is highly robust to the presence of distracters (see Figure 7(a)). Despite the number of distracters added to the input, the proposed model stays focused on the input target (see Figure 6).

On the other hand, as was stated before, the CNFT visual attention model also provides highly acceptable results when the target is perturbed by distracters. However, the accuracy of the CNFT model starts to decrease when the number of distracters added to the background is greater than 10 (see Figures 7(b)).

Again, the time that distracters remain unchanged modifies the accuracy of the model, see Figures 7(a As for the noisy conditions, the accuracy is better with rapidly changing distracters (e.g. changing every computational step), but still highly acceptable when the positions of distracters are updated every two seconds, whereas the CNFT-based model accuracy decreases more rapidly with slightly more inertial distracters.

From these two first sets of experiments, we can state 3) Robustness of the model with different values of γ: In this set of experiments, we modified the value of the γ factor in order to see its influence on the behavior of the proposed model.

All previous experiments were performed using γ = 10. In this section the same experiments are studied while setting γ with 1, 5, 15 and 20. Figure 8 shows the experimental results obtained using different values of γ: each bar shows the average accuracy of the model computed when the target and noise or distracters are updated at different rates.

The behavior of the model when the input is altered by noise and using different values of γ is shown in Figure 8(a). Fig. 6. Trajectories followed by the bubble of activity in the focus map when the target changes its position each ten computational steps. Each column presents the trajectory followed by the bubble when some distracters are added (3, 6, 12 and 24 distracters). Each row presents the trajectory followed by the bubble when distracters are updated at different rates (every 1, 5, 10 and 20 computational steps). The first fact that we have to point out is that when γ = 1 and the input stimulus lacks of noise, the neurons of the model do not spike at all; when some noise is added to the input stimulus the neurons of the model spike but the model is not able to focus on and track the input target. On the other hand, the accuracy of the model starts to decrease when the noise added to the input is greater that 0.6 and γ is greater than 15.

The behavior of the model when several distracters are added to the input with different values of γ is shown in Figure 8(b). In these experiments we observed that no matter the number of distracters, if γ = 1 the neurons of the model do not spike, whereas the behavior of the model is rather similar for all values of γ that are strictly greater than 1. count the number of computational steps that the model needs before the first spike emerges.

The number of computational steps required by the model in order to generate the first spikes are shown in Figure 9. When the value of γ is small, the model requires more computational steps to generate the first spikes in the focus map, see Figure 9(a). Nevertheless, this influence remains weak, except for the case γ = 1.

V. CONCLUSION

A simple visual attention model based of leaky integrateand-fire neurons has been described in this paper. Through several experiments, this model has been proved to be very robust and able to track one moving target in the presence of noise with very high intensity or in the presence of a lot of distracters.

The experimental results obtained with the proposal are comparable to those provided by previously defined CNFTbased and LIF-based models, in terms of both behavioral properties and robustness. However, the level of robustness is greatly improved with respect to the initial CNFT-based model, and, compared against the previous LIF-based model, no filtering technique is used and the computational resources reduce to only one neural map of LIF neurons.

Our model introduces a gaining factor that increases the influence of the input stimulus. The experimental study shows that its influence remains limited, except for its lowest values. Nevertheless, it could have a more significant influence when assembling several spiking neural maps to perform more complex visual tasks such as overt attention.

Though reducing the topological constraints of future hardware parallel implementations thanks to 1-bit communications between neurons (spikes), this model still is not able to be mapped onto hardware devices with a fully parallel approach because of the dense lateral connections. Current efforts aim at exploiting the robustness of the spike-based approach so as to reduce the range of the lateral connection kernel while maintaining highly satisfactory attentional properties.

Fig. 1 .

 1 Fig. 1. Input is a bell-shaped curve centered around (xc, yc) representing an external input. This information is received by the input map which is directly connected to the focus map. (a) Noiseless input. (b) Neurons firing in the focus map as a response to the noiseless input.

Fig. 2 .

 2 Fig. 2. Noisy input. The added noise is assumed to be independently drawn from a Gaussian distribution. (a) Noisy input with σ = 1. (b) Neurons firing in the focus map.

  ) and 7(b).

Fig. 4 .

 4 Fig. 4. A zero-mean Gaussian noise with different variances from 0 to 1 has been added to the input stimulus. The target changes its position every ten computational steps (one second) and the noise changes at differents rates (every 1, 5, 10, 15 or 20 computational steps of 0.1 second, as represented by the bars from left to right for each level of noise). (a) Accuracy of the proposed model in terms of the error level as a measure of the distance to the target using a focus map with n = 50 (using a normalized distance, i.e. computed in the [0, 1] × [0, 1] continuous field that is discretized onto the n × n neurons). (b) Accuracy of the CNFT model in terms of the error level as a measure of the distance to the target using a focus map with n = 50.

Fig. 5 .

 5 Fig. 5. Distracters are exact copies of the input Gaussian patch, but they lack spatio-temporal continuity. (a) Input with 30 distracters (b) Neurons firing in the focus map.

Fig. 7 .

 7 Fig. 7. Zero to 30 distracters (with same width and intensity as the original stimulus) were added to the stimulus. The target changes its position every ten computational steps (one second) and the distracters' positions change at different rates. (a) Accuracy of the proposed model in terms of the error level as a measure of the distance to the target using a focus map with n = 50. (b) Accuracy of the CNFT model in terms of the error level as a measure of the distance to the target using a focus map with n = 50.

4 )Fig. 8 .

 48 Fig.8. Average accuracy of the proposed model in terms of the error level as a measure of the distance to the target using a focus map with n = 50 and different values of γ (1, 5, 10, 15 and 20, as represented by the bars from left to right). The target changes its position every ten computational steps. (a) A zero-mean Gaussian noise with different variances from 0 to 1 is added to the input stimulus. The noise changes every ten computational steps. (b) Zero to 30 distracters (with same width and intensity as the original stimulus) are added to the stimulus. The distracters' positions change every ten computational steps.

Fig. 9 .

 9 Fig.9. Computational steps required by the model in order to generate the first spike using different values of γ. The target changes its position every ten computational steps. (a) A zero-mean Gaussian noise with different variances from 0 to 1 is added to the input stimulus. The noise changes every ten computational steps. (b) Zero to 30 distracters (with same width and intensity as the original stimulus) are added to the stimulus. The distracters' positions change every ten computational steps.

The architecture of[START_REF] Chevallier | Visual focus with spiking neurons[END_REF] is based on a one-to-one correspondance between both maps of LIF neurons. The computational cost mainly lies in the constant update of neuron potentials in both maps. Therefore a rough approximation of the computational cost of[START_REF] Chevallier | Visual focus with spiking neurons[END_REF] is twice the computational cost of our model.