Bernard Ans

Yves Coiton

Jean-Claude Ûilhodes

Jean-Luc Velay

A Neural Network Model for Temporal Sequence Learning and Motor Programming

Keywords: Dynamic memory, Motor program, Motor shape, Neural network, Pointing movement, Temporal sequence, Temporary memory, Tracking movement

A neural network mode/ for fast learning and storage of temporal sequences is presented. The recall of a learned sequence is triggered by the occurrence of an item relating to its identity, and one of the main distinctive features of this mode! is that the speed at which a sequence is repeated can be freely modulated by a contrai subsystem. The possible applications of the mode! are illustrated by applying it to the production of motor forms. It is shown that any spatial shape memorized in exteroceptive terms can be reproduced in terms of movement by any of the effector systems of the body, and in particular by a simulated jointed arm, at any point in its working space and at any suitable size scale. Our theoretical approach reinforces the idea that the structures responsible for planning a movement in the central nervous system might be large/y independent of the motor systems performing this movement.

INTRODUCTION

Most of the existing neural network models for asso ciative memory (Hinton & Anderson, 1981;Rumelhart & McClelland, 1986;[START_REF] Grossberg | Nonlinear neural networks: Principles, mech anisms, and architectures[END_REF][START_REF] Kohonen | Se{forganization and associative memory[END_REF] have been of a static type. They have generally been designed to produce a single learned response to a given trigger stimulus, even if several processing cycles may be necessary for the expected response to stabilize completely. On the other band, there exists a family of memory models of a more dynamic type, in which a specific temporal sequence of expected events is recalled in response to a single trigger stimulus or from a "seed ing" of the system. However, this family of neural net work models for temporal sequence learning and recall ( e.g., [START_REF] Kohonen | Associative memory: A system theoretical ap proach[END_REF][START_REF] Willwacher | Storage of a temporal sequence in a network[END_REF][START_REF] Jordan | Serial order: A paraUel distributed processing approach[END_REF]Dehaene, Changeux, & Nadal, 1987;[START_REF] Elman | Finding structure in time[END_REF]Ans, 1990b;[START_REF] Reiss | Storing temporal sequences[END_REF] , or dynamic memories in short, is not as large as that of static models because the temporal modelling of neural processes is not easy. This is nevertheless a really fundamental line of re-search, if only because in ecological situations, the con trolled production of a coherent chain of events in re sponse to the requirements of the environmental con text is a regular and by no means an exceptional occurrence. At the more theoretical level, the exact knowledge of the working of a connectionist system consisting of "neurons" and "synapses," which is able to perform a coherent sequence of "instructions" in response to a specific execute command, should be of great potential value as a means of understanding how sequential programs are actually implemented at the neurobiological level.

The existing connectionist models for dynamic memories are varied in regard to their degree of bio logical realism and their performances. One model close to neurobiology is that developed by [START_REF] Dehaene | Neural networks that learn temporal sequences by selection[END_REF]. The architecture of this model is based on for mal synaptic triads that are responsible for the hetero synaptic regulation [START_REF] Heidmann | Un modèle moléculaire de régulation d'efficacité au niveau postsynaptique d'une synapse chimique[END_REF] and the synaptic efficacies change in accordance with Hebb's rule [START_REF] Hebb | The organization of behavior[END_REF]. From the behavioural point of view this model is of the selectionist type, which means that although the network is initially capable of sponta neously producing a very large variety of sequences ( pre-representations), only those reinforced by imita tive learning will persist, and the remainder will tend to disappear. Behaviour of this type bas been studied with special reference to the acquisition of bird sangs [START_REF] Konishi | Birdsong: From behavior to neuron[END_REF]. Although this approach is particularly attractive and well suited to the task in question, it is less appropriate when the sequences to be learned are not part of a "genetically" predetermined repertoire and when the model is constantly faced with navel sit uations that have to be memorized.

The model architecture designed by [START_REF] Kohonen | Associative memory: A system theoretical ap proach[END_REF] is a prototype that bas inspired most subsequent models in some way or another. Kohonen's model in volves an associative memory that stores the relation between the current item in a sequence and the ordered succession of its K predecessors. This recent sequence history is obtained by delaying the output from the associative memory by introducing K lag registers that form a spatial buffer. The length of the temporal span, that is, the size of the time window encompassing the current sequence history within the spatial buffer, is therefore fixed and determined by the number of lag registers used. In addition, the associative layer records the general context surrounding the sequence being processed that will subsequently serve to trigger the sequence recall as well as to salve some of the problems inherent to distributed memory systems, centering on the interferences between the sequences. Kohonen clearly set out the main principles underlying his one layer model, without specifying, however, exactly which type of learning rule be uses. lt is hence difficult to assess the efficiency of his model.

Among the descendants of this model, in which the sequential production of items is governed by their re cent history and their surrounding context, the one de veloped by [START_REF] Jordan | Serial order: A paraUel distributed processing approach[END_REF] is characterized by the method used for short-term storage. The current se quence history is set up within a single layer forming a feedback loop: the state of the layer at instant t depends on its own previous state at instant l n -• and on the item in the sequence at that previous instant. By a recursive process, the time span encompasses a recent history, the length of which depends on the structure of the sequence being currently processed. The author uses a layer of hidden units in which the current sequence history is combined with the context involved in the processing, which is conceived of as a "plan." The weights of the connections in hidden and output layers change according to the back propagation algorithm [START_REF] Le Cun | Leaming process in an asymetric threshold net work[END_REF][START_REF] Rumelhart | Learning internai representations by error propagation[END_REF]. The model proposed by Jordan, like that by [START_REF] Elman | Finding structure in time[END_REF], which is an interesting variation on the former, is efficient in view of its long-term storage ca pacity. This quality is in fact due to the learning algo rithm on which the processing is based, which bas been intensively used elsewhere precisely because of its ef fi ciency, despite the fact that it is reputed not to be a very realistic representation of neurobiological pro cesses [START_REF] Grossberg | Competitive learning: From interactive ac tivation to adaptive resonance[END_REF][START_REF] Crick | The recent excitement about neural networks[END_REF]Hinton & Shal lice, 1991;[START_REF] Reiss | Storing temporal sequences[END_REF] . And in the present paper, where we propose a neuromimetic model for temporal sequence learning and recall, we do not use a learning method ofthis type because one of our main aims here is to keep as close as possible to the well known basic neural processes.

This principle ofneurobiological plausibility is par ticularly adopted in [START_REF] Reiss | Storing temporal sequences[END_REF] where back propagation is not consequently used. The authors propose bath a theoretical analysis and a set of simu lations of a model for temporal sequence storage in which the main distinctive feature is that the short term storage of the current sequence history is obtained without using spatial buffering or a single layer with recurrent connections. lt is built up within a layer of leaky integrator neurons, with a range oftime constants, that store directly on their membrane a representation of the succession of their recent inputs. This model features a number of required properties, in particular: the sequences are quickly learned because there are no hidden layers to slow down learning; the system leads to recall in which there is no distortion of the tempor ality of the patterns forming sequences ( the lengths of time each pattern was presented during the learning phase) and in which disambiguation is achieved ( in that the sequence ABCBD . . . can be reliably recalled in spite of the fact that the pattern B (ambiguous) could lead either to C or to D). The other purpose of this paper is to evaluate the relevance of the model to un derstanding hippocampal structure and activity.

There is one natural ability that surprisingly bas rarely been taken into account in the literature, namely, the ability to recall a sequence at a speed that can be modulated in the course of time. This property is nev ertheless perfectly commonplace: we are able, for in stance, to sing a tune we have learned with a variety of different tempos, or we can even choose to hold a single note in the sequence for a fairly long time, and subsequently, resume the normal course of the recall. The model of [START_REF] Reiss | Storing temporal sequences[END_REF] would be able to account for the recall of a sequence at a modulated speed, and to continue a sequence that had been "held" at some point, except at ambiguous pattern points. The neural network model for temporal sequence storage, which is presented here, possesses the same set of gen eral required properties as found in the model ofReiss and Taylor ( we note, in particular, the use of the same realistic learning rule that achieves the same storage properties); but furthermore, the model we propose can recall a sequence at a freely modulated speed even at ambiguous pattern points, and gives an explicit neural implementation of this commonplace ability. This can be achieved because ail the layers of our model are made up of"winner take ail" (WTA) clusters with robust self-sustained states whose updating frequency is regulated by a common contrai module. In the event of the output of the system being maintained during recall at a given pattern, then the specifi c history of this pattern is self-sustained and entirely preserved in a temporary memory layer. On the basis of this nonre-stricted temporal context, recall can resume whether the held pattern was ambiguous or not. In addition one of the other aims of this paper is precisely to show that the explicit neural modelling of the output speed control of a temporal sequence may be crucial in understanding the basic processes that determine the size of the spatial forms produced by living sensorimotors systems.

In Section 2 we give a detailed description of our neural network model for temporal sequence storage, leaving aside its potential applications; this model is partly based on an original architecture briefl y de scribed in a previous paper ( Ans, 1990b).

The general dynamic memory model will be applied to the production of motor sequences in Section 3.

Most of our everyday gestures require the partici pation of numerous effectors that are coordinated in both time and space. Sorne of these movements, such as those involved in locomotion are highly automatized [START_REF] Shik | Control of walking and running by means of electrical stimulation of the mid-brain[END_REF] : in this case they call on a specifi c set of effectors and are largely con trolled by subcortical structures. Motor activities of other types, such as writing and drawing, for example, rely more strongly on cortical structures and are not as strictly linked to one particular set of effectors. Movements of the latter type are probably based on centrally represented motor programs ( Keele, Cohen, & Ivry, 1990), the spatiotemporal characteristics of which are invariable [START_REF] Viviani | Space-time invariance in learned motor skills[END_REF]). These "motor forms" can moreover be produced in various formats and sizes without undergoing any change in their characteristic basic geometrical proportions [START_REF] Bernstein | The coordination and regulation of movement[END_REF]. Last, these motor sequences can be repeated using a different set of effectors from those with which the movement was initially leamed. For example, a child who bas leamed to write on a hori zontal sheet of paper, using mainly bis fi ngers and wrist, will also be spontaneously able to write on a vertical blackboard, using bis wrist, elbow, and shoulder mus cles. These motor behaviours are in accordance with the hypothesis that the structure responsible for plan ning a movement may be largely independent of the motor system performing this movement.

Dynamic memories seem to provide a suitable rep resentation for the structures that may be at the root of the movement planning. The undedicated dynamic memory model, described in Section 2, was subse quently coupled with a sensorimotor system capable of performing goal-directed arm movements. This sys tem was given the neuromimetic sensorimotor archi tecture we developed separately in a previous modelling study on goal-directed movements [START_REF] Coiton | A neural network mode) for the intersensory coordination involved in di rected movements[END_REF]Gilhodes, Coi ton, & Velay, 1991 ) . It consists of a Kohonen layer [START_REF] Kohonen | Se{forganization and associative memory[END_REF] where sensory information of two types is combined in a functional map. This sensory layer controls a motor layer that drives the effectors of either a numerically simulated arm or an artifi cial jointed arm. After a leaming phase, the arm is able to perform movements aimed toward either a fixed target ( pointing move ments) or a moving target ( tracking movements). The coupling between the dynamic memory model and the sensorimotor system was achieved by means of a neu romimetic interfacing module.

A NEUROMIMETIC MODEL FOR DYNAMIC MEMORY

General Description of Model

The model, which is schematized in Figure 1 F( t) of layer F is transmitted topographically to layer S ( these two layers both have the same structure) via strong specific excitatory connections, which leads to S(t) = F(t). Hence the temporal sequence S(t) oc curring in layer S is a copy of F( t) and conveys in a recoded form the original input sequence Z(t). Due to the intervention of the time-lag modules R, the cur rent state of layer Bis a global representation B(t) of the states preceding the current state S(t) of layer S within a variable time window: layer B is a kind of temporary memory that preserves the current history of the sequence S(t). Layer G combines this current history B( t) with the constant activity pattern C identifying the sequence being processed. Then layer S stores the association between its present state S(t) and the pattern G(t) rep resenting its history in the light of the sequence iden tifier. Severa! sequences, each presented with its own separate identification key, can be stored in this way in the permanent memory S after being processed several times.

In the sequence recall phase, the input layer Fis no longer activated and it is the occurrence of the identity pattern ID that triggers the recall of its associated se quence reproduced at the output S(t) in a different format from that of the original sequence Z(t). The permanent memory S progressively reconstructs the successive items by means of the maintained identifier and the temporary memory, which itself is built up dynamically, exactly as in the learning phase.

As we shall see below, the pattern of activity of each of the five main layers of the model is preserved in a robust self-sustained state that cannot be modified un less a resetting is induced by an inhibitory control pro cess (A, I) that periodically updates the whole archi tecture.

Z(I) = Zj [F) 0 0 (a)

Structure and Working Principles of Layers

Each of the layers F, S, G, C, and B (Figure l) is composed of a set of nonoverlapping and unconnected clusters of elementary neuron-like units. The numbers of units inside the clusters are not necessarily the same and the number of clusters composing each layer may also change in the different layers. These numbers are structural parameters that have to be chosen in simu lations. Each unit within a cluster activates itself and inhibits all other units in the cluster ( see also Figure 2a showing an example of a cluster structure), and this fixed prewiring subserves a competitive process between the units inside each cluster. [START_REF] Grossberg | Contour Enhancement, short-term memory, and constancies in reverberating neural networks[END_REF]Grossberg ( , 1987a) ) has developed a neurobiologically plausible mathe matical model for this process where the continuous variation in time in the activity of the competing units can be obtained from a system of differential equations. This author ( Grossberg, 1987a) has proposed that this competitive process, which is costly to model in detail, can be simulated using a simple choice algorithm: within a cluster only the unit receiving the largest ex ternal input activation ( the winning unit) achieves its maximum value, whereas ail other units in the cluster become silent (WTA cluster). We have adopted here this contrast-enhancement process that behaves ap proximately like a binary switching and is currently in use in neural network models, particularly in those with special references to neurobiological correlates ( De haene, Changeux, & Nadal, 1987;Dehaene & Chan geux, 1989, 1991;[START_REF] Strong | A solution to the tag assignement problem for neural networks[END_REF]. The temporal properties of the intracluster competitive process, in the scope of the time sequence processing, will be detailed in the case of the input layer F ( see Section 2.2. l ) that has the same basic working prin ciples as that of the four other layers. The input connectivity to all the layers, except that concerning the specifi c wiring from layer F to layer S, is of the distributed type, that is, each single unit in a given source layer ( or each component of the multi dimentional external inputs Z or ID) is connected to all the units in the corresponding target layer. It is in particular the same for the spreading connections to layer B, but in this case the activity patterns originating from the source layers ( i.e., from layer Sand from layer B itself through recurrent connections) are fi rst delayed in time by lag modules R ( see Section 2 .2 .2 ). In fact, the model does not necessarily require such a complete connectivity and a random partial wiring may suffice when the five layers contain large numbers of clusters and units per cluster.

The one exception referred to above is the connec tivity between layers F and S, which is of a specifi c type. Here each unit belonging to a given cluster in F is connected only to the corresponding unit in the ho mologous cluster in S. These one-to-one excitatory links, which have the same fixed positive synaptic weights, mean of course that the two layers both have the same structure, that is, the same number of clusters and the same number of units in each cluster. The other interlayer distributed connections, and those involving the external inputs, are also all excitatory. They are weighted with positive synaptic efficacies initially cho sen at random, and remain subsequently unchanged, except for the weights of the connections from G to S that can be modifi ed according to a synaptic plasticity rule.

.. 1. Layer F. Figure a shows one of the clusters constituting the input layer

F. An input sequence Z is represented by a time-varying vector Z(t) = [z 1 (t), z 2 (t), ... zj(t), ... zdt)], whose particular forms of time variation will be discussed in Section 2 .3 .3 . This input pattern activates all the units i in all the clusters constituting the layer, via links weighted with fi xed, positive synaptic efficacies w li . In the simplest case, where vector Z is a single pattern with no tem porality, and none of the units in the layer have yet been activated, the competition within the cluster can be expressed in more formai terms as follows. Within a cluster, only the unit receiving the strongest external input activation a; = i 1 w li z 1 (the winning unit) bas an active output (J; = 1 by convention), whereas the other units ( the losing units) have a null output. Hence a given item Z is transformed into a binary valued pattern by all the clusters in layer F, so that for example F = [(010)(001) ... (010)( 100)], assuming by conven tion ( as we shall do throughout Section 2 ) that a layer and its output have the same name. Provided that the size of input Z and the number of clusters in the layer are not too small, the minimum hypothesis consisting of choosing the weights w; 1 at random should yield a wide range of distinct states Fin response to a large set of different vectors Z.

In the most general case, that where an input Z (t) changes constantly with time, it is necessary to intro duce an external means of controlling the recoding process in layer F. The competitive mechanism is a very fast one, and at time t = t I for instance, layer F "captures" the instantaneous state Z ( t i) of the input according to its specific format F( t 1 ) • The winning unit in each cluster subsequently freezes, however, into a robust, self-sustained state, and the layer becomes in sensitive to any further changes in the sequence Z(t): for t > t 1 , layer F remains in the steady state F(t 1 ). This insensitivity bas been tested on low-level simula tions using the system of differential equations proposed by Grossberg ( 197 3 , 1987a), which expresses compe tition. For the clusters to become receptive again to the input sequence, they have to be reset by an input /(1 2 ) that at t = t 2 delivers a brief but intense inhibitory signal to all the units in the layer. Just after the end of this short inhibition, the layer becomes able to quickly pick up the current state Z(t 2 ) in the sequence in the form F(t 2 ). The process then continues in the same way, so that the sequence Z(t), whether its values are graded or not, is sampled at each instant t = t n whenever the updating signal l(t n ) occurs:

F(t) = constant = F(t n ) for l n < t < l n+I , corresponding to the capture of the current state Z(t n ).
The reset signal is triggered by a permanent source of inhibition /, which in turn is under the inhibitory control of a subsystem A ( Figure 1 ) : when A is active, source / is inhibited (inactive) and conversely, when A is inactive, source / is released (active). Figure 2 b il lustrates this opposition by showing that the intermit tent updating signal resulting from / is generated by short breaks in the activity of A. The control device A ( which was not modelled here) is assumed to be capable ofmodulating the frequency of the breaks, and thus to be responsible for determining the resolution at which the temporal sequence Z(t) is sampled. This control device is initially at rest, so that layer Fis spontaneously inhibited and thus maintained in the inactive state. When an input sequence Z ( t) occurs, module A needs to be activated, and the inhibitory infl uence of / on layer Fis therefore suppressed so that the fi rst sample F(t,) ofan early state Z(t 1 ) of the input sequence begins to take place.

... Other Layers.

The basic principles we have just outlined in the case of layer F also apply to the other layers. The inhibition / that is necessary for zeroing to occur bas to be conveyed to all the layers in the archi tecture if we want them to be simultaneously updated and the information to circulate throughout the whole network. The speed at which the information circulates depends entirely on the frequency of the updating sig nal l(t).

The role of layer C is similar to that of F: it recodes and samples the identity pattern ID, which has to be kept constant over time, C(t n ) = C, while the associated sequence Z(t) is being processed.

Layer S is similar in structure to layer F because both layers are linked together by specific connections: these are assigned with weights that are large in com parison with those assigned to the distributed connec tions from G to S. Consequently, when F and G si multaneously activate S, the outcome of the competi tion inside the clusters in S will depend only on the influence exerted by F. The winning cells in S will therefore be homologous with those in F, and hence

S(t n ) = F(t n ) at each updating l(t n )-
Layer B is distributively activated by the delayed output from layer Sand by its own delayed activity through self-recurrent connections. The lag modules R, which have not been explicitly modelled, are as sumed to transmit a change in their input to their out put after an interval that can be fairly short because the state information is picked up quickly in the layers. At each updating of the system at l = l m layer B acquires a state that depends both on the previous state S( l ni ) of layer Sand on its own previous state B(t n -d, cor responding to the previous updating that occurred at l n -t • lfwe take <f> to denote the layer B output function, this yields the following expression:

B(tn) = «t>[S(ln-d, B(tn_i)].
This equality can be successively written as:

B(tn) = «t>[S(tn-1), «t>[S<tn-2), BUn-2)]], B(ln) = «/>[S(ln-1), «t>[S(ln-2), «t>[SUn-3), B(ln-3)]]],
and so on,

B(ln) = «t>[S(tn-d, «t>[SUn-2), <f>[S(tn-3), • • • <f>[S(tn-k), B(tn-d] • • .]]],
where it can be seen that at each updating l n , layer B contains a representation of the current history of the sequence S(t), the length of which is limited because the diminishing influence exerted by a remote past state of S will disappear. This layer is a temporary memory, the state of which can remain preserved as long as the control module A continues to be active without any interruptions. The size of the time window encom passing the current sequence history ( the span size of the temporary memory) depends on several factors, including the ratio between the strength of the self-ac tivation of layer B and that of layer S, which is mainly a question of the relative sizes of these two layers. Here a parameter p weighting the self-activation of B was introduced as a means of dealing with this source of variation in the simulations. This overall parameter can be taken to account for example for variations in the density of the axonal branches that convey feedback information to the temporary memory. If pis very small or equal to zero, for instance, the temporary memory will contain only the most recent previous state in the sequence (span size = 1 ), because only the influence of S is taken into account in the competitive process; when p increases, a concomitant increase in the size of the time span occurs. If p becomes too large, however, the competitive process will no longer be affected by the activation of S, and layer B will become merely a sequence generator bearing no relation to the input Z ( t). On the other hand, the current size of the time span also depends on the structure of the sequence. These performances, which can easily be predicted at the theoretical level, were checked by carrying out sim ulations.

In an early version of the model ( Ans, 1990b), the temporary memory consisted of a series of K layers, each of which was specifi cally activated by the lagged output from the previous layer. One advantage of a structure of this kind can be that a trace of each indi vidual item is stored in the temporary memory, but there are also disadvantages because the time span has a fixed size that depends entirely on the number K of layers ( spatial buffer oflength K). In the present version, we fe lt it would be preferable to deal with a succession of separate events that could be represented in a single layer, and with an adjustable time span: layer Bis there fore now a kind of "distributed buffer." The model de veloped by [START_REF] Jordan | Serial order: A paraUel distributed processing approach[END_REF] also involved a temporary memory consisting of a single layer with recurrent con nections: but this layer ( and the others) is not based on a local competition principle capable of maintaining a frozen state for some time, and the processing system proposed has a set speed that cannot be regulated by an external module.

Layer G combines the current sequence history B(t n ) with the constant pattern C identifying the sequence being processed, which can be written G(t n ) = ip[ C, B(t n )L where it, denotes the output function of layer G. Here again a factor JI, playing a similar role to that of p, is introduced to facilitate matters when it cornes to dealing with the relative influences of layers C and B on G. As in the case of layer F, these three layers have to be large enough to be compatible with a wide range of different states.

Learning Rule in Layer S.

Last, layer G is con nected to layer S, through synapses of modifiable weight. Figure 3 shows one of the clusters in layer S, where the external activation applied to each unit i consists on the one band of the output J; from the ho mologous unit in the input layer F, and on the other band, of the elementary components gj oflayer G having a positive weighting m u . The latter distributed activation is denoted U; = L j m u g j . The changes in the synaptic weights m u with time are governed by the following synaptic plasticity rule ( of the error correction type), or learning rule ( Ans,l 990a,b):

[S]

2.3.2.

Sequence Reca/1. Because input Fis not acti vated at the sequence recall phase, it is the onset of the identity pattern ID which triggers the recall of its as sociated sequence at output S. Te mporary memory B is initially empty, and permanent memory S recon structs the first item of this sequence on the sole basis of the identifi er. During the successive passages through the loop B, G, S ( at each updating ln), memory S pro gressively builds up the successive items of the recalled sequence based on the maintained identifier and the temporary memory, which itselfis built up dynamically, exactly as in the leaming phase. The production speed can be freely adjusted by varying the frequency of the intermittent signal emanating from the control module A. In particular, a given output item S(t n ) can be sus tained as long as module A maintains its activity with out interruption; the normal course of events can then be resumed as required, because the history of item S(t n ) is also held and preserved in temporary memory B. It is also worth mentioning that the sequences learned should theoretically ail be given the same ter minal symbol, which could then be used to stop the closed-loop production process. The occurrence ofthis particular item at the production phase would then serve to inactivate the control device A and hence to bring the activity of the whole system to a stop. This stopping device was not actually modelled here, how ever.

2.3.3. Simulation. The following simple simulation of the processing of two sequences of formai items serves to illustrate the ability of our model to deal with long repeated subsequences that may be the source of in trasequence and intersequence ambiguities. In this simulation, there is no initial recoding of the input items Z and ID: a time sequence is expressed directly in the form of a series of states F(t n ), and its corresponding identifier is expressed in the form of a constant state C(t n ) = C. The two following sequences, identifi ed by items Cl and C2, were processed:

C 1: ABCDEFGHIJKLMNOP ABCDEFGHIJKLMNOPM C2: ABCDEFGHIJKLMNOP RRRRRRRRRRRRRRRRRRRRE.
Here the letters forming sequences constitute special configurations ofwinning units in layer F, which were picked up at the various instants l n at which the control module A performed the successive updatings. The same is true for C 1 and C2, which are two distinct constant states oflayer C. The spaces within sequences have no temporal significance but have been inserted as a means of clearly demarcating the subsequences. The letters in bold print indicate the fi rst successor of a preceding subsequence. The first sequence C 1 consists of two identical subsequences each consisting of 16 items, the fi rst of which is followed by item A and the second by M. At the recall phase, the permanent mem ory S must be able to reconstruct a given item on the basis of this item's history, which is contained in the temporary memory B, in the light of the sequence identifier C 1. Because this identifier is held constant during the recall of the whole sequence, it cannot have any intrasequence discrimina tory power. Hence for the two items A and M to be unambiguously reconstituted on the basis of two distinct histories, the current span in the temporary memory must encompass at least the 17 previous items. lfit is not the case, for example with a time window of size 16, or less, items A and M will have the same history and there will be intrasequence ambiguity. The second of the above sequences, C2, be gins with the same subsequence as C 1, but this first part is followed by R instead of A. Hence these two items have the same history, say B A = B R = B, within the temporary memory B, whatever the current span size. But layer G, which combines the history of an item with the sequence identifier, provides two distinct combined histories for items A and R, respectively G A = it,[ C 1, B] and Ga = it,[ C2, B], which will avoid an intersequence ambiguity during the recall of these two items. It should also be noted that item R is repeated 20 times in C2 before the occurrence of E. For E to be correctly recalled, it is necessary for the current memory span to encompass at least the 20 previous items.

When the above example was simulated numerically, each of the clusters constituting all the layers of the model was taken to contain five competing units, and layers F and S to contain four clusters. A state corre sponding to the letters in the above two sequences is given for example by [ ( 00001 )( 00010)(00001 ) ( 01000)]. Layers B, G, and C contain 15, 30, and 10 clusters, respectively. Fairly similar compositions be tween identity items C 1 and C2 were chosen in order to demonstrate that their intersequence discriminatory power does not necessarily depend on the existence of strong differences. With the type of coding process adopted, involving 10 clusters of fi ve units, these two identity patterns have five clusters in the same state ( the same winning unit) and five in different states. In layers B and G, the fixed weights assigned to each unit are chosen at random on the basis of a uniform law in the range 0-1, and then normalized. Parameters p and Ji are taken to be equal, hence p = Ji = 5. In layer S the adjustable weights assigned to each unit are chosen initially at random on the basis of a uniform law in the range 0-0.005. In the synaptic plasticity rule ( 1 ), a = 0.03, f3 = 1, and 'Y = 0.1.

With these parameters, sequences C 1 and C2 can be learned very quickly: after being introduced only six times each, they can be perfectly recalled in response to their identifier. Any within, or between, sequence ambiguities can thus be resolved in the case of the pres ent diflicult example, which means that the constraints imposed on the span size of the temporary memory have been satisfied.

The above simulation was chosen because it is most representative of the model efliciency. A large number of other simulations, performed with sequences of var ious lengths and structures, have led to satisfactory re sults. This efliciency arises, in particular, from the length of the current history preserved in the temporary memory; this history can be very long hence allowing the resolution of complex ambiguity problems. Sorne general features of the model and some particular se quence structures have to be examined however.

The input temporal patterns to be stored, Z(t), have frequently been referred to as sequences in the foregoing because, in storage modelling, these temporal signais are usually simple successions of distinct patterns of variable duration ( square waves principally). Actually, our mode) can cope with time-varying multidimen sional signais Z ( t), with either graded or discrete values, whose temporal form may be rather general: analog signais, continuous or not, smooth or not. This vector signal is sampled and recoded by the input layer F in the form of a sequence of patterns, the components of which are quantified signal with binary values; it is that recoded sequence F(t), and not the original input Z(t), that is actually stored in the model and recalled at the output S(t).

ln those cases where it is necessary to recall a time-varying vector in its original form Z ( t), a conversion module playing the opposite role to that of layer F should be able to reconstitute the original input from the output sequence S( t). This will in fact be done in Section 3 where this conversion module simply consists of an associative memory. The reconstituted signais at the output of this neuromimetic digital to analog con verter will be all the more accurate because the sampling frequency, during learning, of the original time-varying vector Z(t) is high. This high frequency condition is especially required when the original vector to be re constructed bas components that are smoothly chang ing signais; in this case the converted outputs will appear as fi nely quantifi ed signais. But if we suppose that the neuron-like units composing the neural converter have time constants, then these quantifi ed signais will con sequently be smoothed by the persistent activity of the converter units. On the other band, the converted out puts will be more or less temporally contracted or ex panded in comparison with the original input Z(t), and that according to the production speed of the dy narnic memory controlled by module A. This temporal similarity (here used in the sense oftemporally homo thetical forms) will preserve, for instance, the relative durations of the patterns making up the usual sequences that are simply composed of distinct patterns of variable duration.

Another time variation type that bas to be examined is that where the input time-varying vector Z(t) con tains gaps, that is, time intervals of variable length where all the components of the vector are null. In this case, the model will show some working difficulties un less it is supposed that an input gap is in fact represented by a non-null specifi c pattern 0 in the input layer F. This is easily achieved if, during the learning of an input Z ( t), layer Fis jointly activated by a constant activity pattern H. When an input gap occurs, only pattern H activates layer F and hence the specifi c state 0 is in duced. With this simple additional assumption, which was not mentioned in the foregoing for the sake of sim plicity, we corne back to an easily tractable situation, where gaps of different lengths within Z(t) are respec tively coded in layer F by subsequences of distinct sizes composed of the repeated item 0; as previously, the original input Z ( t) containing gaps can be reconstituted by a converter. This additional hypothesis and the time constants of the converter units will not be taken into account in the following application.

APPLICATION TO PRODUCTION OF MOTOR FORMS

The general dynamic memory model was coupled with a sensorimotor system. A combined model was thus developed for learning spatial shapes, which are ex pressed in terms of temporal sequences of relevant ex teroceptive quantities, and producing these shapes in terms of motor activity. Hence we assumed that the format in which a given shape is memorized need not be the same as that into which it is translated in motor terms by the effector systems.

Sensorimotor System

A neuromimetic model for [START_REF] Kohonen | Se{forganization and associative memory[END_REF], served here mainly to im plement both a plurimodal signal integration and a nonlinear sensorimotor coupling. Generally speaking, it can be used as a hidden layer in the learning ofhighly arbitrary associations [START_REF] Ans | Learning arbitrary associations: A neuromimetic model[END_REF].

The sensorimotor module is organized during a learning period, during which the arm randomly ex plores its own working space. This exploration is in duced by applying a forcing activity M on the cells constituting the motor layer by means of a random ac tivity generator RG producing diversity. The move ments performed during the learning period are not directed toward any particular goal: they are simply "blindfold" exploratory movements. It is assumed that during this phase, the exteroceptive signais will inform the network about the position of extremity D of the arm in the working space. At each new arm position, a pair of sensory signais (E, P) contributes to the changes in the weights of the input adaptive connections in layer K. Kohonen's simplified self-organization al gorithm [START_REF] Kohonen | Se{forganization and associative memory[END_REF] the activity of the units in this layer. On the other band, sensory layer K participates in developing the adaptive links with motor layer AMI, which is an associative memory, and the sensorimotor coupling is thus built up. Here a simple classical error correction rule is used, where the required output is induced by the specifi c activation M delivered by the random generator RG.

Because layer K bas a localized activity, the positive sign of the synaptic weights does not change in layer AMI when an error correction rule is applied. In this highly particular case, using this simple rule is not in compatible with the requirements of neurobiological realism, which is not generally true ( cf. Section 2.2.3). At some stage in the exploration of the arm's working space, the single activity focus in layer K begins to spe cifi cally encode the bimodal information consisting of the arm posture P and position E of its extremity D.

Any changes in the arm position are accompanied by changes in the site of the activity focus in the layer. This layer moreover undergoes some self-organization as the result of which it refl ects the topological relations im plicitly contained in the inputs it receives. At the end of the learning phase, the random generator is removed and the connections lose their previous plasticity. At the subsequent operational phase, the coordinates of any target Tin the working plane can be specifi ed on the basis of the exteroceptive information. In this case, the arm movement is such that the distal extremity D gradually approaches the target T. Because of the mod el's structure, movements of the arm toward the target consist of a series of successive steps. If the target begins to move, the arm then performs a tracking movement.

To make this system a little more natural, it was decided to add a switching device, simulated in the model by "valve" V, with which any irrelevant extero ceptive information can be gated off. Without this switch, the arm systematically performs the movement whenever the exteroceptive information is conveyed to any point other than extremity D of the arm. The switching device is assumed to be under the control of an attentional system that decides whether or not it is desirable to reach a given target. with origin O and ( x, y) components. The shape can therefore be defined by the temporal sequence of events El, E2, E3, El, assuming for the sake of simplicity that the form is scanned starting at vertex El, in the direction indicated by the arrows. Let us imagine that it is then proposed to store this sequence in a dynamic memory capable of delivering it as and when required via an appropriate interface to activate the sensorimotor module, which would consequently be activated by a series of virtual targets. The arm would respond by performing a tracking movement, and could thus be said to have produced the previously learned shape "from memory." Because the events constituting the sequence studied are absolute positions in a fixed Cartesian reference frame unrelated to the object, the arm should be unable to repeat the same figure con sistently at various points in its working space. What we intend to demonstrate here is precisely that one same learned shape can be reproduced as required whatever the initial arm position in space, and in a whole range of sizes. To achieve this, it is assumed that the exteroception, which up to now simply specified the absolute positions E, is now in addition capable of providing transient information oE, in terms of quan tities proportional to the differences between successive positions. It is then assumed without any loss of gen erality that these quantities are simply equal to these differences. In the example given in Figure 5, for in stance, when the exteroception successively specifies points El and E2, it will concomitantly also generate the transient information oEl, which can be formally expressed as the difference between vectors E2 and El: oEl = E2 -El = (ox, oy). Now the components of a vector difference can sometimes be negative, as occurs here in the case of the component ox = x 2 -x,, and there exists no justifi cation for assuming a priori in a neuromimetic meta phor that the sign of a signal being transmitted along a nerve fibre ( which is analogous to a spike frequency having no sign) might be variable. In order to overcome this impediment to neurobiological realism, it is as sumed that these elementary algebraic differences are encoded by two pairs of associated increment captors: (Ax + , Ax-) for the x axis and (Ay + , Ay-) for the y axis. When ox is positive, then ax + = ox and ax-= 0, and when ox is negative, then ax + = 0 and ax-= 1 ox 1, and likewise in the case of the two associated difference captors on the y axis. A shape segment is therefore defined by the quadruplet .1.E = ( ax + , Ax-, ay + , ay-).

Learning and Motor Production of Shapes

l. Temporal Encoding of Spatial Shapes.

Leaming the contour of an object will then consist of sequentially specifying its most singular points in the exteroceptive modality E and storing in the dynamic memory the resulting temporal sequence .1.E( t). In the case of the example in Figure 5, the sequence oEl, oE2, oE3 will therefore be replaced by the corresponding sequence of patterns .1.El, .1.E2, .1.E3. classification will be. The architecture and working principles of the other DM layers remain unchanged.

The interface module schematized in Figure 7 con tains two processing stages. The first is an associative memory AM3, which bas previously undergone a learning period at the time when the self-organization oflayer F of the DM module was being set up. At that time, layer AM3 learned to convert the localized format of the output S ( which was a replica of F) delivered by module DM into a graded activity conveyed by the ex teroceptive signal /lE; here each channel is processed separately by this digital to analog converter. Taking any one channel as an example, the graded component dx + imposes the output of the first cell inAM3, which is adaptively connected by all the units in the cluster expressing this same difference component in a local ized way. Changes in the adaptive connections are gov erned by a classical error correction rule satisfying the constraint whereby no change in the signs of connec tions should occur because the associative information is localized. A similar process takes place in all the other three channels. After this learning phase, layer AM3 is able to reconstruct the information /lE when it receives no specific forcing sensory input. The ac curacy of this digital to analog conversion depends on the number of units in the clusters belonging to S.

The second stage in the interface module is an ad ditive device denoted � assumed to have prewired input connections and which additively combines the two types of exteroceptive information ( absolu te and dif ferential positions). The first cell � x in layer� therefore calculates the sum x + dx + -dx-, where the sub tractive part is given by an inhibitory connection. Be cause the signais dx + and dx-are mutually exclusive (when the one is active, the other is null), cell �x ac tually calculates the algebraic sum x + ox; the proce dure is the same with unit �y-In this way layer� carries out the vectorial sum E + oE, and it is this sum, which is of a purely positional nature, that will actually ac tivate layer Kin the sensorimotor module.

The associative memory AM2 implements a new relation between sensory layer K and the exteroceptive position information E. Learning in this layer is as sumed to take place concomitantly with that in the whole sensorimotor module, and is set up in a very similar way to that of the symmetrical layer AMI on the motor side. The specific forcing input here is the exteroceptive sensory information E, whereas the dis tributed adaptive input is a collateral of the localized activity in layer K. Changes in the weighting are subject to the same error correction learning rule and to the same requirement that no change in the signs of syn aptic weights should occur. Introducing this layer can prove to be useful in situations where the exteroceptive sensory flow is interrupted by the attentional switch V: when this occurs, the layer continuously reconstructs the virtual exteroceptive information about the position E of the distal extremity D of the artifi cial arm on the basis of the currently active focus in layer K.

Results

l. Shape Learning.

Once the prestructuring of the various modules bas been completed, the whole system is ready to learn spatial shapes and to produce them in motor terms. Te mporal sequence /lE( t) is processed in the dynamic memory DM as described in Section 2, except that now we have Z(t) = dE(t). The learning of a given shape occurs when the temporal sequence llE(t) induced by exploring the contour ofthis shape ( with attentional valve V open) is repetitively fed into input layer F. The control device A must be made to operate periodically at a constant frequency adapted to the fineness of the transient signal /lE sampling. In parallel, the activity oflayer C must be sustained by an item ID that encodes the identity of the shape being learned, such as triangle, square, etc.

The relation between this identity item and the se quence being learned is encoded in terms of changes in the connections in the permanent memory S of DM, which are now the on/y permanently adaptive links in the whole architecture. During the learning period, whatever arm movements are induced have no effect on the memorizing of the shape. lt should be mentioned however that these arm movements generally tend to approximate the contour of the shape being learned.

3.3.2. Motor Shape Production. Let us take the most interesting case, where the sensory flux arising from the exteroceptive detectors is broken by the attentional switch V. In this case, the point in space being encoded at that moment by the exteroceptive sensors is not taken into account and bas no effect on the arm posture, which is of no importance at the beginning; whereas the virtual exteroceptive information E reconstructed by AM2 gives the current position of the arm's extrem ity D.

The motor production by the arm of a shape mem orized in exteroceptive terms t:.E occurs as follows. The occurrence of an item ID identifying a given shape to be generated triggers the production of the correspond ing temporal sequence S( l n ), where l n denotes the suc cessive updatings of the module DM; the speed of these updatings is regulated by the control module A. The associative memory AM3 converts the sequence of lo calized activities S(t n ) into a graded sequence t:.E(t n ), which is then added to the position information E in the summator 2:. At the beginning, the sensorimotor module is in an equilibrium confi guration [P(O), E(O)] corresponding to any arm posture whatever. When the fi rst element in sequence t:.E(t i ) begins to reach the summator 2:, sensory layer K receives the input activation E(O) + oE(t i ), specifying the position of a virtual target, and an arm movement toward this target is triggered. The directional pattern t:.E(/ 1 ), or oE(t 1 ), is no longer a transient signal generated as in the learning stage be cause it can now be sustained for some length of time under the control of module A. As mentioned in Section 3.1, the arm movement toward a fi xed target is per formed step-wise, which means that several iterations of the sensorimotor loop are necessary for a target to be reached. At the first iteration r 1 in the sensorimotor module, the distal extremity of the arm will reach po sition E( T i ), midway between E(O) and position E(O)

+ oE(l i ) of the virtual target. The information E(t i ) is reconstructed by memory AM2, and at the subse quent iteration r 2 , if the activity t:.E(l i ) is sustained, layer K is activated by the sum E( T i ) + oE(t i ), spec ifying a new virtual target that bas shifted in the direc tion oE(t i ). The extremity of the arm continues to move in this direction and reaches position E( r 2 ), midway between E( r i ) and the new position of the virtual target E( T i ) + oE(l i ), and so on. As long as the activation t:.E(t i ) is sustained, layer K is activated at the subsequent iterations T, by the successive inputs E(r,_ i ) + oE(l i ), and the arm's extremity thus con tinues to track the elusive target as it continues to move in the same direction oE(t i ). As the other directional vectors t:.E(l n ) are sequentially generated by the sub sequent updatings l n of the dynamic memory, the arm continues to track the virtual target as previously, but makes a change of direction whenever the vectors t:.E(t n ) do so.

In this way, an identified shape will be sequentially produced at the motor level in a size that depends on the frequency at which the dynamic memory is up dated. This frequency must be maintained constant throughout the motor shape production period to en sure that the resulting product is not distorted. If the updating of module DM takes place at a low frequency, the final shape will tend to be large because the arm will continue to track the moving virtual target in the same direction for long periods. Conversely, if the up dating frequency is high, the final shape will tend to be small sized. Generally speaking, it is via the control exerted by module A on the dynamic memory's output speed that the size of the motor figure produced in fine can be adjusted as required.

The velocity of the arm movement can furthermore be controlled. First, it should be mentioned that when a given directional item t:.E(t n ) is produced by the dy namic memory, the velocity of the arm movement in duced step-wise in the appropriate direction depends on the timer controlling the iterations T. in the senso rimotor loop. The setting of this timer, which up to now was implicitly assumed to be constant over time, is in fact itself liable to be modulated in sensory layer K. This layer, the internal functioning of which involves a competitive process similar to that implemented in the dynamic memory clusters, is destined also to enter robust self-sustained states. Strictly speaking, it should also be updated by a system (A', f), which is analogous to system (A,/). By introducing this further control device, it would be possible to modulate the frequency of the updating in layer K, and hence to adjust the velocity of the arm movement as required.

Several simulations have been carried out with the combined architecture. The following two very simple simulations clearly illustrate the main behavioural properties of the model. After a learning period during which a given shape was presented to the model at a single point in space and in a single-sized format, the artificial arm was able to draw this shape anywhere in its working space in whatever size was required. The first property in question, which we have called "trans lation invariance," is illustrated in Figure 8, and the second, which we have called "size invariance," in Fig ure 9.

CONCLUSION

The dynamic memory neuromimetic model described in Section 2 features an essential distinctive property: it is able to recall a learned sequence at a freely adapt able speed. This ability is fairly commonplace among living creatures, but bas rarely been dealt with in the connectionist literature. Apart from the fact that this ecological feature bas to be accounted for by a general dynamic model, the tlow control property can have other more far-reaching effects. We have demonstrated for example, by applying the model to the production of motor shapes, that the dynamic memory's ability to contract or expand the lime scale when recalling a given sequence learned in exteroceptive terms resulted at the motor level in the production of forms that could be either reduced or enlarged on the spatial scale. Another important property of our dynamic mem ory is its ability to rapidly leam sequences of temporal events. This aptitude is rare among the present day models, and yet it is essential for both living and arti ficial systems that constantly encounter new situations that have to be learned rapidly. Fast leaming is a char acteristic feature of single-layer memory models ( with no hidden layer), as in [START_REF] Reiss | Storing temporal sequences[END_REF] and in the present paper, but this feature is especially enhanced in our model where the memory layers are based on a competitive learning procedure. Indeed it takes much less time to approximate a set of adaptive weights re sulting in only one particular unit in a cluster becoming the winner than to finely calculate, in the case of models involving no competition, the exact weights required to produce accurate, possibly graded outputs.

Generally speaking, the larger the layers constituting the model, the more efficiently they perform: the size of layer G is a particularly decisive factor in this respect. The relations stored in permanent memory S between a current item and its history, which is represented in layer G, can be highly arbitrary; and the synaptic plas ticity rule ( 1 ) , which was chosen mainly because of neurobiological plausibility, is unable to store in the distributed form a large number of sequences contain ing many items unless the dimension of the patterns G is sufficiently large. This constraint is not necessarily too severe to be compatible with the neuronal resources of a real-life brain.

Our model involved a minimal approach based on a set of synaptic weights most of which were chosen at random and fixed, and on a synaptic plasticity rule that is fairly simple and yet satisfi es the most elementary natural constraints. To achieve even greater biological realism, each ofthese basic units with a strong autoex citatory connection, so designed in order to simplify the formalism, could in fact be easily replaced by a micropopulation of neurons densely interconnected by excitatory synapses. The basic principle adopted here, according to which the format in which a shape is memorized is not nec essarily the same as that into which it is translated and produced in motor terms, gave rise to some important behavioural properties emerging from the mode!. The fi rst two properties, which were mentioned above, have been called "translation invariance" and "size invari ance": a shape which bas been learned at a single point in space and in a given size can be drawn by the arm anywhere in its working space and in a whole range of sizes.

Another consequence of this basic principle is that the memorized shape can be repeated by other effector systems that have never participated in the shape learn ing because the exteroceptive temporal information produced on the output of the dynamic memory is, precisely, not related to the particular sensorimotor or ganisation of the simulated limb performing move ments: the classic example here is the ability to write one's signature with the tip of one's foot [START_REF] Bernstein | The coordination and regulation of movement[END_REF]. To better understand this commonplace behav iour, suppose that distinct sensorimotor modules, each composed of a set of layers ( K, AMI, AM2, �) similar to that depicted in Figure 6, have first separately ac quired, on the basis of the common exteroceptive in formation giving absolu te position E, the basic ability to drive the pointing movements of their corresponding specific limbs. Consider now that the output b..E of a unique and common dynamic system, which is com posed of the dynamic memory DM and its associated converter layer AM3, connects ail the layers � belonging respectively to the different sensorimotor modules ( as defined above). The output of AM3, which conveys relative position information b..E(t), temporally defin ing a given shape in a common exteroceptive format, will be added in the different layers }; together with the current absolute positions E of the respective distal ex tremities D of the corresponding limbs. Hence these distinct effector systems will be able to perform similar motor figures in response to the same exteroceptive shape stored in a unique dynamic memory.

The above behavioural properties, which originate from a realistic neural network model in which they are not implemented a priori, reinforce the idea that the structures responsible for planning a movement in the central nervous system might be largely indepen dent of the motor systems performing this movement. Our model proposes that these structures ( here only the dynamic memory) would store generic programs, dynamically defining "shape template," which would then be translated in various ways in the same or in several motor systems, even in the effector systems that have never been recruited during shape learning. This single entity giving rise to diversity is related to Schmidt's idea ( 1988) of a "generalized motor pro-gram" which would contain only one unique represen tation of the essential invariant features for a particular type of action and whose expression could be varied depending on the choice of certain parameters. And our contribution here was to propose how these abstract programs could actually be implemented at a quasi neural level.

FIGURE 1 .

 1 FIGURE 1. The architecture of the temporal sequence-leaming model. Each rectangular module is a layer consistlng of units ( small circles) arranged in separate clusters ( enclosed units). To lllustrate this, the number of units per cluster and the number of clusters in each layer were arbitrary chosen. ln each cluster, only one unit is active ( black circle) and the others are sllent ( white circles). The input connectivity to all the layera is of the distributed type, except for that from F to S, which is of a speciflc type (heavy arrow).

  FIGURE 2. (a) Activation of one of the clusters in layer F by applying the multidimensional temporal sequence Z(t). The vertical line above unit l is a simplified dendritic tree receiving the horizontal inputs z 1 to which a synaptic weight w, has been assigned. The connectivity within the cluster subserves a competitlon mechanism. l(t) is an inhibitory signal that serves to reset the whole layer. ( b) Diagram showing the opposition between the adjustable frequency signal emitted by the control device A and the inhibitory resetting signal l(t).

FIGURE 3 .

 3 FIGURE 3. Activation of one of the clusters in layer S by a highly welghted specific input F ( weighting not shown here) and by distributed associatie connections with modifiable welghts m,. ln this case, which arises during the leaming period, the out come of the competition within a cluster depends only on the strongly weighted specific input, and therefore S = F.

  a sensorimotor system was developed in a previous study in the framework of re search on the organization of goal-directed movements ( Coi ton et al., 1991; Gilhodes et al., 1991). The point ing or target tracking movements in question were either simulated numerically or performed by an artifi cial arm. The network drives the arm movements on the basis of proprioceptive and exteroceptive information picked up by sensory receptors. The proprioceptive in formation is given by sensors that code the angular val ues at the joints. The exteroceptive information specifies the position of any point in the arm's working space. Simulations were carried out on arms with two or three degrees of freedom moving in two-and three-dimen� sional space. Among the various versions of the model we have studied, one bas been chosen here that is now briefly described. A very schematic diagram of this sensorimotor module is given in Figure 4. The sensory layer K consists of cells forming a two-dimensional network. The cells are interconnected in such a way that each unit is linked via fixed excitatory connections to its nearest neigh bours and via fixed inhibitory connections to the cells in the less immediate environment. In this competitive situation, a single activity focus emerges, the site of which depends on the input afferent patterns. The cells in this layer have a binary activity ( the active cells in the winning focus score 1, and the others, 0). The structure of this layer, which was based on the descrip tion by

FIGURE 4 .

 4 FIGURE 4. Diagram of the sensorimotor module. A simulated ann moves in a plane and its proprioceptive detectors encode its joint angles 8 1 and 8 2 in the fonn of a graded activity pattern P = (8 1 , 8 2 ). The exteroceptive infonnation is given in the form of a graded activity pattern E = (x, y), the components of which are positive quantities giving the cartesian coordinates of any point T in the ann's working space referred to the arbitrary origin O. This exteroceptive information can possibly be gated off by attentlonal valve V. The ann is controlled by motor layer AM1 , consisting of two cella, each of which activetes the eflector responsible for moving one of the joints. The motor command Mis a graded pattern of activity, the two components of which specify the pairs of jOint angles (8 11 8 2 ) to be adopted by the ann. The• proprioceptive and exteroceptive infonnation P and E is distributed among all the cella in sensory layer K, which is itself fully connected to motor layer AM1. These connections are all of the adaptive type during a leaming period when the arm actively explores its working space under the control of the random activity generator RG.

  FIGURE 5. A simple exemple of a shape ( a triangle) drawn on the working plane. The contour of the triangle is encoded ln temporal terme by the exterocepe modality E that sequentlally explores the vertices E1, E2, and E3.

FIGURE 8 .

 8 FIGURE 8. The general architecture of the model. The left-hand skie mainly processes the exterocepe differen infonnati AE in the dynamic memory module DM. The right-hand side deala mainly with the exterocepe position lnfonnatlon E withln the sensorimotor module. The interface module, whlch is shown here in a dotted frame, addltively combines the two types of exterocepve information, the sensory flow Of whlch can be interrupted by attentional valve V.

FIGURE 7 .

 7 FIGURE 7. Working scheme of the interface module.

FIGURE 8 .

 8 FIGURE 8. An example showing the translation invariance property. (a) The shape leamed. (b) and (c) The trajectories taken by the arm's extremity while producing the leamed shape starting from two different points in space.

□FIGURE 9 .

 9 FIGURE 9. An example showing the size invariance property. (a) The shape leamed. ( b) The trajectory taken by the extremity of the ann whlle producing the learned shape. ( c) The trajectory taken by the ann's extremity while produclng the learned shape with a larger amplitude.

  Micropopulations of this kind ( with two persistent states of activity) and the clusters within which they are engaged in competitive relation ship ( neural assemblies) have been roughly assimilated to the micro and macrocortical columns, respectively (Dehaene et al., 1987; Dehaene & Changeux, 1989, 1991; Strong & Whitehead, 1989). The combined model described in Section 3, in which the dynamic memory module was linked up with the sensorimotor module, operates on the information of the proprioceptive and exteroceptive types. The pro prioceptive sensory detectors were assumed to pick up purely postural information, although the messages conveyed by human muscle spindles are known to relate both to posture and to the velocity of the joints involved in the movement ( Matthews, 1972). This additional velocity information, which is essentially transient, can in fact also be directly taken into account by the model. Another way of dealing with this problem is to assume that the purely postural information that activates sen sory layer K of the model is conveyed by an intermediate structure responsible for preprocessing the composite message originating from the sensory detectors. It is worth mentioning here that we have previously de scribed models, some with neuromimetic features ( Ans & Gilhodes, 1983; Ans, Gilhodes, & Hérault, 1983; Hérault & Ans, 1984; Ans, Hérault, & Jutten, 1985), and others with a differential formalism ( Gilhodes, Coiton, Roll, & Ans, l 993 ), including intermediate structures of this kind where composite afferent mes sages are decomposed into their position and velocity components. We have used the term exteroceptive when referring to the sensory information of the second type dealt with in the model. In humans, exteroceptive information is mainly of the visual type. lt is usually by means of vision that a target to be reached in extra-personal space is detected and located. In this case, the absolute po sition E might result from the combination of two sig nais: the target retinal coordinates and the extra-retinal signal, from either central or peripheral origin, inform ing on the eye-in-orbit position. The relative position t:.E might be obtained in two different ways: it might either result from the central processing of the retinal error ( the distance and the direction of a target image relative to the fovea), or it might be processed from the extra-retinal signal changes occurring during the movement the eye performs to go from one point to another. The precise processings involved in generating the two kinds of exteroceptive information were not modelled, this being beyond the topic of the present paper.

Acknowledgements: We would like to thank Dr. Jessica Blanc for the English revision of the manuscript. This research was supported by the French Ministry for Research and Technology, Cognitive Sci ences Grant 90C0076, and by the National Center for Scientific Re search (CNRS) URA 372-URA 665.Requests for reprints should be sent to Bernard Ans, Université Pierre Mendès France, Laboratoire de Psychologie Expérimentale, BP 47, 38040 Grenoble Cedex 9, France.


Paul De Buyer
email: debuyer@math.cnrs.fr

A PROOF OF THE BUNKBED CONJECTURE FOR THE COMPLETE GRAPH AT p = 1 2

Keywords: 1991 Mathematics Subject Classification. 82B43, 60K35. Key words and phrases. Percolation, Bunkbed graph, Combinatorics, Bunkbed Conjecture

The bunkbed of a graph G is the graph G × {0, 1}. It has been conjectured that in the independent bond percolation model, the probability for (u, 0) to be connected with (v, 0) is greater than the probability for (u, 0) to be connected with (v, 1), for any vertex u, v of G. In this article, we prove this conjecture for the complete graph in the case of the independent bond percolation of parameter p = 1/2.

Introduction

Percolation theory has been widely studied over the last decades and yet, several intuitive results are very hard to prove rigorously. This is the case of the bunkbed conjecture formulated by Kasteleyn published as a remark in [START_REF] Van Den | A correlation inequality for connection events in percolation[END_REF]) which investigates a notion of graph distance through percolation.

A bunkbed graph of a graph G = V , E is the graph G = (V, E) = G × {0, 1}, to which we have added the edges that connect the vertices (x, 0) to (x, 1) for all vertices x ∈ V , see Figure 2.1. It is natural to distinguish vertices whether they are on the lower level, the vertices (x, 0), or on the upper level, the vertices (x, 1).

The bunkbed conjecture (see [START_REF] Häggström | Probability on bunkbed graphs[END_REF] for a more general setting) suggests that two vertices u = (x, 0) and v = (y, 0) on the lower level are closer than u and v ′ = (y, 1). Closeness of two vertices has to be understood through the probability of the existence of an open path in the sense of percolation.

The percolation model is defined as follow. We open each edges of E independently with probability p and close them with probability 1 -p and we write P p the law associated to this percolation model. We call a configuration, an element ω = (ω e ) e∈E ∈ {0, 1}

E corresponding to the bond percolation model where ω e = 0 means that the edge e is closed and 1 means that the edge e is open. We call an open path a path of open edges and for two vertices x, y ∈ V , we write x ↔ y if there exists an open path between x and y. By convention, for any configuration, a vertex is always connected to itself, i.e. x ↔ x. For a general introduction on percolation, see [START_REF] Grimmett | Percolation since Saint-Flour[END_REF].

In this article we prove the bunkbed conjecture for the complete graph when the percolation parameter p is equal to 1/2. Theorem 1.1. Let G be the bunkbed graph of the complete graph G = K n . For all vertices x, y of K n :

P 1 2 ((x, 0) ↔ (y, 0)) ≥ P 1 2 ((x, 0) ↔ (y, 1))
In the previous works of S. Linusson and M. Leander, the conjecture has been proven for the outerplanar graphs and the wheels graphs using the so-called method of minimal counter-example, see [START_REF] Leander | Sjalvstandiga arbeten i matematik[END_REF][START_REF] Linusson | On percolation and the bunkbed conjecture[END_REF]. However, this method might not be suitable for the complete graph due to the its geometric nature. We have chosen to study the conjecture for the complete graph because we think that it should be easier to show the following proposition than the bunkbed conjecture itself: "If the bunkbed conjecture is verified for a graph G, then it is verified for the graph G\ {e} where e is an edge of G".

One can note that the bunkbed conjecture is true whenever p is small enough. Indeed, in these cases, only the shortest paths can be open to connect (x, 0) with (y, 0) or (y, 1); since the shortest path from (x, 0) to (y, 1) is longer than (x, 0) to (y, 0), the conjecture is proven. Note d (., .) the usual graph distance, then one can prove that for all vertices u, v and w, d (u, v) > d (u, w) ⇒ P p (u ↔ v) > P p (u ↔ w) for sufficiently small p.

Finally, we underline some related works on bunkbed graphs. In the random walk field, an analogical problem of the reaching time of a random walk has been studied, see [START_REF] Bollobás | Random walks and electrical resistances in products of graphs[END_REF][START_REF] Häggström | On a conjecture of bollobás and brightwell concerning random walks on product graphs[END_REF][START_REF] Van Den | Some conditional correlation inequalities for percolation and related processes[END_REF]. In the random directed graph field, it has been shown that it is equivalent to study random orientation of edges on the graph and percolation on the graph with Bernouilli paramater 1/2 (as it is in our case), see [START_REF] Richard | The transitive closure of a random digraph[END_REF][START_REF] Linusson | On percolation and the bunkbed conjecture[END_REF][START_REF] Mcdiarmid | General percolation and random graphs[END_REF].

A first approach of the problem would be to study the ratio of the probability of connection, see (1.1), and to study the derivative according to p, the Bernouilli parameter associated to the probability of opening an edge. Let u = (x, 0) and v = (y, 0) be two vertices on the lower level and define the vertex v ′ as v ′ = (x, 1), the vertex above v, then the ratio of the probability of connection is written as:

(1.1)

P p (u ↔ v ′ ) P p (u ↔ v)
As a result of the previous remark whenever p tends to 0, the ratio of (1.1) tends to 0, and clearly when p is equal to 1, this ratio is equal to 1. Because the events {u ↔ v} and {u ↔ v ′ } are increasing events, the derivatives can be studied using Russo's formula. However, even using Russo's formula, see [START_REF] Aleksandrovich | Probabilistic characteristics of graphs with large connectivity[END_REF][START_REF] Russo | On the critical percolation probabilities[END_REF], it reveals itself strenuous to study it. One can notice that the derivative of P p (u ↔ v) cannot be always greater than the derivative of P p (u ↔ v ′ ), since they are both equal to 0 when p = 0 and equal to 1 when p = 1. However, we conjecture here that the derivative of the ratio is increasing, meaning that for all 0 < p < 1:

P p [u ↔ v ′ ] ∂ p P p [u ↔ v] P p [u ↔ v] ∂ p P p [u ↔ v ′ ]
From now on, we note G = (V, E) the bunkbed graph of the complete graph and we label the vertices of V from 1 to 2n such that

V = {s i , i ∈ [1; 2n] ∩ N} and ∀i, j ∈ [1; n] ∩ N, s i ∼ s j and s i ∼ s i+n where ∼ is the neighbour relation, x ∼ y ⇔ {x, y} ∈ E.
We can consider that the vertices labelled from 1 to n are the vertices of the lower level (or level 1) and the vertices labelled from n+1 to 2n are the vertices of the upper level (or level 2). In the complete graph, two vertices play the same role. Therefore, it is enough to prove the bunkbed conjecture for two vertices, here s 1 and s n . Moreover, it is trivial to see that P (s 1 ↔ s 1 ) = 1 ≥ P (s 1 ↔ s n+1 ).

Our approach, based on combinatorics since p = 1/2, is to decompose the graph into different appropriate classes which solve the bunbed conjecture. We count the number of ways to connect s 1 to s n and to connect s 1 and s 2n . The idea of the proof is the following. We define the main component as the connected component that contains the vertex s 1 and all the vertices that are connected to s 1 by an open path. Then, we distinguish different classes of main component depending on the number of vertices on the lower level, on the upper level, and depending on the number of parallel vertices, notion that will be defined later. Lemma 2.1 will give the number of ways to connect the vertices of a main component. Lemmas 3.3 and 3.4 will give the number of configurations containing a main component with x vertices on the lower level, y vertices on the upper level and a key argument to properly add them together. Finally, we prove the main Theorem in section 4.

Covering Graphs

We recall that a covering graph of a graph

G = (V, E) is a subgraph G ′ = (V ′ , E ′ ) such that G ′ is connected, V ′ = V and E ′ ⊆ E.
From now on, G will be used to refer to the bunkbed graph of the complete graph. We classify the subgraphs of G according to the nomber of vertices in the upper and lower level, as well as the number of parallel vertices. We define G x,y,z the set of connected subgraph of G such that ∀G ′ = (V ′ , E ′ ) ∈ G x,y,z :

(1) ∃λ 1 , ..., λ x ∈ {1, ..., n} , ∃µ 1 , ..., µ y ∈ {n + 1, ..., 2n} such that ∀i ∈

[1; x] s λi ∈ V ′ and ∀j ∈ [1; y], s µj ∈ V ′ , ∪ i {s λi } ∪ j {s µj } = V and #V = x + y (2) ∃λ 1 , ..., λ z ∈ {1, ..., n} such that ∀i, j z, λ i = λ j and {s λi , s λi+n } ⊂ V ′ (3) ∀x, y ∈ V ′ , {x, y} ∈ E ′ iff {x, y} ∈ E Graphs of G x,
y,z can be seen as extraction of subgraph of the bunkbed graph G. Condition 1 insures that there are exactly x vertices on the lower level and exactly y vertices on the upper level. Condition number 2 insures that exactly z vertices among the y vertices on the upper level are above the x vertices of the lower level. Finally, condition 3 insures that vertices present in the extraction comes along with the corresponding edges. We will say that a graph of G x,y,z has x vertices of level 1 and y vertices of level 2 and z parallel vertices (e.g. figure 2.1). Moreover, one can see that two graphs G 1 , G 2 of G x,y,z are isomorphs.

We define the function GC : N 3 → N which gives the number GC (x, y, z) of covering graphs of a graph in G x,y,z .

Lemma 2.1. Fix x, x ′ , y, y ′ , z ∈ [0; n] ∩ N such that x + y = x ′ + y ′ and z min (x, x ′ , y, y ′ ). If |x -y| |x ′ -y ′ | then GC (x, y, z) GC (x ′ , y ′ , z)
Remark 2.2. The function GC is symmetric in its first two coordinates, meaning that for all x, y, z, we have GC (x, y, z) = GC (y, x, z).

Proof. To prove this lemma, by iteration, it is sufficient to prove the inequality GC (x + 1, y, z) GC (x, y + 1, z) for x > y z. For this matter, we need to give an upper bound and a lower bound of GC (x, y, z). As an upper bound we use a trivial one: we bound the number of covering graphs by the number of possible graph knowing that one vertical edge need to be open. Since in a complete graph with n vertices, there are n (n -1) /2 edges, therefore we have:

GC (x, y, z) 2 x(x-1) 2
(2 z -1) 2

y(y-1) 2
As a lower bound, we consider the only case where we connect by at least one vertical edge a covering graph K x and a covering graph of K y :

GC (x, y, z) GC (x, 0, 0) × (2 z -1) × GC (0, y, 0)
Therefore, one has:

GC (x + 1, y, z) GC (x, y + 1, z) GC (x + 1, 0, 0) × (2 z -1) × GC (0, y, 0) 2 x(x-1) 2 (2 z -1) 2 y(y+1) 2 = GC (x + 1, 0, 0) × 2 -x(x-1) 2 × GC (0, y, 0) × 2 -y(y-1) 2 × 2 -y
By [START_REF] Flajolet | Analytic combinatorics[END_REF], the number of covering graphs or number of connected labelled graph with n vertices is given by the following approximation:

GC (n, 0, 0) = 2 n(n-1) 2 1 -2n2 -n + o 2 -n
We slightly modify this approximation in the case n 7:

GC (n, 0, 0) 2 n(n-1) 2 1 -3n2 -n
Then, we split the problems into different cases. First case, whenever x 10 and y 7. Since x y + 1, one has:

GC (x + 1, y, z) GC (x, y + 1, z) 2 x-y 1 -3x2 -x 1 -3y2 -y 2 × 1 -3 × 10 × 2 -10 × 1 -3 × 7 × 2 -7 1
Second case, when x 10 and y < 7, then x y + 4:

GC (x + 1, y, z) GC (x, y + 1, z) 2 x-y-1 1 -3x2 -x 2 3 × 1 -3 × 10 × 2 -10 1
Finally, when 10 > x > y, we have computed the result by computer which ends the proof.

Result on the Size of the Classes

We define G 1 the set of connected subgraphs of G containing the vertices s 1 and s n , and G 2 the set of connected subgraphs of G containing he vertices s 1 and s 2n . Moreover, we define the set of graph G 1

x,y,z = G 1 ∩ G x,y,z and the set G 2

x,y,z = G 2 ∩ G x,y,z . We define the functions q 1 : N 3 → N and q 2 : N 3 → N such that q 1 (x, y, z) = #G 1 x,y,z and q 2 (x, y, z) = #G 2 x,y,z . Finally, we define the function q : N 3 → N such that (3.1) q (x, y, z) = q 1 (x, y, z) -q 2 (x, y, z) + q 1 (y, x, z) -q 2 (y, x, z) if x = y q 1 (x, x, z) -q 2 (x, x, z)

if x = y
Before giving the main result on the function q, we give two preliminary results on the exact value of the functions q 1 and q 2 . Lemma 3.1. For all x, y z 1 such that x + y -z n

q 1 (x, y, z) = (n -2)!x (x -1) (x -z)!z! (n -x -y + z)! (y -z)!
Proof. First, if z = 0 and y > 0, then a graph cannot be connected. Moreover, s 0 and s n have to be in the set of vertices of the graph of G 1

x,y,z , x has to be greater that 2 otherwise G 1

x,y,z is an empty set. Then, we have to choose the x -2 vertices of level 1 among the the n -2 vertices left, distribute z vertices of level 2 on top of the x vertices previously chosen, and choose y -z vertices among the n -x vertices left. Therefore, we can write:

q 1 (x, y, z) = n -2 x -2 × x z × n -x y -z × 1 x 2
Then, one can note that for all k ∈ N, the following equality holds:

(3.2) 1 (x -k)! 1 x k = 1 x! k i=0 (x -i)
So we can write:

q 1 (x, y, z) = (n -2)! (n -x)! (x -2)! × x! (x -z)!z! × (n -x)! (n -x -y + z)! (y -z)! 1 x 2 = (n -2)!x (x -1) (x -z)!z! (n -x -y + z)! (y -z)! Lemma 3.2. For all x, y z 1 such that x + y -z n: q 2 (x, y, z) = (n -2)! (xy -z) (x -z)!z! (n -x -y + z)! (y -z)!
Proof. First, we notice that for any graph in G 2

x,y,z , s 0 and s 2n belong to the set of vertices. Then, to count the number of graphs G in G 2

x,y,z , we distinguish 4 different cases: either s n and s n+1 belong to G; either s n belongs to G but not s n+1 ; either s n does not belong to G but s n+1 does; either s n and s n+1 don't belong to G. We can write:

q 2 (x, y, z) = n -2 x -2 × x -2 z -2 × n -x y -z × 1 x 2,y 2,z 2 + n -2 x -2 × x -2 z -1 × n -x y -z × 1 x>max(1,z),y<n + n -2 x -1 × x -1 z -1 × n -x -1 y -z -1 × 1 x<n,y>max(1,z) + n -2 x -1 × x -1 z × n -x -1 y -z -1 × 1 max(1,z)<x<n,max(1,z)<y<n
Therefore, using (3.2), we have:

q 2 (x, y, z) = (n -2)! (x -z)!z! (n -x -y + z)! (y -z)! × z (z -1) + (n -2)! (x -z)!z! (n -x -y + z)! (y -z)! × z (x -z) + (n -2)! (x -z)!z! (n -x -y + z)! (y -z)! × z (y -z) + (n -2)! (x -z)!z! (n -x -y + z)! (y -z)! × (x -z) (y -z) = (n -2)! (x -z)!z! (n -x -y + z)! (y -z)! (xy -z)
Using lemmas 3.1 and 3.2, we have that:

q 1 (x, y, z) -q 2 (x, y, z) = (n -2)! x 2 -x -xy + z (x -z)!z! (n -x -y + z)! (y -z)!
Recall the definition of the function q in (3.1), we have for all x z 1:

q (x, x, z) = (n -2)! (z -x) (x -z)! (x -z)!z! (n -2x + z)!
And for all x, y z 1:

(3.3) q (x, y, z) = (n -2)! x 2 -2xy + y 2 -x -y + 2z (x -z)! (y -z)!z! (n -x -y + z)! Note that: (3.4) q (x, y, z) 0 ⇔ x ∈ y + 1 - √ 8y -8z + 1 2 ; y + 1 + √ 8y -8z + 1 2
Lemma 3.3. For all k z, the following inequality holds:

k-z i=0 q (k + i, k -i, z) = 0
Proof. To prove the theorem, it is actually easier to prove that:

k-z i=1 q (k + i, k -i, z) = -q (k, k, z)
Using as arguments of q the triplet (k + i, k -i, z), some factors of (3.3) become independent of i. Indeed, we get the following equality:

q (k + i, k -i, z) = 4i 2 -2k + 2z (k + i -z)! (k -i -z)! × (n -2)! z! (n -2k + z)!
Therefore, proving the Lemma is equivalent to prove that:

k-z i=1 4i 2 -2k + 2z (k + i -z)! (k -i -z)! = k -z (k -z)! (k -z)!
Then, it is enough to see that:

k -z (k -z)! (k -z)! = 4 -2k + 2z (k + 1 -z)! (k -1 -z)! + 3 (k + 2 -z) (k + 2 -z)! (k -2 -z)! 1 k-z 2
and that for all k -z > i 2:

(2i -1) (k + i -z) (k + i -z)! (k -i -z)! = 4i 2 -2k + 2z (k + i -z)! (k -i -z)! + (2i + 1) (k + i + 1 -z) (k + i + 1 -z)! (k -i -1 -z)! Whenever i = k -z, then: (2i -1) (k + i -z) (k + i -z)! (k -i -z)! = 4 (k -z) 2 -2k + 2z (2k -2z)!
which ends the proof.

Lemma 3.4. For all k z, the following inequality holds:

k-z i=0 q (k + i + 1, k -i, z) = 0
Proof. The proof goes in the same way as lemma 3.3. Indeed, it is enough to prove that:

k-z i=1 2i 2 + 2i -k + z (k + i + 1 -z)! (k -i -z)! = k -z (k + 1 -z)! (k -z)!
Then, we have:

k -z (k + 1 -z)! (k -z)! = 4 -k + z (k + 2 -z)! (k -1 -z)! + 2 (k -z + 3) (k + 3 -z)! (k -2 -z)! 1 k-z 2
Then, for all k -z > i 2, the following equality holds:

i (k + i + 1 -z) (k + i -z)! (k -i -z)! = 2i 2 + 2i -k -z (k + i -z)! (k -i -z)! + (i + 1) (k + i + 2 -z) (k + i + 1 -z)! (k -i -1 -z)! 8 U V V ′ U V V ′ Figure 4.1. Decomposition of a configuration Whenever i = k -z, i (k + i + 1 -z) (k + i + 1 -z)! (k -i -1 -z)! = 2 (k -z) 2 + 2 (k -z) -k + z (2k -2z + 1)!
which ends the proof.

Proof of the main theorem

Recall that G = (V, E) is the bunkbed graph associated with the complete graph K n . We recall here the idea of the proof of the main theorem. We split the configuration depending on the number of vertices of the main component and then split again depending on the number of vertices on the lower/upper level. Figure 4.1 gives a decomposition of a configuration and we give some explanation about the figure. For simplicity we have not drawn a complete graph. Then, edges drawn in solid lines are open edges and edges drawn in dotted lines are closed edges. Green vertices and green edges correspond to the main component. Red edges are the adjacent edges of the main component that need to be closed, if not, the main component would expand. Blue vertices and blue edges correspond to the "outside" of the main component. In this sense, we define O (x, y, z) the number of "outside" edges when the main component has x vertices in the lower level, y vertices in the upper level and z parallel vertices. A simple computation gives that:

O (x, y, z) = n (n -x -y) + 1 2 x 2 -x + y 2 -y + z
As well as the following relations:

O (x, y, z) = O (y, x, z) O (x + 1, y, z) -O (x, y + 1, z) = x -y
The second relation can be understood in the spirit of Lemma 2.1 through the following statement: x + y = x ′ + y ′ and |x -y| |x ′ -y ′ | implies O (x, y, z) O (x ′ , y ′ , z). All the quantities developped before allows us to express the probability of connection between two vertices. First, note that since the Bernouilli parameter of the percolation p is equal to 1/2, therefore every configuration has the same probability, i.e. for any configuration ω, P 1/2 (ω) = 2 -#E . Moreover, knowing that the main component of s 0 , noted M C (s 0 ), is a subgraph of a graph in G 1

x,y,z , there are GC(x, y, z) way for it to be connected, and O (x, y, z) outside edges which won't affect the connectivity of s 0 and s n . Furthermore, there are q 1 (x, y, z) to choose the subgraph corresponding to the main component. Thus, we have the following equalities:

P 1/2 (s 0 ↔ s n ) =
x,y,z MC(s0) subgraph of G∈G 1

x,y,e

2 O(x,y,z) P (ω) = 2 -#E

x,y,z

2 O(x,y,z) GC (x, y, z) q 1 (x, y, z)

= 2 -#E k 0 z 0 i∈Z
2 O(k+i,k-i,z) GC (k + i, k -i, z) q 1 (k + i, k -i, z)

+2 -#E k 0 z 0 i∈Z 2 O(k+i+1,k-i,z) GC (k + i + 1, k -i, z) q 1 (k + i + 1, k -i, z)
The third equality is obtained by an operation of renumbering. In the same way, we have that:

P 1/2 (s 0 ↔ s 2n ) = 2 -#E k 0 z 0 i∈Z 2 O(k+i,k-i,z) GC (k + i, k -i, z) q 2 (k + i, k -i, z) +2 -#E k 0 z 0 i∈Z
2 O(k+i+1,k-i,z) GC (k + i + 1, k -i, z) q 2 (k + i + 1, k -i, z)

For all k and z, because of the symmetry of the function GC and the function O, one has: i∈Z 2 O(k+i,k-i,z) GC (k + i, k -i, z) (q 1 (k + i, k -i, z) -q 2 (k + i, k -i, z))

= i 0
2 O(k+i,k-i,z) GC (k + i, k -i, z) q (k + i, k -i, z)

Recall that q (k + i, k -i, z) might be negative, see (3.4), and because of lemma 2.1 for all k, there exists an i 0 such that for all 0 i i 0 :

GC (k + i, k -i, z) GC (k + i 0 , k -i 0 , z) GC (k + i + 1, k -i, z) GC (k + i 0 + 1, k -i 0 , z) O (k + i, k -i, z) O (k + i 0 , k -i 0 , z) q (k + i, k -i, z) 0
and for all i i 0 :

GC (k + i, k -i, z) GC (k + i 0 , k -i 0 , z) GC (k + i + 1, k -i, z) GC (k + i 0 + 1, k -i 0 , z) O (k + i, k -i, z) O (k + i 0 , k -i 0 , z) q (k + i, k -i, z) 0

Figure 2 . 1 .

 21 Figure 2.1. The bunkbed graph of K 5 and an element of G 3,4,2

By lemma 3.3:

And by lemma 3.4:

This concludes the proof since:

The author thanks Cyril Roberto, Julien Bureaux and Florent Barret for the many insightful conversations about the topic as well as Joseba Dalmau and Anna Bonnet.