Keywords: neural fields, pattern classification, dynamical systems, computational neuroscience, spatiotemporal sequences champs neuronaux, reconnaissance de motifs, systémes dynamique, neuroscience computationnelle, séquences spatio-temporelles

In this work we present the parameter optimization of a distributed model for the classification of temporal sequences. The model we present codes information with a population of units, and it can be applied to discriminate between several spatiotemporal sequences. The implementation of this classification technique strongly depends on how the the connection weights are set according to the sequence we want to code. This "learning" phase depends on several parameters, for which we present a detailed analysis in this work. In the first part, we give some examples and we retrieve the model parameters in 1D by extending other existing studies. Later on we extend the analysis into 2D, proposing at the end of the report a strategy to code any given spatiotemporal sequence that can be described as a set local spatiotemporal trajectories. The derivation of parameters is performed analytically and numerical simulations are performed to verify our results.

Introduction

In this report we present the details of a bio-inspired model for the coding and classification of spatiotemporal patterns based on the Continuous Neural Field Theory (CNFT). In particular we report, the way in which this model can be tuned in order to obtain coding and classification properties.

A spatiotemporal pattern can be understood as a volume in space and time, where space is a generic dimension depending on the problem (position, frequency or features in general). In computer vision, action recognition from video sequences is an example of classification of spatiotemporal patterns. Similarly, speech recognition in the audio processing field is another example of spatiotemporal pattern classification. Both problems have in common that is not just to recognize (and to code), the right picture or phoneme, but also to verify the right temporal sequence at a specific speed of presentation.

In the Continuous Neural Field Theory (or CNFT), we code space whatever its nature, as a population of units that represent it locally (close units code for close stimuli in space) and we study the temporal activity of these neurons with no transmission delay among units [START_REF] Taylor | Neural "bubble" dynamics in two dimensions: foundations[END_REF] and a linear dependency with the RR n°7543 input activity. The advantage of this kind of model is a distributed representation of space suitable to be deployed in parallel implementations and even tolerant to failures: for example if some of the units that represent the space "fail". The CNFT provides also a macroscopic description for the behavior of massive populations of units presenting an interesting bridge of discussion with neuroscience, from a theoretical point of view.

This work presents a variant of the CNFT theory able to code spatiotemporal patterns, considering them as a temporal sequence of spatial patterns. For this model that we are introducing, we show both analytically and experimentally that it can be effectively tuned to code and retrieve simple, yet generic, temporal sequences.

Spatiotemporal patterns

Spatiotemporal patterns can be found in different contexts, as we have mentioned. The main property of these patterns is the temporal dependency (there is a sequence), while the spatial dimension may change depending on the problem: for video processing it is usually the 2D space, for audio processing it is usually the Fourier domain. We will begin the presentation of our work with an illustrative example from the image processing domain for the coding and classification of spatiotemporal patterns. After this example, we will focus on a special kind of pattern, suitable to be described as a set of local spatiotemporal features. Later, we will show how it can be modeled with the help of the CNFT, and we derive the precise parameters required for it.

An example pattern: video sequences

In image processing, any video sequence can be seen as a temporal stack of images, where each image is a static pattern. Once we have seen a given sequence, we can ask, given a new sequence, if it corresponds to one of the sequences we have already seen. If we consider a special kind of videos: human motion, we can think of sequences such as to walk, to box or to wave, see Figure 1, and we could ask for a new video sequence, which of the actions we have already seen is the most similar. This kind of classification task must deal with the generalization/over-fitting problem [START_REF] Cherkassky | Learning from Data: Concepts, Theory, and Methods[END_REF].

Coding with snapshots

In general, a given spatiotemporal pattern will not have any structure, and we will be forced to completely record each "2D" spatial pattern (or snapshot, in the video example). Also, considering cyclic movements, the phase is not necessarily 0, i.e. the sequence does not necessarily start at the same point in time.

Suppose we want to code a simple on-to-off sequence as in Figure 2(a). The spatial dimension we are considering is light intensity (ℓ), a continuous variable that for the on-to-off sequence goes from 0 to 6. If after black (no light) it comes white again and between each light intensity there is a fix time, we can consider it as a cyclic sequence. We want to answer the question: is a given series of n images with speed of presentation v, in the on-to-off sequence?, see RR n°7543 Let us consider that the solution to this problem should first verify for each image of the set n, if it matches some image in the sequence. Only after this, we verify the temporal sequentiality. We will study this kind of solution, and propose a practical implementation using a variant of the CNFT.

Complete Snapshots coding (1D ACNFT)

The classification problem can be solved with several methods [START_REF] Cherkassky | Learning from Data: Concepts, Theory, and Methods[END_REF] from statistical decision theory such as LDA and Fisher's LDA, regression based techniques as MLP, RBF and SOM, nearest-neighbor or prototypes methods like k-nearestneighbor and LVQ, without forgetting SVM's. Yet, if we require the solution to give partial answers while the input is displayed and also to answer in case of time distortions1 (see Figure 2(b)), then the solution becomes less evident, even more if we want a robust distributed representation. In this case, an interesting

P k k , using d(I l , P k k ) = exp(-(I l -P k k ) 2 )
approach to consider is by Giese et al. [START_REF] Giese | Neural mechanisms for the recognition of biological movements[END_REF], where they consider a similarity function d between an input image (I) and a prototype image (P k ): in our discrete on-to-off example, we need 7 light intensity prototypes P k (see 2(a) below). Note that in general the input (I) and prototypes (P k ) are images and as in the on-to-off sequence, images are single colored, we note I l and P k l for the light intensity l of: the input image and the k th prototype image respectively. Also, as we can choose freely where to start to count the prototypes we impose P k k .

For each input image at any time we will have an answer from all prototypes, as the pattern may start at any light intensity, in our example. We can see the profile of responses for three prototypes with respect to light intensity of different inputs (I l ), in Figure 3(b). The shape of the profiles in Figure 3 will be given by the similarity function, but we still need to verify that images appears at precisely the right order and speed. To illustrate how the mechanism work we will explain it recursively. Suppose that we are verifying that image I at time t (or I(t)) belongs to the sequence, we know that until time t -1 the sequence is correct and we also know that next image in the sequence must be P n . In this case we need to verify that image I(t) corresponds to the prototype P n . If both elements: history and current image are correct, we can stand that the image I(t) correspond to the sequence.

In the CNFT, we represent the spatial dimension (light intensity) as units, where we describe the variation of the activity for each unit as a differential equation with a decay term (for the on-to-off sequence there are 7 units), see the left part of Eq. 1. In this specific case we consider the input as the similarity function that we mention, and we require the interaction between units to account for the verification of the sequence. All of which can be summarized in Eq. 1.

τ ∂m(k, t) ∂t + m(k, t) = k ′ w(k ′ -k)m(k ′ , t) + D(I(t), P k ) + (1)
RR n°7543

where we note D(I, P k ) = d(I l , P k k ) to simplify the notation, [ ] + is the nonlinear rectification equivalent to max( , 0). The function w(k ′ -k) codes the sequence and the speed the light should change. Here, w must be asymmetric with respect to k ′ -k, to account for the sequentiality. The precise degree of asymmetry (the parameters of w) codes for the speed and it requires to determined, as it will be explained further in Section 3.1. To the the variation of the CNFT with an asymmetric kernel (w) we call it Asymmetric CNFT, or ACNFT.

To answer if an input image sequence is in the on-to-off sequence with a given speed, we look at the value of,

A(t) = k,t m(k, t) = 2π t r 0 (t) (2)
which should be maximal for the on-to-off sequence at the speed for which we tune the system. In Eq. 2, r 0 (t) accounts for the total instantaneous activity over space. Any other spatiotemporal sequence either with different spatial patterns or temporal order should deliver a lower value of A. This model allows to correctly verify a given sequence, and to answer as input images become available, not necessarily with the complete sequence.

There are several metrics we can use as d function, in general any RBF as other authors have proposed [START_REF] Casile | Critical features for the recognition of biological motion[END_REF]. However, for the exact shape of w in terms of the input speed v there is no analytical result to our knowledge. The authors of [START_REF] Casile | Critical features for the recognition of biological motion[END_REF] report manual tuning by using data directly. In the next section we will show how to effectively tune the model for a given input speed v, in such a way to guarantee a maximal value of A for the right sequence.

Complete Snapshots coding, an example

Following the on-to-off example, we show now how Eq. 1 provides an effective way to code and classify spatiotemporal patterns.

To check the sequentiality, we will consider two possible input sequences: the on-to-off and the exact inverse off-to-on sequence and only one population of units coding the on-to-off sequence. The values of A as function of time t, with A defined as in Eq. 2, can be seen in Figure 4(a), where it can be observed that the ACNFT is able to distinguish the on-to-off sequence before the complete pattern is presented (at t = 6) as the value of A(t) is higher with the on-to-off sequence as input than with the off-to-on sequence, converging quickly to the stable state of the system as can be seen in Figure 4(b),

Until now, we have introduced a variant of the CNFT, able to code spatiotemporal patterns, mainly developed by [START_REF] Casile | Critical features for the recognition of biological motion[END_REF][START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF][START_REF] Hansel | Methods in Neuronal Modeling: From synapses to networks, chapter Modeling Feature Selectivity in Local Cortical Circuits[END_REF]. In Section 3.1 we present the theoretical analysis for the optimization of parameters in the 1D case, to make the model viable to be implemented as a classification system. Details about the parameters and methods of the numerical simulation, can be found in the Appendix E.

Coding with local features

As we have seen, we can code an arbitrary spatiotemporal sequence as a series of spatial patterns keeping full record of each spatial pattern, and for the temporal part, we can use a differential equation to represent the speed the sequence should appear. To perform this we require a distance function d, to compare the input pattern with the prototypes that compose the sequence stored in advance (P k ).

In fact, we do not need the metric d and we do not need to store the complete spatial patterns if the spatiotemporal pattern can be described as a sparse set of local features in the spatiotemporal volume. Many sequences can be decomposed as a set of "trajectories" in time: in human movements if we enlight the joints, the sequences can be described with a few trajectories, see Figure 1. The sparse local feature coding that we propose in 2D using the ACNFT, takes advantage of this idea.

Sparse local feature coding (2D ACNFT)

The idea to code a spatiotemporal volume using local features has been addressed before with the local-feature description, specially in the image processing domain [START_REF] Laptev | On space-time interest points[END_REF]. In these methods common issues are: how they respond to variations in speed (time compression/dilation), the lack of response for partial patterns and phase-dependent recognition. For those reasons, a model that combines both elements (the advantages of local feature detection with a distributed mechanism to compare patterns improving these issues) appears as an interesting and appealing idea.

To code spatiotemporal patterns in 2D, we consider only those that can be described as a set of trajectories (joints trajectories for human movement in Figure 1). The main restriction for the local-features is that they have to be "close" in time, forming continuous trajectories in the space-time volume. We apply the ACNFT model to the set of trajectories, but without loss of generality we first focus on a single trajectory.

The 2D ACNFT directly uses a local feature detection g over the image (see Eq. 3). The difference with the metric d in 1D is the absence of prototypes, performing an isotropic feature detection in g, in other words an implicit representation of the prototypes. Furthermore, we modify the kernel w, to code the location where the trajectory goes through and its corresponding speed. This can be summarized as,

τ ∂m( x, t) ∂t + m( x, t) = Ω w( x ′ , x)m( x ′ , t)d x ′ + g(I( x, t)) + (3)
Eq. 3 differs from Eq. 1 by the spatial variable x of the pattern and by the input defined as g(I( x, t)), where g(•) is transformation independent from the sequence we want to code (local descriptors [START_REF] Laptev | On space-time interest points[END_REF], see also [START_REF] Edelman | Representation and recognition in vision[END_REF] for a discussion of existing pattern coding schemes). Also, the w function no longer depends only on the distance, but also on the position of the unit (because we code the precise trajectories). To answer if an input image sequence belongs to a previously coded sequence with a given speed, we look at the total value of Eq. 2 as in the 1D case.

Sparse local feature coding, an example

To visualize the kind of pattern we could potentially code, we consider for example several moving trajectories that correspond to the positions of the joints for a moving person in a video sequence, see Figure 1. The first example we will consider is the simple case of one left-to-right and one right-to-left trajectory moving at the same speed, where we use the same ACNFT population to discriminate between both. We can see the results of the 2D ACNFT in Figure 5(a), where the value A(t) is larger for the speed the system codes for, and the convergence to the stable state is fast, see Figure 5(b). In this example, the input speeds where {5, -5} in rad/frames (same conditions as in the 1D example), and the system had the same parameters for both inputs, see Appendix E for more details on the numerical simulation. In this particular example, we use a version of the 2D ACNFT detailed in Section 3.2 able to code only one trajectory at a time.

RR n°7543

Parameter tuning

The model presented in the first part of the previous section and introduced by [START_REF] Giese | Neural mechanisms for the recognition of biological movements[END_REF][START_REF] Casile | Critical features for the recognition of biological motion[END_REF][START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF], is able to code any video sequence by saving snapshots (complete pictures) at each time instant (the 1D ACNFT). In the second part, we introduced a new model able to code any 2D sequence that can be decomposed into a set of local features (the 2D ACNFT). However, for both cases, in order to implement them, we need to precise the shape of w and link it to the speed of the input sequence v. The following analysis has been inspired by the work of [START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF] in 1D and extended to the 2D case.

1D ACNFT

To determine the shape of w, we will use the form proposed by [START_REF] Hansel | Methods in Neuronal Modeling: From synapses to networks, chapter Modeling Feature Selectivity in Local Cortical Circuits[END_REF]

see Fig- ure 6(b), w(k ′ -k) = J 0 + J 1 cos (k ′ -k + β) ( 4
)
where k is the spatial variable. The w function should be asymmetric, periodic and continuous at least for the analysis, but this precise form only simplifies the analysis. This function has its highest value at k ′ -k = -β. This implies k > k ′ if β > 0, or for the on-to-off sequence that this function gives a maximal positive value to units placed at distance β to the left of the unit k (i.e. before in terms of time of the sequence).

As we have mentioned, to code different speeds of appearance for the stack of images, the β parameter in Eq. 4 should be tuned accordingly to the input speed v. To derive β(v), this should maximize A as define in Eq. 2 when the input is at speed v. To start the analysis we rewrite A in the continuous case as m(k, t)dk ′ dt = 2π r o (t)dt, and we impose also a system with stable response, i.e. dr 0 /dt = 0, allowing us to write A = (t f -t 0 )r 0 .

RR n°7543

In fact, r 0 is the first Fourier component of m; to simplify the calculations we consider k to be an angular variable, to account for cyclic patterns and to avoid border conditions,

r 0 (t) = π -π m(k, t)(2π) -1 dk (5)
r 0 is calculated in [START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF][START_REF] Hansel | Methods in Neuronal Modeling: From synapses to networks, chapter Modeling Feature Selectivity in Local Cortical Circuits[END_REF] Considering the single bump assumption, introducing k c (t) and imposing simultaneously that dr 0 /dt = 0 and dr 1 /dt = 0, r 0 can be written as (see details in Appendix A),

r 0 (t) = S -J 0 -cos(k c )/f 0 (k c ) (6)
where f 0 is a increasing function of k c as defined in [START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF][START_REF] Hansel | Methods in Neuronal Modeling: From synapses to networks, chapter Modeling Feature Selectivity in Local Cortical Circuits[END_REF], S is the contrast of the sequence parameter (or the rate between the minimum and maximum values, see Figure 6(a)). As we want β(v) to maximize r 0 , in Eq. 6 this is equivalent to minimize k c (J 0 < 0 and S > 0). To minimize k c , we use the constraint of a single bump of activity in Eq. 7, first introduced in [START_REF] Hansel | Methods in Neuronal Modeling: From synapses to networks, chapter Modeling Feature Selectivity in Local Cortical Circuits[END_REF] and detailed in Appendix A,

S ′ = J 0 f 0 (k c ) + cos(k c ) J 2 1 f 2 1 cos(∆) 2 -2J 1 f 1 cos(∆) cos(∆ + β) + 1 (7)
here S ′ is a parameter that depends on the shape of the input sequence, f 1 is a function of k c and ∆ = arctan(τ v). Deriving Eq. 7 with respect to v and asserting dk c /dv| v=vm = 0 (see details in Appendix B), we obtain the following expression for the optimal speed (v m ) as function of β (here our analysis goes beyond [START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF]):

v m = J 1 f 1 (k c )τ -2τ cos(β) -J 2 1 f 2 1 (k c )τ 2 -4τ 2 J 1 f 1 (k c ) cos(β) + 4τ 2 2τ 2 sin(β) (8)
all the terms in Eq. 8 are known except for k c (t), but the solution we look for must be independent of the input if we want to apply it to other sequences and link v m with β. Considering the limit when the input width tends towards a localized input, this implies k c tends towards zero, simplifying v m ,

v m (β) = 1 -cos(β) τ sin(β) (9)
To verify this result, we took the known no-contrast limit of [START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF] (or width of the input → ∞) and our high contrast limit at Eq. 9 (or width of the input RR n°7543 → 0), and we ran simulations to check that inputs with medium contrast are better represented by our high contrast limit, see Figure 7. In Figure 7 we see that as the input contrast (ǫ) is higher our approximation of v m (β) is better, or in other terms, for a given input speed we can derive more precisely the right β so that the mean activity r 0 is maximal. By consequence, we maximize A as function of β. We can also notice that in the high contrast limit the relation between the optimal β and the input speed v is almost linear.

2D ACNFT

In 2D, the ACNFT codes a set of trajectories in space with the same population, where each unit has a local kernel w. We will consider in this analysis the case where w has the size of the input image, i.e we can code only one trajectory with our system (one joint at a time, in Figure 1). If features are sparse "enough" this will be the case in general.

w( x, x ′ ) = p(y, y ′ )q(x, x ′ ) = A (1 + cos(y ′ -y)) (J 0 + J 1 cos (x ′ -x + β)) ( 10
)
where x = (x, y) and x ′ = (x ′ , y ′ ). In Eq. 10 we impose w( x, x ′ ) = p(y, y ′ )q(x, x ′ ) where the asymmetry will be only in the q function as in the 1D case, and the p(y, y ′ ) function is a RBF or something close (cosine in our analysis). The separability of w helps with the analysis, since as along the x-axis we have a system similar to the 1D case, see Figure 8(b). The input we consider is again a spatial bump, moving in the x-direction without loss of generality, see Figure 8(a). The first Fourier component of Eq. 3 needs to be computed as in the 1D case,

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r 0 (t) = π -π π -π m( x, t)(2π) -2 d x ( 11
)
where the spatial variable y is also considered as angular and (x,

y) ∈ [-π, π] × [-π, π].
Using the separability of variables of w and the single bump restriction as for the 1D case, we can derive an expression for r 0 (see Appendix C for details).

r 0 (t) = (C + D 2 ) -AJ 0 -(2 cos(x c ) + cos(x c ) 2 ) /g 0 (x c ) (12)
The main difference is that now the first Fourier component (r 0 ) or the total activity depends on the size of the bump of activity in both axes. The size can be expressed as x c for the width in the x-axis and y c for the height in the y-axis.

As in the 1D case, we want to tune parameter β of this system so as, to maximize r 0 , see Eq. 12. The value of r 0 depends on g 0 (x c ) which is an increasing function of x c and y c (see details in Appendix D), which is equivalent to say that r 0 is a decreasing function of x c and y c . The next step is to use the single bump constraint of the solution, and to make dx c /dv| v=vm = 0 and dy c /dv| v=vm = 0. We thus obtain (see details in Appendix D),

v m = AJ 1 g 1 (x c , y c ) -2 cos(β) -(AJ 1 ) 2 g 2 1 (x c , y c ) -4AJ 1 g 1 (x c , y c ) cos(β) + 4 2τ sin(β) ( 13
)
RR n°7543 v m is the magnitude of the speed, as the input is defined to be moving only in the x direction. The difference with the 1D case is the g 1 function, defined in Appendix 13. For v m we can again use the high contrast limit (g 1 (x c , y c ) → 0),

v m (β) = 1 -cos(β) τ sin(β) (14)
which is the same results as in Eq. 9. In a similar way for the the low contrast limit v m = tan(β) τ can be found in the 2D case, as obtained by [START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF] in the 1D case. To verify both limits in the 1D case, we perform simulations, see Figure 9, where it can be seen that at different input contrast levels (in the 2D case, the contrast is defined by C and D) the optimal value of β is close to the high contrast limit we obtained, yet is not as precise as the 1D case, probably due a to several constraints we assume that do not completely hold in general, see the end of Appendix C for details.

Generic 2D trajectories

Until now we have considered straight line trajectories in 2D, moving along a single axis with constant speed. Yet, curved trajectories can still be coded considering them locally as rectilinear ones. Concerning trajectories where dv/dt = 0, they can still be coded with our model if it is possible to consider them as piece-wise functions with constant speed.

In the general case, if we want to code a given trajectory we have each position available in space x i = (x i , y i ) in pixels and its derivative v i = (v i x , v i y ), in pixels per frame. Due to implementation issues the size of the kernel should be much smaller than the total size of the field σ ≪ Σ, and obviously this size limits the maximal speed the local kernel will be able to code as |β| < σ/2. On the other hand, the minimal speed is constrained to the equivalent of 1 pixel/frame, or |β| > 2π/σ. The position in pixels within the trajectories identifies the unit which kernel should be modified directly as x i = (x i , y i ). Then, the direction of movement θ can be calculated as θ = arctan(v i y /v i x ), this angle is the rotation necessary to put the x-axis in the direction of motion. The magnitude of the speed in pixels | v i | must be transformed into radians per frame with v i r = | v i |2π/σ, then used so as to calculate the asymmetry in the axis of movement to obtain β i .

x θ = (x -x i ) cos(θ) -(y -y i ) sin(θ) ( 15
)
y θ = (x -x i ) sin(θ) + (y -y i ) cos(θ) (16) w( x θ ) = exp - 1 2σ 2 (x θ -β i ) 2 + y 2 θ cos 2πx θ σ + β i (17)

Conclusions

In this work we have presented an analytic optimization of the parameters in the asymmetric CNFT, where we assume a precise form of the input and the kernel. More precisely, we maximize the amplitude of the activity, for the stable state as function of the asymmetric parameter β in both, 1D and 2D scenarios. The optimization in 1D and the complete analysis in 2D are our main contributions, and allow us to propose a new classification model based in the decomposition of spatiotemporal volume into set of local trajectories, see Figure 10.

The results we have shown indicate that as the input stimuli gets more and more localized in space (less noisy), the relation between the asymmetry parameter (β) and input speed (v) becomes almost linear, far from the intrinsic velocity of the system in the absence of stimuli. A simple relation like this for the asymmetric CNFT is interesting: as other inputs (with different shapes) get more localized, they will have a function β(v) close to linear so as, to maximize the amplitude. Following this idea this we propose how to generalize the ACNFT for generic trajectories, see Figure 10, in order to build a classification model based in our ideas.

The kind of analysis we have presented is different to some other approaches, for example numerical ones [START_REF] Frédéric Alexandre | On practical neural field parameters adjustment[END_REF][START_REF] Alecu | Application-driven parameter tuning methodology for dynamic neural field equations[END_REF], where no assumptions about the input are performed. Yet, as we have shown in this work, even if we use a specific input function the results may hold in general cases, for example taking the appropriates limits. Nevertheless, it will be interesting to compare numerical approaches as the ones proposed by [START_REF] Frédéric Alexandre | On practical neural field parameters adjustment[END_REF][START_REF] Alecu | Application-driven parameter tuning methodology for dynamic neural field equations[END_REF], to see how the solutions differ (or not).

Appendices A Analytical expression of r 0 (t)

This first analysis has been performed elsewhere (see [START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF][START_REF] Hansel | Methods in Neuronal Modeling: From synapses to networks, chapter Modeling Feature Selectivity in Local Cortical Circuits[END_REF]), and we only make explicit and clarify some of the steps. Given Eq. 1, we first look for a closed form for the mean activity (first Fourier component), as defined in Eq. 5. To achieve this we use the same input as in [START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF][START_REF] Hansel | Methods in Neuronal Modeling: From synapses to networks, chapter Modeling Feature Selectivity in Local Cortical Circuits[END_REF],

I(k, t) = C [1 -ǫ + ǫ cos(k -vt)] -T ( 18
)
at t = 0 this represents a bump around k = 0, shifting towards the right at speed v, where the parameters C, ǫ and T control the ratio between maximal (C -T ) and minimal (C -T -2ǫ) activity. We also need the dynamics of the second Fourier component r 1 (t) defined in Eq. 19, because r 0 (t) depends on it, as we will see later on.

r 1 (t) = π -π m(k ′ , t)(2π) -1 e i(k ′ -ψ(t)) dk ′ (19)
In Eq. 19 ψ(t) is an unknown function of time, that makes r 1 (t) a real and positive number, in other words for a single bump solution, this is the peak of the bump position (or phase in the complex plane). Introducing Eq. 18 and Eq. 4 into Eq. 1 and identifying r 0 as defined in Eq. 5 we can rewrite Eq. 1 as,

τ ∂m(k, t) ∂t + m(k, t) = J 0 r 0 (t) + C(1 -ǫ) -T + Cǫ cos(k -vt) + J 1 π -π m(k, t)(2π) -1 cos(k ′ -k + β)dk ′ + (20)
here the point is to simplify as much as possible any dependency in the space, and to impose a single bump solution. To do this, we can rewrite the right-side

RR n°7543
of Eq. 20 in the complex plane to simplify calculations (considering only the real part) and then using Eq. 19,

J 0 r 0 + C(1 -ǫ) -T + Cǫe -i(k-vt) + J 1 π -π m(k, t)(2π) -1 e i(k ′ -k-β) dk ′ J 0 r 0 + C(1 -ǫ) -T + Cǫe -i(k-vt) + J 1 π -π m(k, t)(2π) -1 e i(k ′ -k-β+ψ-ψ) dk ′ J 0 r 0 + C(1 -ǫ) -T + Cǫe -i(k-vt) + J 1 π -π m(k, t)(2π) -1 e i(k ′ -ψ) e i(-k-β+ψ) dk ′ J 0 r 0 + C(1 -ǫ) -T + Cǫe -i(k-vt) + J 1 r 1 (t)e i(-k-β+ψ) (21)
To ensure a bump solution, we assume a single cosine shape as total input (all the terms inside [ ] + ) introducing another variable: φ(t), which represents the spatial position of the total input center in Eq. 21 (not in m as ψ(t)). Taking only the real part of Eq. 21, and introducing φ -φ we obtain,

I 0 (t)+ cos(φ -k)(J 1 r 1 cos(ψ -β -φ) + Cǫ cos(vt -φ))+ sin(φ -k)(J 1 r 1 sin(ψ -β -φ) + Cǫ sin(vt -φ)) (22)
where I 0 (t) = J 0 r 0 (t) + C(1 -ǫ) -T . The single bump shape for the solution translates then into imposing Eq. 23.

J 1 r 1 sin(ψ -β -φ) + Cǫ sin(vt -φ) = 0 (23)
Using Eq. 22, the condition in Eq. 23 and the auxiliary variable I 1 (t) = J 1 r 1 cos(ψ -β -φ) + Cǫ cos(vt -φ), we can rewrite Eq. 1 as:

τ ∂m(k, t) ∂t + m(k, t) = [I 0 (t) + cos(k -φ(t))I 1 (t)] + (24)
Until now, we have imposed a shape for the solution (single bump) and thus we have obtained a restriction (Eq. 23). Rewriting the system, we have derived Eq. 24. The next step is to deal with the non-linearity [ ] + and to show the existence and the stability of the solution, i.e. dr 0 (t)/dt = 0 and dr 1 (t)/dt = 0. In order to simplify the non-linearity we use variable k c (t) (the unknown width of the total input bump). When k c = k -φ, the right side of Eq. 24 is zero and we obtain I 0 (t) = -I 1 (t) cos(k c ), this allows us to write,

τ ∂m(k, t) ∂t + m(k, t) = [I 1 (t) (cos(k -φ(t)) -cos(k c ))] + (25)
Before using dr 0 (t)/dt = 0 and dr 1 (t)/dt = 0, the last step is to transform Eq. 25 into the Fourier domain by integrating each term in Eq. 25 with π -π (2π) -1 dk for the first Fourier component, and with π -π (2π) -1 e ik dk for the second component. The second component derives into two equations since it is complex, describing the dynamics of the system with a total of 3 coupled equations in the Fourier domain, see Eqs. 26, 27 and 28.

RR n°7543 τ ∂r 0 (t) ∂t + r 0 (t) = I 1 (t)f 0 (k c ) (26) τ ∂r 1 (t) ∂t + r 1 (t) = I 1 (t)f 1 (k c ) cos(φ -ψ) (27) τ r 1 (t) ∂ψ(t) ∂t = I 1 (t)f 1 (k c ) sin(φ -ψ) ( 28
)
f 0 and f 1 are increasing functions of k c defined as in [START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF][START_REF] Hansel | Methods in Neuronal Modeling: From synapses to networks, chapter Modeling Feature Selectivity in Local Cortical Circuits[END_REF]. The system is coupled because I 1 (t) depends on r 1 (t). Now, we can impose dr 0 (t)/dt = 0, dr 1 (t)/dt = 0 and dψ(t)/dt = v (input and solution for m(k, t) move at the same speed), to finally obtain the closed form of r 0 and r 1 ,

r 0 (k c ) = f 0 (k c ) S -J 0 f 0 (k c ) -cos(k c ) (29) r 1 (k c ) = f 1 (k c ) S -J 0 f 0 (k c ) -cos(k c ) cos(∆) (30)
where S = (C(1 -ǫ) -T ) and ∆ = arctan(τ v). This solution exists (and if it exists we know it is stable) if the condition in Eq. 23 can be achieved. To check this we inject Eqs. 29 and 30 into Eq. 23, obtaining:

S ′ = J 0 f 0 (k c ) + cos(k c ) J 2 1 f 2 1 cos(∆) 2 -2J 1 f 1 cos(∆) cos(∆ + β) + 1 (31)
where

S ′ = (1 -(Cǫ/C -T ) -1 ) 2 .
Then the system has a solution, which is stable, only if Eq. 31 can be verified for a given set of parameters. This verification cannot be performed analytically (we do not know k c ), but numerically changing v and β and fixing the other parameters there is a range of parameters where Eq. 31 can be verified.

B Optimization of v(β)

The analysis detailed now is original from this work and it differs from [START_REF] Hansel | Methods in Neuronal Modeling: From synapses to networks, chapter Modeling Feature Selectivity in Local Cortical Circuits[END_REF] in the asymmetry and in the results we obtain and the limits we verify from [START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF].

Once the shape of r 0 (t) has been determined, see Appendix A for details, we want to optimize it as a function of the input speed v. In other words, we want β(v) to maximize r 0 for stable solutions, i.e. that verify Eq. 31.

First, it can be noticed from Eq. 32 that maximizing r 0 (k c ) is equivalent to minimizing k c in Eq. 29 as f 0 is an increasing function of k c , J 0 < 0 and S > 0.

r 0 (k c ) = S -J 0 -cos(k c )/f 0 (k c ) (32)
In the other hand, Eq. 31 can be derived in terms of v and setting the extrema at zero i.e. ∂k c /∂v| v=vm = 0, we can obtain the minimum k c in terms of v, or v m . If we apply the operator ∂ ∂v in Eq. 31, we obtain two solutions for v m :

v m = J 1 f 1 (k c )τ -2τ cos(β) ± J 2 1 f 2 1 (k c )τ 2 -4τ 2 J 1 f 1 (k c ) cos(β) + 4τ 2 2τ 2 sin(β) (33)
In order to choose between both solutions, we verify that at the limit J 1 f 1 → 1 cos(kc) , the solution must verify the only admissible solution v = tan(β)/τ to the system, see details on how to obtain this limit in [START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF]. This makes us choose the "-" solution. It can be noticed that as the optimization starts with a stable solutions, the solution we found belongs to this regime. This remark is important, as this kind of system usually gets unstable as the input speed is more distance from the intrinsic speed of the system (usually as lurching waves, see [START_REF] Folias | Stimulus-locked traveling waves and breathers in an excitatory neural network[END_REF]). To determine the range of operation (where activity is stable), and to obtain the small perturbations analysis a linearization of the system must be performed to obtain its eigenvalues, for more details check the chapter on small perturbations in [START_REF] Goldstein | Classical Mechanics[END_REF], and the application to this case in [START_REF] Folias | Stimulus-locked traveling waves and breathers in an excitatory neural network[END_REF][START_REF] Hansel | Methods in Neuronal Modeling: From synapses to networks, chapter Modeling Feature Selectivity in Local Cortical Circuits[END_REF][START_REF] Xie | Nonlinear dynamics of directionselective recurrent neural media[END_REF].

C Analytical expression of r 0 (t), 2D

The following 2D analysis has not been performed elsewhere to our knowledge. The idea is to follow the 1D idea, and to find an expression for the first Fourier component, or the mean total activity and 3 other Fourier components needed to express the dynamics of the system. The first step is to assume an input function,

g(I( x, t)) = E( x, t) = D 2 [1 + cos(y)] [1 + cos(x -vt)] + C ( 34
)
where D and C are parameters controlling the bump highest activity and the spontaneous activity level. If x ∈ [-π, π] and y ∈ [-π, π], then E( x, t) defines a single bump moving from left to right (v > 0). For the kernel function we consider the same equation as in Eq. 10,

w( x, x ′ ) = A [1 + cos(y ′ -y)] [J 0 + J 1 cos (x ′ -x + β)] ( 35
)
where the asymmetry is only in the x-axis, therefore we are considering only sequences moving on that axis. If y ∈ [-π, π], then w is a decreasing function of y. As in 1D we start by writing the dynamics of m( x, t) in terms of quantities that are space independent (Fourier components). We know the definition of r 0 (t) in 2D from Eq. 11, but we require also higher Fourier terms:

r 1 (t) = π -π π -π m( x, t)e i(x-ψ(t)) (2π) -2 d x (36) r 2 (t) = π -π π -π m( x, t)e i(y-Ω(t)) (2π) -2 d x (37) r 3 (t) = π -π π -π m( x, t)e i(x+y-λ1(t)) (2π) -2 d x (38) r 4 (t) = π -π π -π m( x, t)e i(x-y-λ2(t)) (2π) -2 d x (39) RR n°7543
where the unknown functions of time ψ(t), Ω(t), λ 1 (t) and λ 2 (t) are such that they make each quantity real and positive. Before using the different Fourier components, we express the dynamics of m using Eq. 35,

τ ∂m( x, t) ∂t + m( x, t) = E( x, t) + AJ 0 r 0 (t) + AJ 0 4π 2 cos(y ′ -y)m( x ′ , t)d x ′ + AJ 1 4π 2 cos(x ′ -x + β)m( x ′ , t)d x ′ + AJ 1 4π 2 cos(y ′ -y) cos(x -x ′ + β)m( x ′ , t)d x ′ + (40)
All the right side of Eq. 40 is the total input, and the [ ] + operator must be applied to it. Now, we rewrite the total input in the complex plane to simplify,

= E( x, t) + AJ 0 r 0 (t) + AJ 0 e i(y-Ω) 4π 2 e -i(y ′ -Ω) m( x ′ , t) x ′ + AJ 1 e -i(x-β-ψ) 4π 2 e i(x ′ -ψ) m( x ′ , t) x ′ + AJ 1 2 e i(x+y-λ1+β) e i(x ′ +y ′ -λ1) 4π 2 m( x ′ , t) x ′ + e i(x-y-λ2+β) e i(x ′ -y ′ -λ2) 4π 2 m( x ′ , t) x ′ + ( 41
)
where we omit the limits of the integrals as they are all either π or -π. Introducing now Eqs. 36, 37, 38 and 39 we can write more compactly Eq. 41 as, = E( x, t) + AJ 0 r 0 (t) + AJ 0 e -i(y-Ω) r 2 (t) + AJ 1 e -i(x-β-ψ) r 1 (t)+

AJ 1 2 e i(x+y-λ1+β) r 3 (t) + e i(x-y-λ2+β) r 4 (t) + (42)
In the complex plane, we will impose a total input with the shape of a single bump moving along the x direction. This implies that the modes associated to x + y and x -y must have the same coefficients, i.e. r 3 = r 4 and λ = λ 1 = λ 2 . Using this and Eq. 34, the total input can be written as:

= I 0 (t) + e i(x-φ) I 1 (t) + e -i(y-Ω) I 2 (t) + I 3 (t)e -iα (e i(x+y) + e i(x+y) ) + (43)
where -vt) , AJ 0 e -iΩ r 2 (t) + D 2 and I 3 (t) = (AJ 1 e i(β-λ+α) r 3 (t) + D 2 e i(α-vt) )/2. In Eq. 43 we have also introduced two variables φ(t) and α(t), that are used later to ensure a single bump. Eq. 43 can be simplified into Eq. 44 using the identity e i(x+y) + e i(x-y) = 2 cos(y)e ix .

I 0 (t) = AJ 0 r 0 (t) + C + D 2 , I 1 (t) = AJ 1 e i(β-ψ+φ) r 1 (t) + D 2 e i(φ
= I 0 (t) + e i(x-φ) I 1 (t) + e -i(y-Ω) I 2 (t) + I 3 (t)2 cos(y)e i(x-α) + (44)

RR n°7543

As in the 1D case, we want a single bump for the total input and we impose I 1 (t) to be real and to ensure that we introduce φ(t) similarly for I 3 (t) and α(t). Also in 2D, we want Ω(t) = 0 as the total input should remain static in the y-axis. Furthermore, the total input expressed in Eq. 44 requires ψ(t) = λ(t) and φ(t) = α(t) to be a single bump, which implies r 1 (t) = r 3 (t), otherwise two bumps could exist. Using these assumptions, and taking back Eq. 44 into the Real domain, we obtain,

= [I 0 (t) + I 1 (t) cos(x -φ) + I 2 (t) cos(y) + I 1 (t) cos(y) cos(x -φ)] + (45)
There is one aspect of Eq. 45 that requires to be addressed; to have a symmetric single bump we must have I 1 (t) = I 2 (t) or, using both definitions, A(J 0 r 2 -J 1 r 1 cos(β +ψ +φ)) = D 2 (cos(φ-vt)-1), in other words r 1 (t) ∝ r 2 (t). This property will help to obtain r 0 (t). Using this assumption Eq. 45 can be simplified into Eq. 46

= [I 0 (t) + I 1 (t) cos(x -φ) + I 1 (t) cos(y) + I 1 (t) cos(y) cos(x -φ)] + (46)
The total input is a single bump in 2D of width x c and height y c , using the symmetry of the total input we get x c = y c . At point (x c , x c ) we know that the total input is zero or I 0 = -I 1 (2 cos(x c ) + cos(x c ) 2 ), allowing us to write: = I 1 (t) [cos(x -φ) + cos(y) + cos(y) cos(x -φ) -2 cos(x c ) -cos(x c ) cos(x c )] + (47) Eq. 47, implies that we can describe the dynamics of m in 2D, using two Fourier components: r 0 (t) and r 1 (t) as in the 1D case, and integrating Eq. 47 by d x ′ /(2π) 2 and e ix d x ′ /(2π) 2 :

τ ∂r 0 (t) ∂t + r 0 (t) = I 1 (t)g 0 (x c ) (48) τ ∂r 1 (t) ∂t + r 1 (t) = I 1 (t)g 1 (x c ) cos(φ -ψ) (49) τ r 1 (t) ∂ψ(t) ∂t = I 1 (t)g 1 (x c ) sin(φ -ψ) ( 50
)
where the only differences with the 1D case are functions g 0 and g 1 that we define in Eq. 51 and Eq. 52. The set of equations 48, 49 and 50 describes the dynamics of m( x, t) and as we want a stable activity, we impose dr 0 /dt = 0, dr 1 /dt = 0 and dψ(t)/dt = v and use I 0 = -I 1 (2 cos(x c ) + cos(x c ) cos(x c )) to obtain,

r 0 (x c ) = (C + D 2 )
-AJ 0 -K(x c )/g 0 (x c ) (53) in this expression the main difference with the 1D case is the term with g 0 , yet r 0 (t) is still a decreasing function of x c as g 0 is an increasing function of x c and K(x c ) is a decreasing function (C, D, A > 0 and J 0 < 0), making the term K(x c )/g 0 (x c ) a decreasing function of x c .

The final expression we have obtained in Eq. 53 for r 0 holds if several conditions can be verified, in particular if the total activity is one symmetric bump. We have not verified the symmetry condition, but numerically the shape of the bump seems symmetric. However, in the 2D case is common to observe small asymmetries in the shape of the bump.

D Optimization of v(β), 2D

In this appendix we derive the expression of the asymmetry parameter β in terms of the input speed v, to maximize the total activity r 0 . Considering the expression for the total activity for the population r 0 of units m obtained in Appendix C for the 2D case, this is equivalent to minimize x c (the size of the activity bump). To impose this we use the constraint of a single bump for the total input of activity, AJ 1 sin(β -ψ + φ)r 1 + D 2 sin(φ -vt) = 0 (54)

Using the expression we derived for r 1 in the 2D case (analogous to r 0 in Eq. 53) in Eq. 54, we can derive and expression for the existence of the stable solution, S ′ = J 0 g 0 (x c ) + K(x c )

J 2 1 g 2 1 cos(∆) 2 -2J 1 g 1 cos(∆) cos(∆ + β) + 1 (55)
As in 1D this expression can be checked numerically for a given set of parameters as we do not know x c . Yet, deriving by ∂/∂v and imposing ∂x c /∂v| v=vm = 0 we can find an expression for v m (β),

v m = J 1 g 1 (x c ) -2 cos(β) ± J 2 1 g 2 1 (x c ) -4J 1 g 1 (x c ) cos(β) + 4 2τ sin(β) ( 56
)
where there is still a dependency with x c expressed by g 1 (x c ). Checking the expression and choosing the appropriate solution (-), we obtain the low x c → ∞ (or J 1 g 1 → 1 cos(β) ) and high contrast x c → 0 (or g 1 → 0) limits, re-obtaining the known solution from the literature in the 1D for the first case, and the expression we obtained in Appendix B for the high contrast expression,

v m (β) = 1 -cos(β) τ sin(β) (57)
RR n°7543

E Numerical Simulations

The numerical simulation of Eq. 1 and Eq. 3 have been performed with the RK4 method (Runge-Kutta 4th order). Here we will give the parameters and discretization of the simulation in the 1D case, an only the extra parameters for the 2D case. First, we rewrite Eq. 1 as,

∂m(k, t) ∂t = 1 τ (-m(k, t)+ k ′ w(k ′ -k)m(k ′ , t)dk + C [1 -ǫ + ǫ cos(k -vt)] -T + )
(58) where we applied directly the RK4 method, see [START_REF] William | Numerical recipes in C (2nd ed.): the art of scientific computing[END_REF], to discretize the integral we use a trapezoidal rule. Here we use the parameters: dt = 0.1, dk = 2π/60, τ = .15, J 0 = -9.8, J 1 = 13.5, C = 5, T = 4.9 and the total activity was computed over 1000 iterations. For the 2D case the parameters where: A = .2, dk = 2π/21 and the total activity was computed over 100 iterations. Numerically, the system is quite robust to simulate and we where able to obtain similar results using the Euler method. Both implementations where performed in Matlab and are available from the first author website.

Figure 1 :

 1 Figure 1: Human movements (a) to walk, (b) to fight and (c) to wave. In white we mark more relevant locations in terms of the information they can contribute to be differentiated from other sequences. (d) The set of trajectories in time for the joints in the walking sequence.

Figure 2 :

 2 Figure 2: Continuous and discrete light intensities. Also slow and fast discrete on-to-off sequence.

Figure 2 (

 2 Figure 2(b) for different speeds of presentation. Additionally, we require answers with partial inputs (n < 7) and we will suppose that sequences do not always start at the same point in time (different phase).Let us consider that the solution to this problem should first verify for each image of the set n, if it matches some image in the sequence. Only after this, we verify the temporal sequentiality. We will study this kind of solution, and propose a practical implementation using a variant of the CNFT.

Figure 3 :

 3 Figure 3: The similarity function for P 3 3 , and the sequence of similarities that corresponds to a part of the on-to-off sequence. Here we plot d(I l , P k k ) for several P k k , using d(I l , P k k ) = exp(-(I l -P k k ) 2 )

Figure 4 :

 4 Figure 4: Temporal evolution of: (a) the total activity A(t) and (b) the instantaneous activity r 0 (t) for the 1D example.

Figure 5 :

 5 Figure 5: Temporal evolution of: (a) the total activity A(t) and (b) the instantaneous activity r 0 (t) for the 2D example.

Figure 6 :

 6 Figure 6: (a) Input function, at difference contrast levels. (b) Asymmetric kernel function for β = 1.5, considering a neuron at position 0, only neurons at position -1.5 or close will have positive weights.

Figure 7 :

 7 Figure 7: The β(v) function for the maximal mean activity (r 0 ) at different values of ǫ. Continuous lines correspond to theoretical results in low (ǫ = 0) and high contrast limits (ǫ = ∞), the others tree dotted curves to simulations.

Figure 8 :

 8 Figure 8: (a) Input function, the deplacement is only in the x-axe. (b) Asymmetric kernel function for β = 1.5 for a neuron at (0, 0), only neurons at x = -1.5 or close will have positive weights.

Figure 9 :

 9 Figure 9: The β(v) function for the maximal mean activity (r 0 ) at different values of the contrast. Continuous lines correspond to theoretical results in low and high contrast limits, the others tree dotted curves to simulations. Note that we use C = 4D 2 -1, to have an amplitude of 1 in all the cases.

Figure 10 :

 10 Figure 10: Schematic view illustrating training and evaluation stages for the classification task with our proposed method.

  x) + cos(y) + cos(y) cos(x)-K(x c ))d x x) + cos(y) + cos(y) cos(x) -K(x c )) cos(x)d x (52) using K(x c ) = 2 cos(x c ) + cos(x c )2 . The function to be integrated in Eq. 51 represents a bump, with the same form in the four quadrants (symmetry). More precisely l(x c ) = arccos[(K(x c ) -cos(y))/(1 + cos(y)].

  for the 1D system assuming that the expected activity in m(k, t) evolve as one single "bump" moving at input speed v and that this front has width 2k c (t), where k c (t) is an unknown function of time. Using k c (t), we can deal with the non-linearity [ ] + in the right side of Eq. 1, which is non-zero only in the interval [φ(t) -k

c (t), φ(t) + k c (t)]), where φ(t) is the position for the center of the bump.

usually called time-warping RR n°7543

RR n°7543


René Sandretto
email: rene.sandretto@univ-lyon2.fr

K Eywords

F rançois Per roux, a precursor of the cur rent analyses of power

Keywords: Asymmery, power, domination, influence, coercion, general equilibrium

Without any doubt, François Perroux is one of the most important European economists of the 20st century. Today, in English-speaking countries, François Perroux is recognized mostly for his original contributions in the field of spatial economics and economics of development. However, taking into account the variety of his contributions, Perroux deserves certainly more attention. This paper underlines a less well-known aspect of Perroux's works: his illuminating views on asymmetry, domination and power which can be considered a source of works carried out in this field half a century later, such as the American realist and neo-realist approaches to power or the concept of structural power developed by Susan Strange.

Introduction

Unquestionably, François Perroux is one of the most original European economists of the 20st century. He was a prolific author. His innovations are many, ranging from spatial economics in which he analyzed the economics of location, the forces of agglomeration and dispersion of polarization and diffusion [START_REF] Perroux | Les Macrodécisions[END_REF], to the analysis of economic development, including an interesting attempt to develop an alternative approach to the general equilibrium, based on the recognition of the active role of agents [START_REF] Perroux | Unités actives et mathématiques nouvelles: révision de la théorie de l'équilibre économique général[END_REF]1976). Despite a considerable work, François Perroux is not really appreciated to its true value. Perroux' ideas reached the apex of their popularity during the 1960s, namely in Europe and Latin America. Benjamin Higgins, for example notes that he was nominated several times for the Nobel Prize (Higgins, 1988). In his famous book: The Development of Economic Thought, W. Spiegel considers Perroux to be "the only living economist who developed a theoretical system rivaling conventional equilibrium economics" [START_REF] Spiegel | The Development of Economic Thought[END_REF]. Similarly, John B. Parr underlines that during this decade, the perrouxian concepts of "Growth poles" and "development poles" were adopted enthusiastically by regional economy analysts and planners [START_REF] Parr | Growth-pole Strategies in Regional Economic Planning: A Retrospective View[END_REF]. A similar judgement was given more recently by Stephen [START_REF] Meardon | Modelling Agglomeration and Dispersion in City and Country: Gunnar Myrdal, Francois Perroux, and the New Economic Geography -Critical Essay[END_REF]. Even though the assessments of Higging, Spiegel and many others do not represent a clear consensus they clearly suggest that at that time Perroux was held in high regard by many of his peers.

Since then, Perroux' thinking gradually sank into oblivion, namely in Englishspeaking countries.

Part of the reason for that is related to the fact that Perroux did not publish a lot in English.

Actually, he published in many different languages: Spanish, Italian, German, Portuguese, Polish, Hungarian, Rumanian, Japanese, even in Esperanto. But, among the 689 publications registered in his less incomplete bibliography compiled by Michel Chai 1 less than 3% were initially published or subsequently translated or reprinted in English. Surprisingly, works such as La Coexistence pacifique [START_REF] Perroux | La coexistence pacifique[END_REF], Economie et Société. Contrainte, échange et don [START_REF] Perroux | Économie et Société, Contrainte, Échange[END_REF], Pouvoir et Economie (1973) and even his masterpiece L'Economie du XXe siècle (1961) have never been translated into English.

However, more recently, some of the main ideas that Perroux defended so forcefully seem to experience renewed interest. The Nobel prize awarded to Krugman with whom Perroux is frequently associated can be considered a proof of this revival.

Among the various topics to which Perroux made important contributions, the most insightful and not so well recognized one is his "domination effect" and his analysis of power. This paper aims at underlining Perroux's innovative views on asymmetry, domination and power. More precisely, my objective is to bring to light how and why his contributions can be considered a source of works carried out in this field half a century later.

When, in 1948, François Perroux wrote his famous article "Esquisse d'une théorie de l'économie dominante" [START_REF] Perroux | Esquisse d'une théorie de l'économie dominante[END_REF](Perroux, , 1950)), he was obviously influenced by the economic and political context of the times. In 1945, the United States had become the dominant economy in the world, and it seemed destined to remain so for many years to come [START_REF] Perroux | Leadership économique des États-Unis et analyse moderne des échanges extérieurs[END_REF]. A superficial reading of this seminal article might lead one to believe that Perroux had restricted himself to constructing a theoretical architecture, based on this single concrete example. But Perroux's range of interest was considerably broader.

In the present as in the past, an analysis of the effect of domination on international relations tightly conditions our knowledge of current affairs. After the Second World War, lopsided power relations induced trade imbalances and generated an irreversible orientation of trade and financial flows that was apparently incompatible with the creation of a new international economic multilateralism. At the same time, the chronic shortage in international liquidities raised formidable threats to the viability of the Bretton Woods arrangement. Creditors seemed to remain perpetual creditors; debtors became more and more indebted.

The domination effect and the concept of the internationally dominant economy offer particularly penetrating views of the new world that was being built in the early second half of the 20th century. The scope of Perroux's analysis, however, is of a completely different nature. As this paper shows, it is much larger and more ambitious.

Our purpose, in the following pages, is to conduct an indepth analysis of the content of Perroux's concepts of domination, dominance and power and to examine how these seminal ideas were precisely implemented in Perroux's works, empirical and theoretical. To that end, I want first to stress the operational character of Perroux's concepts of asymmetry, domination (and dominance), and power. Even if some of his writings can be considered highly theoretical and philosophical, the major and permanent concern of Perroux throughout his entire work was to design analyses leading to a better understanding of the world in which we live. His entire body of work bore witness to the imperative upon which he unremittingly insisted: a symbiosis between the "theoretician's discourse" and the "observer's discourse" (section 2). Then I describe more thoroughly the relations between structures and power (section 3) and the difference introduced in Perroux's later publications between domination and dominance (section 4).

The most interesting aspect of Perroux's thinking is that the domination effect represents a common denominator and a key to interpreting a multiplicity of apparently heterogeneous phenomena (section 5). This concept is probably what confers upon Perroux's work its overarching character and its pioneering feature (section 6). The components of power relations called asymmetries, inequalities, and irreversibilities therefore constitute central themes and structural elements. For Perroux, they are the main building blocks of any attempt to the renovation of the general equilibrium theory which he considered a necessity along with a new way of looking at human beings and society (section 7).

T he Concept of Power and the Domination E ffect

Starting with Max Weber, Perroux defines power as "the probability that an agent within a social relation will be in a situation to make his own will effective, in spite of resistances (…) The essence of power, sought after, used, expected, or predicted, is an asymmetric relationship that stems from unequal actions and reactions" (Perroux, 1973: 30). This asymmetry of influence, which he calls the domination effect, can exist between various units: individuals, firms, social groups, nations, and others.

It is useful at the outset to quote Perroux's own definition: "To consider just two economic units, we would say that A exercises a domination effect over B, when, in the absence of any particular intention, A exerts a certain influence on B without the reverse being true or true to the same degree. A dissymmetry or irreversibility of principle or degree is constitutive of this effect" (1948: 248). As an imperfect form of reciprocity and interdependence, the domination effect is therefore not necessarily the expression of a desire for power. It may also be the consequence of an existing situation -the result of differences in size and disparities in structure.

According to F. Perroux, the domination effect has three essential components. The first one is the relative sizes of the units, including factor components, assets, respective contributions to supply and demand, information available, and so on. The second one is the bargaining powerthe units' respective abilities to negotiate conditions and exchange rates, or even to set the rules of the game. Lastly, a unit's domination effect results from its belonging to an active zone, the position that it occupies in that zone, and the nature of the activity that it exercises. Of the three above components, the last one is the least intuitive. It is, however, fundamental. The implication is that the strength or domination of one unit is not reduced exclusively to its individual features, but results from synergies and externalities -from the connection between the unit and its "environment." For this reason, Perroux' multifaceted concepts of dominance and power have a lot of empirical applications. At the level of the firm, among several possible illustrations, the perrouxian growth poles analysis is based on the central role of the propulsive firms whose innovative capacity and leadership exert an asymmetrical (but stimulating) influence on other firms (VázquezBarquero A., 2002). This analysis gave rise to growth pole policies. At the macroeconomic level, one possible illustration of the domination effect is the "macro-decisions" of the governments (see below 2.2). But, obviously, the international level is the most important application field of power relations (see below 2.1 and 3.).

Unintentional Domination

Unit A exerts a dissymmetrical influence on other decision-making centers, even though it has no desire to act on them, by increasing or decreasing its production or by changing its prices. The other units must adjust to A's decision by changing, in turn, their level of activity and/or their prices, without these reactions affecting or influencing A to the same degree. This effect can be clearly observed when we examine, for example, how changes in the monetary or budgetary policy of the United States affect the circumstances of its partners. The domination effect may take the form of a cumulative sequence of asymmetrical influences in concentric rings. For instance, a restrictive monetary policy conducted in the United States to reduce domestic inflationary pressures forces other industrialized countries to align themselves by raising their interest rates. Attempts at disconnection almost always prove to be impractical, except through rigorous control of the exchange rate (this is the first circle of asymmetry). The corresponding rise in international interest rates then induces a more strongly asymmetrical influence on developing countries, which are indebted at a floating rate, by increasing the service of their debt (this is the second circle of asymmetry).

An internationally dominant economy thus appears to be a "relatively autonomous" conjunctural centre of momentum, which, independent from any hegemonic intentions, influences the situations of its partners. In a world that is de facto increasingly globalized, the interplay of irreversible and dissymmetrical influences develops both directly and indirectly along more or less complex chains (or sequences) of domination effects.

The market for Eurodollars is a cogent example. During the critical period of dislocation of the international currency system (late 1960s and early 1970s), the Eurodollar market undeniably served as a communication channel for an indirect domination effect. Unintentional at first, it relayed the influence of the American monetary market to other national monetary markets, especially with regard to determination of interest rates and pressures on exchange rates. This asymmetry in the constraints exerted on (or exerted by) foreign monetary markets was to some extent determined by the comparative sizes (masses) of the respective markets at that time (see figure 1). Analogous linkage, distribution, attraction, and polarization effects -or, on the contrary, inhibition, braking, or blockage effects -can easily be observed between a variety of units: first, between competing firms through innovations, industrial or financial concentration, choice of production diversification, and so on; secondly, between industries, for which interdependences are never symmetrical. Some industries exert asymmetrical induction effects on costs, prices, productivity, and the dynamics of technological change on activities both upstream and downstream. (Static inter-industrial matrices of goods and services record these effects only very imperfectly). Third, these effects are observable between regions that are unequally able to mutually supply net consumption or investment flows (taking account of their respective weights and urban agglomerations), and also have an unequal capacity to induce growth and attract material, human, and financial resources. Fourth, they are observable between nations, which are unequally capable of causing cyclic variations. Lastly, to these effects are added asymmetries that may arise between units of different nature, such as between large firms and their regions, between the capital city and the regions, between urban areas and countryside, and so on.

Intentional Domination

The domination effect may also be exerted deliberately and in an even more obvious way. In contrast to unintentional domination, the study of intentional asymmetrical actions involves the subject's plan, the advantages it seeks and obtains, its desire for power, and its field of action. [START_REF] Perroux | Pouvoir et Économie[END_REF] 31 and seq.) distinguishes three modalities of power relationships corresponding to three kinds of intentional domination effects.

-Influence. Influence means that Unit A tries to change the behaviour of unit B without force, through persuasion, using its authority to encourage participation or imitation. Thus, A exerts over B a power "beyond its capacity to force the other to do or not to do" (Perroux, 1973: 31).

The phenomenon of the leader firm, as has long been observed, for example, in global nickel and aluminum oligopolistic markets, is characteristic of this type of influence. Without being forced, firms explicitly or tacitly align their decisions regarding prices, production, investment, and other activities with choices made by one of them, which plays the role of barometric firm [START_REF] Stigler | The Kinky Oligopoly Demand Curve and Rigid Prices[END_REF]. A similar situation arises with regard to social conditioning of consumption linked to Duesenberry's ratchet effect and the respective effects exerted by "influence merchants" (through advertising and the media) on consumers and citizens.

-Imposition or coercion. Through the use of force or violence, A forces B "to do" or "not to do."

The field of action may be more or less extensive. It may be manifested between decision-making units or between groups of social actors or classes. It is seen primarily in, for example, relations between the metropolis and its colonies and is most transparently the result of macro-decisions [START_REF] Perroux | Les Macrodécisions[END_REF][START_REF] Perroux | L'Économie du XXe siècle[END_REF] made by the government, as a "monopolizer of legitimized violence" (Perroux, 1961: 377). It may also take place between private actors, particularly in the form of "production prices" (set and imposed unilaterally for a time by a firm or group of firms).

However, it is most visible in international relations (embargoes, blockades, prohibitions, reprisals, retaliation, and the like).

-Subordination. According to Perroux, "subordination involves the duration of the coercive action by A (individual or group) on B (individual or group). This is the area of structural influence, institutionalized or not" (Perroux, 1973: 32) -that is, an orbiting process linked to the combination of all of A's asymmetrical actions with regard to B, which tend to substitute (more or less completely) the decision-making power of the former for that of the latter [START_REF] Perroux | Indépendance de l'économie nationale et interdépendance des nations[END_REF]. 10

Structures and Power

Of course, the distinction between intentional and unintentional domination effects does not mean that these two "varieties" are mutually incompatible or independent. Although Eugen von [START_REF] Böhm-Bawerk | Macht oder ökonomisches Gesetz[END_REF], in Macht order ökonomisches Gesetz rejects the subordination of human beings to a structural determinism that destroys their quality as actors, he tries to show that the "influence of power" (Machteinfluss) is exerted even more easily and durably when it is based on a "position of power" (Machtstellung) that respects and uses economic laws. Perroux's asymmetry analysis provides a powerful tool, with the domination effect, towards a better understanding of the complex relations between "structures" and "power". Two examples can illustrate these relations.

-Trade in raw materials. Industrialized countries and raw material producing countries have unequal capacity to influence trade and international markets. Indisputably, raw material producers would be less vulnerable and their negotiating power would be stronger if: (i) they were not so hyper-specialized (sometimes mono-specialized), as opposed to the diffused specialization of industrial countries. (ii) If industrialized countries were not so dangerous competitors in the export of raw materials. Contrary to a received idea, the most important exporters in raw materials (agricultural and mineral) are developed countries. (iii) If they were not sometimes forced to sell at any price, and were less dependent on markets in industrialized countries and on foreign firms. Transnational firms also intentionally structure relationships.

-Location strategy of transnational firms. In some particular industrial sectors, transnational groups sometimes apply what may be called a strategy of "specialization/diversification" in which each of their subsidiaries is made into an ultra-specialized plant. In this strategy, the transnational firm spreads the geographic distribution of its subsidiary plants in different countries for each stage of the production process. For example, one subsidiary of a car manufacturer will produce only brake pads, and this part will be manufactured simultaneously in Mexico, Brazil, and Asia. This particular strategy is obviously a case of intentional implementation of structural power relations. It leads, among other things, to a productive and spatial structure that generates domination effects between the firm's management and the government and social groups in the host country. It minimizes the effects on the firm of potential technical, political, or social incidents. It dissuades host countries from nationalizing foreign firms established on their soil, since each "plant" is viable only within the industrial group upon which it is totally dependent. It also protects the industrial group against the consequences of strikes, which are not internationally coordinated. On the other hand, this structure makes the group's threat or counterthreat of lock-out or closure credible to both unions and local governments, since a local production stoppage does not disrupt the group's activity [START_REF] Sandretto | Le commerce international[END_REF].

These two examples clearly illustrate that power relations do not come out of the blue, but that they may be structurally determined.

Domination and Dominance

In his later writings, Perroux has refined his analysis of the domination effect with the distinction between 'domination' and 'dominance' depending on the intensity and extent of asymmetrical relations [START_REF] Perroux | Unités actives et mathématiques nouvelles: révision de la théorie de l'équilibre économique général[END_REF]:

-Dominance is a degree of asymmetry that cannot be evaded over a given period, resulting from the asymmetry in the respective influences, which is such that Unit A exerts an influence on B, while B exerts an influence on A but not at the same degree.

-At the opposite, domination (strictly speaking) is corresponding to an intrinsic dissymmetry of principle (total irreversibility, always within a given period). In other words, domination would correspond to total dominance. Perroux also sometimes used the term domination to designate crystallized dominance in structures and institutions.

Analytically, dominance is more interesting than domination, as it has a greater range. As F. Perroux wrote in a personal correspondence with the author "Total dominance of A over B destroys (economically) duality" [START_REF] Perroux | [END_REF] because it suppresses B's quality as an actorits decision-making capacity, individuality, and integrity. This may be why Perroux abandoned the expression of domination (and "domination effect") for the more general one of "dominance" in his later writings.

The main problem resides in the difficulty of measuring the intensity and effectiveness of the dominance effect. Expressing it in the most general terms, Perroux characterized the search for the most useful result possible in the behaviour of two units, A and B, in a relationship of pure trade. This point is reached when A and B equalize the marginal return obtained and the marginal return ceded:

1 ceded return Marginal obtained return Marginal 
The impact of the dominance effect, according to Perroux, is evaluated by the external advantage of pure trade -by the proportion within which the respective performances of A and B may deviate from the above standard of optimality and equity. The breadth of the dominance effect may thus be quantified by the price supplement that A imposes on B above the marginal cost of the good that it is selling or by the difference between the remuneration to the holder of a production factor and that factor's maximum marginal productivity.

Following a similar approach, the international neoclassical equilibrium model described by Heckscher, Ohlin, and Samuelson (the HOS model) can be interpreted as the standard of equal trade, conflictually neutral, from which all dominance relations are axiomatically excluded [START_REF] Sandretto | Inégalités transnationales. Une application de la théorie des jeux[END_REF]. Measurement of economic power by deviation from the model of pure trade (or the ideal model of perfect competition) is both simple and attractive on the conceptual level.

But is it the most operative and significant in practice? According to Perroux, the answer is no, because it refers not to a particular concrete situation, but to an unrealistic model -a fictitious one.

Power, keystone of the Per rouxian analysis

In L'Économie du XXe Siècle [START_REF] Perroux | L'Économie du XXe siècle[END_REF], Perroux shows the importance of the dominance effect within his work. The structure itself of this book reflects this prominence.

This major work begins, in fact, with a section devoted to the dominant economy (Part 1). It is followed by a study of spatial, linkage, and agglomeration asymmetries -in other words, the formalization of the growth poles which are a direct application of the dominance theory (Part 2).

Then comes the concept of macro-decisions, a clear example of the structural decision-making power of a dominant actor: with the help of the irreversible and asymmetrical influence that it is capable of exerting, the macro-decision imposes the rules of the game on and shapes the environment of the other units (Part 3). The analysis then turns to the process of development and underdevelopment, which is dependent on the coupling mechanisms that link the various parties in the global economy and the asymmetries that it supports (Part 4). This process leads to the identification of the forces of dynamism and progress, the main one of which is the search for power (a motivation "as powerful as the search for well-being or monetary gain" [Part 5]).

Economic movements are accomplished by human beings, innovation, and power. They result from the intersection of the behaviours of "diversely and unequally active" units (Part 6). We can thus see that dissymmetries and irreversibilities of power relations, far from being dispersed instruments in an analytic toolbox, are essential aspects of an overall interpretation.

In this work -no doubt Perroux's most important synthesis -the dominance effect appears to be the common denominator in most of his key concepts. Later, it became the bedrock of an overall theoretical reformulation and of his last project: the construction of a theory of general equilibration [START_REF] Perroux | Unités actives et mathématiques nouvelles: révision de la théorie de l'équilibre économique général[END_REF].

The concept of economic power elaborated by Perroux is deduced from his analysis of asymmetries, irreversibilities, and constraint "in a universe of actors." It is no exaggeration to state that this concept of dominance is the cement that gives his complex body of work its coherence.

A prefiguration of the most recent analyses of power

Perroux's concept of power can be compared to more recent concepts of power. In the American neo-realist approach, power is relational. First, in a context of strong asymmetries, power can be applied through constraints. In the contemporary American neo-realist analysis, power is defined as the capacity of unit A (state, firm) to directly force unit B to do what it would not have done spontaneously, in order to better satisfy the interests of unit A. This is called hard power, because it relies on the comparative strengths of units A and B: military power, natural resources, demography, capacity to inflict damage on another or to avoid such damage. The "hard power" is one of the key concepts of the American neo-realist school of thought, whose one of the most prominent representative is Kenneth Walz (1979). In contrast to this, the American liberal institutional approach has emphasized the quite different concept of soft power based on cultural ideological and political attraction, and on the attractiveness of the way of life of the leading country(ies) (Nye 1990: 167). Actually, in the globalized multipolar and complex world in which we are living, the reinforcement of global interdependences creates a situation such that hegemonic (hard) power is increasingly difficult to exert. Force cannot always easily impose power. Power then tends to change in nature. Leading states attempt more and more frequently to exert influence through other instruments than military force or economic constraint, using state's capacity to seduce, convince, or yet induce through cultural and/or ideological attraction. More than traditional means, soft power is based on intangible assets (culture, way of life).

Another important recent contribution to power analysis has been achieved by Susan Strange.

Similarly to François Perroux, Susan Strange suggested a structural approach of power, that is a concept of power basically linked to structures and, more generally, to the environment in which agents' actions take place. She defines power as the capacity of an individual or a group to influence a situation in such a way that the preferences of the individual or group have priority over others' preferences [START_REF] Strange | The Retreat of the State: The Diffusion of Power in the World Economy[END_REF]. In other words, power is the capacity to conceive, legitimize, implement, and control rules for individual and collective action. This amounts to the capacity to shape structures within which other actors will have to evolve. The Strangeian concept of power is also structural in so far as it relies on structural components. Indeed, it is easy to recognize the hard power as a modern expression of imposition or coercion, while the soft power is quite similar to influence in the Perrouxian typology. Similarly, structural power, as described by Strange, is reminiscent of unintentional dominance along with the Perrouxian suggestion that intentional dominance can rely on unintentional dominance. In both

Perroux and Strange we can also find the idea that power relationships are structurally conditioned.

Conclusion

Perroux's acceptance at the outset of asymmetrical and irreversible actions explains why his analysis was so strongly distinct from the major traditional currents of economic analysis. It may also explain why his analysis sank into a relative oblivion during so many years. In several publications, Perroux criticized the various processes whereby the mainstream economic analysis eludes the power relationship, namely:

-the waterproof partitioning between economics (often considered a science of means) and politics (as a science of ends), which leads to a tendency to associate power only with government power and extract it from the economic sphere (belonging to the private sector),

-the rudimentary separation of economic relations (superficially conceived as merchant relations governed by private profitability) and social relations (often trivially considered an obstacle or a corrective to economic activities), -the separation of economic variables (prices and quantities) from extra-economic data (rules of the game, institutions, ownership regimes, power relations, and so forth).

Finally, Perroux was fundamentally opposed to reductionist simplifications -the neoclassicists' irenicism (total absence of conflicts) and the Marxists' polemicism (unavoidable confrontation between two inherently antagonistic classes) [START_REF] Perroux | Les Conceptualisations implicitement normatives et les limites de la modélisation en économie". Économies et Sociétés[END_REF](Perroux, : 2271)). For Perroux, all exchanges are of a hybrid nature: they mix free and reciprocal transfers and power relations [START_REF] Perroux | Pouvoir et Économie[END_REF].

Economic relations, as any social relations, intimately blend conflict and co-operation.

One of the main questionings that we find in several publications of Perroux was: can the neoclassical model be amended? François Perroux believed that the very foundations of the model were defective. In his opinion, the neoclassical theory would not be improved by the introduction of power, for it would certainly then be ruined. In short, according to an aphorism that Perroux borrowed from Oskar Morgenstern: "there is no road from L. Walras to reality" (1988b, p. 83). Thus, he considered that a complete reconstruction on other foundations was necessary. Of course, this conclusion is debatable. However, it still remains that despite important recent improvements related namely to the game theory, the general equilibrium theory did not really succeed in taking into account this "rebellious exile" as Perroux described power.

The integration of power in the heart of economic analysis is still a challenge.

Perroux tried to perform this reconstruction by replacing equilibrium with equilibration of active units' energy for change: "Equilibration is a chain of decisions and acts over a period of time.

Notes

1. Unpublished document.

2. In 1974, the size of the American monetary market was around 20 times as big as the Eurodollar market. At that time, we could consider the Eurodollar to be some kind of second Federal Reserve System. Since then, the situation has completely changed: the Eurodollar grew much more rapidly than the American monetary market and the respective masses reversed in Page 20

Figure 1 .

 1 Figure 1. Relative masses of monetary markets 2

  the field of finance and the environment) -evolve. Consequently, S. Strange considered that if the most important global risk in the long run is related to environment, the closest risk of global crisis will have to do with the world financial system and a risk of systemic financial global crisis. Obviously, the current financial crisis has vindicated this forecast.The above brief contextualization of approaches to power shows the extent to which Perroux's own approach was seminal. Half a century earlier, Perroux had framed an approach that incorporates American neo-realism, liberal institutionalism and the Strangeian approaches, each of them appearing as specific cases of this Perrouxian unified theoretical framework, since this last one includes both relational and structural powers. Perroux' point of view suggests that hard power, soft power and structural powers can be regarded as different aspects of a more general conceptualization.

	Strange suggests four basic power structures:
	1. Security structures, the way in which each society organizes and manages its protection
	against various risks (wars, terrorist threats, natural catastrophes, epidemics, food and
	environmental security)
	2. Production structures (e.g., multinationals, innovation)
	3. Financial structures (with financial globalization)
	4. Knowledge structure, a domain in which asymmetries of acquisition and holding of
	knowledge have increased dramatically with recent changes in communication technology
	On the basis of these concepts, Strange puts forward two major theses: first, she contests the
	thesis of a U.S. decline. Second, all states (including the United States) see their capacity to act

on society and the economy decline in comparison with private actors. Therefore, zones of ungovernance -spheres in which neither state nor non-state actors can master events (in particular in

18

The state that we call equilibrium is the more or less durable persistence of sequences conveying, under explicit structural constraints, the cross-compatibility of actors' projects and activities" (Perroux, 1975: 148). Thus reformulated, the general equilibration of an economic set is attained when the sum of the actors' energies for change is close to zero.

Unfortunately, this theoretical reconstruction project is both uncompleted and incomplete. As a result, it leaves the reader frustrated not to find something more revolutionary which would really replace the Walrasian approach. As Stephen J. Meardon judiciously points out the model developed by Perroux in 1975 "is not specified mathematically". Perroux' representation has nothing to do with "a mathematically specified general equilibrium model that fulfills the same functions as the Walrasian model, but in addition captures the exercise of market and extra market power that the Walrasian model entirely misses".

The fact that Perroux failed to achieve a complete theoretical alternative reconstruction to the general equilibrium model does not condemn the project itself and its necessity. It does not invalidate in any way Perroux's ideas. His last book published in 1975 is full of intuitions and suggestions aimed at transforming the concept of general economic equilibrium from its initial description as a balance of forces applying to objects (to robots) towards a representation which incorporates the active role of economic agents and their power relations. In other words, towards the modelling of a process which tends to make compatible agents' plans as well as the change energies that they develop individually and collectively.

The fact that this project has not been brought to a successful conclusion is a limitation indeed.

However, in one sense, it also remains a potential: an invitation to present and future generations to resist mental laziness and follow and explore the rocky path that Perroux opened up.