Fast Optimization Scheme for the Muscular Response to FES Stimulation to Design a Smart Electrostimulator

Sandrine Gayrard 1 * Abstract-In this article, we present the design of a smart electrostimulator for muscle rehabilitation or reinforcement, using fast computations, in order to control the muscular force. The Ding and al. model allows to predict and to optimize the muscular force response to functional electrical stimulation. [START_REF] Ding | Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains[END_REF]. We analyze the estimation of the Ding and al. parameters using an approximation of the force response [START_REF] Bakir | Finite Dimensional Approximation to Muscular Response in Force-Fatigue Dynamics using Functional Electrical Stimulation[END_REF] which depends upon the 6 parameters of the Ding's model and we derive optimization scheme which bypass the time computational expensive integration of the dynamics of the Ding and al. equations.

I. INTRODUCTION

A. Industrial Project

The aim of the study project is the design of a smart electrostimulator, in the isometric case (without displacement of the muscle), for muscle rehabilitation or reinforcement based, in particular, on modern sensors and fast computations using embedded electronics. The idea is to design training programs to track a force of reference [START_REF] Doll | Optimization of a stimulation train based on a predictive model of muscle force and fatigue[END_REF] or to maximize the muscular force. Each training program can be converted into an optimization problem. The mathematical equations presented below are the tools to the design of this smart electrostimulator.

B. Mathematical equations

The Ding and al. model [START_REF] Ding | Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains[END_REF] allows to predict and to optimize the muscular force response to functional electrical stimulations and was validated experimentally [START_REF] Wilson | Force response of locust skeletal muscle[END_REF]. The system is described as follow.

The input u of a pulse train is defined, for t ∈ [0, T ], by: u(t) = ∑ n i=0 δ (tt i ), with 0 = t 0 < t 1 < • • • < t n < T the impulse times with n ∈ N fixed. This physical control allows us to obtain the signal E(t), which permits to calculate the force response following a train of electrical pulses. The dynamics is given by (1) :

Ė(t) + E(t) τ c = 1 τ c n ∑ i=0 η i R i δ (t -t i ), (1)
with E(0) = 0, and depending on the response time τ c and the function R i defined by

R i = 1 if i = 0 and R i = 1 + (R 0 - 1) exp(- t i -t i-1
τ c ) elsewhere . It allows the visualization of the tetany phenomenon, which corresponds to the memory of the effect of successive pulses. η i is the amplitude of the electric pulse stimulation. The FES (Functional Electrical Stimulation) signal allows to determine the evolution of electrical conduction as a function of the dynamics describing the evolution of the concentration c N of the calcium ions (Ca 2+ ), which plays an important role in muscle contraction.

ċN (t) + c N (t) τ c = E(t) (2)
The others equations of the Ding and al. model are described below and provide the muscular force response.

m 1 (t) = c N (t) K m + c N (t) , m 2 (t) = 1 τ 1 + τ 2 m 1 (t) , (3)
with m 1 the Michaelis-Menten (1913) function. The muscular force response satisfied the Hill-Huxley's models dynamics.

Ḟ(t) = -m 2 (t) F(t) + m 1 (t)A, (4)
with τ c , R 0 , A 0 , K m , τ 1 , τ 2 the set of parameters described in [START_REF] Ding | Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains[END_REF].

C. Motivation

The controls are the frequency and amplitudes of the electrical stimulations. They fit in the sampled-data control category [START_REF] Bakir | Finite Dimensional Approximation to Muscular Response in Force-Fatigue Dynamics using Functional Electrical Stimulation[END_REF]. The goal is to maximize the force response or to track a force of reference [START_REF] Doll | Optimization of a stimulation train based on a predictive model of muscle force and fatigue[END_REF].

In previous references [START_REF] Bakir | Pontryagin-type conditions for optimal muscular force response to functional electrical stimulations[END_REF], [START_REF] Bakir | Direct and indirect methods to optimize the muscular force response to a pulse train of electrical stimlation[END_REF] the problem was analyzed using direct or indirect optimal control schemes which are computationally expensive due to the numerical integration of the dynamics. In a more recent work we derive an approximation of the force response in [START_REF] Bakir | Finite Dimensional Approximation to Muscular Response in Force-Fatigue Dynamics using Functional Electrical Stimulation[END_REF] which depends upon the 6 parameters of the Ding's model and which can be used to derive a fast optimization scheme in this category by bypassing the time computational expensive integration of the dynamics.

II. APPROXIMATION OF THE MUSCULAR FORCE RESPONSE TO FES

We present a brief recap of the [START_REF] Bakir | Finite Dimensional Approximation to Muscular Response in Force-Fatigue Dynamics using Functional Electrical Stimulation[END_REF]. We derive an approximation of the force response, which gives an explicit form of the dynamics of the force as a function of time.

1) Definition 1: The muscular force, with F(0) = 0, is defined in [t k ,t k+1 ], k = 0, . . . , n, by

F(t) = A 0 M(t) t 0 M -1 (s)m 1 (s) s, (5)
with

M(t) = exp - t 0 m 2 (s) s .
We define a finer partition of (t i ) 1≤i≤n noted (t i+ j/p ), i = 0, . . . , n, j = 0, . . . , p -1, with p ∈ N * the number of the new points inserted between t i and t i+1 . Using (5), we construct the approximation of the force F on [t k+ j/p ,t k+( j+1)/p ], k = 0, . . . , n, j = 0, . . . , p -1, by :

F(t) ≈ A 0 M(t) t 0 M-1 (s) m1 (s) s, ∀t ∈ [t k+ j/p ,t k+( j+1)/p ],
with M(t) = exp -t 0 m2 (s) s , et mi , i = 1, 2 an approximation of m i defined by : mi (t) = a i j,k (tt k+ j/p ) + b i j,k , for t ∈ [t k+ j/p ,t k+( j+1)/p ] and k = 0, . . . , n, j = 0, . . . , p -1.

The constants a i j,k , b i j,k , j = 0, . . . , p -1 are calculed at specific times, depending on the choice on the approximation of m i .

As ṁ2 = τ 2 ṁ1 /(τ 1 + τ 2 m 1 ) 2 , then ṁ2 is equal to 0 when c N is maximal. On [t k ,t k+1 ], m 2 can be approximated by mk 2 (t) = a i j,k (t -t k+ j/2 ) + b i j,k , k = 0, . . . , n and j = 0, 1, with

t k+1/2 = argmax t∈[t k ,t k+1 ] cN (t)
. By calculation, we end up with

a i j,k = m k i (t k+1/2 ) -m k i (t k ) t k+1/2 -t k , b i j,k = m k i (t k ).
2) Proposition 1: The approximation of the muscular force F on [t k t + j t /p ,t k t +( j t +1)/p ], k t = 0, . . . , n, j t = 0, . . . , p-1 is given by :

F(t)/A 0 = k t -1 ∑ i=0 p-1 ∑ j=0 t i+( j+1)/p t i+ j/p M(t) M-1 (s) m1 (s) s + j t -1 ∑ j=0 t k t +( j+1)/p t k t + j/p M(t) M-1 (s) m1 (s) s + t t k t + j t /p M(t) M-1 (s) m1 (s) s. (6)
3) Proposition 2: From a practical point of view, we just need to estimate the values of the force F for t k+ j/p , which gives :

F(t k+ j/p )/A 0 = k-1 ∑ i=0 p-1 ∑ j=0 t i+( j+1)/p t i+ j/p M(t k+ j/p ) M-1 (s) m1 (s) s + j-1 ∑ j=0 t k+( j+1)/p t k+ j/p M(t k+ j/p ) M-1 (s) m1 (s) s (7)

III. SMART ELECTROSTIMULATOR

A. Industrial Project

Let's remind that the goal is the design of smart electrostimulator for muscle rehabilitation or reinforcement, using modern sensors and fast computations using embedded electronics, in order to control the muscular force [START_REF] Doll | Optimization of a stimulation train based on a predictive model of muscle force and fatigue[END_REF].

B. Materials and methods

The measurements of the muscular force were carried out under isometric conditions, i.e. without displacement of the muscle. The measurements were performed at the biceps brachii (BB) level. The stimulation protocol and data acquisition are also presented. The subject is placed on an adjustable chair. His/her arm is placed on a L-shaped platform so that the arm is in a 90 degree position and the chair is adjusted so that the arm is completely on the platform and is "glued" to it to allow stabilization with a strap hooked around the biceps. The experimental setup is presented Fig. 2. Interlink Eletronics FSR 402, 406). The resistance of the sensor decreases as the force applied to its surface increases. The pressure sensor is placed between the strap and the biceps, in such way that when the biceps contracts, the biceps compresses the sensor, in a perpendicularly direction.

The range of force sensitivity varies according on the chosen model. The sensor is connected to an Arduino UNO to sample the output voltage and to process it.

Finally, a premilinary calibration to grade the relation between the sensor output voltage and the induced force, is performed.

2) Stimulation: Electrical stimulation is provided by a commercial electrostimulation device (ref : Beurer EM 49 Digital TENS/EMS unit, see Fig. 2) designed for EMS (Electrical Muscle Stimulation) to promote muscle regeneration. Its adjustable parameters are the duration of the stimulation train t stim (fixed at 15 ms), the duration of rest t rest (fixed at 10ms), the width of the pulse (fixed at 150µs). The stimulation frequency f is chosen according to the objective of the experiment and the amplitude is modifiable during the experiment, as the controls.

C. Data acquisition

During the muscle contraction, determined by its stimulation amplitude, the biceps compresses the force sensor. Isometric force is sampled at a sampling rate of 1kHz and recorded through a serial port driven by a Matlab program.

D. Training program : algorithm

Each training program can be translated into an optimization problem. The several steps are described below and resumed in Fig. 3 • A succession of pulse trains is send to the biceps. Each train (of frequency f ) lasts t stim and is followed by a rest time of t rest . • During the pulse train sent to the biceps, the sensor measures the induced force. • At the end of the pulse train, the resting time starts.

During this resting time, the force measurements are recovered and processed, by a Matlab program running on a standard computer. • The force model parameters are estimated by fitting the measured data to the force prediction model described by the muscle force approximation. • From the force prediction model and the parameters estimated before, the optimal control is calculated according to an optimization problem that allows to track a force of reference or to maximize the muscular force. • This new calculated optimal command is then applied to the muscle during the next impulses train, at the end of the resting time. The two main parts are : 1) the muscle parameter's estimation; 2) the computation of the optimal control, using the approximation model force. They are both computed during the resting time t rest , which reinforces the need for fast computations.

IV. ESTIMATION OF THE DING AND AL. PARAMETERS

The muscle parameters of each person are estimated during the resting periods with low frequency stimulation using the piezoelectric force sensor. A. Data-fitting 1) Model: The explicit form of the force approximation gives the force in terms α a set 6 parameters of Ding and al. More precisely, [START_REF] Mathworks | Curve Fitting Toolbox User's Guide[END_REF] gives the approximation of the force in terms of a set α of 6 parameters and t k+ j/p with k = 0, . . . , n (n number of pulses during the train), j = 0, . . . , p -1 and p ∈ N * the number of the new points added between t k and

t k+1 . α = [τ c , R 0 , A 0 , K m , τ 1 , τ 2 ].
2) Data: Force data were recorded using the experimental set-up. The input, a n th pulse train with a stimulation frequency f gives the muscular response with as many bounces as pulses. For each bounce k, the average value is calculated using the data value F data :

F[t k ,t k+1 ] = 1 t k+1 -t k t k+1 t k F data (t) dt (8)
The vector of data force, which is going to be used to fit the model and estimate the 6 parameters of Ding and al. can be written for each impulse k ∈

[1, n -1] by F data-f it = [F data (t k ), F[t k ,t k+1 ] ] (9
) and is represented Fig. 4.

F data-f it = [F data (t 1 ), F[t 1 ,t 2 ] , . . . , F data (t n-1 ), F[t n-1 ,t n ] ] (9)
All the information of the model (the 6 parameters) is contained in the muscular response of the impulse train of size n. However, because of too high calculation time, only the muscular response to 5 or 6 impulses is taken into account, thus requiring a compromise between amount of information and computing time. Fig. 4. At the right, F f it with the dashed curves corresponding to the times t k+ j/p . At the left, F data (t) and F data-f it the vector of values which contains the data that will be fitted to the model. p = 2 and n = 6.

3) Cost Function: Each value of F data-f it fits the model F(t k+ j/p ) which corresponds to the approximation force calculated for each impulse time t k and for t k+ j/p , the intermediate times between 2 pulses (see Fig. 4). For k ∈ [1, n], the values F data (t k ) fit F(t k ) and the values F[t k ,t k+1 ] fit F(t k+ j/p ) (p = 2 and j = 1).

The cost function to minimize is described below :

J(α) = N ∑ i=1 ∥ Fi (t k+ j/p , α) -F i,data-f it ∥ 2 2 (10)
with F the approximation force that fits the values of force

F data-f it . α = [τ c , R 0 , A 0 , K m , τ 1 , τ 2 ]
. N the size of the data and the fit vectors (N = n.p).

B. Practical estimation 1) Computation:

The use of a non-linear solver, helps to solve non-linear curve fitting problems of the form m α inJ(α) with α the elements and α 0 the initial vector of the parameters. The solver starts at α 0 and attempts to find a local minimum α of the function described by J.

The associated problem is solved using a quasi-newton method, with Matlab [START_REF] Mathworks | Curve Fitting Toolbox User's Guide[END_REF], on a standard computer. We initialize the parameters with the Ding and al. parameters [START_REF] Ding | Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains[END_REF].

2) Experimental results: The estimated parameters are calculated for one subject with 4 different frequencies and various amplitudes. They are given in Fig. 6 and in Table I. For each frequency, identical stimulation have be send to the muscle BB and the parameters have been estimated for each train : the represented values correspond to the average values calculated using all the data, for each frequency. The corresponding curves for 3 various frequencies are depicted Fig. 5. 3) Observation and discussion: The errors between measured data F data-f it and predicted force values increase at lower pulse frequencies because the approximating force model considers force at a given time t k+ j/p but does not take into account the dynamic behavior of the force with bounces, whose peak-to-peak amplitude increases for lower frequency.

The parameters τ c , R 0 , τ 1 and τ 2 show a standard deviation with the Ding and al. parameters, respectively less than 26%, 3%, 12% and 1.5%. The difference appears for the parameters K m and A 0 with standard deviations greater than 50% and increases with the frequency of stimulation.

These two parameters correspond to the non-fatigued model. The experimental results show difficulties to predict results depending on the muscle fatigue. The biceps can be subjected to a significant effort without the subject having performed it, thus disturbing the measurements. Sources of error can be a lack of hydration, general fatigue or skin's condition. To overcome this problem, it is necessary to multiply the experiments, on the same subject and on other subjects.

The difference obtained with the parameter A 0 with the Ding's values comes from the fact that the latter depends on the calibration of the force measurements. It is important to note that Ding et al. performed experiments mainly on the quadriceps. [START_REF] Ding | Two-step, predictive, isometric force model tested on data from human and rat muscles[END_REF] shows that parameters varied with muscle length and fatigue.

A statistical analysis was done to understand the difference in data for each group of parameters for different frequencies. Using an R algorithm and after checking that the measurements fits either a normal distribution or a 'poisson' distribution, it was observed, using the ANOVA (analysis of variance) test that:

• For the parameter τ 1 , there are significant errors (p ¡ 0.05) between the group of measurements for frequencies: 5Hz -1Hz with p = 0.00014 and 5Hz -2Hz with p = 0.012. In other words, the group of τ 1 values at 5Hz is significantly different than at 1Hz and 2Hz. • There are no significant errors between measurement groups of estimated parameters τ c , τ 2 , A 0 , R 0 and K m (p > 0.05) for each frequency, which means that for each parameter, the groups of parameter meurements can be considered identical. This also shows that these 5 factors do not vary with stimulation frequency. The most surprising result is the statistical analysis of A 0 : this parameter describes muscle fatigue and decreases when the muscle is fatigued. The table of values shows a decrease in A 0 as the frequency increases, which is consistent with the fact the higher the frequency the more the muscle is subjected to electric impulses, the more it fatigues the muscle. However, the statistical analysis of these values describes that there is no significant differences between these values packages for the three different frequencies studied. It was also shown in the analysis of Ding et al. [START_REF] Ding | A predictive model of fatigue in human skeletal muscles[END_REF] that the parameters τ 1 and K m change with muscle fatigue. A significant error was found between the parameters at 5 Hz versus 1 Hz and 2 Hz for τ 1 . But no significant errors (p > 0.05) were found for K m . Another comment, only experiments with constant stimulation frequency were performed because, firstly, the stimulation equipment imposes this constraint and secondly it was shown that parameters vary significantly between constant and non-constant frequency stimulation, which is part of the limitation of the force model described by Ding in [START_REF] Ding | Two-step, predictive, isometric force model tested on data from human and rat muscles[END_REF].

Amplitudes are fixed at one. Impulse times are the optimization variables (x = (t 1 ,t 2 , ...,t n )). They belong to the finite dimensional input-space I ∈ [0, T ] n . They are subjected to two constraints :

• 0 = t 0 < t 1 < ... < t n < t n+1 = T , with n a fixed integer, representing the number of pulses in the train. • t i+1t i > ε = 10 ms for i = 0, ... , n.

A. Optimal sampled-data control problem

Using the equation of the approximation force [START_REF] Mathworks | Curve Fitting Toolbox User's Guide[END_REF], we can generate the explicit expression of the approximation of the force, in terms of the optimization variables x. J1 and J2 constitute the approximated cost functions of the optimization routine. For each case, the associated problem which gives the approximation J of the cost function J based on the approximation F of the variable F described before, is solved using an interior point method, with the function "fmincon" in Matlab, on a standard computer. For the first case, we initialize the amplitudes of the pulse train to random values on [0, 1] and for the second case, we initialize the times of the pulse train to random values, respecting the two constraints described before.

1) Numerical results: 1.1) The optimal solution η i * and the force response F corresponding are depicted in Fig. 7. 1.2) The optimal solution t i * and the force response F corresponding are depicted in Fig. 8.

Fig. 1 .

 1 Fig.1. Construction of the approximation of F(t) and F(t k+ j/p ) (red curve) with k = 0, . . . , n, j = 0, . . . , p -1 and p ∈ N * the number of the new points added between t k and t k+1 . The black curve is the force calculated with an Euler integration scheme. The constants are p = 2 and n = 6 impulses.

Fig. 2 .

 2 Fig. 2. Experimental setup

Fig. 3 .

 3 Fig. 3. Successive steps of the training program.

Fig. 5 .

 5 Fig.5. The black curve corresponds to the force data measured with the FSR sensor. The dashed curve is the average force data measured at t k+ j/p with k = 0, . . . , n, j = 0, . . . , p -1 and p ∈ N * the number of the new points added between t k and t k+1 . The red curve represents the approximation of the force that fits the measured data. Values of the constants are n = 5 and p = 2. The computation time is less than 2s.

Fig. 6 .

 6 Fig. 6. Estimated experimental parameters for one individual biceps, nonfatigued case, compared to the Ding and al. parameters, for 4 different frequencies : f = [1, 2, 5, 10] Hz.

  V. OPTIMIZATION PROBLEM The training programs are being translated into an optimization problem. Both problems fit in the sampled-data control category. a) Endurance program: We consider a single train [0, T ] on which the corresponding problem is : ) -F re f | 2 dt (11) The amplitudes are optimization variables (x = (η 1 , η 2 , ..., η n )), which belong to the finite dimensional input-space I ∈ [0, 1] n+1 defined by amplitude constraints : η i ∈ [0, 1], i = 0, ... , n, with n a fixed integer, representing the number of pulses in the train. T is the time of the end of the pulse train. The reference force F re f is adjusted in relation with the user. b) Punch program: We consider a single train [0, T ], T free, on which the corresponding problem is :

∥ 2 2

 2 Fk (t k+ j/p , α, η i ) -F re f ∥ Fk (T, α,t i )) (14) with α = [τ c , R 0 , A 0 , K m , τ 1 , τ 2 ].

Fig. 7 .

 7 Fig. 7. The optimal solution (η * 0 , ..., η * n ) minimizes the cost J1 from (13). F+ and Fare respectively the upper and the lower approximation of F, described in the article [6]. Values of the constants are τ c = 20ms, K m = 0.1030, R 0 = 1.14, A 0 = 3.14 N/m, τ 1 = 54ms, τ 2 = 127ms, n = 8 and F re f = 0.1 kN.

TABLE I ESTIMATED

 I EXPERIMENTAL PARAMETERS FOR ONE INDIVIDUAL BICEPS FOR DIFFERENT STIMULATION FREQUENCY.

	α	τ c (ms)	R 0	K m	A 0 (N/ms)	τ 1 (ms) τ 2 (ms)
	1Hz	19.83	1.14	4.38	1.31	50.91	124.39
	2Hz	19.86	1.14	4.82	1.14	50.88	124.40
	5Hz	19.87	1.13	4.87	0.77	50.83	124.39
	ᾱ	19.82	1.13	5.56	0.94	50.86	124.39
	α Ding	20	1.14	0.1030	3.009	50.95	124.40

ACKNOWLEDGMENT

The work presented in this article was performed in the frame of a PhD "Design of a smart muscle stimulator", as a CIFRE agreement between the Burgundy university (Dijon, France) and Segula Matra Technologies (Trappes, France) under the direction of Pr. Toufik Bakir, Pr. Bernard Bonnard and Mr. Mokrane Abdiche.

Fig. 8. The optimal solution (t * 0 , ...,t * n ) maximize the cost J2 from (14). F+ and Fare respectively the upper and the lower approximation of F, described in the article [START_REF] Bakir | Finite Dimensional Approximation to Muscular Response in Force-Fatigue Dynamics using Functional Electrical Stimulation[END_REF]. Values of the constants are τ c = 20ms, K m = 0.1030, R 0 = 1.14, A 0 = 3.14 N/m, τ 1 = 54ms, τ 2 = 127ms, n = 8.

B. Smart electrostimulator

Only the endurance program has been implemented into the smart electrostimulator because of the hardware constraint which imposes a constant frequency during the stimulation train. Besides, maximizing muscle strenght during the experimental tests generates pain. The training program, translated into an optimization problem, has been tested using the algorithm presented in Fig. 3. Optimal controls are the amplitudes η of the stimulation train. In the case of the smart electrostimulator and for simplification purposes, we impose optimal solution η i * constant for all i ∈ [0, n] with n the number of stimulation impulses.

1) Experimental results: The optimal solution η * i , the predicted force response F and the measured force F data are depicted in Fig. 9 and 10. 2) Observation and discussion: The error between measured data and predicted force values increases when the pulse frequency is lower. Indeed, the dynamic behavior of the force with bounces whose peak-to-peak amplitude increases for a lower frequency, makes the tracking more difficult. The most problematic fact comes from the computation of the error between measured force data and F re f reference force. The error has to be compute either from the lower peak of force bounces or the higher peak of force bounces. If we introduce a total error margin of 5% around the reference value, the measured force are out of this range from 28% at the bottom and 71% with a low frequency. This error decreases when the stimulation frequency is higher. But a higher frequency means the perception of pain increasing and more importantly, the fatigue increases rapidly which means that the muscle can't maintain the reference force.

VI. CONCLUSIONS

Our study shows how to estimate the set of 6 parameters of the Ding's model using an approximation of the force response, that allows to bypass the time computational expensive integration of the dynamics. This study is about the design of a smart electrostimulator with fast computations and modern sensors, allowing us to fit the model to the sensors values and to compute the optimal trajectory.


Sebastian M Büttner

Monika Eigmüller

Nikola Tietze

Camille Noûs

Space, territories, and borders in Europe

Space, territories, and borders "are created in action and […] embodied in institutions that pre-structure action" (Löw 2016: xiv). The dual movement from structure to action and from action to structure, which is generally a feature of spatial constructions, is exemplified in the European Neighborhood Policy (ENP) implemented by the European Union (EU) in 2004.

With the creation of the ENP, the governments of the EU member states established structures for a space in the sense of the French word espace, which not only refers to distance and time but also to "place," "site," "area," "surface," or "region" (Shields 2006: 147). Drawing on this spatial construction, the ENP regulates, finances, and legitimates (as well as delegitimates) actions, for example, in the realm of cross-border economic and workplace relations or in infrastructural networks such as rail transport or waste water collection systems connecting the EU and a number of nation-states that border on the EU to the south, southeast, or east

(with the exception of Turkey and Russia). Furthermore, this EU program is an essential element of the European border regime. The neighborhood space establishes the framework for administering and controlling migration on the so-called external borders of the EU and creates interaction between the EU member states and the nearby states. By creating such interaction, the ENP highlights the general features of political constructions of space that define areas, territories, and borders. Such definitions determine who belongs and has access to rights and thus who is included or excluded; they also legitimate domination and the sovereignty of power. With respect to these general features of political constructions of space, the EU is characterized by a unique additional trait: the EU is founded on the principle of supranational governance. In other words, the Union's spaces intersect with the territorial sovereignties of the EU member states. As a result, in many policy areas, the competence of the European communities and the EU and those of the member states do not constitute "distinctly separate spheres" (Patel & Röhl 2020: 217). This complex interplay also results in flexible redefinitions of state borders.

For sociologists investigating Europeanization, space, territories, and borders are revealed in the codification of laws, in the architecture of institutional regimes, in the organization and regulation of markets, in the ways infrastructure is generated, in the coordination of social Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 2 security systems, in the harmonization of criteria, standards, and administrative procedures. In the EU, legal, institutional, administrative, economic, and social processes, practices, and structures reveal specific ways in which space construction overlap with respect to the member states' integrity and identity and to fluid EU borders [START_REF] Jureit | Postsouveräne Territorialität. Die Europäische Union als supranationaler Raum[END_REF].

Nonetheless, EU spatial governance is built on "phantom geographies"-"the traces left in contemporary societies by defunct territorialities" (Hirschhausen 2017: 107). Such phantom geographies intersect with contemporary EU spatial constructions, be it in the realm of the European neighborhood policy, with respect to the enlargement of the European communities [START_REF] Vobruba | The Enlargement Crisis of the European Union: Limits of the Dialectics of Integration and Expansion[END_REF], issues related to labor migration, to Spanish cities in Morocco, to French départements in the Carribbean, or to privileged economic cooperation, for example between Polish and Ukrainian companies.

Space, territories, and borders are not only objects of social science research; they are also categories of sociological observation and explanation. As such they offer opportunities for discovering and elucidating the interdependencies, interactions, and discontinuities between various realms of actions related to "phantom geographies," as well as between distinctive territorial scales of political and economic power or between different domains of governance. This perspective focuses on social relations that are ordered by spatial constructions, territorialization, and the drawing of borders. These three spatial categories for observation reveal inequalities and tensions that generate the architecture of a political order or an actor's activities that run counter to such architectures, for example, in the form of illegal border crossings.

In this chapter, the double character of space, territories, and borders-first, as an object of study in the sociology of Europeanization and, second, as a category for observing and explaining Europeanization-is outlined in two steps. After briefly sketching the historical origin of the three categories of spatial governance just mentioned, this chapter will focus on what spaces, territories, and borders have been established in the EU by means of law, institution building, and markets. The following sections consider the spatial categories of sociological observation and explanation and discuss two methodological approaches: the scalar structuration approach [START_REF] Brenner | The Limits to Scale? Methodological Reflections on Scalar Structuration[END_REF]) and the jeux d'échelles approach (Revel 1996).

Box 1: Space as a concept in the social sciences and humanities

Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 3 In his book La production de l'espace (1974), which appeared in English in 1991 as The Production of Space, the French philosopher and sociologist Henri Lefèbvre referred to space as a complex social construction consisting of meanings, ideas, and practices. As meanings, ideas, and practices, spatial concepts become the basis for action and instruments for structuring interactions. According to Lefebvre, they are thus tools for social control and the enforcement of domination and power. Lefebvre's thinking has had a major impactfirst in France and, from the late 1980s, in the English-speaking world-in particular on human and economic geography, which studies the interactions and interdependencies between social, cultural, economic, and political activities and the environments in which they occur. This is reflected in the work of critical urban theorists, among others, who address urbanization and capitalist governance from the point of view of social justice [START_REF] Brenner | The Urban Question and the Scale Question: Some Conceptual Clarifications[END_REF][START_REF] Brenner | Henri Lefebvre on State, Space, Territory[END_REF], Döring & Thielmann 2008). A continent, a place, a mountain range, a country, a district, a river, a border, etc. are terms used in the social sciences and humanities to denote relations: (i) between actors and structures, (ii) between different activities, and (iii) between activities, actors, and the earth's surface. In this sense, space is a form of order, observation, and communication [START_REF] Jureit | Das Ordnen von Räumen[END_REF]). In the realm of political relations, where power, domination, and control are at stake, the concept of space has found its expression historically in territories and borders. This means, for example, that the designation of a "natural" border (for instance the Pyrenees) naturalizes a relationship of domination. Legal regulations, control devices, and the othering found on either side of the border materialize the definition of the border; physical features on the earth are not what materializes a border.

Space, territories, and borders from the perspective of the history of political ideas

The term space, which is used in this text in the sense of the French term espace as outlined above refers to geographical features, that is, "specific arrangements of objects and people" (Löw 2016: xii) and sets social interactions in relation to a specifically constituted environment. The political meaning of space emerged in the early modern period and refers to the territorialization of state rule. In the late Middle Ages in Europe, rulership was tied to a place that was the seat of a ruler such as a castle, fortification or marketplace. But this was frequently accompanied by uncertainties about the limits of domination. Such uncertainties accelerated a process in which governance was territorialized. With the development of the Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 European territorial states, territory became the basis for the organization of rule and its expansion.

Since the 19th century, political spatiality has become intrinsically intermeshed with the spatiality of national territory and of state sovereignty. One result of this development is the emergence and ongoing significance of the sovereignty of nation-states. According to Charles S. Maier, territoriality represents the central concept of European governments in the 20th century. "Territoriality means simply the properties, including power, provided by the control of bordered political space, which until recently at least created the framework for national and often ethnic identity" (Maier 2000: 808). The "control of bordered political space" has not only led to the establishment and consolidation of border regimes (Eigmüller & Vobruba 2016). In the course of the 19th century and in the European context, national territory also increasingly framed economic relations, industrialization and infrastructure policies, resource extraction and the energy supply, and, last but not least, the development of social security and health care systems.

In the 20th century, the "control of bordered political space" found its expression in two different forms of territorial domination, which nevertheless refer to one another: the European nation-state and European colonial empires; that is, the attempts to establish regional spheres of influence or, in Carl Schmitt's terminology, Großräume [large spaces] [START_REF] Jureit | Das Ordnen von Räumen[END_REF]. The conflicts as well as the cooperation of European governments after World War I were based on the understanding that a nation-state consists of and is a territory. Thus, the integrity of a state is damaged when its borders are crossed without permission or when other ruling groups hinder it in exercising its sovereignty. An almost corporeal understanding of law underpins the right to self-determination of peoples, which was of fundamental importance for decolonization processes until well after 1945. The European states, which simultaneously drew on and were in competition with the nation-state understanding of territoriality, nevertheless consolidated and expanded their political power on the basis of their colonial empires and the creation of regional spheres of influence. The first half of the 20th century was therefore also a period in which hegemonic theories of large spaces like Carl Schmitt's highly controversial theory were taken up and developed. These theories, which are rooted in particular in European imperial traditions, have been instrumental in demarcating an imagined Europe of civilization or culture that was perceived as different from other regions.

Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 The concept of Europe gained political significance in the course of the 19th and 20th centuries in a context marked by the tension between the two territorial forms of rule that enacted competing national and ethnic identities-the European nation-state, on the one hand, and the expansion of their spheres of influence in the form of colonial empires and metropolitan policies, on the other. The competition among nation-states for economic and technical progress as well as for social modernization, cultural development, and the realization of perceived standards of civilization measured in terms of nation-state parameters shaped the spatial experience of European political and economic actors and established the spatial image of a Europe that was seen as being too crowded. Moreover, within the framework of the European nation-state model and in the aftermath of World War I, the notion emerged and was reinforced that social relations can be planned and spatially ordered on the basis of scientific arguments, for example by population resettlement or transfer [START_REF] Bernhardt | Governance, Statehood, and Space in 20th Century Political Struggles: An Introduction[END_REF][START_REF] Rosental | Géopolitique et Etat-Providence: Le BIT et la politique mondiale des migrations dans l'entre-deux guerres[END_REF]). With Europe perceived as "too crowded" to allow for the competition among European nation-states that was deemed necessary to achieve progress and modernization, compensation was found in the expansion of colonial empires, in the establishment of colonial trading posts and settlement projects, in the transfer of civilization and culture, and so on. Constitutive for this policy was the idea and production, with the use of force, of "empty areas," "espaces de peuplement," areas of exploitation," and of "frontiers" as transition spaces (Jureit 2016).

The contradictions and violent conflicts in the interwar period can be seen as responses to profound changes in global power balances. New nation-states with claims to global political and economic leadership such as the United States had entered the international arena, and South American states that became independent in the 19th century demanded a say in international organizations and negotiations. Moreover, economic interdependencies intensified and expanded to the extent that economic processes could no longer be controlled on the basis of national territoriality. These developments, which continued long after the end of World War II, have meant that the market, and above all the "world market," has become increasingly relevant for the territorial organization of state rule.

Against the backdrop of these changes, the establishment of the first European communities in the 1950s reflects the attempts of Western European states to respond to global economic interdependencies as well as to wartime devastation by creating common European markets, asserting their place in the emerging bloc formation of the Cold War, and countering the Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 6 consequences of decolonization (not least of their own colonial empires) for acquiring raw materials, for trade, and for labor markets [START_REF] Milward | The European Rescue of the Nation-State[END_REF]. With this attempt, the governments of the founding states of the European communities at the same time clearly rejected the historical models of hegemonic and violent control of a bordered political space [START_REF] Wobbe | Die Verortung Europas in der Weltgesellschaft: Historische Europasemantik und Identitätspolitik der Europäischen Union[END_REF]-not only on the discursive level, but above all by forming a body of European law and building European institutions.

As the history of political ideas outlined here suggests, the sociology of Europeanization encompasses issues that transcend by far, in both temporal and spatial terms, the founding of the European Communities and the European Union. However, in the following I will limit myself to addressing spatial constructions, territoriality, and definitions of border regimes in the context of these legal institutional arrangements.

Spatial constructions, territories, and borders: Three interdependent objects of sociological research on Europeanization

The European Coal and Steel Union (1951), founded in 1951, and especially the European Economic Community (EEC), founded in 1957, laid the groundwork for the "area without internal frontiers," as it is called in Article 13 of the Single European Act (1986). On the one hand, this spatial construction of a European area overcomes the model of the European nation-state on the basis of a contractually guaranteed and shared sovereignty in certain policy areas. These policy areas are constantly expanding in the course of the European integration process, which has accelerated as a result of the Single European Act. This act replaced the principle of unanimous decision making by member states with qualified majority voting rules. The member states have thus abandoned the paradigm of harmonization of regulations in social relations and facilitated legislative processes on the European level. The area without internal frontiers is founded on European laws that aim to combine and coordinate divergent national legal orders (Patel & Röhl 2020: 81-91). On the other hand, this European area consists of the sum of the national territories of the members of the European Communities and today's European Union (EU). Article 4 of the Consolidated Version of the Treaty of the European Union states as follows: "The Union shall respect the equality of Member States before the Treaties as well as their national identities.… [and] their essential State functions, including ensuring the territorial integrity of the State, maintaining law and order and safeguarding national security." Thus, the structural core of the EU's current spatial Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 7 construction is the way in which both nationally and supranationally guaranteed claims to power and sovereignty and nationally and supranationally guaranteed economic and social orders and forms of solidarity overlap. On the basis of these overlaps, a complex and flexible European border regime has emerged at multiple locations, some of which are extraterritorial; this regime has various instruments of control at its disposal [START_REF] Eigmüller | Grenzsicherungspolitik. Funktion und Wirkung der europäischen Außengrenze[END_REF][START_REF] Mau | Liberal States and the Freedom of Movement: Selective Borders, Unequal Mobility[END_REF]. What is more, this regime created opportunities for the European Communities and the EU to expand their sphere of influence, following the logic of "tiered levels of integration" [START_REF] Vobruba | Grenzen des Projekts Europa: Von der Expansionsdynamik zur abgestuften Integration[END_REF]. Under this regime, implementation of classic forms of border control on the periphery of the member states' territories has become a sign that states of emergency are in effect (for example, following terrorist attacks or during pandemics).

Law and institutions

The overlapping of supranational and national claims to power and sovereignty and of economic and social orders constitutes "spatialities of Europeanization" [START_REF] Clark | The Spatialities of Europeanisation: Territory, Government and Power in 'EUrope[END_REF] that shape and influence the relationship between the territories of the member states as well as the territorial frames of social relations and interactions in the area without internal frontiers. The spatialities of Europeanization-which are marked, from a sociological perspective, by the dialectic between Europeanization as a process of opening and as a process of national closure [START_REF] Münch | Die Konstruktion der europäischen Gesellschaft: Zur Dialektik von transnationaler Integration und nationaler Desintegration[END_REF])-are based to a decisive extent on the legal codifications set out in European primary law, i.e. on European treaties, and secondary law, i.e. European directives, and regulations. The member states that signed the Treaty of Rome in 1958 agreed to four fundamental principles: free movement of goods, free movement of capital, freedom to provide and establish services, and the free movement of persons. By establishing these four liberties, they outlined a European legal space [START_REF] Vauchez | Brokering Europe: Euro Lawyers and the Making of a Transnational Polity[END_REF]. At the core of this legal space is the prohibition of "any discrimination on grounds of nationality" (Article 7 EEC). These principles ensure equal access to the national legal orders of the member states for companies, investors, service providers, and employees and also form the basis for the development of a constitutional framework for the European Union. The legal space thus not only frames and legitimates the supranational regulation of relations between the sovereign powers of the member states and the hierarchies between European norms and national legal systems. It also enables the institutions that represent this space to define fundamental legal norms, install European citizenship, and enact laws and monitor compliance with them. From a sociological perspective, the European legal space generates interactions and interdependencies between representatives of the respective member states' governments and public authorities, but also among various groups of non-governmental actors such as market participants, experts, workers, persons covered by different national security schemes, students, etc. The tensions and conflicts associated with these interactions and interdependencies, due to clashes of interests, claims to hegemony, or uneven social developments, shed light on the fractures and discontinuities in the area without internal frontiers. One indicator of these shifting fractures and discontinuities is the extent to which mobile EU citizens do or do not have access to tax-funded social benefits in other member states. In various rulings handed down since 1998 (judgement of the ECJ C-86/96 Sala v.

Freistaat Bayen), the Court of Justice of the European Union has recognized that EU citizens who reside, because of their job or education, in a member state other than the state whose nationality they have are entitled to social benefits. Such claims to social benefits have nonetheless repeatedly been challenged by national social welfare offices and social courts.

Since 2014 (judgement of the ECJ C-333/13 Dano v. Jobcenter Leipzig), the European Court of Justice has itself negated such claims in three cases [START_REF] Blauberger | ECJ Judges Read the Morning Papers: Explaining the Turnaround of European Citizenship Jurisprudence[END_REF][START_REF] Tietze | Legal Imagination am Europäischen Gerichtshof: Erzählungen europäischer Richter über Gleichbehandlung und Kategorisierungen[END_REF]).

These conflicts of solidarity [START_REF] Farahat | Der Europäische Gerichtshof in der Eurokrise: Eine konflikttheoretische Perspektive[END_REF] highlight contradictions that have emerged in the legal space of the EU area without internal frontiers as a result of inconsistencies between the right to free movement for workers and trainees and their right to social security, on the one hand, and the social rights guaranteed for mobile EU citizens, on the other. Tax-funded social benefits as a core element of welfare state policies have remained within the domain of member states [START_REF] Ferrera | European Integration and National Social Citizenship: Changing Boundaries, New Structuring?[END_REF]. Allocation of benefits to mobile EU citizens who become unemployed or require assistance during their training is subject to the European process called coordination among member states [START_REF] Pataut | Securité sociale, assistance sociale et libre circulation: remarques sur les frontières de la solidarité en Europe[END_REF], which means that, with directly elected deputies since 1979 in Strasbourg and Brussels-and the establishment of normative institutions such as the prohibition of discrimination on the basis of nationality, the portability of social rights and benefits, and the prohibition of double taxation of mobile EU citizens are all part of these processes.

The European institutional order, in which the area without internal frontiers intersects with the territories of the member states, regulates the formal relationship between the EU institutions and the institutions of the member states, and determines which decision-making competence is assigned to what authority. A complex regime of different layers of regulation structures the European area without internal frontiers and determines the details of, for example, the "free movement of persons" that is constitutive of European citizenship and the area without internal frontiers. A differentiated system of EU and member states' regulations stipulates who is included in or excluded from the area without internal frontiers under what conditions, for what period of time, and with what options for taking action. The various categories of mobile workers (cross-border workers, posted workers, workers who reside in another member state than their nationality, third-country workers, etc.) shape both contrasting mobilities taking place within this area and distinctive access to the rights that establish it. In this respect the European legal space and the institutional orders associated with it lead to multiple unequal "spatialities of Europeanization"-not only in the domain of the labor relations within the European area.

Markets and social protection

Such differences and inconsistencies in Europeanization contradict the unity and especially the uniformity suggested by the spatial constructions of the European Union and its predecessors, the European Communities. An example of such a spatial construction that suggests uniformity is the single market. The latter was established with the Single European Act 1986 and builds on the free trade zone (1960) and the customs union (1968), which were also spatially constructed. The introduction of the Euro on January 1, 1999 highlighted the symbolic unity of this single market. In reality, however, it established a new space within the single market-the Eurozone. The "spatial image" of the market invokes the "dream" of a society that bases its cohesion on egalitarian exchange relations and the interdependencies of supply and demand. 2 In this respect, the political decisions of the member states and the legal Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 10 codifications that materialize the single market represent, among other things, the attempt of the governments of the member states to reduce the complexity of global economic interdependencies and to find an adequate response to economic globalization processes beyond territorially constituted national economies.

"Speaking as one voice, the EU carries more weight in international trade negotiations than each individual member would," according to an official statement on the EU website (European Union 2019). The EU trade balances, which Eurostat compiles annually on the basis of exports and imports of member states to and from countries outside the EU, demonstrate the existence of the single market and the associated global flows of goods. They also reinforce the importance of the single market for European and non-European economic actors such as companies, investors, national governments, employees, etc. Eurostat statistics and data materialize the construction of space in a market economy by classifying political economic action on the basis of European categories and simultaneously create reference points for political and economic action. In this respect, the statistical office of the European Union reproduces the dual history by linking statistics to the establishment of the European nation-state model and the public policies decided on in the name of the nation-state. On the one hand, statistics serve decision-makers as evidence of the sphere in which specific policies apply, and, on the other hand, as an instrument of justification for their scope [START_REF] Desrosières | The Politics of Large Numbers: A History of Statistical Reasoning[END_REF]Bartl, Papillaud & Terracher-Lipinski 2019.) Eurostat statistics and data reveal the basic structure of the single market, which consists of the sum of the economies of the member states [START_REF] Fink | Ungleichheiten harmonisieren: Lohngleichheit und Sozialstatistiken in den Europäischen Gemeinschaften (1957-1978)[END_REF] Despite its member-state structure, the single market has undoubtedly transnationalized corporate structures, labor markets, infrastructure development, knowledge and technology production, educational pathways, and also individual biographies within the EU, thus Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 11 contributing significantly to a convergence of living standards as well as to converging GDPs.

The Commission notes in a report published in 2007 an "increase in welfare of € 518 per head in 2006 compared to the situation without the single market (corresponding to a 2.15% increase of GDP over the period 1992-2006)" (SEC (2007) 1521). However, the growth in wealth generated on the single market has both produced and reinforced an uneven distribution of resources and opportunities to participate in the market exchanges. This development could be observed before the EU enlargements of the 2000s, as well as in the aftermath of the economic and financial crisis of [START_REF] Münch | Die Konstruktion der europäischen Gesellschaft: Zur Dialektik von transnationaler Integration und nationaler Desintegration[END_REF][START_REF] Beckfield | Remapping Inequality in Europe, The Net Effect of Regional Integration on Total Income Inequality in the European Union[END_REF][START_REF] Beckfield | Remapping Inequality in Europe, The Net Effect of Regional Integration on Total Income Inequality in the European Union[END_REF]; Heidenreich

2016). The European Observation Network for Territorial Development and Cohesion

(ESPON), for example, observed two dividing lines on the single market in 2011, in particular with respect to competitiveness. These divisions were between regions in the East of the "marketplace" and those in the West and between regions in the South and those in the North (see for example "Lisbon performance and regional economic development, 2001-2011," ESPON Atlas, Territorial Observation No.3, https://mapfinder.espon.eu/mapfinder/?p=2432).

Furthermore, the increase of prosperity and the converging living standards in the EU have not abolished the large differences between member states in the portions of their populations that are at risk of poverty. These relative differences have increased significantly in the wake of the economic and debt crisis since 2010. The territorial cohesion of the single market and social inclusion into the European area without internal frontiers are accompanied by large social inequalities, which follow the two dividing lines just mentioned and intersect the prosperous zones of economically strong member states. Figure 1, Regional at-risk-of poverty (ARoP) rates in 2011, illustrates how economic competitiveness follows regional dividing lines and highlights the uneven topography of converging standards of living. This uneven topography is also found in countries with generally more equal levels of income and distribution of income, in particular in Germany, and countries with large intra-national disparities such as Italy, where economic inequalities reflect underlying weaknesses in the construction of the Italian nation-state. Also, the map illustrates that the single market in part ends at the borders of the member states. Although several border areas-for instance, the region along the Rhine between France and Germany, the area between the Belgian city of Antwerp and the Dutch city Rotterdam, or the Spanish and French Basque region-have low ARoP rates, the ARoP rates of other border areas such Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 13 as the Catalan region or the German-Polish Pomerania region point to the impacts of national borders on the capacity of regional actors to take part in competitive single market exchange.

The intra-European asymmetries that the single market generates, perpetuates, or deepens are an indication of the unequal bargaining power of the member states. These inequalities are, to a large extent, a result of the member states' different shares of intra-European trade in goods. The interstate framework of the single market contributes to promoting asymmetric power relations between the member states, due to unequal participation in intra-European trade leads, which in turns leads to uneven bargaining power [START_REF] Lechevalier | Social Europe and Eurozone Crisis: The Divided States of Europe[END_REF]). As a result, economic and social inequalities between member states tend to be reproduced, because the basic structure of the market construction in the member states for the most part prevents processes in which political differences are balanced out at the level of other territorial differentiations.

Regulation of the market is largely in the hands of the governments of the member states and is therefore based on the power of a government to assert itself in negotiations on market regulation in intergovernmental councils. Furthermore, decisions on market regulation are Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 14 made in relation to other policy areas, which are differentiated according to the division of competences between the EU and the member states. These policy areas include business and industry, competition, education, training and youth, employment and social affairs, energy, agriculture, environment, regional investment and solidarity and others. The interdependencies and interactions between the European single market, the Eurozone, the European area without internal frontiers, the Schengen area in which border controls have been abolished, and the European legal space of fundamental rights form a blind spot in the institutionalized organization of intergovernmental EU decision-making. However, they give rise to inequalities of inclusion and to uneven territorial cohesion. Indeed, the austerity policies implemented from 2010 on in the wake of the Euro and debt crisis highlight this political deficit of European spatial constructions.

Various funds-including the European Social Fund (ESF) (1958), the European Regional Development Fund (1972), or the Cohesion Fund (1994)-have been established at the EU level to "correct" social and economic "disparities" and "imbalances" between regions and member states and to mitigate the problems associated with unemployment, as the Commission explains. But these funds do not initiate processes that counteract or dismantle the power hierarchies between member states and/or implement the redistribution of social resources on a supranational and or intranational level. Moreover, a significant cause of the uneven bargaining power of the member states is the policy context in which these funds are implemented. Since the late 1970s, measures taken to enhance social welfare and mitigate economic inequality between structurally weak and economically strong regions have more and more frequently been designed to serve the market [START_REF] Lechevalier | Social Europe and Eurozone Crisis: The Divided States of Europe[END_REF]. Policy approaches that run counter to market logics or uphold the need for the protection of social welfare and the leveling of regional differences as by-products of an increase in the GDP and converging living standards have lost influence in the single market. Instead, the member states and the European Community as well as the EU define social and infrastructure policy in terms of the goal of increasing the competitiveness of companies and business locations. This shift became apparent from the 1990s on, when the ESF programs, for example in the field of vocational training and retraining, and regional and cohesion policy funding assistance were placed within the ideological framework of "social investment" and "activation". In the context of "market-driven human resource management" (Bode 2019: 136), regions and urban areas, like companies and employees, are perceived as individual entrepreneurs who manage their own Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 man-and womanpower. In this perspective, the resources of regions, urban areas, and employees must and should be "activated" with the help of business management instruments.

Observing and describing Europeanization: Space, territories, and borders as perspectives for sociological research

The previous section referred to statistics and maps to elucidate EU spatial constructions, the intergovernmental relations between member states, and the redefinition of border regimes.

Statistical categories and cartography are part of the toolkit employed in the social sciences and historical research to observe and describe social realities. However, the intention behind the use of statistical and cartographical instruments in research is to avoid reproducing the power relationships embedded in space, territories, and borders by reflecting on and comparing statistics and maps. By interpreting statistics and maps, social scientists aim to study the processes triggered by the social and political constructions of space, territories, and borders as well as the domination, hierarchization, differentiation and de-differentiation, and, last but not least, (in-)equalities associated with these processes. When space, territories, and borders become devices for the sociological investigation of Europeanization, then such work focuses on the actors who act out and realize the intersections of space, territories, and borders-for instance, mobile workers, street-level bureaucrats in the authorities that deal with applications for political asylum, or experts who establish and elaborate categories for Eurostat or European guidelines [START_REF] Büttner | Mobilizing Regions, Mobilizing Europe: Expert Knowledge and Scientific Planning in European Regional Development[END_REF][START_REF] Hassenteufel | Still the Sound of Silence? Towards a New Phase in the Europeanisation of Welfare State Policies in France[END_REF][START_REF] Kazepov | Rescaling Social Policies towards Multilevel Governance in Europe: Some Reflections on Processes at Stake and Actors Involved[END_REF][START_REF] Lahusen | A European Field of Public Administration? Administrative Cooperation of Asylum Agencies in the Dublin system[END_REF]. Utilizing spatial categories for sociological observation and description enables researchers to relate processes on the macro-theoretical and on the microtheoretical level to one another. The sociological challenge is to integrate the multiple layers of interactions, mobilities, and regulations that inform social relations and power relationships within the EU into a coherent interpretive framework.

Describing the "politics of scales"

The scalar-structuration approach proposed by geographer and political scientist Neil Brenner, drawing on Fernand Braudel and Henri Lefebvre, offers a means of studying these multiple layers. This approach focuses on the question of how geographical scales create vertical orders and in doing so (re-)hierarchize market economic exchange between states or regional or urban actors in the context of capitalist globalization [START_REF] Brenner | The Limits to Scale? Methodological Reflections on Scalar Structuration[END_REF][START_REF] Brenner | The Urban Question and the Scale Question: Some Conceptual Clarifications[END_REF]. A Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73 "geographical scale" represents a site for economic and political action and, according to Brenner, should be distinguished from other social science spatial concepts such as space, place, locality, and territory. In building descriptions of the construction of the European legal space, of the institutionalization of the area without internal frontiers, or of the single market on the analysis of "scaling processes" and "politics of scales," researchers shed light on the "changing positionalities" in Europeanization (Brenner 2001: 603) and on the multiple "spatialities of Europeanization" [START_REF] Clark | The Spatialities of Europeanisation: Territory, Government and Power in 'EUrope[END_REF]. The scalar-structuration approach facilitates researchers' efforts to identify the opportunities that emerge (or fail to emerge) for economic and political actors when their activities interact with European border definitions or with the European intersections between transnational spaces, on the one hand, and respect for the "national identities" of "Member States" (art. 4 TEU), on the other. By taking changing positionalities seriously, such approaches bring into focus the uneven distribution of bargaining power of EU member states as well as the hierarchies and power relations that exist, for example between multinational companies and medium-sized or small companies or between metropolitan areas and border areas or areas with low population density. The sociological observation and description of rescaling processes makes it possible to highlight shifts in power that result, for example, from the re-or decentralization of financing structures in a member state or in the Euro area.

Taking up the actors' "jeux d'échelles"

The jeux d'échelles [games of scales] approach developed by French historians and anthropologists as a tool for observing and describing interconnections in social relations also creates a perspective for the study of multiple layers in the processes of Europeanization and their discontinuities (Revel 1996;Marung & Middell 2019). In contrast to Brenner's concept of scales, proponents of the jeux d'échelles perspective conceptualize scales as yardsticks for gauging action. These yardsticks are not only based on territorially defined scales, like administrative levels of public action, but also on scales defined by professional socialization, by income, by social or legal status, or by historical experience. Depending on the configuration of the social relations, the interactions involve a certain variety of scales that mirror the multilayered character of social relations. According to the jeux d'échelle approach, the significance that an EU institution, a single-market freedom, the law of a member state, trade unions, the action of an NGO, etc. assumes in the process to be described arises in each case from experiences on different territorial scales and from the actors' socialization in specific contexts of interaction. In this perspective, the historical, anthropological, or sociological observation of the interdependencies and the structures of interactions is based, as a methodological principal, on the intentional variation in the scale focused on, in order to consider the layered structure of the social. This methodological principle is of interest for research on Europeanization in three ways. First, processes on the micro-and the macro-theoretical level can be empirically and analytically conceptualized as interconnected. For example, on the basis of an intentional scale variation, it is possible to relate the construction of the European legal space to the practices of freedom of movement or to the organization of a company, to the work in a local authority, or to access to social benefits in actual practice. Second, an intentional scale variation makes it possible to observe how territorial levels-for instance, European guidelines or the European production of expert knowledge-affect various occupational groups in different ways and to a different extent.

Third, the methodological principle of scale variations is associated with a "mobile perspective" [START_REF] Amelina | The Shadows of Enlargement: Theorising Mobility and Inequality in a Changing Europe[END_REF]. This perspective enables researchers to reflect on the flexible ways in which spatial constructions, territorial levels, and border definitions are related to the "immobile;" that is, they can employ this approach to structure the character of political regimes and take into account the regulation of social interactions as something that is always situated in a specific place. Last but not least, the jeux d'échelles approach brings into focus the relations between mobile elites in the EU and population groups that are tied to "locales." The methodological principle of utilizing various focal points and scales induces researchers to reflect on their own position in the research process as sociologists of Europeanization and to integrate their position vis-à-vis the respective objects or subjects of their work into the analysis of over-lapping spaces, territories, and borders.

Concluding remarks

Both the intersections of EU spatial constructions and respect for the identity of member states and their territorial integrity (Article 4, Treaty on European Union), on the one hand, and the overlapping of new shifting border definitions and traditional member states' borders, on the other hand, create variations and differentiations in the process of Europeanization. As this chapter has elucidated, the multiple forms of Europeanization result from asynchronous processes of legal and economic integration and transnationalization of criteria and standards for public policies. At the same time, the multiple forms of Europeanization are an expression of contradictions in the process of dismantling and constructing borders. The EU asylum regime is one of the most Europeanized policy areas. The construction of European external borders at times resemble a frontier-for instance, within the European Neighborhood Policy-, at times a control site in the style of historical comptoir colonial [colonial trading posts], and at times a border fence. Together they constitute a series of "spatialities of Europeanization" inside and outside of the EU. Another contradiction emerges from the discontinuities between transnational harmonization in the domain of the four freedoms (free movement of goods, free movement of capital, freedom to provide and establish services, and the free movement of working persons) and interstate coordination in welfare state affairs. For example, whereas regulations pertaining to the internal market are "harmonized" across EU territorial borders, social security systems are differentiated and coordinated within the framework set by member states' borders and last but not least by national egoism. One of the consequences of these contradictions is the threat that Europeanization, as a political integration project, will increasingly be limited to a focus on market efficiency and performance.

Key learning points:

• Political space is mirrored in the territorialization of governance and the establishment of border regimes. Spatial constructions serve as a means of structuring social interaction and as such become an instrument of social control, domination, and power. Space, territories, and borders define membership and access to rights, determine inclusion and exclusion, and legitimate the sovereignty of rule.

• The EU is based on a specific spatial construction, in which nationally and supranationally guaranteed sovereignty as well as nationally and supranationally guaranteed economic and social orders and forms of solidarity are intertwined. The result is that, on the one hand, the member states of the EU share their sovereignty in certain policy areas, which have been steadily increasing in the course of political integration. On the other hand, the EU consists of the sum of the territories of the member states.

• A complex regime of different layers of regulations structures social interactions in the EU. This leads to multiple and uneven "spatialities of Europeanization." These "spatialities of Europeanization" result, among other things, in unequal opportunities for participating in the "area without internal frontiers" and in the "single market," as well as in unequal protection of the right to free movement through social policies. The

  since 2014, access to benefits for mobile EU citizens not recognized as workers or trainees depends more and more on discretionary decisions reached by individual member states. Nevertheless, precisely the tensions and conflicts in the area without internal frontiers highlight the potential of law to Europeanize political, economic, and social relations and to institutionalize European forms of conflict management. Both the founding of institutions with a material presence in a specific locale-such as the establishment of the Court of Justice of the European Union in Luxembourg (as part of the Treaty of the European Coal and Steel Community of 1952) and the Common Assembly that has become the European Parliament Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51-73

  . For example, the external trade balances not only inform us that, between 2003 and 2017 (a period in which several Eastern European countries joined the EU and which also includes the 2008-2009 economic and debt crisis), extra-EU exports increased by 117.9%, while extra-EU imports increased by 98.7%. Data also shows that, during this period, Germany was the largest EU exporter with €532 billion, far ahead of the United Kingdom (€ 204 billion), Italy (€ 199 billion), France (€ 195 billion), and the Netherlands (€ 143 billion) (Bundeszentrale für politische Bildung 2019).

Fig. 1 :

 1 Fig. 1: Regional at-risk-of-poverty (ARoP) rates, 2011 (TiPSE Project, 2014, https://mapfinder.espon.eu/mapfinder/?p=2520) copyright ESPON, origin of information and data ESPON EGTC. (The interpretation of ESPON material does not necessarily reflect the opinion of the ESPON 2020 Monitoring Committee)

Figure 2

 2 Figure 2 summarizes the differences in import and export among the EU member states in 2019.

  Preprint Version of the contribution to the textbook edited by Sebastian M.Büttner, Monika Eigmüller and Susann Worschech: Sociology of Europeanization, De Gruyter Oldenbourg 2022, p. 51- 73 8

interstate framework of EU spatial constructions, on the one hand, and the almost exclusively market-based understanding of territorial and social cohesion, on the other hand, prevent measures aimed at mitigating the "multiple uneven spatialities of Europeanization."

• Sociological observation and description in spatial categories focus on interactions and interdependencies between actors, their actions, and a specifically constituted or structured environment. Spatial approaches offer opportunities for researchers to relate processes on the macro-theoretical to those on the micro-theoretical level. When space, territories, and borders become devices used in sociological investigations of Europeanization, the focus is on the actors who act out and realize the intersections of space, territories, and borders, such as mobile workers, street-level bureaucrats who deal with asylum applications, or experts establishing and elaborating categories for Eurostat or European guidelines. The sociological challenge is to integrate the multiple layers of interactions, mobilities, and regulations informing social relations and power relationships in the EU into a coherent interpretive framework.

Central texts and recommended further reading:

•

Sources for current data:

Preprint Version of the contribution to the textbook edited by Sebastian M. Büttner

Glossary:

• Border: the main instrument for implementing a monopoly on defining legitimate modes of movement and for governing transnational interaction

• Geographic scale: a site for economic and political action (e.g. Brenner 2001)

• Jeux d'échelles: the interplay of yardsticks for gauging action (Revel 1996)

• Phantom geographies: the "traces left in contemporary societies by defunct territorialities" (Hirschhausen 2017: 107)

• Space: refers, in the sense of the French word espace, not only to distance and time but also to "place," "site," "area," or "region" [START_REF] Shields | Knowing Space[END_REF] • Territory: represents, in the 20th century, the central concept of European governments for framing economic relations, industrialization, infrastructure policies, and welfare protection (e.g. [START_REF] Maier | Consigning the Twentieth Century to History: Alternative Narratives for the Modern Era[END_REF]