domain: spi phys math info

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M Brunot

A Janot

A New Recursive Instrumental Variables Approach for Robot Identification

Keywords: closed-loop identification, robot identification, recursive instrumental variables

The work presented in this paper focus on robot identification and presents a method based on the use of instrumental variables (IV). When dealing with en-bloc and offline identification of robots, the instrumental matrix constructed with the inverse dynamic model (IDM) and simulated data obtained from the simulation of the direct dynamic model (DDM). In this paper, a new recursive IV approach relevant for robot identification is presented. The instrumental matrix is constructed with the IDM and the references and their derivatives which are previously filtered by the transfer function of the position closed loop. This new way of building the instrumental matrix avoids the simulation of the DDM and offers some perspectives for online identification and real-time implementation. This recursive IV method termed IDIM-RIV (Inverse Dynamic Identification Model Recursive Instrumental Variables) is experimentally validated on the two degrees-of-freedom SCARA robot. Finally, some hints for real-time implementation are provided.

INTRODUCTION

Dynamic models used in robotics are continuous-time models resulting from law of Newton and, like all mechanical systems, robots exhibit a double-integrator behavior. They must be identified while they are operating in closed loop [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF], [START_REF] Swevers | Dynamic model identification for industrial robots[END_REF]) and [START_REF] Gautier | A new closedloop output error method for parameter identification of robot dynamics[END_REF] among others. The standard identification method makes use of the inverse model (IDM) which is linear in relation to the dynamic parameters and the Least-Squares (LS) estimation. This standard method is termed IDIM-LS (Inverse Dynamic Identification Model Least Squares). Although good results can be obtained provided that an appropriate data filtering is used [START_REF] Swevers | Dynamic model identification for industrial robots[END_REF]) and [START_REF] Gautier | A new closedloop output error method for parameter identification of robot dynamics[END_REF], the approach based on an instrumental variables (IV) method proposed in (Janot et al. 2014 a), termed IDIM-IV, should be preferred. Indeed, it is known that LS estimates are biased when the system is identified in closed loop (Van den Hof 1998) and [START_REF] Gilson | Optimal instrumental variable method for closed-loop identification[END_REF]. However, both the IDIM-LS and IDIM-IV methods mentioned above are en-bloc techniques and their recursive variants were not discussed nor mentioned.

For robot identification, the Extended Kalman Filter (EKF) is one the most popular algorithms for recursive identification of the dynamic parameters, see e.g. [START_REF] Gautier | Extended Kalman Filtering and Weighted Least-squares Dynamic Identification of Robot[END_REF], [START_REF] Lightcap | An Extended Kalman Filter for Real-Time Estimation and Control of a Rigid-Link Flexible-Joint Manipulator[END_REF], [START_REF] Ulrich | Extended Kalman filtering for flexible joint space robot control[END_REF] and [START_REF] Masi | An Application of the Extended Kalman Filter for a Sensorless Stepper Motor Drive Working With Long Cables[END_REF]. However, as pinpointed out in [START_REF] Gautier | Extended Kalman Filtering and Weighted Least-squares Dynamic Identification of Robot[END_REF], the EKF does not outperform the standard approach while the friction parameters are not well identified. This may due to the fact that using the direct dynamic model (DDM) in order to estimate the dynamic parameters is not relevant because of the lack of sensitivity of the simulated joint positions, velocities and accelerations with respect to parameters' variations. This was emphasized in (Janot et al. 2014 b). In automatic control, recursive IV schemes have been deeply investigated and applied to linear systems i.e. systems being linear in relation to both the states and parameters [START_REF] Young | An instrumental variable method for real-time identification of a noisy process[END_REF][START_REF] Young | Parameter estimation for continuous-time models -a survey[END_REF], [START_REF] Söderström | Instrumental Variable Methods for System Identification[END_REF], [START_REF] Söderström | System Identification[END_REF] and [START_REF] Young | Recursive Estimation and Time-Series Analysis: An Introduction for the Student and Practitioner[END_REF]. Unfortunately, such recursive approaches are not popular in robotics. This may be explained by the fact that the systems considered in these references are linear which means that the researchers and engineers do not see a clear and straightforward way to apply them. In [START_REF] Boeren | Enhancing feedforward controller tuning via instrumental variables: with application to nanopositioning[END_REF], the authors present an interesting recursive IV approach that makes use of the IDM in order to identify the mechanical parameters of a nanopositioning system and they made some relationships with (Janot et al. 2014 a). However, the authors focused on a batching strategy i.e. the IV estimates are calculating over a whole trajectory and not at each time sample. Further, the system was still linear with restect to states and the parameters.

Motivated by the results presented in [START_REF] Boeren | Enhancing feedforward controller tuning via instrumental variables: with application to nanopositioning[END_REF], a recursive IV approach relevant for robot identification termed IDIM-RIV (Inverse Dynamic Identification Model Recursive Instrumental Variables) is presented in this paper. The instrumental matrix makes use of the IDM filled with the reference trajectories and their derivatives which are previously filtered by the transfer function of the position closed loop. Because simulated data are not used, this way of constructing the instrumental matrix avoids the simulation of the DDM and this offers interesting perspectives for online identification and real-time implementation. The experimental results obtained with a 2-DOF planar SCARA robot for online and online executions show the relevance of the IDIM-RIV method.

The paper is organized as follows: Section 2 reviews the background of robotics while Section 3 presents the IDIM-RIV method. The experimental results are given in Section 4 and Section 5 gives some concluding remarks.

BACKGROUND OF ROBOTICS

Inverse dynamic model of robots

The IDM of robots with n moving links calculates the ( )

1 n ×
joint torques vector τ as a function of generalized coordinates and their derivatives [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF] ( ) ( )

, = + τ M q q N q q ɺɺ ɺ , (1)
where q , q ɺ and q ɺɺ are respectively the ( )

1 n ×
vectors of generalized joint positions, velocities and accelerations;

( ) M q is the ( )

n n × inertia matrix; ( )
, N q q ɺ is the ( ) )

1 n × vector of centrifugal,
, , = τ IDM q q q β ɺ ɺɺ , (2) where ( )
, , IDM q q q ɺ ɺɺ is the ( )

n b
× matrix of basis functions of bodies dynamics; and b is the number of base parameters. Relation (2) represents the Inverse Dynamic Identification Model (IDIM). It is recalled that the base parameters are the minimum number of the dynamic parameters from which the IDM can be calculated. They are obtained from standard dynamic parameters by regrouping some of them with linear relations (Gautier & Khalil 1990). The standard parameters of a link j are:

Direct dynamic model of robots

The DDM of a robot expresses the joint accelerations as function of the joint torques, positions and velocities [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]. From Newton's law, we have ( ) ( ) , = -M q q τ N q q ɺɺ ɺ .

(3) Relation (3) shows that the DDM is nonlinear in relation to the states and the dynamic parameters. This is the reason why the DDM is rarely used in robot identification (Gautier et al. 2013).

Control and data acquisition

In robotics, Proportional-Derivative (PD) and Proportional-Integral-Derivative (PID) controls are often used to identify β . The joint j torque, j τ , can be thus written as ( ) ( )

j j j j j j r j g g C s q q τ τ τ ν = = - , (4)
where ( ) j C s is the transfer function of the joint j controller; j r q is the joint j position reference; j q is the joint j position; j g τ is the joint j drive gain; ν the joint j control signal; and s is the Laplace's variable. The data available from the controller are q and the ( ) [START_REF] Gautier | A new closedloop output error method for parameter identification of robot dynamics[END_REF].

1 n × vector of the control signals ν ,

The IDIM-LS identification method

The IDIM given by ( 2) is sampled at a measurement frequency deonted m f while robot is tracking the reference trajectories ( ) , , r r r q q q ɺ ɺɺ . In (2), q is estimated with q obtained by filtering the measurements of q through a lowpass Butterworth filter in both the forward and reverse directions. ( ) ˆ, q q ɺ ɺɺ are then calculated with a central differentiation algorithm of q . Finally, τ being perturbed by high-frequency disturbances, a parallel decimation procedure is applied to eliminate torque ripples. All the details are given in [START_REF] Gautier | A new closedloop output error method for parameter identification of robot dynamics[END_REF]. After the data processing described previously, the following over-determined linear system is obtained

( ) ( ) ˆ, , = + y τ X q q q β ε ɺ ɺɺ , (5)
where ( ) y τ is the ( )

1 r × measurements vector built from actual torques τ ; ( ) ˆ, , X q q q ɺ ɺɺ is the ( ) r b × observation matrix built from ( )
ˆ, , IDM q q q ɺ ɺɺ ; ε is the ( )

1 r × vector of error terms; e r n n
= ⋅ is the number of rows in (5); and e n is the number of samples of each joint j . The IDIM-LS estimates are given by ( )

1 ˆT T LS - = β X X X y . ( 6
)
The statistics of the IDIM-LS method is detailed in (Janot et al. 2014 a).

The IDIM-IV method

The IDIM-LS method can be improved by adopting an IV approach that requires the construction of an instrumental matrix denoted Z . To construct Z , the DDM given by ( 3) is simulated with the previous IV estimates denoted as 1 ˆit IV β and assuming the same references and the same control law structure for both the actual and the simulated robots. S q ɺɺ , the vector of the simulated joint accelerations, is given by

( ) ( ) 1 1 ˆ, , , it it S IV S S S S IV - - = - M q β q τ N q q β ɺɺ ɺ where , S S
q q ɺ are respectively the ( )

1 n×
vectors of the simulated joint positions and velocities calculated by numerical integration of the DDM while S τ is the ( )

1 n× vector of simulated torques with j S
τ , the j th element of S τ , is given by ( ) ( )

j j j j S j r S g C s q q τ τ = - . The instrumental matrix Z is given by ( ) 1 , , , it S S S IV - = Z X q q q β ɺ ɺɺ , (7)
where ( )

1 , , , it S S S IV - X q q q β ɺ ɺɺ is the ( ) r b × sampled matrix of ( ) 1 , , , it S S S IV
-IDM q q q β ɺ ɺɺ . At iteration it , the IV estimates are given by ( )

1 ˆit T T IV - = β Z X Z y . ( 8
)
This iterative process is run until convergence. The statistics of the IDIM-IV method is detailed in (Janot et al. 2014 a).

A RECURSIVE INSTRUMENTAL VARIABLES APPROACH FOR ROBOT IDENTIFICATION

In this section, a Recursive IV approach, termed IDIM-RIV, relevant for robot identification is presented. This section is the theoretical contribution of this paper.

Choice of the set of instruments

To be implementable online, the IDIM-RIV must avoid the simulation of the DDM. Otherwise, an IV variant based on a batching strategy must be preferred and adopted, as in (Boren et al. 2017).

To do so, let us first consider a joint j control illustrated in

( ) ( ) , , n j j i i j i j p M q N q q ≠ = - -  q ɺɺ ɺ where ( ) , j i
M q is approximated by a constant inertia denoted j J given by ( ) ( )

, max j j j j i j j q J ZZ Ia M ZZ Ia = + + - - q
. Let us now consider the closed-loop relations that are given by ( ) ( ) ( ) From relation (6), j nf q , the noise-free part of j q , is given by ( ) ( )

j j j j j j r p j q q H s q D s p D s ε ε = + + , ( 9
j j j nf j r p j q H s q D s p = + , ( 10
)
and it follows that j nf q given by ( 10) must be utilized to construct Z . However, because j p still depends on the base parameter, so does j nf q . It follows that choosing (10) to construct Z still requires the simulation of the DDM or at least the calculation of j p .

Let j n ω be the bandwidth of the joint j position closed loop.

In robotics, the gains within ( ) j C s are tuned such that the influence of j p proves to be negligible in order to obtain a good tracking [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]. Hence, below j n ω , the following approximation holds ( )

j j j f nf j r r q H s q q ≈ = , (11)
leading to construct Z with (7) where s q and its derivatives is replaced with f r q , the ( ) 1 n× vector of filtered references and its derivatives. It yields that for the IDIM-RIV method, the instrumental matrix is then given by ( )

, ,

f f f r r r = Z X q q q ɺ ɺɺ . ( 12
)
Loosely speaking, the instrumental matrix constructed with ( 12) is the IDM filled with the sampled vector of reference trajectories filtered by the position closed loop. Like the IDIM-IV method, the IDIM-RIV approach still makes use of the IDM. However, unlike the IDIM-IV method, the IDIM-RIV technique does not require the simulation of the DDM. Indeed, the controller ( ) j C s as well as the value of j J being known, the reference trajectories being designed and imposed by the user, and the basis functions of the IDM being literal expressions that can be explicitly calculated with the SYMORO+ software, it is therefore no longer necessary to simulate the DDM to construct Z . It results that choosing j f r q defined by (11) instead of j S q allows the user to construct Z before running experiments which means that Z is constructed once and for all. As we shall see later, this result is particularly promising for real-time implementation of the IDIM-RIV method. Interestingly, in (Boren et al. 2017), the same kind of instrumental matrix is obtained in a somewhat different context.

Scheme of the IDIM-RIV algorithm

The IDIM-RIV method is based on a standard recursive IV algorithm described by the following equations (see e. -update of the gain at time k ( )

1 1 1 1 1 1 T T k k k n k k k - - - - - - = + k P z I x P z , (13)
-update of the covariance matrix at time k

1 1 1 k k k k k - - - = - P P k x P , (14)
-prediction of the parameters at time k ( )

1 1 1 1 ˆˆk k k k k k - - - - = + - β β k y x β . ( 15
)
β is initialized with the Computer Aided Design values (CAD values) which are usually given by the manufacturers.

It is also possible to get an initial value of β by running the IDIM-IV method which is not sensitive the initialization of β . This way of doing is quite usual, see e.g. [START_REF] Young | Recursive Estimation and Time-Series Analysis: An Introduction for the Student and Practitioner[END_REF]) p. 173.

Some hints for real-time implementation for embedded systems

As indicated in the previous sections, the IDIM-RIV method does not require the simulation of the DDM resulting in an easy-to-implement recursive IV approach suitable for embedded systems such as electric motors, mobile robots UAV's... Indeed, most of today microcontrollers are equipped of large-size volatile and non-volatile memory blocks as well as Floating Point Unit (FPU). It is recalled that volatile memory requires power to maintain the stored information (typically Random Access Memory -RAM) whereas non-volatile memory does not require power to maintain the information (typically Read Only Memory -ROM). Since the user has all the information needed to construct Z in advance, it is thus possible to store Z in a non-volatile memory and once the system powered on, to relocate it into a block of RAM in order to get an effective computing time. This relocation is needed because there are some wait states (or delays) when fetching the information stored in the non-volatile memory. This produces latencies while the code is running. Naturally, microcontroller must be chosen accordingly to the size of the matrices and vectors that are involved in the recursive IV algorithm; the larger the sizes of those matrices and vectors are, the more powerful microcontroller must be.

In addition, instructions including matrix operations (addition, multiplication, inversion…) can be dealt with optimized libraries that are provided by the manufacturers. However, when the number of parameters is limited (typically less than 5 which is the number of base parameters of a 1-dof robot), the instructions can be expanded in order to obtain classical algebraic expressions. This way of doing is quite usual in real-time programming and limits the number of jumps that produce some delays. It should be stressed that nonlinear functions (e.g. sine, cosine, square and sign that are often present in the basis functions of the IDM) are also treated with appropriate and optimized libraries delivered by the manufactures. It is no longer required to approximate those nonlinear functions by a Taylor series.

As an indication, let us consider the TMS320F28069 manufactured and sold by Texas Instruments which has a clock rate of 90 MHz and is equipped of a FPU. The equations ( 13), ( 14) and ( 15) were implemented and executed in order to simulate the behavior of the EMPS prototype which has the following features 4 b = and 1 n = . It takes only 80.10 -6 s to execute the RIV algorithm.

EXPERIMENTAL VALIDATION

This section presents the experimental validations obtained with the SCARA robot. This section contents the experimental contribution of this paper.

Presentation of the SCARA robot

The SCARA robot presented Fig. 2 is a 2-dof planar robot. The SCARA robot is controlled with a simple PD controller whose the desired natural frequency The data measurement frequency is m f = 200 Hz. The torque data are calculated with (4) while the positions are obtained through incremental encoders (2000 and 5000 (lines/rev) for joint 1 and 2, respectively) with a 4-fold subdivision of each encoder line (8000 and 20000 (pulses/rev) for joint 1 and 2 respectively).

The reference trajectories, ( ) , , r r r q q q ɺ ɺɺ , are fifth order polynomials and their duration is 12s. Since one obtains ( ) ( ) ˆ, , 25 cond = X q q q ɺ ɺɺ , the parameters are well excited [START_REF] Gautier | Exciting trajectories for the identification of the inertial parameters of robots[END_REF].

Offline experimental results

The IDIM-RIV approach is evaluated offline by comparing its results with those obtained with the en-bloc IDIM-IV method. The two methods are carried out with the filtered positions, q ˆ, calculated with a 10 Hz forward and reverse Butterworth filter and with the velocities, q ɺ , the accelerations, q ˆɺ ɺ , calculated with a central differentiation algorithm of q ˆ. The maximum bandwidth for the second joint being 10 ) for the IDIM-IV method (resp. IDIM-RIV method). The IDIM-IV estimates and the average values of the IDIM-RIV estimates calculated over the 10 last seconds i.e. when the variations of the IDIM-RIV estimates prove to be negligible are gathered in Table 1. Likewise their relative deviations and the relative errors, they are averaged over the ten last seconds. The IDIM-RIV estimates and their 95% interval bounds are plotted in Fig. 3 while in Fig. 4 are plotted the actual joint torque and the torque predicted with the IDIM-RIV estimates (direct comparisons). The IDIM-RIV estimates being comparable with the IDIM-IV estimates, the relative error being less than 5%, the IDIM-RIV estimates converging rapidly to their steady-state values and the reconstructed torques matching the measured ones, it can be concluded that the IDIM-RIV method provides satisfactory results. The IDIM-RIV is now evaluated online. Because forward and reverse filters cannot be used, the data captured at time k i.e.

the ( ) 2 1 × vector of positions, q , and the ( ) 2 1 × vector of control signal, ν , are filtered with a stable low-pass filter with an unitary gain and a bandwidth of =8 fq ω Hz. ( ) ˆ, q q ɺ ɺɺ are then calculated with a backward differentiation algorithm of q i.e.

( )

1 ˆˆk k k dt - = - q q q ɺ
where dt is the sampling time with 1/ m dt f = . We still have m f = 200 Hz. The IDIM-RIV estimates are initialized with the CAD values while 6 0 8 10 = P I . Finally, the IDIM-RIV algorithm is executed via a Dspace card.

The results obtained with the online IDIM-RIV method are very close to those exposed in Section 4.2.: the IDIM-RIV estimates are comparable with the IDIM-IV estimates, the relative error are less than 5%, the IDIM-RIV estimates converge rapidly to their steady-state values and the reconstructed torques match the measured ones. They are not re-shown here for sake of clarity. The IDIM-RIV method executed online provides satisfactory results which means that the IDIM-RIV method presented in Section 3 can be applied for online identification of robots.

CONCLUSION

In this paper, a recursive instrumental variables approach suitable for identification of rigid industrial robots, termed IDIM-RIV, has been presented and validated offline and online with a 2 degrees-of-freedom SCARA robot. The instrumental matrix makes use of the inverse dynamic model and is constructed with the references and their derivatives filtered by the transfer function of the position closed loop. In so doing, the instrumental matrix is constructed once for all since the simulation of an auxiliary model is no longer necessary. This makes the IDIM-RIV method suitable for embedded systems. The experimental results showed that the IDIM-RIV provides estimates that are comparable with those obtained with the en bloc IDIM-IV method and has a rapid convergence. Future works concern the use of the IDIM-RIV method to identify a 6-DOF industrial robot as well as its implantation and real-time programming for embedded systems. Finally, statistics will be addressed in order to obtain estimates with variances as small as possible.

  Coriolis, gravitational and friction torques. The modified Denavit and Hartenberg (DHM) notation allows the user to obtain an IDM which is linear in relation to the ( ) 1 b × vector of base parameters β , i.e.

(

Fc

  the viscous and Coulomb friction parameters of joint j .

Fig. 1 .

 1 Fig. 1. In robotics, it is convenient to consider the nonlinear plant of robots as decoupled linear models. The joint j decoupled linear model denoted ( ) j P s is a double-integrator

Fig. 1 .

 1 Fig. 1. Actual joint j

  g.[START_REF] Söderström | Instrumental Variable Methods for System Identification[END_REF] or[START_REF] Young | Recursive Estimation and Time-Series Analysis: An Introduction for the Student and Practitioner[END_REF] for the details) where kx (resp. k z ) is the kth ( ) n b × line of X (resp. Z ), k k is the ( )

  driving capacity without saturation of the joint drive. For this robot, we obtain a full bandwidth with 1 . It is worth noting that several controls including PID control and feedforward velocity/acceleration which give better tracking accuracy were tried and similar results have been obtained. This shows that the IDIM-RIV method is not sensitive to the control structure.

Fig

  Fig. 2. SCARA robot

  is important to note that we can use a forward and reverse filter because the methods are evaluated offline; such filters cannot be used for online schemes. Finally, we choose a 10 Hz cutoff frequency. The parallel decimation is carried out with a lowpass Tchebyshef filter with a cutoff frequency 2 we choose a 4 Hz cutoff frequency. The sample rate m f is divided by a factor d n =20. The IDIM-IV method starts with the regular initialization. The IDIM-RIV estimates are initialized with the CAD values while diagonal element of the covariance matrix of the IDIM-IV or IDIM-RIV estimates. Finally, the relative errors denoted ε y are computed with ˆIDIM IV

Fig. 3 .

 3 Fig. 3. Convergence of the parameters ZZ1, FV1, FC1, ZZ2, MX2, MY2, and FC2 -95% interval confidence

Table 1

 1

	. IDIM-IV estimates and IDIM-RIV estimates averaged over
		the 10 last seconds
	Parameters	IDM-IV	σ	j % β	IDM-RIV	σ	j % β
	ZZ1	3.446	1.2%	3.445	1.3%
	FV1	0.010	59.5%	0.005	231.0%
	FC1	0.833	2.5%	0.815	5.3%
	ZZ2	0.063	2.1%	0.062	2.2%
	MX2	0.249	1.2%	0.249	1.4%
	MY2	0.010	452.3%	0.015	942.1%
	FV2	0.021	6.3%	0.020	9.5%
	FC2	0.129	5.1%	0.135	6.2%
	ε y	3.9%			4.8%


Joseph Vidal-Rosset

NO VICIOUS CIRCULARITY WITHOUT CLASSICAL LOGIC

This paper proves that the paradoxes called "vicious circles" like Russell's, require classical logic, since the double negation elimination plays a key role in their derivation.

It seems that, at least confusedly, [START_REF] Quine | Philosophy of Logic[END_REF] recognized the existence of a relationship between the paradoxes of set theory and the use of classical logic when he wrote:

Moreover, the paradoxes of set theory put an added premium on constructivism; for what we accomplish within its restraints is pretty clearly immune to the threat of contradiction that lingers outside. (Quine, 1986, p. 88) The rejection of the excluded middle being an expression of constructivism, if the adoption of constructivism excludes the paradoxes of set theory, it is of course possible to suppose that this principle of classical logic can be one of the culprits of these paradoxes. But, of course, this suspicion does not explain the role of classical logic in the expression of these paradoxes.

To unravel this question, let us recall, first of all, that Russell's paradox starts from the definition of a set whose all elements are sets that do not belong themselves, a definition that can be translated in turn by this first order formula:

∃y∀x(x / ∈ x ↔ x ∈ y)
(1) and, by instantiation, an absurdity follows:

∀x(x / ∈ x ↔ x ∈ a) (2) a / ∈ a ↔ a ∈ a (3)
The trouble is that it is provable in intuitionistic logic that (1) implies a contradiction, and it probably explains why Tennant claimed that "excluded middle plays no essential role in the derivation of the paradoxes" (Tennant, 2017, p. 285) in disagreement with [START_REF] Field | Saving Truth From Paradox[END_REF] and [START_REF] Priest | The Logical Paradoxes and the Law of Excluded Middle[END_REF] on this question. The analysis made by Tennant is all the more disturbing because, in spite of his philosophical commitment in favor of intuitionism, he disagrees with Brouwer by exempting the law of excluded middle of any responsibility in the derivation of paradoxes of set theory. Therefore, Tennant's analysis constrains to clarify this question: to what extent are the logical laws specific to classical logic involved in the derivation of these paradoxes? (1) The distinction between use and mention could be useful here. The intuitionist logician who seeks to understand Russell's paradox translates the excluded middle as meaning that, for any set, it is decidable that it is not an element of itself or that it is an element of itself, to point out that the excluded middle does not provide any method of construction to build the set of all sets that are not elements of themselves. On the other hand, for classical logicians like Russell and many others, this definition is robust, because it is based on the law of the excluded third, hence the fact that Russell mentioned this law twice, explaining the form of this paradox. :

If x be a predicate, x may or may not be predicable of itself. Let us assume that "not-predicable of oneself " is a predicate. Then to suppose either that this predicate is, or that it is not, predicable of itself, is self-contradictory. The conclusion, in this case, seems obvious: "not-predicable of oneself " is not a predicate. (Russell, 1903, p. 102) That is this last expression of Russell's paradox that has been translated by the famous Grelling-Nelson paradox published by [START_REF] Grelling | Bemerkungen zu den Paradoxien von Russell und Burali-Forti[END_REF].

Here is this paradox. A word is "autological" if and only if it describes itself. For example "short" is a short word, "short" is short. A word that is not autological is heterological. Now, let us see how the law of excluded middle and the rule of double negation elimination are used in the derivation of Grelling-Nelson paradox. Definition 0.1 (Rule aut ). -For any term t, the assumption that t is autological entails the following disquotation: "t" is t. Definition 0.2 (Rule het ). -For any term t, the assumption that t is heterological entails the following negation and disquotation: "t" is not t.

(1) The famous distinction made by Ramsey (Ramsey, 1990, pp. 183-184) between logical paradoxes and epistemological paradoxes is not taken into account in this paper, where I wonder only on the logic that is necessary to the derivation of paradoxes described by [START_REF] Russell | The Principles of Mathematics[END_REF].

Proposition 0.3. -Any term that is not heterological is autological (from the definiens of "heterological" and the double negation elimination). Proof. -A consequence of proposition 0.4 is that the term "heterological" is either heterological or not heterological. This instance of the excluded middle offers a choice of two incompatible assumptions. The trouble is that it is provable that each assumption entails its negation, hence the paradox.

-Assumption 1: "heterological" is heterological. Therefore, via rule het "heterological" is not heterological (4)

But ( 4) is the negation of Assumption 1.

-Assumption 2: "heterological" is not heterological. Therefore "heterological" is autological ( 5) by proposition 0.3 (double negation elimination). Therefore "heterological" is heterological (6) via rule aut. But ( 6) is the negation Assumption 2.

Kleene (Kleene, 1952, p. 37) and Schwartz (Schwartz, 1991, p. 28) pointed out that Russell's paradox lies in the fact that each member of the alternation (a / ∈ a) ∨ (a ∈ a) is proved, while clearly these two formulas contradict each other. Grelling-Nelson paradox has the same structure. Is classical logic necessary to express exactly this form of paradox? Pace Tennant, it is provable that classical logic is indeed necessary to translate the famous "vicious circle" on which [START_REF] Russell | The Principles of Mathematics[END_REF] insisted. Proposition 0.6. -No vicious circularity as such can be proved without help of classical logic, that is without the rule of double negation elimination.

Proof. -The vicious circularity of paradoxes related to Russell's can be described as a complete reversal of an instance of excluded middle:

A ∨ ¬A (7) becomes ¬A ∧ A (8)
because from assumption A, ¬A is proved, and, from assumption ¬A, A is proved in turn. The biconditional

A ↔ ¬A (9)
is exactly the form of the conclusion of Russell's paradox as well as Grelling

A ↔ ¬A ⊥ (11)
is provable in minimal logic, (2) but there is a crucial difference between assuming a formula and deriving it (von Plato, 2013, p. 83), and it is unquestionable that the derivation of ( 9) is not possible without the double negation elimination rule of classical logic.

It must therefore be admitted that the disturbing beauty of vicious circles can not be unveiled without the rule of double negation elimination. Quine might have concluded that this was ultimately one of the signs of the superiority of classical logic over intuitionistic logic; nevertheless, he wrote: Intuitionism antedates Godel's proof that there can be no complete proof procedure for number theory. This great result of Godel's adds force, however, to the intuitionist protest. The excess of admitted questions over possible answers seems especially regrettable when the questions are mathematical and the answers mathematically impossible. (Quine, 1986, p. 87) But Quine never considered that this "excess of admitted questions over possible answers" was caused by the classical interpretation of the excluded middle. [START_REF] David | Introduction à la logique: théorie de la démonstration[END_REF] (2) To get the proof of (11) in normal form see von [START_REF] Von Plato | A Problem of Normal Form in Natural Deduction[END_REF] and (Tennant, 2017, pp. 289-293).

  Proposition 0.4. -Any term is either heterological or not heterological (excluded middle). Proposition 0.5. -With Definitions 0.1 and 0.2, Propositions 0.3 and 0.4 entail a contradiction.

  But if the first member of this conjunction is reducible to ¬A in minimal logic, the second member is not reducible to A without the use of classical logic. Indeed, if A is deducible from ¬A, it is because ¬¬A is provable, but A is deducible from ¬¬A only in classical logic, because of the rule of double negation elimination. Therefore, no vicious circularity as such can be proved without help of classical logic.

		-
	Nelson paradox: (9) is a deducible vicious circle, that is a conjunction of two
	conditional formulas
	(A → ¬A) ∧ (¬A → A)	(10)
	that are both provable.

Remark 0.7. -It is clear that classical logic is not required to deduce ⊥ from (

9

) as open assumption, because the sequent


Maud Frot
email: maud.frot@univ-lyon1.fr

Michel Magnin

François Mauguière

Luis Garcia-Larrea

HUMAN SII AND POSTERIOR INSULA DIFFERENTLY ENCODE LASER STIMULI

Keywords: CO2 laser, Evoked potentials, Posterior Insular cortex, Intra-cerebral recordings, SII area, Thermal and painful stimuli

The SII area and the posterior insular region are activated jointly by thermal stimuli in functional imaging studies. However, controversy remains as to a possible differential encoding of intensity between these two contiguous areas. Using CO2 laser stimulations, we analysed the modifications induced by increasing thermal energy on evoked potentials recorded in epileptic patients, with electrodes implanted within SII and posterior insular cortices. Although increasing stimulus intensities enhanced both SII and insular responses, the dynamics of their respective amplitude changes were different. SII responses were able to encode gradually the intensity of stimuli from sensory threshold to barely painful levels, but tended to show a ceiling effect for increasing pain intensities. In contrast, the posterior insular cortex failed to detect non-noxious laser pulses but encoded faithfully stimulus intensity variations at painful levels, without showing saturation effects for intensities above painful threshold. According to these results, one can assume that insular cortex could be more involved in the triggering of affective recognition of, and motor reaction to noxious stimuli, whereas SII would have a more dedicated role in finer-grain discrimination of stimulus intensity, from non-painful to painful levels.

Introduction

According to numerous electrophysiological and functional imaging studies conducted in humans in the recent years, pain perception is subserved by the coordinated activity of multiple cortical regions. Although the activated areas are not strictly the same throughout studies, and their relative importance may be dependent upon the specific stimulus parameters and the experimental conditions used, it appears clearly that the cortical regions located in the upper bank of the lateral sulcus, including the second somatosensory area (SII), and the insular cortex, are the sites most consistently activated by all kind of nociceptive stimuli (reviews in [START_REF] Peyron | Functional imaging of brain responses to pain. A review and meta-analysi[END_REF][START_REF] Schnitzler | Neurophysiology and functional neuroanatomy of pain perception[END_REF], Treede et al 2000[START_REF] Garcia-Larrea | Brain generators of laser-evoked potentials : from dipoles to functional significance[END_REF][START_REF] Vogel | Dipole source analysis of laser-evoked subdural potentials recorded from parasylvian cortex in humans[END_REF].

Neurophysiological studies in monkeys have demonstrated the existence of nociceptive neurons in both SII and the insula (Robinson and Burton 1980a,b, Dong et al. 1989[START_REF] Gracely | Studies of pain in normal man[END_REF], Dostrovsky and Craig 1996[START_REF] Zhang | Monkey insular cortex neurons respond to baroreceptive and somatosensory convergent inputs[END_REF], and nociceptive regions of thalamus in non-human primates send axons to the parietal operculum, the mid-and posterior insular cortex as well as the retroinsula [START_REF] Burton | The posterior thalamic region andits cortical projection in new world and old world monkeys[END_REF], Jones and Burton 1976[START_REF] Mufson | Thalamic connections of the insula in the rhesus monkey and comments on the paralimbic connectivity of the medial pulvinar nucleus[END_REF][START_REF] Mesulam | The insula of Reil in man and monkey[END_REF][START_REF] Burton | Second somatic sensory cortical area (SII) in a prosimian primate, Galago crassicaudatus[END_REF], Friedman and Murray 1986[START_REF] Stevens | Spinothalamocortical projections to the second somatosensory cortex (SII) in squirrel monkey[END_REF], Craig 1995).

In humans, regions containing nociresponsive neurons in and around the thalamic ventral caudal nucleus [START_REF] Lenz | Neurons in the area of human thalamic nucleus ventralis caudalis respond to painful heat stimuli[END_REF][START_REF] Lenz | A population of cells in the human thalamic principal sensory nucleus respond to painful mechanical stimuli[END_REF] have been shown to project to the insular cortex and the parietal operculum [START_REF] Van Buren | Variation and connections of the Human thalamus[END_REF]. These supra-sylvian nociceptive-related areas appear also implicated in the processing of innocuous somatosensory information. In functional imaging studies, there is a substantial overlap in activity evoked by noxious and innocuous stimuli within SII [START_REF] Coghill | Distributed processing of pain and vibration by the human brain[END_REF][START_REF] Chen | Differentiating noxious and innocuous related activation of human somatosensory cortices using temporal analysis of fMRI[END_REF], and our previous data recorded with intra-cerebral electrodes in humans showed that both noxious CO 2 laser and innocuous electrical stimulations evoke responses within the same sub-regions of SII [START_REF] Frot | Responses of the supra-sylvian (SII) cortex in humans to painful and innocuous stimuli. A study using intra-cerebral recordings[END_REF]. One recent study has provided fMRI evidence suggesting that a posterior region within SII could be specifically involved in the processing of noxious stimuli [START_REF] Ferretti | Functional topography of the secondary somatosensory cortex for nonpainful and painful stimulation of median and tibial nerve: an fMRI study[END_REF].

However, these authors used electrical stimuli that activated simultaneously noxious (A-delta) and non-noxious (A-Beta) afferents [START_REF] Gracely | Studies of pain in normal man[END_REF]; therefore, no definite conclusion could be drawn as to whether these separate areas also participate to the encoding of inputs coming from specific activation of nociceptors.

A major obstacle for the understanding of this region's role in pain perception is our limited knowledge on response properties of perisylvian nociceptive neurons [START_REF] Treede | Cortical representation of pain: functional characterization of nociceptive areas near the lateral sulcus[END_REF]. Previous studies, including ours, usually compared stimulations of different modalities (mostly electrical or tactile vs noxious heat), and therefore could not assess specifically whether the SII-insular cortex is able to encode intensity within the thermo-algesic modality. The possibility remains that, in these three patients, the supra-sylvian opercular and insular cortices could have shown some

Materials and Methods

Patients

Electrode implantation

Intracerebral electrodes were implanted using the Talairach's stereotactic frame. As a first step, a cerebral angiography was performed in stereotactic conditions using an X-ray source located 4.85 meters away from the patient's head, thus eliminating the linear enlargement due to X-ray divergence, so that the films could be used for measurements without any correction. In a second step, the relevant targets were identified on the patient's MRI, previously enlarged at scale one-to-one. As (AC-PC) plane (see also Frot and Mauguière 1999, 2003, Frot et al. 1999, 2001) .

In the SII region, electrodes were implanted

Stimulation procedure, recording and signal averaging

The LEP recordings were performed between 10 and 15 days after electrodes implantation.

During the recordings, the patients laid relaxed

Amplitude measurements

Given the high signal/noise ratio obtained in intracortical recordings, the LEP amplitudes at insular and opercular sites could be measured on individual single sweeps, without the need of averaging. A total of 112 single responses were analysed for each intensity level.

Statistical analysis

Amplitudes and latencies of responses were submitted to repeated measures analysis of variance (ANOVA, Statistica 6®), with 3 within-

Results

Psychophysical responses

By definition (see Experimental Procedures), all the subjects rated 0 (no sensation) on the visual analogue scale (VAS) when the intensity was under the perception threshold (I0).

Subjective intensity rates to I2 corresponded well to a barely painful sensation (boiling water drop on the skin), while I3 (maximal intensity)

was unanimously considered as painful and quite unpleasant, albeit tolerable (mean 5.4 /10). This latter stimulus intensity being at the upper limit of tolerance, higher intensities were not used for evident ethical reasons. There

Polarity, latency and voltage of SII-insular LEPs

Two distinct evoked potential components contralateral to the stimulation site were recorded along all the electrode tracks implanted in SII, anterior and posterior to the rolandic fissure. They consisted of a negative wave (Nop, for "negative-opercular") followed

Stereotactic localization of the SII-insular LEPs

The maximal amplitude of the N/P deflection was taken to determine the electrode contact likely to be the closest to the source. 2 and Figure 4). The contacts recording these responses with maximal amplitude were distributed between 29 and 38.25 mm from the median line (x coordinates) (Table 2 and Figure 4).

Statistical analysis

Effect of electrode localization on SII-insular LEPs

In the patients whose SII or posterior insular cortices were explored by several electrodes

Effect of stimulus intensity -on source localisation

For each patient, the electrode contacts yielding maximal SII or insular responses were the same for all intensities. Therefore, the sources location of these responses did not appear to be modified by the intensity changes. However, due to our restricted spatial sampling, especially along the antero-posterior (y) and vertical (z) axes, we cannot draw any definitive conclusion on this point.

-on SII and insular LEPs latencies and amplitudes

Repeated-measures ANOVA showed no significant effect of intensity on the latencies of insular or SII LEPs (Table 3). There was no effect of epoch order on response latencies either, i.e., for all intensity conditions both insular and SII response latencies remained stable between consecutive epochs (Table 3).

Repeated-measures ANOVA showed a significant effect of stimulus intensity on SII and insular LEP amplitudes (Table 3). For a given stimulation intensity, there was no effect of the epoch order on response amplitudes, i.e., both insular and SII response amplitudes remained stable between consecutive epochs (Table 3). SII and insular responses showed a highly significant increase of their amplitude between the two extreme intensities (I0 and I3;

t-tests, p<0.001) (figure 5).

-on the dynamics of SII and insular

LEP amplitudes change

Although recordings. This point deserves some considerations since, contrary to our study and Bornhovd's data, several previous studies described activations within insular cortex for non noxious stimuli [START_REF] Coghill | Distributed processing of pain and vibration by the human brain[END_REF][START_REF] Craig | Functional imaging of an illusion of pain[END_REF][START_REF] Derbyshire | Exploring the pain "neuromatrix[END_REF][START_REF] Davis | Functional MRI study of thalamic and cortical activations evoked by cutaneous heat, cold and tactile stimuli[END_REF][START_REF] Becerra | Human brain activation under controlled thermal stimulation and habituation to noxious heat: an fMRI study[END_REF][START_REF] Maihöfner | Temporo-spatial analysis of cortical activation by phasic innocuous and noxious cold stimuli -a magnetoencephalographic study[END_REF]. The processing. In monkeys, SII cortex receives its major thalamic input from the ventroposterior inferior thalamic nucleus (VPI) (Stevens et al. 1993, Friedman andMurray 1986) whereas the posterior insula (granular and dysgranular parts of insular cortex) is afferented by a variety of thalamic nuclei including, in addition to the VPI, the suprageniculate-limitans complex, the basal ventromedial, the medial pulvinar and the posterior nuclei [START_REF] Burton | The posterior thalamic region andits cortical projection in new world and old world monkeys[END_REF][START_REF] Mufson | Thalamic connections of the insula in the rhesus monkey and comments on the paralimbic connectivity of the medial pulvinar nucleus[END_REF], Friedman and Murray 1986). Neuronal units in monkey's VPI are known to be essentially somatic, and contain a majority of nonnociceptive and wide-dynamic-range (WDR)

Discussion

neurons that could encode gradually the stimulus intensity from non-painful to painful ranges [START_REF] Apkarian | Properties of nociceptive neurons in the lateral thalamus of the squirrel monkey[END_REF][START_REF] Apkarian | Squirrel monkey lateral thalamus. I. Somatic nociresponsive neurons and their relation to spinothalamic terminals[END_REF]. Conversely, thalamic nuclei sending projections on insular cortex contain much more polymodal neuronal units that respond to a variety of stimuli including not only somatosensory, but also auditory and visual stimulations [START_REF] Berkley | Response properties of cells in the ventrobasal and posterior group nuclei of the cat[END_REF][START_REF] Hicks | Organization and properties of visually responsive neurones in the suprageniculate nucleus of the cat[END_REF][START_REF] Benedek | Visual, somatosensory, auditory and nociceptive modality properties in the feline suprageniculate nucleus[END_REF] 1985,1996). This probably explains that insular responses do not show a saturation effect for intensities above painful threshold. It is not surprising that posterior insular cortex, which is directly connected to areas contributing to the emotional processing of painful events [START_REF] Büchel | Amygdala-hippocampal involvement in human aversive trace conditioning revealed through event-related functional magnetic resonance imaging[END_REF][START_REF] Bornhövd | Painful stimuli evoke different stimulus-response functions in the amygdala, prefrontal, insula and somatosensory cortex: a single-trial fMRI study[END_REF] and the orienting reactions towards the noxious stimulation [START_REF] Büchel | Dissociable neural responses related to pain intensity, stimulus intensity, and stimulus awareness within the anterior cingulate cortex: a parametric singletrial laser functional magnetic resonance imaging study[END_REF] encodes for stimuli well above painful threshold.

According to our results we can assume that the encoding of thermal stimuli above pain threshold includes a maximal response in SII and a pain level related response in the insula.

It can be hypothesized that pain sensation results in part from this coupled activation of both areas. This assumption is supported by the fact that when SII and insular data were pooled together, the stimulus-response function of the coupled areas yielded a linear function similar to that of the subjective intensity perception (compare figure 1 and the right part of figure 6).

For evident ethical reasons, we fixed the highest stimulus intensity (I3) at 20% over the pain threshold level and thus did not explore subjective pain levels higher than 6-7/10 on VAS ratings. Consequently we cannot ascertain whether the 'ceiling effect' that we observed for painful pulses in SII remained so for very highly painful intensities. A similar ceiling effect was however suggested by [START_REF] Ferretti | Functional topography of the secondary somatosensory cortex for nonpainful and painful stimulation of median and tibial nerve: an fMRI study[END_REF] in a recent fMRI study.

These authors showed that the activity of an anterior part of SII, largely overlapping the location of our SII electrodes, failed to increase between non-painful and painful stimulations. LEPs recorded on two depth contacts located in SII (1) and insular (2) cortices of one patient, for each level of stimulus intensity (I0, I1, I2, I3). These LEPs were recorded in referential mode. Note that in SII (1) we recorded a late negative response indicated by a black star. This component did not appear to be generated in SII but rather corresponded to the diffusion of the Ni component of the insular LEPs, due to the proximity of the contacts 1 and 2. This was confirmed by recordings in bipolar mode where this late SII negative component disappeared (see figure 5). A similar phenomenon is present on insular recordings where a positive peak occurs at about 180ms (black star on 2).

  the contiguous SII and posterior insular cortices has proven very difficult in both functional imaging and electrophysiological studies. Most human imaging pain studies have considered the SIIinsular region, especially in its posterior extent, as a functional entity with common response characteristics (see Peyron et al 2000, Garcia-Larrea et al 2003 for reviews). However, Ferretti et al. (2004) suggested a functional dissociation within opercular sub-areas, and Frot and Mauguière (2003) demonstrated by intracortical recordings that the responses of the opercular and the contiguous insular cortices could be distinguished on the basis of their different latency and morphology. Therefore, their functional capacities to encode stimulus intensity may also be different. Studies of the response properties of SIIinsular activities have been notoriously inconsistent. Timmerman et al. (2001) and Bornhövd et al. (2002) studied the response modes of SII and insula to progressively increasing stimulus intensities, and did not disclose any difference in their respective modes of response; they considered therefore that both areas responded exclusively to painful stimuli. On the contrary, Davis et al (1998) found that innocuous thermal stimuli often activated the posterior insula, but never SII. In contrast to all of them, Coghill et al (1999) found consistent SII activation at innocuous intensities (35 to 46 °C), and described a significant correlation between thermal intensity and contralateral SII activity (while the anterior insula tended to be activated only at painful levels). Very recently, Iannetti and coworkers (2005) described a positive relationship between the amplitude of SII-insular responses and the subjective pain magnitude to laser pulses. However, these authors used exclusively stimuli in the painful range, and therefore could not investigate whether pre-pain sensations were also coded by these early responses. In this study we attempted to characterize response properties of the SII and posterior insular regions by studying local intracortical evoked potentials directly recorded within these two cortices, and using increasing levels of thermal energy, ranging from non-noxious to noxious levels. Stimuli were delivered by means of a laser beam, thus avoiding

  All of the ten patients included in this study had refractory temporal lobe epilepsy and were investigated using stereotactically implanted intracerebral electrodes before functional surgery. Among other sites, these patients had electrodes chronically implanted in the SIIinsular cortex for the recording of their seizures. The decision to explore this area resulted from the observation during scalp video-EEG recordings of ictal manifestations suggesting the possibility of seizures originating in SII and/or insula (see[START_REF] Isnard | The role of the insular cortex in temporal lobe epilepsy[END_REF][START_REF] Isnard | Clinical manifestations of insular lobe seizures: a stereo-electroencephalographic study[END_REF] for a complete description of the rationale of electrode implantation). This procedure, performed routinely before epilepsy surgery in patients implanted with depth electrodes, is completed by the functional mapping of potentially eloquent cortical areas using evoked potentials recordings and cortical electrical stimulation (see[START_REF] Ostrowsky | Representation of pain and somatic sensation in the human insula : a study of responses to direct electrical cortical stimulation[END_REF][START_REF] Mazzola | Somatosensory and pain responses to stimulation of the second somatosensory area (SII) in humans. A comparison with SI and insular responses[END_REF] for a description of the stimulation procedure). According to French regulations concerning invasive investigations with a direct individual benefit, patients were fully informed about electrode implantation, stereotactic EEG (SEEG), evoked potentials recordings, and cortical stimulation procedures used to localize the epileptogenic and eloquent brain areas and gave their consent. The CO 2 laser stimulation paradigm was submitted to, and approved by, the local Ethics Committee. Two patients out of the 10 recorded were excluded from the study because paroxysmal epileptic discharges originated in the recorded SII and/or insula. For the other patients (8 cases) several spontaneous seizures could be recorded during the SEEG, all of which originated in the mesial structures of the temporal lobe. In these patients ictal discharges propagated outside the mesiotemporal cortex and involved most frequently the temporal pole, the temporal neo-cortex, the cingulate gyrus, and the orbito-frontal cortex. In three patients the supra-sylvian operculum showed a rhythmic spike-wave activity during the spread of the discharges and in two of these patients this type of activity was also observed in the insular cortex.

  used to perform the implantation within the stereotactic frame of[START_REF] Talairach | Co-planar stereotaxic atlas of the Human brain. 3-Dimensional Proportional System : An approach to Cerebral Imaging[END_REF]. The electrode tracks and the contacts of each electrodes could be plotted onto the appropriate MRIs slices of each patient (MRIcro® software;[START_REF] Rorden | Stereotaxic display of brain lesions[END_REF].Each of the contacts was then localized in the Talairach space using its stereotactic coordinates: x for the lateral medial axis, with x=0 being the coordinate of the sagittal interhemispheric plane; y for the rostro-caudal (anterior-posterior) axis, y=0 being the coordinate of the vertical anterior commissure (VAC) plane and z for the inferior-superior axis, z=0 being the coordinate of the horizontal anterior commissure-posterior commissure

  caudal and rostral to the VAC plane (y=0). The deepest contacts of the electrodes implanted in SII or the first temporal gyrus explored the insula proper. Four patients were implanted by a single opercular electrode exploring either the pre-(2 cases) or the post-rolandic (2 cases) SII cortex. In the 4 other patients both the frontal and the parietal SII were each implanted by one electrode. Five patients had one electrode implanted in the first temporal gyrus, the deepest contacts of which exploring the insula proper. Thus our data were collected using a total number of 17 electrodes, 12 of them having contacts in SII and 15 of them in the posterior insula (figure 2 and table 2). Thirty contacts explored the posterior insular cortex, distributed along the rostro-caudal axis, 14 mm rostral and 19 mm caudal to the VAC plane (y coordinates). Sixty-three contacts explored the SII area, distributed along the rostro-caudal axis, 14 mm rostral and 23 mm caudal to the VAC plane (y coordinates).

  on a bed in a quiet room. Cutaneous heat stimuli were delivered by a CO 2 laser (10.6 µm wavelength, beam diameter 3 mm, Optilas®, Evry, France) thus avoiding skin contact and co-activation of mechanoreceptors. Therefore the cortical activations linked to laser stimuli could be safely ascribed to the specific stimulation of epidermal thermo-nociceptors. CO 2 laser pulses were applied at 4 different intensities in each subject. The power output being fixed, the amount of thermal energy delivered depended on the duration of the pulse. Pulse duration was set up according to subjects' subjective reports, rated on a visual analog scale (VAS) with an anchor point corresponding to pain threshold. The printed scales consisted of 10-cm horizontal lines where the left extreme was labelled "no sensation" and the right extreme "maximal pain", and an anchored level 4 was at pain threshold (Lickert-type scale). duration: 15-45 ms, mean energy density: 19 mJ/mm 2 , producing a detectable non painful sensation reported for more than 90% of stimulations; For 1/3 of patients this sensation was a warmth sensation and for the others 2/3 a slight non painful pinprick sensation; VAS 1.6 ± 1.09); (c) I2 (Intensity 2) : pain threshold (pulse duration: 25-80 ms, mean energy density: 33 mJ/mm 2 , producing a pricking sensation, like a hair pulling or a drop of hot boiling water on the skin; VAS 3.9 ± 1.46); (d) I3 (Intensity 3) : 20% above pain threshold (pulse duration: 35-110 ms, mean energy density: 46 mJ/mm 2 , producing a pricking sensation described as clearly painful; VAS 5.4 ± 1.6). The subjects were instructed to draw a vertical mark at the appropriate position on the VAS to indicate the perceived pain intensity. This procedure mostly aimed at differentiating between pain threshold (I2) and clearly painful sensation (I3). It was checked that VAS rates were significantly higher for this latter intensity (Student's t test for paired data, p<0.05). Two separate runs of 12 to 16 stimulations applied to the superficial radial nerve territory on the dorsum of the hand were delivered at each intensity value, the order of intensities being randomized. The interstimulus interval varied randomly between 10 and 25s. The laser beam was slightly moved between two successive stimuli to avoid habituation and especially to avoid peripheral nociceptor fatigue (Schwarz et al. 2000). On-line recordings were performed using a sample frequency of 256 Hz and a band pass filter of 0.03-400 Hz (Micromed®, St Etienne des Oullières, France) both in bipolar and reference modes. The reference electrode was chosen for each patient on an implanted contact located in the skull. Epoching of the EEG, selective averaging and record analysis were performed offline using the Neuroscan® software. The continuous EEG was cut in epochs (each epoch of EEG began 100 ms before the stimulus and ended 900 ms after). A 100 ms prestimulus baseline correction was performed. Analysis was performed both on single epochs and on averages. Averaging was performed to reduce the background EEG noise so as to facilitate analysis of stimulus-locked activity (evoked potentials); epoch averaging was done after rejecting epochs with epileptic transient activities. Finally, the two runs for a given stimulation intensity were averaged after having checked that the averaged waveforms were reproducible.

  subject factors : Intensity (I0, I1, I2 and I3), localization (Pre vs post-central operculum for SII, and pre vs post vs ventral insula for the insular cortex) and epoch order. The Geisser-Greenhouse (G-G) procedure was applied to correct degrees of freedom (Geisser and Greenhouse 1958). The G-G correction was used whenever a significant violation of the sphericity assumption was detected in repeated-measures ANOVA (with more than 2 df) (see Vasey and Thayer 1987). Significance was accepted at p<0.05. Post hoc comparisons t tests were performed with a threshold significance at p<0.05. Correlation between VAS and intensity was assessed using a linear regression model. To define the dynamics of the responses amplitudes as a function of stimulus intensity obtained in SII and insula, date were fitted with a polynomial function [f(x)= b 0 + (b 1 * x) + (b 2 * x 2 ) + (b 3 * x 3 )]. Paired t tests were performed between the coefficients determining the increase (b 1 ) and shape (b 2 and b 3 ) of the fitted curves (see Timmermann et al. 2001).

  was a positive significant linear correlation between the subjective VAS reports and the stimulus intensities (r=0.87, p<0.001) (Fig 1).

  by a positive one (Pop), the latencies of which are given in table 1. Similarly, a biphasic negative (Ni, for "negative-insular") -positive (Pi) components were recorded on contacts located in the posterior insular cortex. Note that absolute latency values could not be used for comparison due to latency prenormalisation across patients (see figures 3 and 5). However, latency differences among electrode plots within a single patient remained valuable despite normalisation. Calculation on relative latencies showed a significant delay of the insular response relative to the opercular one (t tests, p<0.05, Table1). No earlier response peaking before these Nop-Pop and Ni-Pi were observed along the electrode tracks implanted respectively in SII and in the insula (Figures2 and 3).

The

  plane, and between horizontal planes 2 mm below and 21 mm above (z coordinates) the horizontal anterior commissure-posterior commissure (AC-PC) plane. These responses were picked up with maximal amplitude on contacts located between 33.25 and 52.75 mm from the midsagittal vertical plane (x coordinates) (Table

  The insular LEPs were recorded by the 2 or 3 deepest contacts of the electrodes penetrating the opercular and temporal cortex between vertical planes 14 mm rostral and 19 mm caudal (y coordinates) to the VAC plane, and between horizontal planes 4 mm below and 20 mm above (z coordinates) the AC-PC plane.

(

  two or three, see Experimental Procedures) along the antero-posterior axis (y), repeated-measures ANOVA showed no effect of electrode location on the latency or amplitude of LEP components (Table 3). This reflected a certain level of homogeneity of the responses recorded by the different electrode tracks, at least in the sub-regions of SII and insular cortices we explored. In support of this, (i) there was no waveform difference in SII or insular responses recorded along different electrode tracks; (ii) polarity reversals along the different electrode tracks in a given patient always occurred at the same depth, and (iii) the dynamics of SII and insular responses to variations of stimulus intensity (see below) were always similar along the different electrode tracks. Therefore, we considered that when several electrodes with different anteroposterior (y) coordinates were located in SII or in the insula, they all recorded responses originating from the same source.

  increasing stimulus intensities enhanced both SII and insular responses, the dynamics of their respective amplitude changes were different. In SII, a significant increase of the LEP was observed as soon as the stimulus intensity reached the sensory threshold (between I0 and I1, p<0.001), as well as between sensory and pain thresholds (I1 to I2, p<0.001) while amplitudes rapidly reached a plateau for intensities above pain threshold (no significant amplitude difference between I2 and I3, p=0.1). In the insula, no significant amplitude increase was observed for low stimulation intensities (p=0.1 between I0 and I1), LEP amplitudes also increased between sensory and pain threshold intensities (p<0.05 between I1 and I2) and, contrary to what was observed in SII, continued to increase significantly at higher intensities over pain threshold (p<0.001 between I2 and I3).

Figure 3

 3 Figure 5). Note that when SII and insular data were pooled together, the stimulus-response function of the coupled areas yielded a linear function similar to the VAS reports (right part of figure 6).

  ntracranial recordings provide a unique opportunity to explore, with good spatiotemporal resolution, the activity of most cortical structures, even those buried in the depth of sulci and hence of difficult access using scalp or epicortical recordings. Using this technique, we previously showed that the responses to painful stimuli in SII and insula could be distinguished on the basis of response latencies and of stereotactic source coordinates (Frot and Mauguière 2003). In the present work, we further specify that, although increasing stimulus intensities enhanced both SII and insular responses, the dynamics of their responses as a function of thermal stimulus intensity are significantly different. The SII responses were able to encode gradually the intensity of laser thermal stimuli from sensory threshold to barely painful levels, but tended to show a ceiling effect for increasing pain intensities. In contrast, the posterior insular cortex failed to detect responses changes for very low levels of stimulus intensities (around sensory perception threshold), but encoded stimulus intensity variations in the painful range without showing saturation effects for the highest painful intensities used. Very recently, Iannetti and coworkers (2005) described a positive relationship between the amplitude of SII-insular responses and the subjective pain magnitude to laser pulses. These authors founded their analysis on the behaviour of the scalp 'N1' response, which most probably reflects lumped opercular and insular sub-components (Valeriani et al. 2000, Garcia-Larrea et al. 2003); therefore, sustained increase at painful ranges may have reflected the insular, rather than the SII contribution to scalp N1. Indeed, as Iannetti et al. (2005) used exclusively stimuli in the painful range, they could not investigate whether pre-pain sensations were also coded by these early scalp responses. No evoked responses were recorded in the insular cortex explored by our electrodes for stimulus intensities at, or just above, sensory threshold (I1), while clearly recordable insular potentials were recruited at pain threshold (I2). However, as we did not test intermediate intensities, firm conclusions cannot be drawn on the insular encoding properties for intensities between sensory and pain levels. It is however noteworthy that no significant LEPs could be recorded in posterior insular cortex for the non-noxious levels for which clear evoked responses were recorded in the neighbouring opercular (SII) cortex. This is in agreement with the results of an elegant study of Bornhövd et al. (2002) on the dynamics of fMRI signals in response to progressively intense laser pulses. In accordance with our data, their results suggested that neither anterior nor posterior insular responses distinguished among non-painful intensities, but showed a positive relationship for painful trials such as we observed in our insular

  stimulation technique used may be at the origin of this discrepancy since our study and the one of Bornhövd used laser stimulations (brief thermal pulses: 1 to 110 ms; small skin surface stimulated: about 30 mm 2 ), while the others studies performed stimulations of much longer durations (several hundreds of ms up to minutes) involving larger skin areas (thermode: up to several cm 2 ). These diverging results are by themselves very interesting. They suggest that insular activation by non noxious stimuli may be possible if thermal non noxious stimuli are applied long enough on sufficient extended skin areas. This could indicate the necessity of some time/surface dependent recruitment processes in order to obtain some responses within the posterior insular cortex. Since reports of posterior insular activation by nonnoxious stimuli (innocuous warmth) most often used mixed mechanical and thermal stimuli, sometimes moving, it is difficult to ascertain which of stimulus duration, stimulus surface or mechanical contact (or a combination of the three) was the most important contributor to eliciting the insular response. A good compromise to obtain good time/surface recruitment without losing thermal specificity could perhaps be achieved in a near future by specific stimulation of warmth (C-fiber) receptors without skin contact, using a recently described laser technique (Cruccu et al. 2003). However, there have not been to date any reports of intracranial responses obtained using such a technique. Bornhövd et al. (2002) also described similar activation patterns in SII and posterior insula, while our intracranial data showed different dynamics of response between the insula proper and SII responses. In our study, SII responses were able to distinguish between low levels of innocuous stimulus intensities (see figures 3 and 5). Given the limited spatial resolution of fMRI relative to SEEG studies, it is possible that different signals from SII and posterior insular cortices could not be disentangled in Bornhövd et al.'s study (2002), and that their SII-insular fMRI signal reflected almost exclusively the insular response. A similar inability to detect SII response changes to small intensity levels was reported by Timmermann et al. (2001) using MEG. The stimulus-response function described by these authors in SII cortex showed no significant activation change at low stimulus intensities (subthreshold and perception threshold), while a sharp increase in source activation was observed for stimuli above pain threshold. As was the case with fMRI, surface MEG recordings are probably not precise enough to dissociate activation patterns emanating from SII and posterior insula, even when using dipolar modelling. One could assume that Timmermann et al's results may have actually reflected a mixed signal largely dominated by the insular responses. In accordance with this view, a recent meta-analysis of dipolemodelling studies of cortical pain responses indicated that scalp-derived modelled dipoles reflect a 'lumped' activation of several sources in the suprasylvian region, including both SII and the insula (Garcia-Larrea et al. 2003, Apkarian et al 2005). Survey of the literature therefore suggest that separation of SII and insular encoding properties may be beyond the reach of both haemodynamic functional imaging (PET / fMRI) and scalp electrocortical recordings (EEG / MEG), at least until the signals emanating from these two very closely located regions can be reliably segregated by these techniques (see reviews in Peyron et al. 2000, Derbyshire 2000, Jones et al. 2002, Garcia-Larrea et al. 2003, Apkarian et al. 2005). The present work is, to our knowledge, the first demonstrating that the closely located SII and posterior insular cortices differently encode gradual thermal stimulus intensity changes, and thus points out different functional organizations in these two areas. The fact that insular responses continued to increase when SII potentials tampered makes it clear that posterior insula cannot receive thermal information exclusively from the SII regions explored in this study. This is consistent with anatomical and physiological data suggesting that these two cortices are pertain to partially different networks for somatosensory

  These authors also suggested that a more posterior part of SII could be more specifically involved in pain encoding, since the BOLD signal at this location kept increasing between pre-pain and pain stimuli. None of our opercular electrodes was implanted posterior enough to explore this most caudal region in SII, so our recordings may reflect only part of the opercular pain network. On the other hand, contrary to us, these authors did not explore the SII responses to gradually increasing stimuli. Thus, in the eventuality that a posterior SII sub-region would be able to code for stimulus intensities over pain threshold, it would remain to determine whether it is nociceptive-specific, or made of wide dynamic range cells able to code for stimulus intensities both below and above pain threshold.Table 3Statistical analysis (ANOVA) of the effects of intensity, electrode localization and epoch order on amplitude and latency of SII and insular LEPs. Significant results were indicated by a *, were in bold and underlined. df: before the G-G correction.

Figure 1

 1 Figure 1

Figure 2

 2 Figure 2

Figure 3

 3 Figure 3

Figure 4

 4 Figure 4

Figure 5 Figure 6

 56 Figure 5

Tables & Figures

Latencies (ms) and amplitudes (µV) of responses

All the means are given with the standard errors.

Table 2 Coordinates (Atlas of Talairach and Tournoux) of contacts (in mm) where the maximal amplitudes of the N/P deflection in bipolar mode were recorded.

PrC: Pre-central SII cortex, PoC: Post-central SII cortex


R Sahli

J Hem

C Crauste-Thibierge

F Clément

D R Long

S Ciliberto

Relaxation time of a polymer glass stretched at very large strains

The polymer relaxation dynamic of a sample, stretched up to the stress hardening regime, is measured, at room temperature, as a function of the strain λ for a wide range of the strain rate γ, by an original dielectric spectroscopy set up. The mechanical stress modies the shape of the dielectric spectra mainly because it aects the dominant polymer relaxation time τ , which depends on λ and is a decreasing function of γ. The fastest dynamics is not reached at yield but in the softening regime. The dynamics slows down during the hardening, with a progressive increase of τ . A small inuence of γ and λ on the relative dielectric strength cannot be excluded.

Mechanical and dynamical properties of polymers are intensively studied, due to their fundamental and technological importance [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF]. When strained at a given strain rate, beyond the elastic regime in which the stress is proportional to strain, glassy polymers exhibit a maximum in the stress-strain curves (yield point) at a strain of a few percents [START_REF] Nielsen | Mechanical Properties of Polymers and Composites[END_REF] and the deformation becomes irreversible (see Fig. 1). At larger strains, depending on the history of the sample [START_REF] Meijer | [END_REF], the stress drops (strain-softening regime) before reaching a plateau corresponding to plastic ow. Strain-hardening may then occur at even larger strains, depending on the molecular weight and on the crosslinking of the polymer [4]. The key new insights obtained either by numerical simulations [57] or experiments [8 11] are that strain hardening appears to be controlled by the same mechanisms that control plastic ow [1215]. However the microscopic mechanisms leading to such a mechanical behavior are not fully understood [16,[START_REF] Hoy | Polymer Glasses. Chapter: Modeling strain hardening in polymer glasses using molecular simulations[END_REF]. For example it is unclear to what extent the relaxation dynamics in polymer glasses is modied when the sample is stretched into the plastic region (see for example refs. [1821]).

The purpose of this letter is to bring new insight into this problem by presenting the results of experiments in which we performed dielectric spectroscopy of polymer samples stretched till the strain hardening regime. Dielectric spectroscopy, allows the investigation of the dynamics of relaxation processes by means of the polarization of molecular dipoles.

It is directly sensitive to polymer mobility and probes directly the segmental motion. It can be used to quantify the mobile fraction of polymers. Measuring the dielectric response of polymers in situ had been pioneered by Venkataswamy et al. [START_REF] Venkatasway | [END_REF]. It is complementary to other techniques, such as Nuclear Magnetic Resonance [23] and the diusion of probe molecules [2426], used to study the molecular dynamics of polymers under stress. The dielectric spectroscopy has already been used in combination with mechanical deformation to study the dynamics in the amorphous phase of polymer under stress [27,28]. The results of these experiments were limited to the yield point whereas the experimental studies of the microscopic behavior and processes during strain hardening are more scarce. In this letter we present the results obtained by our original experimental apparatus which can measure with high accuracy the evolution of the Dielectric Spectrum(DS) of a sample stretched, till the strain hardening regime, at dierent strain rates. Thus we can precisely compare the DS s obtained as a function of stress at room temperature with those (named DS T ) obtained as a function of temperature in an unstressed sample. This comparison allows us to extract useful informations on the relaxation dynamics under stress at various strain rates. Our experimental results bring new important informations because they clearly show an acceleration of the dynamics which reaches a maximum in the softening regions. Instead the molecular mobility slows down again during the strain hardening regimes.

Our experimental setup is composed by a home-made dielectric spectrometer coupled with a tensile machine for uniaxial stretching of lms [28,29]. The scheme of the experimental setup is presented in the inset of Fig. 1. More details are given in Annex I.1.

The dielectric measurements are performed by conning the polymer lm between two disc-shaped electrodes of 10 cm diameter. The good contact between the electrodes and the sample, during the whole experiment, is assured by an aqueous gel, which has very low electrical resistivity compared to the sample. We checked that the gel does not perturb the sample response because water absorption in our samples is only 0.2% and by comparing the response in the presence of the aqueous gel and in the presence of mineral oil [29].

Dielectric spectroscopy allows the investigation of the dielectric response of a material as a function of the angular frequency ω. It is expressed by the complex dielectric permittivity or dielectric constant: ε(ω) = ε (ω)-iε (ω), with ε (ω) the real component of the dielectric constant which is related to the electric energy stored by the sample, and ε (ω) the imaginary component which indicates the energy losses. The loss tangent is tnδ(ω) = tan δ(ω) = ε (ω)/ε (ω), where δ is the phase shift between the electric eld applied for measuring ε and the measured dielectric polarization.

We investigated an extruded lm MAKROPOL ® DE 1-1 000000 (from BAYER) based on Makrolon ® poly- carbonate (PC) with a T g of about 150 • C. The sample sheets had a thickness of 125 µm. The tensile experiments are performed at T = 25 • C much below T g . We xed four dierent strain rates: 2.5 × 10 -3 s -1 , 2.5 × 10 -4 s -1 , 2.5 × 10 -5 s -1 and 2.5 × 10 -6 s -1 . Typical strain curves at the dierent strain rates are plotted in Fig. 1. Initially, the stress increases almost linearly up to its maximum (yield stress) which is reached at λ 1.08. The stress decreases after the yield strain softening regime for 1.09 < λ < 1.2. For λ > 1.2, the material is inside the strain hardening regime, where the force increases up to the lm crack. In this article we focus on the measurement at λ > 1.09

being the results at λ ≤ 1.08 already discussed in [28].

For each value of the strain rate the measurement of the dielectric and mechanical properties have been repeated on at least 3 samples, in order to check the reproducibility of the results. The maximum uctuations of the results observed in dierent samples is at most 11%. In order to compensate for the change of the thickness of the sample we study ε n (f ) = ε (f, λ)/ε (400Hz, λ) because above 400Hz the only eect of the applied stress on the dielectric measurement is related to the change of the sample thickness induced by the large applied stress. The quality of this compensation can be checked looking at Figs.2 where we plot ε n (f, λ) and tnδ(f, λ) measured at various frequencies as a function of λ. We clearly see that the effect of the stretching on the DS decreases a lot by increasing the measuring frequency and it almost disappears at 60Hz. This means that ε n cancels the dependence of the sample thickness d(λ) and correctly estimates the variation of ε induced by the strain. This gure also shows that both ε n and tnδ reach the maximum in the softening regime then they decrease and remain constant in the hardening regimes. Fig. 3 shows that the eect of the strain on ε increases with γ as already shown in ref. [28]. FIG. 3. Normalized real part ε n (a) and tnδ (b) as a function of λ measured at 7Hz for dierent strain rates : 2.5 × 10 -3 s -1 (black triangles), 2.5 × 10 -4 s -1 (red circles), 2.5 × 10 -5 s -1 (green squares) and 2.5 × 10 -6 s -1 (blue diamonds).

It is useful to study the evolution of the whole DS s measured at γ = 2.5 × 10 -4 s -1 for which the eect of the strain on ε is very pronounced (see Fig. 2) and at the same time the measure lasts enough time to have a good low frequency resolution of the DS. Figures 4 show ε n (f ) and tnδ(f ) as a function of frequency measured at various λ at γ = 2.5 × 10 -4 s -1 . The DS s recorded during the tensile test are compared to the DS T , i.e. the DS measured as a function of temperature in the unstressed sample. This gure summarizes one the most important ndings of this investigation, that we will explain. First we notice that the low frequency part of DS s has roughly the same amplitude as DS T measured at temperatures close to T g . However the tnδ(f ) of DS s and of DS T have a very dierent dependence on f indicating that a simple relationship between γ and an eective temperature ( presented in several articles [26,28,30,31]) cannot be easily established as we discuss later on. However in Figs. 4,2 we notice several other important results. We clearly see that in the stressed sample the maximum am-

1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 1 . 0 0 0 1 . 0 0 5 1 . 0 1 0 1 . 0 1 5 1 . 0 2 0 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 0 5 1 0 1 5 2 0
f ( H z )

1 3 7 °C λ : 1 . 0 0 1 . 0 9 1 . 1 7 1 . 7 0 ε' n 1 5 2 °C ( a ) T a n ( δ) x 1 0 -3 f ( H z ) ( b ) FIG. 4.
The normalized real part ε n (a) and tnδ (b) as a function of frequency measured, at T = 25 • C, during a stretching experiment at 2.5 × 10 -4 s -1 at dierent values of the strain : λ = 1.00 (equilibrium), λ = 1.08 (plastic yield), λ = 1.17 (softening regime) and λ = 1.70 (hardening regime). In the background, continuous orange lines represent ε n (a) and tnδ (b) measured at λ = 1 in the temperature range between 137 • C, (bottom orange curve) and 152 • C (top orange curve) at 1 • C increment [28].

plitude of ε n (f ) and tnδ at low frequencies is not reached at the yield but in the softening regime at λ = 1.17 and most importantly the amplitude of DS s decreases in the hardening regime at λ = 1.70. This is a new and totally unexpected result, i.e. the eect of the strain on the dielectric constant is not monotonous.

In order to have a more detailed description of these experimental observations, we study the behavior of tnδ(f, λ, γ), which is not aected by any geometrical effect. We plot, in Fig. 5, tnδ(f ) versus frequency measured at λ = 1.17 at various γ. We notice that the low frequency parts decrease as a power law which can be tted with tnδ(ω) = 1/(ωτ ef f ) β where τ ef f and β are tting parameters and ω = 2πf . We nd that β 0.50 ± 0.05 is independent of γ (continuous lines in Fig. 5a), with an accuracy of about 10% . The measured time scale τ ef f , which is a function of γ and λ, is plotted in Fig. 5b). Looking at this gure we see that with a very good accuracy τ ef f = a(λ)/ γ where a(λ) has a minimum at λ 1.17 in the stress softening regime.

We can consider the power law behavior of tnδ as the

1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 -3 1 0 -2 1 0 -6 1 0 -5 1 0 -4 1 0 -3 1 0 2 1 0 3 1 0 4 1 0 5 1 0 6 1 . 2 1 . 6 1 0 2 1 1 0 T a n ( δ) f ( H z ) ( a )
. (a) Dependence on frequency of tnδ, measured in the softening regime λ = 1.17, at dierent γ: 2.5 × 10 -3 s -1 (black triangles), 2.5 × 10 -4 s -1 (red circles), 2.5 × 10 -5 s -1 (green squares) and 2.5 × 10 -6 s -1 (blue diamonds). The low frequency parts of each curve are tted by a power law (straight lines) tnδ = 1/(ωτ ef f ) β The time scale τ ef f depends on γ but the scaling power β = 0.50 ± 0.05 is found to be constant within experimental errors. (b) The tted time scale τ ef f as a function of the strain rate γ measured at several strains: λ = 1.17 (softening regime), λ = 1.30 and λ = 1.70 (hardening regime). The relaxation time follows the law τ ef f = a(λ)/ γ (dashed lines) with the coecient a(λ)

plotted in the inset.

high frequency part of the Cole-Cole (Havriliak-Negami) model [32] for the α peak of the dielectric constant. Indeed, calling τ the characteristic time of the α peak, at frequencies ω >> 1/τ the Cole-Cole model takes the form:

ε ε ∞ -1 (ε o -ε ∞ ) ε ∞ cos(βπ/2) (ωτ ) β (1)
and

tnδ (ε o -ε ∞ ) ε ∞ sin(βπ/2) (ωτ ) β (2)
where ε ∞ and ε o are respectively the high and low frequency dielectric constants. In our experiment β 0.5 thus the two equations become:

ε ε ∞ -1 (ε o -ε ∞ ) ε ∞ 1 (2ωτ ) 1/2 , (3) tnδ
(ε o -ε ∞ ) ε ∞ 1 (2ωτ ) 1/2 . ( 4
)
These two equations show that tnδ ( ε ε∞ -1) which is rather well veried by the experimental data for all λ. This indicates that they are a rather good model of the dielectric constant under stress. Using eq.4, we identify the tting parameter as τ ef f = 2τ /∆ 2 where ∆ = (ε o -ε ∞ )/ε ∞ . At this level we cannot distinguish whether the increase of ε and tnδ is induced by a decreasing of τ or an increase of the relative dielectric strength ∆ [START_REF] Frohlich | Theory of Dielectics, Dielectric Donstant and Dielectric Loss[END_REF].

We suppose rst that the stress has no inuence on τ but only on ∆. Keeping for τ the value of the unstressed sample we can estimate the value of ∆ under stress from the previous expression ∆ = 2τ /τ ef f . To give a lower bound on the value of τ at 22 o C we rely upon the high temperature measurements of the dielectric properties of PC and on the estimation based on the WLF law [28,[START_REF] Huang | [END_REF]. For example at T = 130 o C, which is the minimum temperature at which the WLF law for PC can be tested, one nds using WLF τ 10 7 s. Thus, as τ increases by lowering temperature, we simply assume that at T = 22 o C, τ > 10 7 s, which, in addition, is coherent with the fact that our samples are more than one year old [START_REF] Struik | Physical Aging in Amorphous Polymers and Other Materials[END_REF]. From the data of g.5b) we obtain τ ef f

1/ γ at λ = 1.17. Thus in the hypothesis of constant τ , we nd ∆ > 7 at γ = 2.5×10 -6 and ∆ > 223 at γ = 2.5×10 -3 s -1 . This value is 1000 times larger than that of the unstressed sample for which ∆ 0.12. This is nonphysical because ∆ cannot change of such a large amount even at very large strain (see refs. [START_REF] Frohlich | Theory of Dielectics, Dielectric Donstant and Dielectric Loss[END_REF][START_REF] Yee | [END_REF]37] and appendix I.2). Therefore one has to conclude that, although a small increase of the ∆ cannot be excluded, the observations cannot be explained without a decreasing of τ with the mechanical stress, i.e. an acceleration of the dynamics, which agrees with other experimental results based on other techniques and other polymers [24,25,38]. Using our data we can estimate τ 0.5∆ 2 a(λ)/ γ, assuming that ∆ = 0.2, i.e. keeping x the value of the unstressed sample. At λ = 1.17 we see that a(λ)

1 and using γ = 2.5 × 10 -3 one nds τ = 8s, which implies that one would not observe a clear power law at the highest strain rate because the approximation ω >> 1/τ will be not fully satised for the low frequencies of our frequency range. Therefore at the highest γ, one concludes that τ >> 8 s and ∆ > 0.2. At this point we can safely say that the main eect of the strain is a reduction of τ of several orders of magnitude although a small inuence on ∆ is necessary for the consistency of the observations. Finally we compare these results with those that one extracts from the values tnδ(f ) measured in the unstressed sample at dierent temperatures. In the range 137 o C < T < 152 o C, where we observe an overlap of DS s and DS T in g.2 the tnδ(f ) has a power law dependence on f but the exponent β is a function of temperature: specically 0.25 < β < 0.35 for 137 o C < T < 152 o C. The observation that the β exponent measured under stress is larger than the one measured in the unstressed sample at temperature close to T g , has important consequences. Indeed in the Cole-Cole expression the smallest is the value of the exponent β the broadest is the distribution of relaxation times. Thus we conclude that our measurements are compatible with a narrowing of the distribution of relaxation times under stress observed in refs. [25,38] with other techniques in other polymers.

Let us point out that in the frequency window of our measurements there are no other relaxations which may inuence the results. The beta relaxation is not aected by the stress because DS s does not change above 400Hz. The gamma relaxation time is at very high frequencies at room temperature (see ref [START_REF] Yee | [END_REF]) Structural changes that might also inuence the results are absent. Indeed we checked by Dierential Scanning Calorimetry (DSC) that polycarbonate does not crystallize under strain in the range considered here. Microscopic damaging may also contribute to the dielectric response. It has been shown that damaging occurs indeed during the strain hardening regime, but the volume fraction corresponding to these damages is very small (< 10 -4 for cellulose acetate [39] and for polycarbonate [40]) and the corresponding perturbation regarding the interpretation of the results is negligible.

Summarizing, we have studied the DS s of a sample of polycarbonate at room temperature submitted to an applied stress at dierent strain rates covering three order of magnitude range. In our frequency window we observe that, with respect to the unstressed sample, tnδ increases at low frequencies as a function of γ, whereas at high frequencies it remains unchanged. The increase is not monotonous and reaches its maximum in the softening regime. From the data we extract an eective time scale τ ef f ∝ 1/ γ which depends on λ and reaches its minimum in the softening regime to increase again in the hardening. We have also shown that the distribution of relaxation times is narrower under stress than in the unstressed sample at T close to T g . The large variation of τ ef f cannot be explained only by the increase of the ∆, but it implies that the stress induces a shift towards high frequencies of the α peak without inuencing the high frequency part of the DS. Thus we conrm that the stress accelerates the polymer dynamics as already observed in other experiments [2328], which were limited to the yield points. We have extended the analysis to high values of strain in a polymer which presents stress hardening, nding two very important and unexpected results. Firstly the smallest τ ef f is not reached at yield but in the stress softening regime (see Figs. 2,3). Secondly the stress hardening regime is associated to a progressive increase of the eective relaxation time τ ef f = 2τ /∆ 2 . This increase may again be due either to an increase of τ or to a decrease of ∆ of at least a factor of 3 (see g.5), which, on the basis of the previous arguments, is too large. Furthermore this decrease of ∆ will imply a non monotonous dependence of ∆ as a function of λ, which contradicts previous measurements [37,41]. Specically in ref. [41] the authors measured by NMR a segmental orientation for stretched polycarbonate at room temperature, observing a monotonous increase of the orientational order parameter, whereas we observe a non monotonous behaviour of τ ef f versus λ. Thus it is conceivable to say that such a behavior is induced by an increase of τ in the hardening regime.

As a conclusion our experimental results impose strong constrains on the theoretical models on strain softening and hardening. The mechanical part is composed by two cylinders (21 cm long, 1.6 cm of diameter) used to fasten the sample, a load cell of 2000 N capacity, a precision linear transducer to measure the strain over a range of 28 cm, and a brushless servo motor with a coaxial reducer. This device allows the investigation of polymer lm samples of a maximum width of 21 cm. We use dog-bone shaped sheets with the eective dimensions of the area under deformation of 16X16 cm 2 , in order to focus the deformation of the lm between the electrodes used to measure the DS. The sample is stretched at xed relative strain rate γ = d dt (L/L o ) Where L(t) is the length of the stretched sample and L o the initial length at zero applied force F . Measurements are performed at room temperature at constant γ, till a stretch ratio λ = L(t)/L 0 of about 1.8 (80 % strain) is reached. The force F (λ) applied to the sample is measured during all the experiment by a load cell. The value of the applied stress is dened as σ = F (λ)/(W o d o ) where W o and d o are the initial width and thickness of the sample.

We use an innovative dielectric spectroscopy technique (see ref. [29]), which allows the simultaneous measurement of the dielectric properties in a four orders of magnitude frequency window chosen in the range of 10 -2 -10 3 Hz. This multi-frequency experiment is very useful in the study of transient phenomena such as polymer lms deformation (see [29] for more details), because it gives the evolution of the DS on a wide frequency range instead of a single frequency. Specically the device measures the complex impedance of the capacitance C formed by the electrodes and the sample. In all the frequency range the device has an accuracy better than 1% on the measure of C and it can detect values of tnδ smaller than 10 -4 . The real part of the dielectric constant of the sample is ε = Cd(λ)/(Sε o ) where S is the electrode surface, ε o the vacuum dielectric constant and d(λ) the thickness of the sample whose dependence on λ must be taken into account to correctly estimate the value of ε . This compensation is not necessary for tnδ(f ) because it does not depend on the geometry being the ratio of ε /ε . I.2.

Kirkwood factor

The low frequency dielectric constant ε o is related to ∞ via the Kirkwood factor g = 1+z < cos θ i,j > where < . > stands for mean value, θ i,j is the angle between dipole i and dipole j, and z is the number of relevant nearest dipoles, which is usually taken to be about 10 (see ref. [START_REF] Frohlich | Theory of Dielectics, Dielectric Donstant and Dielectric Loss[END_REF]). Specically one nds that ε o = ε ∞ + A g where A is a material dependent factor. Therefore using this relationship, the quantity ∆ dened in the text becomes:

∆ = ε o /ε ∞ -1 = Ag/ε ∞ .
We notice that, independently of the probability distribution of θ i,j , the value of < cos θ i,j > is limited between 0 and 1 and as a consequence 1 < g < 11. This means that, being ∆ = 0.2 in the unstressed sample, the maximum value of ∆ under stress is ∆ = 2.2. This value is much smaller than the values of ∆ estimated in the main text in order to explain the observations, assuming that only ∆ is aected by the strain. Thus on the basis of the experimental data, we have to conclude that τ decreases in the stressed sample, i.e. the dynamics accelerates under strain. However this acceleration is associated to a small increase of ∆ as we discuss in the main text.

FIG. 1 .

 1 FIG.1. Stress evolution over a wide range of λ at several constant strain rates : 2.5 × 10 -3 s -1 (black), 2.5 × 10 -4 s -1 (red), 2.5 × 10 -5 s -1 (green) and 2.5 × 10 -6 s -1 (blue). Several identical specimens have been measured for each strain rate. Inset: tensile machine for dielectric measurements under stress : motor (A), load cell (B), linear transducer (C), sample fastening cylinders with the stretched polymer lm (D) and electrodes (E) connected to the dielectric spectrometer (not sketched).

FIG. 2 .

 2 FIG. 2. Dependence on λ of ε n (a) and tnδ (b) measured at several frequencies during the stretching experiment at γ = 2.5×10 -4 s -1 . In both panels the solid lines (left hand ordinate) represent ε n (a) and tnδ(b) as a function of λ, whereas the dashed line (right hand ordinate) show the corresponding evolution of the stress, which reveals that the low frequency components of DSshave a maximum in the softening regime of the polymer lm.

  FIG. 5. (a) Dependence on frequency of tnδ, measured in