Hafiz Ahmed
email: hafiz.ahmed@bangor.ac.uk

Rosane Ushirobira

| Denis Efimov

Arbitrarily Fast Delayed Signal Cancellation PLL for Grid-Integration of Renewable Energy Sources

In order to mitigate the negative impact of burning fossil fuels, renewable energy sources (RES) are becoming very popular worldwide. Many RES are now price competitive and/or cheaper compared to conventional gas/coal-fired electricity production. This has contributed to a broader societal acceptance of RES and also led to the large-scale integration of RES into the grid. RES, such as solar or wind, are interfaced with the power grid through power electronic converters. When the RES-powered grid-connected converter (GCC) operates in synchronization with the grid, maximum power transfer can be ensured. To achieve this, the control system of GCC requires the information on the grid, e.g., phase, frequency, and amplitude in real-time. This process is commonly known as grid-synchronization [START_REF] Çelik | Lyapunov based harmonic compensation and charging with three phase shunt active power filter in electrical vehicle applications[END_REF][START_REF] Safa | Open Loop Synchronization Techniques Benchmarking for Distributed Energy Sources Connection[END_REF][START_REF] Ahmed | Sliding mode based adaptive linear neuron proportional resonant control of Vienna rectifier for performance improvement of electric vehicle charging system[END_REF][START_REF] Chedjara | A fast and robust reference current generation algorithm for three-phase shunt active power filter[END_REF][START_REF] Chedjara | A New Quasi Open Loop Synchronization Technique for Grid-Connected Applications[END_REF][START_REF] Verma | An improved hybrid prefiltered open-loop algorithm for three-phase grid synchronization[END_REF][START_REF] Meral | A comprehensive survey on control strategies of distributed generation power systems under normal and abnormal conditions[END_REF][START_REF] Verma | A Robust Lyapunov Demodulator for Tracking of Single-/Three-Phase Grid Voltage Variables[END_REF][START_REF] Safa | A pseudo open loop synchronization technique for heavily distorted grid voltage[END_REF], typically performed by the phase-locked loop (PLL).

RES are usually connected to the grid at the distribution network level. Due to the presence of single-and threephase loads, per-phase loading of low-voltage distribution substations are not identical. This uneven connection of particularly single-phase loads causes phase unbalance at the distribution network level [START_REF] Klusacek | Power Symmetrical Components as Grid Usage Indicator for Unbalanced Prosumers[END_REF][START_REF] Pereira | Phase Rebalancing of Distribution Circuits Dominated by Single-Phase Loads[END_REF][START_REF] Fidalgo | Impact of Load Unbalance on Low Voltage Network Losses[END_REF][START_REF] Elyasichamazkoti | Digital filter-based grid synchronization for autonomous microgrids[END_REF][START_REF] Basit | A state-of-the-art comprehensive review of modern control techniques for grid-connected wind turbines and photovoltaic arrays distributed generation systems[END_REF][START_REF] Ahmed | On Biased Harmonic Signal Estimation: Application to Electric Power Grid Monitoring[END_REF]. Integrating RES into the unbalanced distribution network is challenging due to the simultaneous presence of fundamental frequency positive-sequences (FFPS) and fundamental frequency negative-sequences (FFNS) components in the grid voltage. Fast and accurate estimation of these components is essential for grid fault-tolerant control of RES-interfaced power converters. Typical examples are fault-ride through (FRT) control [START_REF] Hamdi | A short review of grid voltage sags and current control techniques of voltage source inverters in distributed power generation systems[END_REF][START_REF] Benyamina | An augmented state observer-based sensorless control of grid-connected inverters under grid faults[END_REF][START_REF] Benyamina | Online current limiting-based control to improve fault ride-through capability of grid-feeding inverters[END_REF], voltage compensation for sensitive loads [START_REF] Biricik | Protection of sensitive loads using sliding mode controlled three-phase DVR with adaptive notch filter[END_REF], and PLL implementation [START_REF] Özer | Enhanced control method for voltage regulation of DSTATCOM based SEIG[END_REF][START_REF] Sevilmiş | Implementation of enhanced non-adaptive cascaded DSC-PLLs for renewable energy systems[END_REF][START_REF] Alturki | A New Hybrid Synchronization PLL Scheme for Interconnecting Renewable Energy Sources to an Abnormal Electric Grid[END_REF], to name a few.

Delayed signal cancellation (DSC) [START_REF] Wang | Grid Synchronization PLL Based on Cascaded Delayed Signal Cancellation[END_REF][START_REF] Wang | Analysis and Digital Implementation of Cascaded Delayed-Signal-Cancellation PLL[END_REF] is a popular method in the literature to estimate FFPS and FFNS components from unbiased and unbalanced three-phase signals. This technique uses quarter-cycle delayed samples of the measured grid voltage/currents to estimate the sequence components. A modified DSC operation with fast convergence is proposed in [START_REF] Ahmed | Robust estimation of fundamental frequency positive-sequence component for gridintegration applications in energy systems[END_REF][START_REF] Contreras | Fast Delayed Signal Cancellation based PLL for unbalanced grid conditions[END_REF][START_REF] Cárdenas | Resonant control system for low-voltage ride-through in wind energy conversion systems[END_REF][START_REF] Cardenas | Fast convergence delayed signal cancellation method for sequence component separation[END_REF] to reduce the estimation delay. An equivalent version of this approach with the same delay but lower computational complexity is proposed in [START_REF] Mellouli | A third-order MAF Based QT1-PLL that is robust against harmonically distorted grid voltage with frequency deviation[END_REF]. However, these modified DSCs cannot reject DC offset without using an additional half-cycle DSC operation. A key limitation of [START_REF] Cardenas | Fast convergence delayed signal cancellation method for sequence component separation[END_REF] and [START_REF] Mellouli | A third-order MAF Based QT1-PLL that is robust against harmonically distorted grid voltage with frequency deviation[END_REF] is that this approach can work only when no DC offset is presented in the grid. DC offset may appear due to a variety of reasons, e.g., analog to digital conversion of measured signals [START_REF] Ciobotaru | Offset rejection for PLL based synchronization in grid-connected converters[END_REF], grid fault [START_REF] Stringer | The effect of DC offset on current-operated relays[END_REF], current transformer saturation, geomagnetically induced currents [START_REF] Nam | Phasor Estimation in the Presence of DC Offset and CT Saturation[END_REF], etc. In the presence of DC offset, the output of PLL has steady-state ripple, which causes undesirable reactive power injection to the grid. In addition, active power oscillation can also be observed, which degrades the power quality [START_REF] Hwang | DC Offset Error Compensation for Synchronous Reference Frame PLL in Single-Phase Grid-Connected Converters[END_REF]. As such, mitigation of this DC offset is essential to ensure high-performance control of grid-connected converters.

To reject measurement bias, i.e., DC offset, cascaded DSC (CDSC) is proposed in [START_REF] Wang | Grid Synchronization PLL Based on Cascaded Delayed Signal Cancellation[END_REF][START_REF] Mellouli | A third-order MAF Based QT1-PLL that is robust against harmonically distorted grid voltage with frequency deviation[END_REF]. In CDSC, an additional delayed signal cancellation block with a half-cycle delay is considered to reject the DC offset, increasing the overall estimation delay and augmenting the required memory to store the delayed samples. An improved DSC operation is presented in this work to overcome this issue. Our approach uses two equidistantly delayed samples of the measured signals to estimate the sequence components with DC offset rejection. The proposed solution is arbitrarily fast and has low computational complexity. Unlike other two-delayed-samples-based FFPS sequence component estimation techniques [START_REF] Du | A fast positive sequence components extraction method with noise immunity in unbalanced grids[END_REF], our solution is generic since it can estimate FFPS and FFNS components. In addition, our method can reject DC offset, unlike [START_REF] Du | A fast positive sequence components extraction method with noise immunity in unbalanced grids[END_REF].

Like various variants of conventional DSCs [START_REF] Cardenas | Fast convergence delayed signal cancellation method for sequence component separation[END_REF][START_REF] Mellouli | A third-order MAF Based QT1-PLL that is robust against harmonically distorted grid voltage with frequency deviation[END_REF], the proposed modified DSC is also affected by voltage/current harmonics. Additional pre-loop and/or in-loop filtering stages or cascaded DSCs [START_REF] Sevilmiş | Implementation of enhanced non-adaptive cascaded DSC-PLLs for renewable energy systems[END_REF][START_REF] Wang | Grid Synchronization PLL Based on Cascaded Delayed Signal Cancellation[END_REF] have to be implemented. This issue is addressed by integrating our method into a CDSC-PLL where the proposed technique replaces the DC offset and FFNS rejection DSC blocks. Reducing the memory requirement, and, consequently making the convergence time faster for DSC-PLL, is this work's main contribution. Compared to existing DSC-PLLs [START_REF] Wang | Grid Synchronization PLL Based on Cascaded Delayed Signal Cancellation[END_REF][START_REF] Cardenas | Fast convergence delayed signal cancellation method for sequence component separation[END_REF], the proposed method F I G U R E 1 An overview of a general grid-connected converter control system with PLL-based grid-synchronization method.

can have up to 26% lower memory requirement, which makes the technique very suitable for real-time implementation in low-cost embedded devices.

The rest of this article is organized as follows: development of the proposed estimator is given in Section 2, experimental results are provided in Section 3, and finally, concluding remarks are given in Section 4.

| ESTIMATOR DESIGN

An overview of solar energy-powered GCC, including the control system block, is given in Fig. 1. As shown in Fig. 1, grid-synchronization through PLL plays an important role in the outer voltage controller. Here, PLL separates the sequence components and extracts the grid voltage phase for the in-phase operation of the GCC with the grid. It is to be noted here that in this work, the focus is on PLL only, and Fig. 1 is used solely to contextualize the location of PLL in the converter control system. To facilitate this, let us consider the unbalanced three-phase grid voltages with measurement offset in the stationary reference frame as given by:

v α (t ) = V α0 + V + cos(ωt + δ + ) + V -cos(ωt + δ -), (1)
v β (t ) = V β 0 + V + sin(ωt + δ + ) -V -sin(ωt + δ -), (2)
where the superscripts + anddenote the positive and negative sequences, θ ± = ωt + δ ± are the total phase, V ± , ω, δ ± , V α0 , and V β 0 represent the amplitude, angular frequency, phase angle, and DC offsets in phases α and β , respectively. The extraction of the FFPS components

v + α = V + cos(θ + ), v + β = V + sin(θ + ); of the FFNS components v - α = V -cos(θ -), v - β = V -sin(θ -)
, and of the DC offsets V α0 ,V β 0 are considered in this article. In the sequel, the time-dependence of a signal is often not explicitly stated for convenience. Moreover, continuous-and discrete-time signals may be mixed for brevity.

| Delayed Signal Cancellation based Sequence Separation

Let us consider the voltage vector of (1), (2) defined as -→ v αβ = v α + j v β . Then, by applying the DSC method, the FFPS and FFNS sequences can be estimated as [START_REF] Svensson | Practical Implementation of Delayed Signal Cancellation Method for Phase-Sequence Separation[END_REF]:

- → v + αβ (t ) = 1 2 - → v αβ (t ) + j - → v αβ t - T n , (3)
- → v - αβ (t ) = 1 2 - → v αβ (t ) -j - → v αβ t - T n , ( 4
)
where T is the nominal signal period and n = 4 is the delay factor. This delay factor represents a quarter-cycle delay.

However, this delay can not block the DC offset. Another DSC operation with n = 2 needs to be connected in series ( [START_REF] Wang | Grid Synchronization PLL Based on Cascaded Delayed Signal Cancellation[END_REF]) to block the DC offset. As such, the total delay of sequence separation operation reaches three-quarters of a cycle. To reduce the quarter-cycle delay of sequence separation in the unbiased measurement case, i.e., without any DC offset, enhanced DSC is proposed in [START_REF] Cardenas | Fast convergence delayed signal cancellation method for sequence component separation[END_REF], requiring less than a quarter-cycle delay. Any arbitrary value of n < 4 can be used for sequence separation, making this method arbitrarily fast. A simplified version of this tool is proposed in [START_REF] Mellouli | A third-order MAF Based QT1-PLL that is robust against harmonically distorted grid voltage with frequency deviation[END_REF]. However, it still requires another DSC operation in-series with n = 2. So, the total delay is less than threequarters of a cycle but more than half-cycle of the signal period. Higher memory requirements and slow convergence issues are addressed below through our technique.

| Proposed Method

To facilitate the calculations, let us write

v τ α = v α (t -τ), v 2τ α = v α (t -2τ), v τ β = v β (t -τ) and v 2τ β = v β (t -2τ)
, where the time delay τ = NT s with N being the number of delayed samples and T s the sampling time. Then, the grid voltage (1), ( 2) and its two equidistant delayed samples v τ α , v 2τ α , v τ β and v 2τ β are given by:

v τ α (t ) = V α0 + V + cos θ + + V -cos (θ -) cos (φ τ ) + V + sin θ + + V -sin (θ -) sin (φ τ ) , (5a)
v 2τ α (t )= V α0 + V + cos θ + + V -cos (θ -) cos (2φ τ ) + V + sin θ + + V -sin (θ -) sin (2φ τ ) , (5b)
v τ β (t ) = V β 0 + V + sin θ + -V -sin (θ -) cos (φ τ ) + V -cos (θ -) -V + cos θ + sin (φ τ ) , (6a)
v 2τ β (t )= V β 0 + V + sin θ + -V -sin (θ -) cos (2φ τ ) + V -cos (θ -) -V + cos θ + sin (2φ τ ) , (6b)
where φ τ = ωτ. Then, the following algebraic relationship between the sequence components, DC offset, measured voltages, and its two equidistant delayed samples can be obtained as:

v = ΦΩ, (7)
where

v = v α v β v τ α v τ β v 2τ α v 2τ β T , Ω = V α0 v + α v - α V β 0 v + β v - β T , Φ =                 1 1 1 0 0 0 0 0 0 1 1 -1 1 c τ c τ 0 s τ s τ 0 -s τ s τ 1 c τ -c τ 1 c 2τ c 2τ 0 s 2τ s 2τ 0 -s 2τ s 2τ 1 c 2τ -c 2τ                 , F I G U R E 2
Proposed arbitrarily fast modified DSC method.

c τ = cos(φ τ ), c 2τ = cos(2φ τ ), s τ = sin(φ τ ), and s 2τ = sin(2φ τ ). From [START_REF] Meral | A comprehensive survey on control strategies of distributed generation power systems under normal and abnormal conditions[END_REF], Ω can be estimated as follows:

Ω = Φ -1 v, (8)
where . indicates estimated value and

Φ -1 =                      1 2(1-cτ ) 0 cτ cτ -1 0 1 2(1-cτ ) 0 1+cτ -c 2 τ -2c 3 τ 2 1-c 2 τ 1+cτ -2c 2 τ 2sτ 2c 2 τ -1 2(1-cτ ) 1-2s 2 τ 2sτ s 2 τ -1 2s 2 τ -cτ 2sτ cτ 2+cτ -2c 2 τ 2 c 2 τ -1 -cτ -2c 2 τ +1 2sτ 1+2cτ -2c 2 τ 2(1-cτ ) 2s 2 τ -1 2sτ 1+cτ -c 2 τ 2 c 2 τ -1 cτ 2sτ 1+2cτ 2sτ 1 -(1+cτ ) sτ 0 1 2sτ 0 -cτ (1+2cτ ) 2sτ 1 2 -c τ cτ (1+cτ ) sτ c τ -cτ 2sτ -1 2 -1+cτ -2c 2 τ 2sτ c τ -1 2 s τ -c τ cτ -1 2sτ 1 2                     
.

By simplifying [START_REF] Verma | A Robust Lyapunov Demodulator for Tracking of Single-/Three-Phase Grid Voltage Variables[END_REF], the sequences can be found as:

v ± α = 1 γ 2s τ c τ v α -v τ α ± 1 -2s 2 τ v β -v τ β ± c τ v 2τ β -v β ± v τ β -v 2τ β + s τ v 2τ α -v α , (9)
v ± β = 1 γ ± 2s τ c τ v β -v τ β + 1 -2s 2 τ v τ α -v α c τ v α -v 2τ α + v 2τ α -v τ α ± s τ v 2τ β -v β , (10)
where γ = 4s τ (c τ -1) . The presented sequence separation equations require few arithmetic operations and are very simple to implement in real-time. As shown in ( 9) and [START_REF] Klusacek | Power Symmetrical Components as Grid Usage Indicator for Unbalanced Prosumers[END_REF], two trigonometric operations are required to implement the equations. The implementation block diagram of the proposed technique is given in Fig. 2. F I G U R E 4 Overview of the experimental setup.

| Application to Phase-Locked Loop

Sequence separation formulas, as developed in Sec. 2.2, require the information on actual grid frequency. To obtain the actual grid frequency, CDSC-PLL [START_REF] Wang | Grid Synchronization PLL Based on Cascaded Delayed Signal Cancellation[END_REF] can be a suitable choice. An overview of the CDSC-PLL is given in Fig. 3 (a).

For more details about this PLL, the interested reader may consult [START_REF] Wang | Grid Synchronization PLL Based on Cascaded Delayed Signal Cancellation[END_REF][START_REF] Ullah | Comparison of Synchronization Techniques Under Distorted Grid Conditions[END_REF] and the references therein. In this approach, several DSC blocks are cascaded to eliminate the effect of harmonics and DC offset. Out of the five different DSC blocks, DSC 2 is used to eliminate the effect of DC offset, while DSC 4 is used to extract the FFPS component. These two blocks can be replaced by our method while the other DSC blocks are used as it is. CDSC-PLL with the proposed sequence extraction formulas is shown in Fig. 3 (c). Directly using the estimated frequency in the DSC blocks can cause instability. A low-pass filter (LPF) is added to ensure stability and to avoid any algebraic loop.

| RESULTS AND DISCUSSIONS

An overview of the experimental setup used in this work is given in Fig. 4. In this setup, a dSPACE 1104 board is used to generate the grid voltage signals. This signal is then passed through digital-to-analog converters (DAC). A Texas Instrument C2000 F28379D digital signal processor (DSP) is used for the practical implementation of the algorithms.

These algorithms were exported from Matlab/Simulink using code generation. A sampling frequency of 16 kHz is considered. Experimental results are exported to Matlab for plotting purposes. As comparative techniques, we have selected CDSC-PLL [START_REF] Wang | Grid Synchronization PLL Based on Cascaded Delayed Signal Cancellation[END_REF] (Fig. 3(a)) and Fast DSC-PLL (FDSC-PLL) (Fig. 3(b)). In the CDSC-PLL, five DSC blocks are Parameters of the synchronous reference frame (SRF)-PLL [START_REF] Kaura | Operation of a phase locked loop system under distorted utility conditions[END_REF] are selected the same for all three techniques. Cut-off frequency of the low-pass filter is selected as 60Hz. To implement the proposed DSC method, τ = T /4 is considered while τ = T /8 is considered for the fast DSC method. Total delays of the comparative techniques are given in Table 1.

For experimental validation, initially, the grid was balanced. At t = 0.02sec., the grid became unbalanced, distorted, and biased. The considered unbalanced grid is composed of - In this test, frequency step change of +1Hz is considered. Despite using the same loop-filter gain for all PLLs, the proposed method has a faster convergence with lower peak overshoot for the frequency estimation, as shown in Fig.

→ v +1 = 0.733∠0 • , - → v -1 = 0.21∠ -45 • , - → v -5 = 0.031∠45 • , - → v +7 = 0.028∠ -45 • , - → v -11 = 0.024∠180 • , - → v +13 = 0.015∠ -180 • ,

(a)

. Similar performance can be observed for the phase estimation error as well. This is particularly important as the estimated phase is used for reference fame transformation in the control system of grid-connected converter. So, a fast convergence would imply improved stability margin, facilitating a large-scale integration of RES in the power grid.

The summary of the results is tabulated in Table 1. Results show that the proposed method has the fastest convergence time and the lowest peak overshoot in all indicators. The estimated FFPS and FFNS amplitudes in Fig. 5 (d) show that fast DSC is sensitive to noise and harmonics. However, despite using lower memory than the conventional CDSC method, the steady-state performance of our method is similar to CDSC. So, it can be claimed that the proposed method can help achieve a faster convergence time without sacrificing the steady-state accuracy. The total memory requirement of the proposed method can be reduced further if τ = T /8 is considered. However, this value amplifies the measurement noise. If the signal is fairly clean and undistorted, we recommend this value. Otherwise, τ = T /4 can be a suitable choice in the presence of noise and harmonics.

| CONCLUSION AND FUTURE WORKS

An enhanced delayed signal cancellation method was proposed in this article for the grid-integration of distributed RES. The developed method can easily be integrated into the conventional cascaded delayed signal cancellation phaselocked loop, making it suitable as a grid-synchronization tool for grid-connected converters. The developed approach can estimate FFPS and FFNS components with a fast settling time. The proposed technique can estimate the signals within a fraction of the delay compared to the conventional counterpart being arbitrarily fast. However, a judicious choice must be made for the practical implementation between the convergence time, noise sensitivity, and peak overshoot. Experimental results showed that a total delay of half-cycle is a good choice for the proposed DSC technique in a distorted grid with noisy measurements. The method proposed in this work will ensure a fast and accurate integration of RES into an unbalanced distribution grid.

PLL is an integral part of the grid-connected converter system. Typically, a small-signal model of PLL is required for comprehensive tuning and stability analysis of the converter control system. Developing a small-signal model of the PLL can be considered as a future work. There is a growing interest in the power system community on studying the harmonic distortion caused by supraharmonics (i.e., between 2kHz and 150kHz). Sensitivity analysis of the proposed PLL to suprahramonics can also be considered as future work.

  Integrating renewable energy sources into an unbalanced distribution network requires fast and accurate extraction of fundamental frequency positive-and negative-sequence components from the unbalanced three-phase grid voltage signals. For this purpose, various methods are already available in the literature. Out of them, delayed signal cancellation (DSC) is prevalent. Conventional DSC can separate the sequence components using a quarter-cycle delay. Fast DSC tools can achieve the same with less than a quartercycle delay. However, neither conventional nor fast DSC can handle DC offset without requiring additional delayed signals. This article addresses this issue by proposing a modified DSC to estimate the sequence components with DC offset rejection and having arbitrarily fast convergence speed, i.e., low memory requirement. Two equidistant delayed samples of the measured grid voltages/currents are required to implement the proposed technique and can easily be applied in a phase-locked loop (PLL). Comparative experimental results demonstrate the suitability of the proposed approach over other DSC methods.

F

  I G U R E 3 DSC-based PLL implementations: (a) conventional CDSC-PLL; (b) Fast CDSC-PLL; and (c) proposed method.

  Estimated FFPS and FFNS amplitudes F I G U R E 5 Comparative experimental results.used in series namely DSC 2 , DSC 4 , DSC 8 , DSC 16 , and DSC 32 . In FDSC-PLL, only the DSC 4 operator of CDSC-PLL is replaced by the fast DSC method proposed in[START_REF] Cardenas | Fast convergence delayed signal cancellation method for sequence component separation[END_REF][START_REF] Mellouli | A third-order MAF Based QT1-PLL that is robust against harmonically distorted grid voltage with frequency deviation[END_REF], whereas the remained DSC blocks of CDSC-PLL are kept as it is. Similarly, in the proposed method, only the DSC 2 and DSC 4 blocks are replaced by the developed single DSC block.

  and -→ v +30Hz = 0.01∠90 • . In addition, DC offsets of 0.15p.u., -0.15p.u, and 0.1p.u. are added to the first, second and third phases of the three-phase voltage signals, respectively. The experimental signal had a signal-to-noise ratio of 38dB. Experimental results are given in Fig.5.

  TA B L E 1 Comparative time-domain performance summary.

	Performance indicators ⇓	CDSC	FDSC	Proposed
	Frequency settling time (±0.1Hz) (sec.)	0.94	0.0914	0.0884
	Frequency peak overshoot (Hz)	0.35	0.26	0.18
	Phase error settling time (±0.2 • ) (sec.)	0.164	0.15	0.115
	Peak phase error ( • )	7.68	6.07	4.62
	FFPS settling time (±0.02p.u.)	0.0168	0.016	0.0133
	FFNS settling time (±0.02p.u.)	0.0172 0.0152	0.0139
	Total delay	31T 32	27T 32	23T 32

Funding information

Sêr Cymru II 80761-BU-103 project by Welsh European Funding Office (WEFO) under the European Regional Development Fund (ERDF)


Pauline Tarot

Laia Castell

Yuki Nakamura

Coline Rulhe

Aparicio Juri

Laura Arias

Federica Cutando

Denis Bertaso

Emmanuel Hervé

Valjent

Juri Aparicio Arias

Laura Cutando

Federica Bertaso

Denis Hervé

Emmanuel Valjent
email: emmanuel.valjent@igf.cnrs.fr

Psychostimulant Drugs Activate Cell-type Specific and Topographic cFos Expression in the Lumbar Spinal Cord

Keywords: psychostimulants, spinal cord, dopamine, cFos-immunoreactive neurons

come   L'archive ouverte pluridisciplinaire

Introduction

Motor behaviors require the activation of spatially distributed and interconnected neural networks in the forebrain, brainstem and spinal cord [START_REF] Grillner | Current Principles of Motor Control, with Special Reference to Vertebrate Locomotion[END_REF]. The sequential recruitment of specific circuits along this axis allows the selection, initiation, execution and termination of motor actions with precision. Drugs exacerbating dopamine (DA) and norepinephrine release profoundly affect the finetuning control of motor behaviors. Indeed, a single exposure to psychostimulant drugs such as d-amphetamine, cocaine or methylphenidate enhances motor behaviors whose manifestations depend on the dose administered [START_REF] Gaytan | Time-dependent differences in the rat's motor response to amphetamine[END_REF][START_REF] Yates | Bimodal effect of amphetamine on motor behaviors in C57BL/6 mice[END_REF]. At low doses, these drugs stimulate fast exploratory activity characterized by an increase in locomotion and rearing [START_REF] Yates | Bimodal effect of amphetamine on motor behaviors in C57BL/6 mice[END_REF][START_REF] Ztaou | Single Dose of Amphetamine Induces Delayed Subregional Attenuation of Cholinergic Interneuron Activity in the Striatum[END_REF].

However, at high doses, psychostimulants promote a variety of repetitive behaviors ranging from in-place stereotypies (continuous sniffing, circling, intense grooming, and self-gnawing) to patterned locomotion and continuous up-and-down rearing motion often accompanied by intermittent episodes of wall licking [START_REF] Yates | Bimodal effect of amphetamine on motor behaviors in C57BL/6 mice[END_REF][START_REF] Ztaou | Single Dose of Amphetamine Induces Delayed Subregional Attenuation of Cholinergic Interneuron Activity in the Striatum[END_REF].

The broad motor repertoire induced by psychostimulants is highly correlated with neuronal activation within the striatum, a brain structure densely innervated by midbrain DA neurons and involved in selecting and initiating motor actions [START_REF] Canales | Patterns of gene expression and behavior induced by chronic dopamine treatments[END_REF]. Indeed, specific patterns of neuronal activation classically identified by evaluating the expression pattern of immediate early genes are recruited following acute exposure to psychostimulants [START_REF] Berke | A complex program of striatal gene expression induced by dopaminergic stimulation[END_REF][START_REF] Graybiel | Amphetamine and cocaine induce drug-specific activation of the c-fos gene in striosome-matrix compartments and limbic subdivisions of the striatum[END_REF][START_REF] Moratalla | Coordinate expression of c-fos and jun B is induced in the rat striatum by cocaine[END_REF]. For instance, cocaine or methylphenidate induce a widespread expression of the proto-oncogene cFos in striosomal and matrix compartments in both sensorimotor and limbic striatal domains [START_REF] Canales | Patterns of gene expression and behavior induced by chronic dopamine treatments[END_REF][START_REF] Graybiel | Amphetamine and cocaine induce drug-specific activation of the c-fos gene in striosome-matrix compartments and limbic subdivisions of the striatum[END_REF]. In contrast, d-amphetamine increases cFos levels prominently in striosomes, a pattern of expression highly predictive of the intensity of motor stereotypies [START_REF] Canales | Patterns of gene expression and behavior induced by chronic dopamine treatments[END_REF]. Psychostimulant exposure also enhances cFos immunoreactivity in the pedunculopontine nucleus and the cuneiform nucleus, two brainstem excitatory nuclei of the mesencephalic locomotor region regulating motor behaviors [START_REF] Geisler | Prominent activation of brainstem and pallidal afferents of the ventral tegmental area by cocaine[END_REF]. However, psychostimulants-induced neuronal activation within executive spinal motor circuits have never been investigated.

In the present study, we take advantage of an immediate early gene-based method to characterize neuronal populations activated in the spinal cord in response to a single exposure to d-amphetamine. By using a variety of reporter mouse lines, we investigated the cell-type specific identity of amphetamine-induced cFos expression.

We finally assessed the ability of different classes of locomotor-stimulating drugs to trigger neuronal activation in the lumbar spinal cord.

Materials and methods

Animals

Male C57BL/6 (n = 65) from Charles River Laboratories, Gad67-eGFP (Tg(Gad2-eGFP)DJ31Gsat) (n = 3), GlyT2-eGFP (Tg(Slc6a5-eGFP)1Uze) (n = 4), VGluT2-RiboTag (n = 3) and Fos-Trap (n = 9) mice were used in the present study. Vglut2-Ribotag mice were generated by crossing VGluT2-Cre (Tg(Slc17a6-icre)1Oki) and RiboTag [START_REF] Sanz | Cell-type-specific isolation of ribosome-associated mRNA from complex tissues[END_REF] mice and Fos-Trap were generated by crossing Fos-2A-iCreER [START_REF] Allen | Thirst-associated preoptic neurons encode an aversive motivational drive[END_REF] and R26-Trap [START_REF] Liu | Cell-specific translational profiling in acute kidney injury[END_REF] mice. Male 8-to 12-week old mice (25-30 gr) were used in the current study. All mice were housed in groups of 2 to 5 per cage (standard sizes according to the European animal welfare guidelines 2010/63/EU) and maintained in a 12h light/dark cycle (lights on from 7:00 am to 7:00 pm), in stable conditions of temperature (22°C) and humidity (60%), with food and water provided ad libitum. In all experiments, arbitrary assignments were used to allocate mice to specific treatments. No randomization was performed to allocate subjects in the study. All animal procedures were conducted in accordance with the guidelines of the French Agriculture and Forestry Ministry for handling animals (authorization number/license B34-172-41) and approved by the relevant local and national ethics committees (authorizations APAFIS#15092 and APAFIS#26765).

Drugs and treatment

Drugs, drug doses and routes are detailed in Table 1. All drugs were administered at doses inducing known behavioral and/or molecular responses in mice (Gangarossa et al., 2019;[START_REF] Valjent | Mechanisms of locomotor sensitization to drugs of abuse in a twoinjection protocol[END_REF]. For in vivo pharmacological experiments, mice were habituated to handling and saline injections three consecutive days before the experiments. Drugs were dissolved in 9 g/L NaCl (saline) and administered on day 4.

All the mice were injected in the home cage and perfused 90 min after injection except for the time-course experiment for which mice were also perfused at 30, 60, and 120 min after d-amphetamine administration. None of these manipulations were classified as painful or invasive.

Fos-Trap experiments

4-hydroxytamoxifen (4-OHT, H6278, Sigma-Aldrich, Saint-Quentin Fallavier, France) was dissolved in 1 ml ethanol (100%). Corn oil (C8267, Sigma-Aldrich) was then added and ethanol was evaporated by incubating the open tubes at 50°C overnight in the dark. The volume of corn oil was adjusted for injecting the dose of active drug at 10 ml/kg. 4-OHT was administered i.p. at the dose of 50 mg/kg of active Z form. D-amphetamine sulfate (Sigma-Aldrich) was dissolved in 0.9% (w/v) NaCl (saline) and injected i.p. (10 mg/kg). During the 6 days preceding treatments, Fos-Trap mice were habituated to the experimental conditions in the aim to limit the Fos gene induction by novelty. The first day, animals were just handled for 5 min.

The next 2 days, they were handled and pricked in the abdomen with a syringe needle without injection. At day 4, mice were handled and received one injection of saline.

The following two days, they received 2 injections of saline solution in the left and right sides of the abdomen. After each manipulation, the animals were immediately placed back in their home cage. On day 7, mice received an injection of 4-OHT (50 mg/kg, i.p.) and immediately after, 5 of them were administered with d-amphetamine (10 mg/kg, i.p.) and the 4 others with saline solution. After injections, mice were returned to their home cage. Seven days later, all the mice were treated with damphetamine (10 mg/kg, i.p.).

Tissue preparation and immunofluorescence

Ninety minutes after drugs administration (or at 30, 60, 90, 120 minutes for the time course experiments), mice were anaesthetized with Euthasol (TVM lab, France) at a dose (340 mg/kg) allowing rapid, efficient and deep anesthesia, which constitute a prerequisite to avoid the dephosphorylation of proteins [START_REF] Gangarossa | Contrasting patterns of ERK activation in the tail of the striatum in response to aversive and rewarding signals[END_REF].

Mice were then transcardially perfused (every 7 min by alternating groups) with 40 g/L paraformaldehyde prepared in 0.1M sodium phosphate buffer (pH 7.5). Spinal cords were extracted, post-fixed overnight in the same solution and stored at 4°C. The lumbar segments were then included in 4% agarose and 40-µm thick sections were cut using a vibratome (Leica VT 1000S, France). Slices were stored at -20°C in a solution containing 30% (vol/vol) ethylene glycol, 30% (vol/vol) glycerol, and 0.1M sodium phosphate buffer, until they were processed for immunofluorescence [START_REF] Bertran-Gonzalez | Opposing patterns of signaling activation in dopamine D1 and D2 receptor-expressing striatal neurons in response to cocaine and haloperidol[END_REF]. Lumbar sections were identified using the Allen Mouse Spinal Cord Atlas (mousespinal.brain-map.org). Free-floating sections were rinsed three times (10 min/wash) in Tris-buffered saline (TBS, 50 mM Tris-HCl, 150 mM NaCl, pH 7.5) before 15 min of incubation in 0.2% (vol/vol) Triton X-100 in TBS. Sections were then rinsed three times (10 min/wash) in TBS before 60 min blocking in a solution of 3% BSA in TBS. Sections were incubated for 72 h at 4°C with the primary antibodies (Table 2) diluted in a TBS solution containing 1% BSA and 0.15% Triton X-100 [START_REF] Puighermanal | drd2-cre:ribotag mouse line unravels the possible diversity of dopamine d2 receptor-expressing cells of the dorsal mouse hippocampus[END_REF]. Following three rinses of 10 min in TBS, sections were incubated for 45 min at 4°C with the secondary antibodies: goat Cy3-coupled anti-rabbit (1:500, Jackson Immuno-Research, Cat# 111-165-003), goat Alexa Fluor 488-coupled anti-chicken (1:500, Thermo Fisher Scientific, Cat# A-11039), and goat Alexa Fluor 488-coupled anti-mouse (1:500, Thermo Fisher Scientific, Cat# A-11001). Following incubation with secondary antibodies, sections were rinsed twice for 10 min in TBS. Mounting was performed using DPX (Sigma-Aldrich, Saint-Quentin Fallavier, France). Confocal microscopy was carried out at the Montpellier RIO Imaging Facility. Single or double-immunolabeled images were single confocal sections acquired using sequential laser scanning confocal microscopy (Leica SP8 or Zeiss LSM780). pS32-cFos-positive cells were pseudocolored in red while GFP-, HA-, CB-, PV-and CR-immunoreactive cells were pseudocolored in green.

Quantification

Cells were considered positive for a given marker only when the nucleus was clearly visible [START_REF] Bertran-Gonzalez | Opposing patterns of signaling activation in dopamine D1 and D2 receptor-expressing striatal neurons in response to cocaine and haloperidol[END_REF]. Quantification was performed manually taking as standard reference a fixed threshold of fluorescence. Histograms in Figures 1,3,4, 5 and 6, represent the number of neurons averaged from 3 to 4 lumbar sections. 2 and3 show the co-expression of pS32-cFos/markers as percentage of pS32-cFos-and GFP-immunoreactive neurons respectively, in the distinct laminae of the lumbar spinal cord. Numbers of positive neurons summed from 3 to 4 lumbar sections per mice are reported in Table 3.

Histograms in Figures

Statistical analysis

Data were analyzed using one-way ANOVA or Student t-test. Significance threshold was set at p < 0.05. One-way ANOVA analysis was followed by Šidák's multiple comparisons test.

Results

Immediate early gene expression-based method is a widely used approach to characterize neuronal activation at the regional and cellular level. Here we tested the ability of various psychostimulant drugs to recruit specific neuronal populations in the lumbar spinal cord by analyzing the spatio-temporal and molecular distribution of cells expressing the phosphorylated form of cFos at Ser32 (pS32-cFos), a posttranslational modification associated with an increased stability, nuclear localization and activity of cFos [START_REF] Sasaki | Spatiotemporal regulation of c-Fos by ERK5 and the E3 ubiquitin ligase UBR1, and its biological role[END_REF].

Acute d-amphetamine increases pS32-cFos expression in lumbar spinal cord

We first investigated whether d-amphetamine promoted neuronal activation in the mouse lumbar spinal cord. To do so, the number of pS32-cFos-immunoreactive neurons was quantified in lamina I to X of the lumbar spinal cord 30, 60, 90 and 120 min after a single intraperitoneal administration of d-amphetamine (10 mg/kg) (Figure 1). In saline-treated mice (90 min post-injection), few pS32-cFos-positive neurons (<20 cells) were detected in lamina II, III, VI and VII (Figure 1a-b). This pattern remained unchanged 30 min after d-amphetamine injection (Figure 1a-b).

However, the analysis of pS32-cFos expression at 60, 90 and 120 min revealed a gradual increase in pS32-cFos-positive neurons in the dorsal and ventral horns,

D-amphetamine activates both inhibitory and excitatory neurons in the lumbar spinal cord

To determine whether d-amphetamine-induced lumbar spinal cord neuronal activation occurred preferentially in inhibitory and/or excitatory neurons, pS32-cFos expression was monitored in Gad67-eGFP, GlyT2-eGFP and VGluT2-Ribotag mice, which allow the immunohistological identification of GABAergic, glycinergic and glutamatergic neurons, respectively (Figure 2). Based on the kinetics of pS32-cFos expression (Figure 1a), all the subsequent analyses were performed in mice perfused 90 min after d-amphetamine administration. First, we analyzed the distribution of pS32-cFosimmunoreactive neurons in Gad67-eGFP mice, which express the green fluorescent protein (GFP) under the control of the Gad67 promoter (GABAergic neurons) (Figure 2a). Our analysis revealed that the highest percentage of pS32-cFos cells coexpressing GFP was found in the superficial laminae (~35% in lamina I; ~18% in lamina II; ~25% in lamina III and ~15% in lamina IV) and in the lamina X surrounding the central canal (~14%) (Figure 2a-b). In contrast, low or no pS32-cFos/GFP co-expressing cells was detected in lamina V (~2%), VI (~1%), VII (~5%), VIII (~5%) and IX (0) (Figure 2a-b). These results indicate that d-amphetamineinduced pS32-cFos expression occurs in a small fraction of Gad67-expressing neurons.

A distinct cell type-specific pattern of pS32-cFos/GFP-positive neurons was observed when the analysis was performed in GlyT2-eGFP mice that allow the identification of glycinergic neurons (Figure 2c-d). Indeed, percentages of pS32-cFos/GFP-immunoreactive cells were high in laminae activated in response to damphetamine reaching ~55% in lamina VI and ~40% in lamina V, and gradually decreasing in lamina VIII, X and VII (~28%, ~14% and ~12%, respectively) (Figure 2c-d). No co-localizations were detected in lamina I, II and IX (Figure 2c-d). In contrast, the percentage of pS32-cFos/GFP co-expressing cells reached ~26% and ~19% in lamina IV and III, respectively (Figure 2c-d). These findings indicate that damphetamine induces pS32-cFos expression preferentially in glycinergic inhibitory neurons.

Finally, the distribution of pS32-cFos expression was analyzed in VGluT2expressing excitatory neurons (Figure 2e-f). To do so, we generated VGlut2-Ribotag mice, which express the ribosomal protein Rpl22 tagged with hemagglutinin (HA) exclusively in glutamatergic VGluT2 neurons. As shown in Figure 2f, percentages of pS32-cFos/HA co-expressing neurons were high in all the laminae in which damphetamine increased pS32-cFos expression (~40% in lamina VII and VIII; ~33% in lamina X; ~28% in lamina VI and ~20% in lamina V) (Figure 2e-f). Altogether, our analyses show that d-amphetamine increases pS32-cFos expression in both inhibitory and excitatory neurons expressing GlyT2/Gad67 and VGluT2, respectively.

Distribution of d-amphetamine-induced pS32-cFos immunoreactive cells among neurons identified by calcium-binding proteins

To further characterize the molecular identity of activated pS32-cFos-positive neurons in the lumbar spinal cord in response to d-amphetamine, we analyzed the percentage of co-localization between pS32-cFos-immunoreactive neurons and those expressing calcium-binding proteins (CBP), calbindin-D28k (CB), calretinin (CR), and parvalbumin (PV) (Figure 2g-l). Overall, our analysis revealed low or no colocalization between pS32-cFos and CB in lamina V, VI and VII (~1%) and lamina VIII and IX (0) (Figure 2g-h). Similarly, pS32-cFos/CR-positive cells represented ~2% of the total pS32-cFos in the lamina V and VIII, and only ~1% in lamina VI and VII (Figure 2i-j). The highest percentages of co-localization were found between pS32-cFos and PV in lamina V (~7%) and VI (~4%), corresponding to 8 and 13 pS32-cFos/PV-positive cells among the 108 and 283 pS32-cFos-immunoreative cells (Figure 2k-l, Table 3). Strikingly, although d-amphetamine failed to significantly increase pS32-cFos in lamina II (see Figure 1), our analysis revealed that a high percentage of the few pS32-cFos-positive neurons co-expressed CR (~31%) and to a lesser extent PV (~9%) and CB (~7%) (Figure 2i-j). Together, these results indicate that pS32-cFos expression induced by d-amphetamine does not preferentially occur in calcium-binding proteins-expressing cells in the lumbar spinal cord.

Reactivation of d-amphetamine-tagged neurons following d-amphetamine reexposure in the lumbar spinal cord

We sought to determine whether lumbar spinal cord neuronal activation evoked by a single exposure of d-amphetamine was more prone to be re-activated by a second administration of d-amphetamine. cFos-CreERT2-Trap mice were injected with saline or d-amphetamine (10 mg/kg) prior to the administration of 4-OHT (day 1), triggering the cFos driven expression of GFP in activated neurons, and re-injected with damphetamine a week after (day 7). No significant changes were detected in laminae I (t 7 = 1.758, p = 0.1222), II (t 7 = 0.8013, p = 0.4493), VIII (t 7 = 1.610, p = 0.1514) and IX (t 7 = 0.1584, p = 0.8786) (Figure 3a-b). In contrast, d-amphetamine-treated mice on day 1 displayed a higher number of GFP-tagged neurons compared to salineinjected mice in lamina III (t 7 = 3.606, p = 0.0087), IV (t 7 = 3.382, p = 0.0117), V (t 7 = 2.540, p = 0.0387), VI (t 7 = 2.683, p = 0.0314), VII (t 7 = 2.949, p = 0.0214) and X (t 7 = 3.085, p = 0.0177) (Figure 3a-b). Interestingly, the analysis of the percentage of co-localization between pS32-cFos-immunoreactive neurons induced by the administration of d-amphetamine on day 7 and those expressing GFP in response to a first injection of d-amphetamine revealed that about one-third of the d-amphetaminetagged neurons were re-activated following d-amphetamine re-exposure (~43-44% in lamina III and X; ~37-38% in lamina VI and IV; ~33% in lamina V and ~29% in lamina VII) (Figure 3c-d).

Various psychostimulant drugs increase pS32-cFos expression in lumbar spinal cord

We next investigated whether two psychostimulant drugs, cocaine and methylphenidate, could trigger the expression of pS32-cFos in the lumbar spinal cord.

Similarly to d-amphetamine, a single administration of cocaine (15 mg/kg) or methylphenidate (15 mg/kg) increased pS32-cFos expression in lumbar spinal cord at 90 min (Figure 4). All tested psychostimulant drugs tested shared the property of increasing the number of pS32-cFos-positive neurons in lamina III (cocaine t 4 = 5.367, p = 0.0058; methylphenidate t 5 = 4.283, p = 0.0078), V (cocaine t 4 = 3.286, p = 0.0303; methylphenidate t 5 = 4.048, p = 0.0098), VI (cocaine t 4 = 7.141, p = 0.0020; methylphenidate t 5 = 3.901, p = 0.0114), VII (cocaine t 4 = 7.906, p = 0.0014; methylphenidate t 5 = 5.615, p = 0.0025) and VIII (cocaine t 4 = 2.286, p = 0.0843; methylphenidate t 5 = 2.46, p = 0.0572) (Figures 1b and4a-d). On the other hand, the increased pS32-cFos expression in lamina I was only observed after the administration of cocaine (t 4 = 3.051, p = 0.038; methylphenidate t 5 = 0.9555, p = 0.3832) (Figure 4a-b), while the increase in lamina X was only produced by methylphenidate (cocaine t 4 = 0.3592, p = 0.7376; methylphenidate t 5 = 4.792, p = 0.0049) (Figure 4c-d). No significant changes were detected in laminae II (cocaine t 4 = 1.908, p = 0.1290; methylphenidate t 5 = 0.4085, p = 0.6998) and IV (t 4 = 1.835, p = 0.1404; methylphenidate t 5 = 1.832, p = 0.1265) At the doses used here, acute d-amphetamine, cocaine and methylphenidate cause strong locomotor responses [START_REF] Gangarossa | Contrasting patterns of ERK activation in the tail of the striatum in response to aversive and rewarding signals[END_REF][START_REF] Valjent | Mechanisms of locomotor sensitization to drugs of abuse in a twoinjection protocol[END_REF]. To determine whether the induction of pS32-cFos in the lumbar spinal cord was the sole consequence of a generalized hyperlocomotion, we analyzed pS32-cFos expression in response to morphine, an agonist of µ, δ and κ opioid receptors, and dizocilpine (MK801), a noncompetitive NMDA receptor blocker. As shown in Figure 5, acute administration of morphine (5 mg/kg) or dizocilpine (0.3 mg/kg) administered at doses capable to enhance locomotor activity [START_REF] Puighermanal | Functional and molecular heterogeneity of D2R neurons along dorsal ventral axis in the striatum[END_REF][START_REF] Valjent | Mechanisms of locomotor sensitization to drugs of abuse in a twoinjection protocol[END_REF] failed to trigger pS32-cFos expression in the lumbar spinal cord (lamina I: F 2, 12 = 0.6433, p = 0.5428; lamina II: F 2, 12 = 1.571, p = 0.2477; lamina III: F 2, 12 = 0.1978, p = 0.8232; lamina IV: F 2, 12 = 0.9372, p = 0.4186; lamina V: F 2, 12 = 1.397, p = 0.2848; lamina VI: F 2, 12 = 0.8756, p = 0.4416; lamina VII: F 2, 12 = 3.906, p = 0.0494; lamina VIII: F 2, 12 = 3.21, p = 0.0764; lamina X: F 2, 12 = 0.6678, p = 0.5309) (Figure 5a-b). Together, these results suggest that the induction of pS32-cFos in response to psychostimulant is not a direct consequence of hyperlocomotion but rather of a specific effect of this class of drugs.

Regulation of pS32-cFos expression in lumbar spinal cord by monoamine reuptake inhibitors

D-amphetamine, cocaine, and methylphenidate increase the extracellular concentration of DA, norepinephrine, and serotonin [START_REF] Torres | Plasma membrane monoamine transporters: structure, regulation and function[END_REF]. Therefore, we investigated whether increasing the extracellular concentration of these monoamines was sufficient to trigger pS32-cFos expression in lumbar spinal cord. We compared pS32-cFos expression in C57BL/6 mice administered with d-amphetamine or different monoamine reuptake inhibitors 90 min after injection (lamina I: F 3, 11 = 1.143, p = 0.3747; lamina II: F 3, 11 = 1.908, p = 0.1867; lamina III: F 3, 11 = 6.276, p = 0.0097; lamina IV: F 3, 11 = 5.729, p = 0.0128; lamina V: F 3, 11 = 16.07, p = 0.0002; lamina VI: F 3, 11 = 19.42, p = 0.0001; lamina VII: F 3, 11 = 7.06, p = 0.0065; lamina VIII: F 3, 11 = 6.325, p = 0.0094; lamina X: F 3, 11 = 4.131, p = 0.0345) (Figure 6). The administration of mixed norepinephrine and serotonin reuptake inhibitors, desipramine (15 mg/kg) or imipramine (10 mg/kg), failed to increase pS32-cFos expression in lumbar spinal cord (Figure 6a-b). In contrast, the selective DA reuptake inhibitor, GBR12783 (15 mg/kg), injected a dose increasing locomotion [START_REF] Valjent | Mechanisms of locomotor sensitization to drugs of abuse in a twoinjection protocol[END_REF] triggered a pattern of pS32-cFos expression that resembled the one induced by psychostimulant drugs, thus indicating that blockade of DA reuptake is sufficient to increase pS32-cFos expression in lumbar spinal cord (Figure 6a-b).

Discussion

The present study unveils that a single administration of d-amphetamine, at a dose inducing motor activity and stereotypies [START_REF] Gangarossa | Contrasting patterns of ERK activation in the tail of the striatum in response to aversive and rewarding signals[END_REF], activates pS32-cFos-expressing neurons in the dorsal and ventral lumbar spinal cord. We also demonstrate that d-amphetamine-induced pS32-cFos occurs in both excitatory and inhibitory interneurons distributed in various laminae associated with sensory processing and motor function. In contrast, no significant changes in pS32-cFos expression were detected in motor neurons or in superficial laminae of the dorsal spinal cord. Finally, we provide evidence that GBR12783, a selective DA reuptake inhibitor, is sufficient to trigger a similar pattern of increased pS32-cFos expression to those induced by all dopaminomimetic drugs.

In rodents, a single administration of d-amphetamine induces a broad repertoire of motor behaviors ranging from increased locomotion (from 2 to 5 mg/kg) to repetitive/stereotyped motor sequences (above 5 mg/kg) [START_REF] Crittenden | Severe drug-induced repetitive behaviors and striatal overexpression of VAChT in ChAT-ChR2-EYFP BAC transgenic mice[END_REF][START_REF] Mansouri-Guilani | VGLUT3 gates psychomotor effects induced by amphetamine[END_REF][START_REF] Valjent | Mechanisms of locomotor sensitization to drugs of abuse in a twoinjection protocol[END_REF][START_REF] Yates | Bimodal effect of amphetamine on motor behaviors in C57BL/6 mice[END_REF]. Our results indicate that the latter are correlated with an increased expression of pS32-cFos in neurons in the ventral horn of the spinal cord, a spinal region harboring neural circuits dedicated to execution and generation of rhythmic motor behaviors [START_REF] Grillner | Current Principles of Motor Control, with Special Reference to Vertebrate Locomotion[END_REF]. Redex laminae delineation-based analysis revealed that damphetamine failed to increase pS32-cFos expression in lamina IX where motor neurons are located. Conversely, pS32-cFos-immunolabeled neurons were observed in laminae VII and VIII suggesting that d-amphetamine might preferentially activate a subpopulation of neurons of the locomotor central pattern generator network [START_REF] Grillner | Current Principles of Motor Control, with Special Reference to Vertebrate Locomotion[END_REF]. The high percentage of pS32-cFos/VGluT2 and pS32-cFos/GlyT2-positive neurons in the aforementioned layers indicates that damphetamine-induced pS32-cFos may occur in a fraction of both excitatory and inhibitory interneurons of the locomotor central pattern generator network [START_REF] Grillner | Current Principles of Motor Control, with Special Reference to Vertebrate Locomotion[END_REF]. The analysis of the distribution of pS32-cFos among the CBPexpressing interneurons constituted a first attempt to clarify the molecular identity of pS32-cFos-positive cells [START_REF] Alvarez | Postnatal phenotype and localization of spinal cord V1 derived interneurons[END_REF][START_REF] Floyd | Characterization of calbindin D28k expressing interneurons in the ventral horn of the mouse spinal cord[END_REF][START_REF] Lu | Molecular and cellular development of spinal cord locomotor circuitry[END_REF].

Thus, the lack of pS32-cFos/CB-positive cells along the ventral-gray-white matter border in laminae VII and VIII indicates that d-amphetamine does not activate Renshaw cells, which are known to mediate recurrent inhibition of motor neurons [START_REF] Alvarez | Postnatal phenotype and localization of spinal cord V1 derived interneurons[END_REF]. Moreover, the negligible percentage of pS32-cFos found in CB/PV-positive neurons detected in lamina VII also suggests that d-amphetamineactivated cells do not recruit V1-derived interneurons [START_REF] Alvarez | Postnatal phenotype and localization of spinal cord V1 derived interneurons[END_REF]. Given the motor patterns induced by d-amphetamine, future experiments will be necessary to determine whether increased pS32-cFos expression occurs in excitatory V2 and inhibitory V0 interneurons, both controlling the left-right coordination of limbs. D-amphetamine induced a prominent increase of pS32-cFos in excitatory (VGluT2 + ) and inhibitory (GlyT2 + and to a lesser extent Gad67 + ) neurons of laminae V and VI of the SC dorsal horn. This spinal region contains premotor neurons also designated as motor synergy encoder neurons [START_REF] Levine | Identification of a cellular node for motor control pathways[END_REF][START_REF] Osseward | Cell type and circuit modules in the spinal cord[END_REF]. These neurons, which derive from discrete dorsal and ventral progenitor domains [START_REF] Lai | Making sense out of spinal cord somatosensory development[END_REF], control the activation of spatially segregated pools of motor neurons, thereby participating in the fine-tuning of complex coordinated motor actions [START_REF] Levine | Identification of a cellular node for motor control pathways[END_REF]. They also integrate convergent proprioceptive and corticospinal signals conveying sensory feedbacks necessary to adjust motor behaviors [START_REF] Hilde | Satb2 Is Required for the Development of a Spinal Exteroceptive Microcircuit that Modulates Limb Position[END_REF][START_REF] Osseward | Cell type and circuit modules in the spinal cord[END_REF]. Thus, by exacerbating the activity of motor synergy encoder neurons through a mechanism that remains to be established, psychostimulants may hijack the aforementioned functions altering somatosensory processing and therefore favoring patterned motor activities as well as in-place stereotyped motor sequences.

DA arising from midbrain and forebrain nuclei modulates the activity of neural networks controlling motor behaviors [START_REF] Grillner | Current Principles of Motor Control, with Special Reference to Vertebrate Locomotion[END_REF]. Indeed, ascending and descending midbrain DA systems have been characterized, all serving distinct but complementary roles in the regulation of movement. For instance, midbrain (A9 and A10 nuclei) DA neurons are known to regulate motor planning and execution indirectly through ascending projections to basal ganglia circuits [START_REF] Grillner | Current Principles of Motor Control, with Special Reference to Vertebrate Locomotion[END_REF]). An increased extracellular concentration of DA within these circuits has been causally associated with the ability of psychostimulant drugs to promote motor hyper-responsiveness as well as neuronal activation [START_REF] Graybiel | Amphetamine and cocaine induce drug-specific activation of the c-fos gene in striosome-matrix compartments and limbic subdivisions of the striatum[END_REF][START_REF] Moratalla | Coordinate expression of c-fos and jun B is induced in the rat striatum by cocaine[END_REF]. Interestingly, increasing evidence suggests that DA might also indirectly control locomotion through A9/A10 and A13 descending DA pathways which innervate the two major brainstem excitatory nuclei of mesencephalic locomotor region, the pedunculopontine nucleus and the cuneiform nucleus [START_REF] Ryczko | A descending dopamine pathway conserved from basal vertebrates to mammals[END_REF][START_REF] Ryczko | Dopamine and the Brainstem Locomotor Networks: From Lamprey to Human[END_REF][START_REF] Sharma | Parallel descending dopaminergic connectivity of A13 cells to the brainstem locomotor centers[END_REF]. Finally, recent works indicate that opto-activation of the descending DA pathway originating from A11 nuclei known to modulate spinal networks involved in the control of rhythmic movements [START_REF] Han | Dopaminergic modulation of spinal neuronal excitability[END_REF][START_REF] Pappas | Lack of D2 receptor mediated regulation of dopamine synthesis in A11 diencephalospinal neurons in male and female mice[END_REF][START_REF] Sharples | A dynamic role for dopamine receptors in the control of mammalian spinal networks[END_REF][START_REF] Sharples | Dopamine: a parallel pathway for the modulation of spinal locomotor networks[END_REF] is sufficient to increase both locomotion and in-place activity [START_REF] Koblinger | Optogenetic Activation of A11 Region Increases Motor Activity[END_REF]. Surprisingly, no evidence has yet indicated that this pathway might participate in the regulation of motor behaviors induced by psychostimulants. Though addressing this question was not the purpose of the present work, our findings raise the question of the contribution of descending DA neurons in the increase of spinal cFos expression by psychostimulants. Thus, although the widespread distribution of DA projections in lumbar spinal cord might at least in part explain the pattern of cFos induction, functional features of A11 DA neurons question the mechanisms by which psychostimulants would promote the increase of spinal DA concentration. Indeed, spinal DA projections arising from A11 nucleus lack the dopamine transporter [START_REF] Koblinger | Characterization of A11 neurons projecting to the spinal cord of mice[END_REF][START_REF] Pappas | Lack of D2 receptor mediated regulation of dopamine synthesis in A11 diencephalospinal neurons in male and female mice[END_REF], suggesting that the DA-dependent neuronal in the spinal cord may not result from the direct action of psychostimulants onto A11 TH-positive terminals but rather from a network effect.

Beside their ability to increase DA extracellular levels, psychostimulants are potent releasers of other biogenic amines including norepinephrine [START_REF] Gatley | Affinities of methylphenidate derivatives for dopamine, norepinephrine and serotonin transporters[END_REF][START_REF] Kuczenski | Effects of methylphenidate on extracellular dopamine, serotonin, and norepinephrine: comparison with amphetamine[END_REF] that also contributes to the regulation of psychostimulantinduced motor hyperactivity [START_REF] Drouin | Critical role of alpha1-adrenergic receptors in acute and sensitized locomotor effects of Damphetamine, cocaine, and GBR 12783: influence of preexposure conditions and pharmacological characteristics[END_REF]. Interestingly, such regulation occurs mainly through the stimulation of alpha1-adrenergic receptors, which are highly expressed in the spinal cord [START_REF] Day | Distribution of alpha 1a-, alpha 1b-and alpha 1d-adrenergic receptor mRNA in the rat brain and spinal cord[END_REF][START_REF] Drouin | Critical role of alpha1-adrenergic receptors in acute and sensitized locomotor effects of Damphetamine, cocaine, and GBR 12783: influence of preexposure conditions and pharmacological characteristics[END_REF]. Whether pontine A7 and A6 (locus coeruleus) descending norepinephrine neurons [START_REF] Bruinstroop | Spinal projections of the A5, A6 (locus coeruleus), and A7 noradrenergic cell groups in rats[END_REF], that co-express DA, contribute to the regulation of cFos expression induced by psychostimulants through this class of adrenergic receptors will require future investigation.

Over the last decade, compelling evidence indicate that cFos-expressing neurons activated by drugs of abuse define region-specific neuronal ensembles which are causally linked to both conditioned-drug effects and drug-seeking behavior [START_REF] Cruz | New technologies for examining the role of neuronal ensembles in drug addiction and fear[END_REF]. Indeed, selective inactivation of cocaine-activated neurons in the nucleus accumbens has been shown to prevent context-dependent locomotor sensitization to cocaine and context-induced reinstatement of cocaine seeking [START_REF] Cruz | Role of nucleus accumbens shell neuronal ensembles in contextinduced reinstatement of cocaine-seeking[END_REF][START_REF] Koya | Targeted disruption of cocaine-activated nucleus accumbens neurons prevents context-specific sensitization[END_REF]. Moreover, distinct cFos-expressing neuronal ensembles in the ventromedial prefrontal cortex encode cocaine self-administration, extinction or cocaine seeking [START_REF] Kane | Fos-expressing neuronal ensemble in rat ventromedial prefrontal cortex encodes cocaine seeking but not food seeking in rats[END_REF][START_REF] Warren | Separate vmPFC Ensembles Control Cocaine Self-Administration Versus Extinction in Rats[END_REF] as well as context-induced relapse to heroin [START_REF] Bossert | Ventral medial prefrontal cortex neuronal ensembles mediate context-induced relapse to heroin[END_REF]. Further work will be needed to establish whether cFos-expressing spinal neurons identified in laminae hosting locomotor central pattern generator network and motor synergy encoder neurons define selective neuronal ensembles and whether a dysfunction of these ensembles is causally linked to the enduring psychomotor tremor and proprioceptive deficits developed in response to psychostimulants exposure [START_REF] Downey | Psychomotor Tremor and Proprioceptive Control Problems in Current and Former Stimulant Drug Users: An Accelerometer Study of Heavy Users of Amphetamine, MDMA, and Other Recreational Stimulants[END_REF]. 3. Numbers of GFP-positive neurons counted are reported in Table 3.

  notably in lamina III (F 4, 10 = 5.997, p = 0.01), and IV (F 4, 10 = 4.597, p = 0.023), V (F 4, 10 = 20.3, p < 0.0001), VI (F 4, 10 = 35.83, p < 0.0001), VII (F 4, 10 = 26.79, p < 0.0001), VIII (F 4, 10 = 25.58, p < 0.0001) and X (F 4, 10 = 10.7, p = 0.0012) (Figure 1ab). No significant changes were detected in laminae I (F 4, 10 = 0.7929, p = 0.5559) and II (F 4, 10 = 4.282, p = 0.0283) whatever the time points analyzed following damphetamine administration (Figure 1a-b). Moreover, no cFos-positive cells were found in lamina IX where motor neurons are located (Figure 1a-b). These results indicate that a single injection of d-amphetamine evokes specific neuronal activation in the dorsal (laminae V and VI) and the ventral horns (laminae VII and VIII) of the lumbar spinal cord.

Figure legends Figure 1 :

 legends1 Figure legends

Figure 2 :

 2 Figure 2: Molecular identification of pS32-cFos-positive neurons in lumbar

Figure 3 :

 3 Figure 3: Reactivation of d-amphetamine-tagged neurons following d-

Figure 4 :

 4 Figure 4: Cocaine and methylphenidate increase pS32-cFos expression in lumbar

Figure 5 :

 5 Figure 5: Morphine and dizocilpine fail to induce pS32-cFos in lumbar spinal

Figure 6 :

 6 Figure 6: Dopamine reuptake inhibitor increases pS32-cFos expression in

Acknowledgments

The authors thank Jean-Antoine Girault and Giuseppe Gangarossa for critical reading of the manuscript and for their insightful comments. We thank the RAM-iExplore facility for their involvement in the maintenance and breeding of the colonies. This work was supported by Inserm, Fondation pour la Recherche Médicale (EQU202203014705), Agence National de la Recherche (DISCOMMODE, ANR-21-CE37-0013; FrontoFat, ANR-20-CE14-0020; ANR SubDOPA, ANR-21-CE16-0028) (E.V.). Laura Cutando was supported by the post-doctoral Labex EpiGenMed fellowship («Investissements d'avenir» ANR-10-LABX-12-01). Laia Castell was supported by the pre-doctoral Labex EpiGenMed («Investissements d'avenir» ANR-10-LABX-12-01). Coline Rulhe is supported by a fellowship from the French Ministry of Higher Education and Research.

Conflict of interest

The authors declare no conflict of interest.

List of Abbreviations

4-OHT: 4-Hydroxytamoxifen CB: Calbindin-D28K CBP: Calcium-binding protein CR: Calretinin DA: Dopamine Gad67: Glutamic acid decarboxylase 67 GlyT2: Glycine transporter 2 GFP: green fluorescent protein HA: Hemagglutinin PV: Parvalbumin TBS: Tris-buffered saline VGluT2: Vesicular glutamate transporter


Andrei Constantinescu
email: andrei@lms.polytechnique.fr

Ky Dang Van

Habibou Maitournam

A unified analysis for low and high cycle fatigue based on shakedown concepts

Keywords: dissipated energy, high cycle fatigue, low cycle fatigue, shakedown. . The microscopic scale of dislocations. . The mesoscopic scale of grains. . The macroscopic scale of engineering structures

diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Current industrial design is concerned with fatigue, as it is the main failure mode of mechanical structures under variable loadings. During a long period, starting with the pioneering work of Wo È hler (1860) and ending in the late 1950s, high-cycle fatigue (HCF) was the most significant topic for engineers and researchers. They were mainly interested in establishing S±N curves (load versus number of cycles to failure) and in determining the fatigue limit for metallic materials. This approach is still used as a design tool in many cases to predict fatigue resistance of mechanical structures. In the 1960s, a special interest occurred in studying low-cycle fatigue (LCF). Instead of developing stress approaches, Coffin and Manson proposed fatigue models based on the strain amplitude or the plastic strain amplitude. The cited approaches showed their effectiveness for structures in the aeronautical and nuclear industry. However, success was obtained when the stress or the strain cycle was uniaxial and simple. This is not always the case, as stresses and strains are often multiaxial and present a complex path during the loading cycle. Application of classical fatigue models to mechanical structures is then difficult and the predicted results do not always match with test results.

Therefore, a sustained effort has been deployed for deriving reliable fatigue computational methods applicable to the different engineering configurations. The actual prediction techniques are generally based on multiaxial fatigue criteria using a stress approach in the HCF regime and on strain or inelastic strain approaches in the LCF domain. In spite of the fact that materials fail in both domains in similar manners, the two ways of modelling the fatigue problem seem so different that it is difficult to imagine any link between them. The present paper discusses a series of criteria from an unified point of view in order to fill this theoretical gap. It is derived from Dang Van et al. HCF criteria [START_REF] Van | High-cycle Metal Fatigue, from Theory to Applications[END_REF][START_REF] Papadopoulos | High-cycle Metal Fatigue, from Theory to Applications[END_REF] from some recent results in thermomechanical LCF [START_REF] Charkaluk | An energetic approach in thermomechanical fatigue for silicon molybdenum cast iron[END_REF][START_REF] Charkaluk | Fatigue design criterion for structures under thermomechanical loading[END_REF] and shakedown theories. 5±8

T H EM U L T I SC A L EA P P R OA CHI NF A T I G U E

In discussing fatigue phenomena we shall distinguish three scales:

The purpose of this paper is to present a unified analysis to both high and low cycle fatigue based on shakedown theories and dissipated energy. The discussion starts with a presentation of the fatigue phenomena at different scales (microscopic, mesoscopic and macroscopic) and of the main shakedown theorems. A review of the Dang Van high cycle fatigue criterion shows that this criterion is essentially based on the hypothesis of elastic shakedown and can therefore be expressed as a bounded cumulated dissipated energy. In the low cycle fatigue regime, recent results by Skelton and Charkaluk et al. show that we can speak of a plastic shakedown at both mesoscopic and macroscopic scale and of a cumulated energy bounded by the failure energy. The ideas are also justified by infrared thermography tests permitting a direct determination of the fatigue limit.

1

Andrei Constantinescu, Ky Dang Van, Habibou Maitournam

LMS -Laboratoire de mécanique des solides

In a simplified analysis we could say that the fatigue phenomena start generally with appearance of slip bands in grains, which broaden progressively during the cycles. The proportion of grains in which slip bands develop increases with the applied load.

The fatigue domains

Let us consider a structure submitted to a cyclic loading and let us represent the lifetime using a classical Wo È hler diagram (Fig. 1). The lifetime should be understood as the initiation of the first visible crack at the macroscopic scale. For increasing loading levels, different fatigue phenomena will appear and contribute to this initiation; they can be classified as follows.

In the high-cycle fatigue regime (HCF), in general no irreversible deformation, i.e., plastic or viscous, is detected at the macroscopic level. The material behaviour seems to be purely elastic, meaning that stresses and strains are related through linear relations. However, at a mesoscopic level, irreversible strains occur in a certain number of grains and generate a heterogenous plastic strain field. Only misoriented crystals undergo plastic slip corresponding to a heterogeneous distribution of microcracks. Within this region we can distinguish two domains. One at small loads where lifetime seems to be infinite denoted as infinite endurance (HCF-IE) and a second one, finite endurance (HCF-FE) at slightly higher load, where lifetime is large, but definitely finite. The low-cycle fatigue regime (LCF) implies significant macroscopic deformations contributing to irreversible deformations. Now, stress and strain fields are generally related through highly nonlinear differential equations. At the mesoscopic level, the metal grains are subjected to plastic deformation in a more homogeneous manner than in the HCF regime. The first cracks at a mesoscopic level appear in the persistent slip bands quite early in the lifetime of the structure.

Within this region, we generally distinguish three domains with increasing load, one where failure occurs due to alternating plasticity (LCF-P), a next one where failure occurs due to accumulation of plastic strain (ratchetting) (LCF-R) and finally a domain where structural collapse occurs directly from unrestricted plastic strain (LCF-U).

The meso-macro passage

In both LCF and HCF, damage phenomena occur in the grains, and therefore the use of mesoscopic fields seems to be relevant for studying these fatigue phenomena.

Let us recall, that in a certain sense the macroscopic fields (stress S, strain E, plastic strain E p ,.) are related to the mean value of the mesoscopic fields (stress s, strain e, plastic strain e p ,...). In the theories of polycrystalline aggregates the macroscopic fields are therefore supposed constant in a small volume, surrounding the point under consideration, called `representative volume element' or RVE. For instance the mesoscopic stress tensor s and the macroscopic stress tensor S are related by the following formula:

s A S r 1
where r is the mesoscopic residual stress and A is the fourth order tensor of elastic stress localization. In the case of equal mesoscopic and macroscopic elastic moduli, A is the identity tensor. The presence of the mesoscopic residual stresses (which depend closely on the characteristics of the loading path) shows that it is in general not correct to use the macroscopic stress S to characterize phenomena at the grain scale.

The evaluation of the local, mesoscopic fields from the macroscopic ones is in general a difficult task. The material is locally heterogeneous and has to be considered as a complete structure when submitted to complex loading histories. Depending on the loading characteristics one can accept different simplifying assumptions, which will permit a solution to the problem. For load levels inducing an elastic macroscopic stress state, the relation (1) can be solved if A is known. For higher load levels, the macroscopic stress and strain state is elastoplastic and can be directly assimilated with the mesoscopic state, only if one assumes that all grains are equally deformed. A simple direct measure of thermodynamically irreversible phenomena is the dissipated energy. Let us therefore, compare the dissipated energy at both mesoscopic and macroscopic scales. It is well known that the total macroscopic work rate is the mean value in the RVE of the local total work rate. However, the equality between the mesoscopic and macroscopic energy does not hold for plastic dissipation as proven by H.D. Bui and recalled in Ref. [1]. The difference between macroscopic plastic dissipation and mean value of mesoscopic plastic dissipation, decreases with increasing plastic strain, as the plastic heterogeneity from grain to grain decreases. This also justifies why macroscopic plastic deformation or plastic dissipation is a reasonable parameter in LCF.

CYCLIC BEHAVIOUR OF AN ELASTOPLASTIC STRUCTURE

In this section, we present a series of mathematical results regarding the elastoplastic behaviour of a structure submitted to cyclic loadings. As such, the following results are completely independent of the microstructure. The main question concerns the existence of an asymptotic response of the internal variables (S, E, E p ,...). We can classify these responses as follows and relate them directly with the mechanical behaviour observed at a macroscopic scale described in The meso-macro passage section.

1 Existence of a limit cycle . Alternating elasticity (elastic shakedown) (in HCF-IE and HCF-FE).

. Alternating plastification (plastic shakedown) (in HCF-FE and LCF-P).

2 No limit cycle

. Ratchetting, i.e., accumulation of plastic strain over all cycles (in LCF-R).

. Unrestricted plastic yield (in LCF-U).

The asymptotic response of a structure under a cyclic loading can be determined after a complete computation of a large number of cycles or can be directly estimated through application of the theoretical results recalled in the next section.

Direct analysis of structures

The theories and methods for the direct analysis of structures, i.e., estimating directly the limit cycle, provide the `live load' multiplier also denoted as a `safety factor' characterizing the limit between the elastic and the plastic shakedown domain of the structure. The techniques used for this purpose are based on a kinematic or a static approach and we shall only present the main results in order to highlight their relation to the underlying fatigue problems. The limit between the plastic shakedown and the ratchetting regions has not been extensively studied. In spite of recent successes in computing directly the limit elastoplastic cycle [START_REF] Peigney | Approche par contro à le optimal des structures e Âlastoviscoplastiques sous chargement cyclique[END_REF][START_REF] Pommier | De Âtermination de cycle limite par une me Âthode cyclique directe, 335±342[END_REF] we still do not have complete theoretical results characterizing this limit.

The original static theorem of Melan gives a sufficient condition for elastic shakedown for a structure made of elastic perfectly plastic material submitted to cyclic loading. Melan's Theorem: If there exist a time instant T and a time independent self-equilibrated residual stress field R (x) and a safety factor m > 1 such that for all points x of the structure and for t > T the following inequality holds: g m S el x; tRx À k 2 0 then the structure will shakedown elastically.

Here g and k denote, respectively, the yield criterion and the yield limit. S el is the purely elastic stress response of the structure under the same external loading. Without getting in the details of the precise mathematical proof let us recall that two main steps are needed. In the first one, it is proven that under the hypothesis of the theorem, the dissipated energy is bounded. In the second step, it is shown that the distance between the real stress S(x, t) and the purely elastic estimation S el (x, t) tends toward a constant value. The key point of the proof is the associative property of the plastic model, i.e., the fact that the plastic flow is normal to the convex plasticity domain. The results do not state anything about the magnitude of the plastic deformation before shakedown, but one can extract the idea of a bounded plastic work in order to ensure a bounded plastic strain, as explained by Koiter: [START_REF] Koiter | General theorems for elastic±plastic solids[END_REF] ªif the total amount of plastic work performed in the loading process is accepted as suitable criterion for assessing the overall deformation, boundedness of the overall deformation may be proven if the structure has a safety factor m > 1 with respect to shakedown.º Melan's shakedown theorem has been extended by different authors to account for more realistic material behaviour. For isotropic and kinematic hardening material, Mandel et al. [START_REF] Mandel | Adaptation d'une Structure Elastoplastique a Á Ecrouissage Cine Âmatique[END_REF] propose another formulation which is a necessary condition of elastic shakedown: Mandel's Theorem: If there exist a time T and a time independent ( fixed ) stress field S * such as for all x and t > T: g S el x; tÀS Ã x À K Ã2 E p eq x 0 the structure will shakedown elastically.

The isotropic hardening parameter K, also interpreted as a yield radius, is an increasing function of cumulated equivalent plastic strain E p eq and K Ã E p eq is the maximum acceptable value of the yield radius before failure.

In fact, as the yield criterion depends only on the stress deviator, only the deviatoric part intervenes in the previous inequality. At shakedown, the deviators of S Ã and K Ã E p eq are, respectively, the centre and the radius of the smallest hypersphere surrounding the loading path defined by the deviator of S el (x, t).

Koiter's reasoning can be also extended to strain hardening materials using the framework of generalized standard materials as introduced by Halphen and Nguyen (see for instance Ref. [8]). This concept recovers a wide class of elastoplastic or elastoviscoplastic materials including classical isotropic and kinematic hardening materials. Beside the usual strain parameters E, E p Nguyen et al. introduced internal parameters including plastic deformation and a set of strain hardening variables denoted symbolically by b. It is then possible to define a potential energy W(E, E p , b) such as:

S ]W ]E ; A p À ]W ]E p ; A b À ]W ]b
The elastic domain is a convex C in the space of generalized forces, defined by:

gS; A p ; A b 0
The generalized stresses A (S, A p , A b ) are said to be plastically admissible if they satisfy the last inequality. As in perfect plasticity, the normality flow rule is verified by the generalized internal variables. The static shakedown theorem for standard generalized materials is formulated as follows: 8 Q.S. Nguyen's Theorem: Elastic shakedown occurs, for all initial states of the structure, if there exist a field of internal parameters (E à p ; b à ) and a security coefficient m > 1 such that the associated field of generalized forces mA à (t), obtained with fixed internal parameters (E à p ; b à ), is plastically admissible whenever t > T.

In this case the boundedness of the dissipation, ensures that the cumulated plastic deformation and strain hardening parameters are bounded.

In the previous discussion we focused only on results based on a static approach. Let us mention that the dissipated energy is equally a key point of the dual results based on a kinematic approach. [START_REF] Koiter | General theorems for elastic±plastic solids[END_REF] More precisely, in a general framework one can admit that the structure will shakedown if the internal plastic work computed for all admissible plastic strain rate cycles is always greater than the upper bound of the work computed from all external loads.

SHAKEDOWN ANALYSIS OF FATIGUE

We propose to use the shakedown theory for the behaviour of the microstructure in order to derive a unified model valid for structural applications from high to lowcycle fatigue.

High cycle fatigue ± application to fatigue limit

In the high-cycle regime, only few grains, badly orientated with respect to the loading, undergo plastic deformation in localized slip bands. Under the fatigue limit, i.e., the threshold between the infinite and finite lifetime, the dissipation at the mesoscopic scale is bounded and therefore the dissipated energy per cycle decreases to become after a while negligible (Fig. 2). On the same figure, point A represents the elastic shakedown limit. Beyond the point A, plastic shakedown occurs which will lead to fatigue failure characterized by the decrease of the stress. The corresponding dissipated energy is bounded. We already pointed out the difficulties to relate the energy dissipation at mesoscopic and macroscopic scale.

In engineering applications, it is therefore not easy to characterize this energy by evaluating directly the accumulated plastic strain, and consequently the dissipated energy cycle by cycle as was done by Papadopoulos. [START_REF] Papadopoulos | High-cycle Metal Fatigue, from Theory to Applications[END_REF][START_REF] Papadopoulos | Fatigue Polycyclique des Me Âtaux: une nouvelle Approche[END_REF] This calculation is rather a theoretical model of fatigue than a practical proposal for industrial applications. It is the reason why two different ways were explored to derive high cycle fatigue criterion, based on the use of stabilized stress parameters instead of dissipated energy. When the loading characteristics correspond to the fatigue limit, the asymptotic stabilized stress state is contained in a yield limit surface defined by the limit radius K Ã . This stabilized state corresponds to point A of Fig. 2. Dang Van 1 generalizing an idea of Orowan, proposed to consider the mesoscopic current stress state at the stabilized (shakedown) state as the relevant parameter in order to formulate a high-cycle multiaxial fatigue resistance criterion. More precisely, the proposed criterion is a combination of the mesoscopic shear t(t) and the simultaneous hydrostatic pressure p H (t). It can be stated as follows: For all instant t beyond T, one has t(t) ap H (t) À b < 0, fatigue will not occur, where a and b denote material constants which can be determined by two simple fatigue experiments on classical test specimens. For instance b, corresponds to the fatigue limit in simple shear. Papadopoulos proposed a theory in which K Ã depends (linearly) on the maximum hydrostatic tension induced by the loading cycle. [START_REF] Skelton | Energy criterion for high temperature low-cycle fatigue failure[END_REF] The general application of this criterion requires (see for details Ref. [1]):

. First to evaluate the mesoscopic stress tensor knowing the macroscopic stress cycle; this can be achieved by application of the previously recalled shakedown theorems in particular Mandel and Nguyen's theorem. For this purpose, one constructs the smallest hypersphere surrounding the macroscopic deviatoric stress-loading path, the centre of which defines (approximately) the local deviatoric residual stress r. Then the mesoscopic stress is known at any time of the loading cycle.

. Secondly, one must consider the plane and the instant for which the set (t(t), p H (t)) is a `maximum' relative to the criterion.

Details of the resulting computational method and some examples of applications can be found in Ref. [1]. This general framework is also used to derive a highcycle anisothermal fatigue criterion with temperaturedependent fatigue limits [START_REF] Maitournam | Approche multie Âchelle en fatigue polycyclique anisotherme[END_REF] and to predict limited lifetime in high-cycle fatigue of welded structures. [START_REF] Van | Assessment of welded structures by a local multiaxial fatigue approach[END_REF]

Low cycle fatigue ± cyclic material behaviour

The low cycle fatigue regime is characterized by extended irreversible plastic or viscous deformations. A direct computation of the mesoscopic strain from the macroscopic strain is not possible due to the material non-linearity.

In this case let us analyse the typical cyclic material behaviour in a strain-controlled experiment, expressed through the macroscopic stress evolution as a function of the number of cycles (Fig. 3). The apparent load evolves through 3 stages defined by the stabilization, the tangent, and the final points. The first phase corresponds to a cyclic hardening or softening. During this period the microstructure is subjected to plastic loading and converges to a plastic shakedown limit cycle generally attained at N sta cycles. At this point, it is generally accepted that the first microcracks with a length scale of several mm appear. [START_REF] Skelton | Energy criterion for high temperature low-cycle fatigue failure[END_REF] After the initial stabilization phase, the material behaviour is stable and evolves. The microscopic crack grows and attains a macroscopic length after N tan cycles. After this cycle, the crack influences the macroscopic fields of the structure and the applied load is decreasing with growing crack length. Failure is now to be expected after a small number of cycles at N fin .

One can remark that the macroscopic cyclic behaviour (Fig. 3) is very similar to the mesoscopic cyclic behaviour of the grain in HCF regime (Fig. 2) discussed previously. This can be explained by that fact that with increasing irreversible strain, the mesoscopic strain becomes more homogeneous and thus closer to the macroscopic state. The endurance criterion in the HCF regime, assumes that the cyclic evolution of the internal state at a mesoscopic level is stopped since an elastic shakedown state is reached before the point A (see Fig. 2).

In the LCF regime, the corresponding point A at N sta cycle in the cyclic material behaviour (see Fig. 3), describes the point where the macroscopic state converges towards a plastic shakedown regime. At the mesoscopic level, the material evolves also in a plastic shakedown cycle (Fig. 2). A short analysis of the classical LCF criteria shows that these are based on the stabilized material behaviour in a plastic shakedown regime, which characterizes the major part of the lifetime during the evolution from the N sta to the N tan cycle. Therefore, these criteria are focused on different parameters characterizing the dimensions of the cycle in order to define the evolution of damage.

In order to distinguish the cases where the strain rate plays an important role, a major trend was to differentiate plastic and creep fatigue. However, in both cases fatigue phenomena are associated with irreversible strain fields and therefore directly related to the energy dissipation. An inspection of the initial work of Manson and Coffin shows that they considered the amplitude of plastic strain as the relevant parameter for fatigue. Their tests were uniaxial and strain controlled. In this case the stress is related to the plastic strain amplitude for the stabilized cycle and therefore the plastic dissipation can be considered directly as function of strain amplitude. However, 3 The cyclic material behaviour at a macroscopic level.

Load

Number of cycles

N f Macroscopic crack in structure N sta N tan Microscopic crack initiation AB Fig.
in case of more complex loading cycles conducting to multiaxial stress/strain paths there is no simple relation between plastic strain and dissipated energy. The idea of dissipated energy per cycle, obtained from following integral: W cycle s de has been appealing and has been involved in a series of classical criteria like Morrow, Garud, Ostergren, Ellyin et al., Bui Quoc et al., ... (see Socie and Marquis [START_REF] Socie | Multiaxial Fatigue[END_REF] and Suresh [START_REF] Suresh | Fatigue of Materials[END_REF] for a review). In these criteria one can find a direct reference to energy dissipation through the interpretation of the fatigue factor involving products of strain and stress. Unfortunately, most of these criteria are not intrinsic and cannot therefore be generalized to multiaxial situations. The more or less tacit assumption of the existence of a stabilized macroscopic cycle has been skipped in the last decades. For applications related with nuclear or aeronautical structures, a series of precise elastoplastic or elastoviscoplastic constitutive equations coupled with damage evolution were proposed by Lemaitre and Chaboche. [START_REF] Lemaitre | Mechanics of Solid Materials[END_REF] Without discussing in detail these proposals let us just that the computation of all cycles during the lifetime of an engineering structure can easily become cumbersome and contribute to extremely high computational times. Next we shall reintroduce the assumption of a stabilized cycle. The reasoning will afterwards be based on an interesting interpretation of the fatigue life using the cyclic material behaviour recently proposed by Skelton. [START_REF] Skelton | Energy criterion for high temperature low-cycle fatigue failure[END_REF] The experimental evidence collected from a series of isothermal and anisothermal LCF tests on steels show that the cumulated dissipated energy to the saturation point, i.e., during the first N sta cycles, is a constant W sta 1±10 J mm À3 . One can interpret these results as a mesoscopic crack initiation criteria or equivalently as the failure of the mesoscopic elementary volume. It is the failure of the same RVE discussed in the Dang Van HCF criterion. However, the endurance limit (HCF-IE) is defined as the maximal load that avoids the failure of the RVE. The fact that the cumulated dissipated energy W sta is constant, has also been justified by complementary work of Skelton estimating the dissipated energy for the failure of an elementary volume in the crack propagation process.

In conclusion one can interpret W sta as an upper bound for the dissipated energy to elastic shakedown in HCF and a lower bound for the plastic shakedown in LCF.

Starting from this remark and from the fact that the dissipated energy per cycle is an intrinsic quantity which can be summed up under complex anisothermal loading conditions and computed from multiaxial strain and stress fields, Charkaluk and Constantinescu 3 proposed an interpretation of the fatigue life of complex structures. Some applications of this criterion, i.e., lifetime predictions of exhaust manifolds and cylinder heads, are presented in Ref. [4]. Let us just summarize here that the discussed problems involve anisothermal loading creating a multiaxial strain±stress path. The subsequent fatigue problems include creep and plasticity. They have considered that the material behaviour is stabilized after a small number of cycles and the global behaviour of the structure is computed directly in this configuration. The dissipated energy per cycle, characterizes the fatigue evolution during the lifetime. It has been shown that the dissipated energy per cycle can be used to predict the failure of the macroscopic representative volume element of specimens and structures. The prediction can be expressed as an upper bound on the maximum admissible energy dissipation during the lifetime of the structure. The importance of the dissipated energy per cycle stems from the possibility to add up irreversible phenomena at low temperature (where material behaviour and damage are characterized by macroscopic plasticity) with phenomena at high temperature (where damage phenomena and material behaviour are characterized by creep, i.e. viscosity). In these cases, classical approaches are not effective since stresses vary with temperature and plastic strains. During the previous analysis, we have combined two main ideas: the existence of a shakedown state and the boundedness of the dissipated energy to failure.

Experimental evidence by infrared thermography

When submitted to HCF, components are elastic at a macroscopic level and dissipation can occur only in some isolated grains at the mesocopic level as already discussed before. Some researchers have shown, that the underlying dissipation phenomena can be captured using infrared thermography. A direct consequence of these results is a rapid evaluation of the HCF endurance limit, instead of using the conventional stair case method.

Infrared thermography is a convenient technique for producing temperature images obtained from the invisible radiant energy emitted by the test specimen submitted to cyclic loading at an adequate frequency. The temperature rise can be thus captured by a thermographic camera. When the load is increased, one can observe simultaneously a temperature increase. As it was shown by Luong et al. 18±20 (Fig. 4), the fatigue damage is revealed by a sudden change of the rate of the dissipated heat which is proportional to the rate of mechanical dissipation at a mesoscopic level. The fatigue limit (HCF endurance limit) is represented by the point of rate change. It is therefore obtained instantaneously by this method using infrared thermography and the results match those obtained in conventional fatigue tests.

Similar sudden changes in the dissipation regime were previously reported by researchers in different countries from observations using other techniques. [START_REF] Lazan | Damping and resonance fatigue properties of materials considering elevated temperatures[END_REF][START_REF] Morrow | Cyclic plastic strain and energy and fatigue of metals[END_REF] Instead of measuring directly the temperature increase in the stabilized state, these authors introduced the concept of internal friction or specific damping parameter which are not so well defined. Regarding the point of sudden change of the rate, Lazan 21 called it referring to damping increase: `the cyclic stress sensitivity limit'. When comparing these points with the fatigue limit, we remark that they lie lower than the fatigue limit. This can be explained by the fact that these points stem from global mechanical measurements over the specimen and refer therefore to a mean of the thermomechanical fields, whereas the infrared thermography provides a local measurement conducting to an accurate rate evaluation related to the critical regions.

It is important to remark, that in spite of the industrial success in applications we still do dispose of as convincing theoretical explanation of the coincidence between the fatigue limit and the point of sudden rate change in thermal dissipation. Intuitively using the present approach for interpreting fatigue phenomena the reason of this coincidence is straightforward. The fatigue limit corresponds to a threshold on dissipative energy between elastic and plastic shakedown at a mesoscopic level. Below this threshold, dissipation (which increases with increasing stress level) is bounded, and as a consequence plastic strain and internal strain hardening parameters are finite. Above the threshold energy is dissipated at each cycle and the upper limit of the cumulative dissipated energy is attained at failure after a finite number of cycles. This has also been proven by the previous shakedown results of Koiter 5 for perfectly plastic materials and extended by Nguyen [START_REF] Nguyen | Stability and Nonlinear Solid Mechanics[END_REF] for generalized standard material.

CONCLUSION

Metal fatigue has been during the last one and a half century the subject of numerous research studies conducted by scientists of different fields: mechanical engineers, material scientists, physicists, chemists. Although progress can be reported in the understanding of the underlying physical phenomena, a series of difficulties still exist to achieve an interdisciplinary consensus of the way to model fatigue crack initiation. This paper presented an overview of the field showing that the main assumptions can be reduced to shakedown theories and to the bounding of the dissipated energy to failure. The discussions are based on examples of industrial structures submitted to complex multiaxial loading which have successfully been solved in the domain of high-cycle fatigue regime as well as low-cycle in plastic or visco-plastic creep fatigue regime. 1±4, 11,13 As a final conclusion, let us recall this sentence written in 1963 by Drucker which summarizes our proposal: [START_REF] Drucker | On the macroscopic theory of inelastic stress±strain-time-temperature behaviour[END_REF] `when applied to the microstructure there is a hope that the concepts of endurance limit and shakedown are related, and that fatigue failure can be related to energy dissipated in idealized material when shakedown does not occur'.

Intrinsic dissipation

Fig. 4 Variation of the rate change in dissipated energy with the increase of the stress in rotating bending for XC55 steel. The HCF endurance limit coincides with the sudden increase of the rate of dissipation. The measurements have been done after 3000 and 6000 cycles at 50 Hz which assures that a shakedown state has been attained (after Luong). [START_REF] Luong | Fatigue limit evaluation of metals using an infrared thermographic technique[END_REF]

Fig. 1 A

 1 Fig.1A schematic Wo È hler curve ( load versus number of cycles to failure) and fatigue domains with increasing load.

Fig. 2

 2 Fig.2Elastic and plastic shakedown at the mesoscopic scale in HCF: the mesoscopic behaviour of the RVE and the limit cycles (left) and their projection on the cyclic behaviour of the yield limit of a slip plane (right).