Timothée Gentieu

Anita Catapano
email: anita.catapano@bordeaux-inp.fr

Julien Jumel

James Broughton

A mean-field homogenisation scheme with CZM-based interfaces describing progressive inclusions debonding

Keywords: Particle-reinforced composites, Homogenisation, Cohesive zone models, Debonding, Finite element method

The objective of the present study is to describe the progressive debonding of inclusions in particle or fibre reinforced composites. To do so, the meanfield homogenisation scheme of Mori-Tanaka is enriched to take into account imperfect interfaces. The interfaces are modelled by a bilinear Cohesive Zone Model (CZM) taking into account normal and tangential effects. Results obtained with this new mean-field homogenisation scheme are compared to 2D FE-based numerical simulations that are used as reference results. The effects of inclusions volume fraction and size are also observed.

Introduction

Reinforced composites are commonly used in various industries such as automotive, aeronautics, or aerospace. The matrix can be reinforced with particles or fibres to satisfy numerous requirements in terms of mechanical properties. The resulting mechanical properties depend on the components properties, the size and shape of the inclusions, and the adhesion between the inclusions and the matrix [START_REF] Weng | Three-dimensional multi-particle fe model and effects of interface damage, particle size and morphology on tensile behavior of particle reinforced composites[END_REF][2][START_REF] Shams | Analysis of particle-matrix interfacial debonding using the proper generalized decomposition[END_REF]. For the purpose of the design process, simple laws describing the material behaviour are preferred because they can easily be implemented in large scale structures. Mean-field homogenisation techniques are then a good way to describe the material behaviour. In the present study, only the Mori-Tanaka homogenisation scheme [4], [START_REF] Benveniste | A new approach to the application of mori-tanaka's theory in composite materials[END_REF] is considered but multiple schemes with different levels of sophistication have been developed in the literature (e.g. self-consistent scheme [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF], [7], differential scheme [8], Lielens [9], etc.).

In order to develop more realistic analytical models some researchers attempted to take into account non-linear effects such as the debonding between the inclusions and the matrix [10] within the framework of homogenisation schemes. This type of model was notably developed for high explosive materials to predict the decohesion between energetic particles in a polymeric binder matrix. They first determined the mechanical properties of particles/matrix interfaces based on a bilinear Cohesive Zone Model (CZM) approach [11]. Then, the cohesive behaviour of interfaces was introduced into High-Dilute and Mori-Tanaka homogenisation schemes under hydrostatic loading conditions [START_REF] Dastgerdi | Mechanical and real microstructure behavior analysis of particulate-reinforced nanocomposite considering debonding damage based on cohesive finite element method[END_REF] and uniaxial loading [13]. Comparisons with numerical results showed good correlation between numerical and analytical models especially concerning the evaluation of particle volume fraction and size effects on particle debonding [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF].

In the present work, a new mean-field homogenisation scheme is presented. This model is an improved version of the Mori-Tanaka model which is enriched to take into account imperfect interfaces through the introduction of a CZM. The main novelty of the present method concerns the integration of the evolution of the compliance of the interface through a convergence loop based on a CZM law. In order to prove the effectiveness of the proposed model, results are compared with those obtained from a FE-based analysis. This comparison let assess the effects of morphological parameters of the composite material (volume fraction and size of the inclusions) that gives valuable information on the behaviour of reinforced composites. The paper is organised as follows: Section 2 focuses on the description of imperfect interfaces in the framework of mean-field homogenisation schemes, Section 3 presents the Mori-Tanaka model with weakened interfaces, Section 4 introduces the theory of CZMs whilst Section 5 describes the new Mori-Tanaka model combined with CZMs for describing the imperfect interfaces. Numerical results are presented in Section 6 and Section 7 ends the paper with some concluding remarks.

Analytical modelling of linear spring-type imperfect interfaces

In this study of the homogenisation of particle reinforced composites, an imperfect elastic bonding between the inclusions and the matrix is considered. Following the concept developed by Qu in [START_REF] Qu | Eshelby tensor for an elastic inclusion with slightly weakened interface[END_REF] and [16], the imperfect interfaces between two phases can be modelled by introducing an infinitely thin layer in which transferred stresses remain continuous and displacements become discontinuous (see Fig. 1). S is used to denote the surface of the interface and n its outward normal vector. S + represents the interface when approaching it from outside the inclusion and S -when approaching it from inside the inclusion. The stress at the interface being continuous, the normal stress jump ∆σ between the phases is null:

∆σ ij n j = σ ij S + -σ ij S -n j = 0 (1)
The displacement jump ∆u is here modelled with a linear spring:

∆u i = u i S + -u i S -= η ij σ jk n k (2)
The second order tensor η ij represents the compliance of the spring/layer interface. It is assumed to be symmetric and definite positive. One can recognise a perfect interface when η ij is set to zero whereas η ij → ∞ implies a complete debond. Two parameters α and β are introduced to model the interface compliance respectively in the tangential and normal direction of the interface (see Fig. 1). The interface compliance tensor is given by the relationship:

η ij = αδ ij + (β -α) n i n j (3)
where α and β represent the tangential and normal compliances of the interface, and δ the Kronecker symbol. Therefore, the tangential and normal displacement jump can be expressed as follows:

∆u i (δ ik -n i n k ) = ασ ij n j (δ ik -n i n k ) (4) ∆u i n i = βσ ij n j n i (5)
Considering Eqs. 4-5, a new relationship between the perturbation strain field ∼ d in the inclusion and the eigenstrain ∼ * can be established:

d ij (x) = S Esh ijkl * kl -C klmn S ∆u k (ξ)G ijmn (ξ -x)n l dS(ξ), (6)
where G ≈ is the Green function, S ≈ Esh is the inclusion Eshelby tensor, and

C ≈
is the stiffness tensor of the matrix.

It is possible to determine a fourth-order tensor that relates the pertur-bation strain to the eigenstrain: the modified Eshelby tensor S ≈ EshM . With the introduction of a compliant interface, the eigenstrain is not homogeneous in the inclusion any more. To overcome this difficulty, [16] calculated a volume averaged modified Eshelby tensor by integrating the modified Eshelby tensor over the inclusion volume.

[16] determined an exact solution of the problem by permutating the derivation and first integral operations of Eq. 6 and then permuting the surface and volume integrals of the same equation. However, Othmani et al [START_REF] Othmani | Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law[END_REF] noticed that this is not a mathematically correct operation as, according to Fubini's theorem, such permutations are only allowed when the integration volume is free from any singularity (which is not the case here as Green's function is singular along the imperfect interface).

To perform the calculation of the volume averaged modified Eshelby tensor, [START_REF] Othmani | Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law[END_REF] got rid of Green's function singularity by translating the calculations in Fourier space. Othmani obtained the following expression of the modified Eshelby tensor (for convenience, here below the volume averaged will be omitted):

S ≈ EshM = I ≈ + Γ ≈ Ω -1 : S ≈ Esh + Γ ≈ Ω , (7)
while Qu has the following one:

S ≈ EshM = S ≈ Esh + Γ ≈ Ω : I ≈ -S ≈ Esh . (8)
For both Eqs. 7-8, the tensor Γ

≈ Ω is expressed as:

Γ ≈ Ω = I ≈ -S ≈ Esh : H ≈ : C ≈ m , (9)
with

H ijkl = 1 Ω S (η ik n j n l + η jk n i n l + η il n j n k + η jl n i n k ) dS, (10) where H
≈ is a fourth-order tensor representing the influence of both the interface compliance and the morphology of the inclusion. Analytical expressions of this tensor are given by Qu for specific cases where the shape of the inclusions enables the calculation of the integrals analytically. Othmani [START_REF] Othmani | Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law[END_REF] claimed that the derivation of this tensor was also wrong in [16], however a comparison of the expressions given by both in the case of a spherical particle reveals the same result. H ≈ can always be decomposed as follows:

H ≈ = αP ≈ + (β -α)Q ≈ . (11)
For specific inclusion geometries, the expressions of P ≈ and Q ≈ can be analytically determined. For a spherical particle of radius a, one gets:

P ijkl = 1 a I ijkl (12)
Q ijkl = 1 5a (2I ijkl + δ ij δ kl ) (13)
For the case of a cylindrical inclusion of radius a, (based on [START_REF] Dinzart | New micromechanical modeling of the elastic behavior of composite materials with ellipsoidal reinforcements and imperfect interfaces[END_REF]) one can obtain the following expressions:

P 1111 = P 2222 = 4P 2323 = 4P 1313 = 2P 1212 = 1 a , (14)
Q 1111 = Q 2222 = 3Q 1122 = 3Q 2211 = 3Q 1212 = 3 4a , (15)
P ijkl = Q ijkl = 0, with i, j, k, l = 1, 2, 3. (16)
The strain localisation tensors obtained by Qu and Othmani are also different. Othmani obtained (and was confirmed by [START_REF] Dinzart | New micromechanical modeling of the elastic behavior of composite materials with ellipsoidal reinforcements and imperfect interfaces[END_REF]):

A ≈ HD i = I ≈ -S ≈ EshM : I ≈ -C ≈ m -1 : C ≈ i -1 : I ≈ + Γ ≈ Ω -1 , (17)
while Qu reported:

A ≈ HD i = I ≈ -S ≈ EshM : I ≈ -C ≈ m -1 : C ≈ i -1 . (18)
In the present work, to check the correction applied to the modified Eshelby tensor and the strain localisation tensor, the strain in an inclusion subjected to a remote hydrostatic loading is numerically determined and compared to the analytical prediction from Qu [16] and from the corrected expressions [START_REF] Othmani | Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law[END_REF] and [START_REF] Dinzart | New micromechanical modeling of the elastic behavior of composite materials with ellipsoidal reinforcements and imperfect interfaces[END_REF]. A compliant interface is introduced between the inclusion and the matrix (α = β = 4 × 10 -5 mm/MPa). Two types of inclusions are used: a spherical particle of radius a = 100 µm and a cylinder of radius a = 100 µm. The size of the unit cell is L = 20×a so that boundary effects can be neglected. The calculations are made under axisymmetric and plane strain assumptions respectively for spherical and cylinder inclusion. Fig. 2 represents the strain in the inclusion (homogeneous because of the hydrostatic applied loading) against the global strain of the material. In both cases, the corrected expressions almost give a perfect match with the numerical simulations while the results obtained with the expressions of Qu diverge. This test validates the corrected expressions of the modified Eshelby tensor and the strain localisation tensor.

Mori-Tanaka model with weakened interfaces

Exploiting the theory presented in Section 2, a single inclusion is considered within an infinite matrix (so that the interactions between the inclusions are completely neglected). The typical relationship of the mean-field homogenisation scheme to get the effective stiffness tensor C ef f , of a composite material made of m + 1 phases, is:

C ≈ ef f = m r=0 φ r A ≈ r : C ≈ r , (19)
where A r and C r are, respectively, the fourth order tensors of strain localisation and stiffness. φ r is the phase volume fraction while the phase r=0 corresponds to the matrix material. Mori and Tanaka [4] tried to take into account the effect of the inclusions interactions. For multiphasic materials, the Mori-Tanaka scheme can be generalised as follows:

A ≈ M T r = A ≈ HD r : n r=0 φ r A ≈ HD r -1 (20) C ≈ ef f = n r=0 φ r A ≈ M T r : C ≈ r (21)
where Eq. 20 represents the strain localisation tensor of the inclusion belonging to the r th phase and Eq. 21 represents the effective stiffness tensor of the composite. The mean-field homogenisation was revisited by Benveniste [START_REF] Benveniste | A new approach to the application of mori-tanaka's theory in composite materials[END_REF]. According to [START_REF] Benveniste | A new approach to the application of mori-tanaka's theory in composite materials[END_REF], the effect of the interaction between the particles can be modelled by replacing the remote loading by the strain inside the matrix phase. The strain localisation tensor of an inclusion becomes:

A ≈ M T i = A ≈ HD i : (1 -φ) : A ≈ HD m + φA ≈ HD i -1 (22)
Benveniste [START_REF] Benveniste | A new approach to the application of mori-tanaka's theory in composite materials[END_REF] demonstrated that the symmetry of the stiffness tensor is no longer certified when:

• more than two phases are taken into account;

• non-aligned ellipsoidal inclusions are considered.

Moreover, the Mori-Tanaka scheme give good results for low to moderate volume fraction of inclusions because interactions between inclusions are not properly described. However, thanks to the explicit formulation, it can be easily and quickly implemented and adapted to the specific volume fraction of the composited to be analysed.

The mean-field homogenisation scheme can be further improved using the modified Eshelby tensor to obtain the macroscopic behaviour of a composite material containing inclusions with imperfect interfaces. Esteva and Spanos [START_REF] Esteva | Effective elastic properties of nanotube reinforced composites with slightly weakened interfaces[END_REF] determined the effective elastic properties of nanotube reinforced composites with slightly weakened interfaces using a modified Mori-Tanaka scheme described thereafter. Lee and Pyo [20] used this approach to describe the evolution of the macroscopic behaviour with spherical particles considering that the progressive debonding could be represented with a reduction of interface stiffness.

Qu re-derived the Mori-Tanaka formulae with imperfect interfaces [16]. According to this procedure, the imperfect interfaces contribute to the evaluation of the strain in two ways [16]. First, the imperfect interfaces affect ∼ i , the average strain, by taking into account the inhomogeneous distribution of strains. This effect can be taken into account by the use of the modified Eshelby tensor. Secondly, the imperfect interfaces introduce a displacement jump that induces another component ∼ inter corresponding to the third term of the decomposition of the global strain E:

E ∼ = (1 -φ) ∼ m + φ ∼ i + φH ≈ : C ≈ i : ∼ i . (23)
The Mori-Tanaka strain localisation tensor for the phase i then becomes:

A ≈ M T i = A ≈ HD i : (1 -φ) A ≈ HD m + φ I ≈ + H ≈ : C ≈ i : A ≈ HD i -1 (24)
Thanks to this strain localisation tensor of Eq. ( 24) and the typical meanfield homogenisation formula [START_REF] Esteva | Effective elastic properties of nanotube reinforced composites with slightly weakened interfaces[END_REF], an estimate of the effective stiffness tensor can be obtained for a reinforced material with imperfect interfaces:

C ≈ ef f = (1 -φ) A ≈ M T m : C ≈ m + φA ≈ M T i : C ≈ i (25)

Nonlinear interfaces based on a Cohesive Zone Model

A CZM is a zero thickness interface separation law where the stress vector at the interface is related to the displacement jump across the interface. Multiple shapes of traction-separation law have been proposed in the literature and a bilinear law was used for this study (based on Alfano & Crisfield work [21] and implemented in the commercial software Ansys). The bilinear traction-separation law is represented on Fig. 3 and the corresponding behaviour is defined by the following equations:

       stage I: σ int = ∆u k σ stage II: σ int = 1 + k σ kσ σ max -k σ ∆u stage III: σ int = 0, ( 26
)
where k σ and k σ are respectively the slopes of stage I and II of the tractionseparation law. Stage I of the curve shows a linear elastic behaviour until ultimate stress is reached. Then, the cohesive stress at the interface linearly decreases as the interface stiffness is irreversibly decreasing proportionally to damage intensity. This cohesive law is governed by three main parameters: interface stiffness k σ , strength σ c and toughness. The initial slope is generally chosen according to numerical criteria. Indeed, the interface stiffness influences the effective stiffness of the RVE. Then, sufficiently high values are chosen so that the global elastic properties of the composite remain stable. The interface strength controls the damage initiation before a linear softening of the stress occurs until the stress completely vanishes and the interface is considered to be completely debonded. The whole elastic loading and damage process is governed by the interface fracture energy G c which corresponds to the shaded area under the traction-separation curves.

For this study, only one set of parameter is employed, and is summarised in Table 1, where the elastic properties of the matrix and the inclusions are also gathered. The stiffness of the interface is set very high (equal to 2.5 10 6 MPa/mm) so that the global compliance of the material is unaffected by that of the interfaces whatever the size of the inclusions.

Introduction of the CZM in the Mori-Tanaka homogenisation scheme

The CZM traction-separation law is used to govern the evolution of the compliance of the interfaces. At each iteration, a new tensor H ≈ is calculated.

For both the cylindrical and spherical geometries of inclusions, it relies on two parameters α and β. Those two parameters are always considered equal for this study. They are computed at every iteration as the compliance of the interface evolves as the inverse of the stiffness of the traction-separation law.

The flowchart summarising the algorithm developed for the proposed homogenisation model is represented in Fig. 4. During the first phase of the loading, one obtains an elastic calculation with compliant interfaces. Then, once the limit strength of the interface is reached, the interfaces are damaged and their compliance increases according to the evolution of the CZM. To numerically simulate this phase, a convergence loop is employed:

• a first value of the stress at the inclusion interface (σ i ) is calculated using the value of the interface compliance at the previous step;

• the associated displacement of the interface is determined and a new value of the inclusion stress based on the CZM (σ i test ) is computed, as well as the associated compliance of the interface (β test )

• β test is selected as the new compliance of the interface and the convergence loop is repeated until a convergence of the interface compliance is reached (|β -β test | < res).

During the third phase of the loading, the inclusions are considered to be completely debonded (no stress transfer to the inclusions) and the inclusions are replaced by voids with the same geometry. The calculation of the modified Eshelby tensor is particularly adapted to hydrostatic loading. To use the model under uniaxial axial tension, some assumptions have to be made (according to the one made in [START_REF] Othmani | Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law[END_REF]):

1. The eigenstrain in the inclusions is assumed to be uniform; 2. Only a hydrostatic debond of the inclusions is considered (the evolution of the interfacial damage is the same in every direction);

The modified Eshelby tensor is averaged over the whole inclusion.

This procedure is based on the work of [16] which integrates imperfect elastic interfaces in the Mori-Tanaka homogenisation scheme. However, in the present work, there are some fundamental differences and novelties which concerns:

• the integration of the correct version of the modified Eshelby tensor developed in [START_REF] Othmani | Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law[END_REF] (which is not been taken into account in [16]). In [START_REF] Dinzart | New micromechanical modeling of the elastic behavior of composite materials with ellipsoidal reinforcements and imperfect interfaces[END_REF] of the correct modified Eshelby tensor is considered only for 2D circular particles ;

• the stiffness of the interface is governed by a CZM ( [START_REF] Othmani | Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law[END_REF] makes use of the same approach but within the highly diluted inclusions homogenisation scheme);

• the convergence loop used to catch the evolution of the stiffness of the interface is explicitly described unlike [START_REF] Othmani | Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law[END_REF] where the procedure is not clearly exposed and can generate misunderstandings.

Comparisons with numerical results -Effect of morphological parameters

In order to evaluate the effectiveness of the propose analytical homogenisation model, the results have been compared with those of a 2D FE-based one (under plane strain assumption). Random 2D RVE with circular inclusions are generated using the Dropping and Rolling algorithm developed in [22]. The numerical implementation and Periodic Boundary Conditions are described in [22]. The FE model presented in [22] is modified in the present work by introducing Cohesive elements at all the interfaces between the inclusions and the matrix to numerically simulate the debonding of the particles from the matrix. Comparisons with numerical results are made for two different loading cases (in-plane bi-axial loading and uniaxial tension) to assess the capability of the model to be representative of the debonding phenomenon in reinforced composites.

The strain tensor considered for the bi-axial loading is the following:

E ∼ bi-axial =   E 0 0 0 E 0 0 0 0   , (27)
while for the uniaxial loading, it is:

E ∼ uniaxial =   E 0 0 0 0 0 0 0 0   , (28)

In plane bi-axial loading

The first loading case is the bi-axial one where equal displacements along the two in-plane directions x and y are applied to the RVE. A RVE with a 30% volume fraction φ of circular inclusions is selected (Fig. 5). All the inclusions have the same size (R = 80 µm). The stress-strain curve along xdirection, obtained using the FE model is represented with dots in Fig. 6 (a). This curve represents the reference values to be met. This curve can be divided into three parts. The first one corresponds to the elastic regime and all the interfaces behaviour belongs to stage I of the traction-separation law, Eq. 26. Then, inclusions start to debond and a global softening of the material is observed. In the final part, all the inclusions are completely debonded (no stress is redistributed to the inclusions) and one obtains another linear behaviour.

The analytical homogenisation model, presented in Section 5, is then employed and the resulting stress-strain curve is also plotted in red in Fig. 6 (a). The same trends corresponding respectively to inclusions completely bonded, debonding, and completely debonded, are also observed when using this model. Moreover, a good fit of the numerical results is observed, especially during the first and last phase. While in the middle part of the curve, i.e. during the debonding phase, only the very beginning of the FE-based results are correctly fitted. Indeed, in this part of the curve, a small load drop is observed with the numerical results whereas this phenomenon is not captured by the analytical homogenisation model.

The stress-displacement values of the interface determined at each corresponding point of the homogenisation model are plotted in Fig. 6 (b). One can see that they are perfectly superimposed with the adopted tractionseparation law (continuous black line), confirming the good convergence of the homogenisation model.

Uniaxial loading

The same RVE of Fig. 5 is employed to test the homogenisation model under uniaxial loading along y-axis. The results from the semi-analytical and the numerical models are represented in Fig. 7 (a). Similar to the biaxial, the behaviour of the material under uniaxial loading can be divided into three parts (inclusions completely bonded, debonding, and completely debonded) for both models. In the first part of the curve (elastic behaviour), the results for both models are almost superimposed. Then, as inclusions start to debond, the semi-analytical model seems to give a more compliant response than the numerical model. However, the starting point of the debonding seems to be accurate. For the last phase (i.e. inclusion completely debonded), different slopes are observed for the semi-analytical and numerical models. They, nethertheless, roughly show the same qualitative behaviour. This can be explained by the fact that under uniaxial loading, when the inclusions are replaced by voids, these finally tend to collapse under the imposed loading. This behaviour is not representative of reality as the presence of the inclusion avoids this squashing effect. With the numerical model, inclusions are not completely debonded (at the equator) and a compressive stress is redistributed to the inclusions by Poisson's effect.

The stress-displacement point of the inclusion interfaces corresponding to the evolution of the loading are plotted in Fig. 7 (b). Once agin, similar to the bi-axial loading, a good match between the stress-strain homogenised curve given by the analytical model and the imposed traction-separation law is observed.

Parametric study: effect of inclusion volume fraction and size

In order to test the homogenisation model under multiple configurations, the effect of inclusion volume fraction is first evaluated and compared to numerical results. Concerning the FE models, RVEs with different inclusion volume fractions (20% to 60%) are generated using the same procedure as described in the previous subsections. All the inclusions are represented by circles of a 80 µm radius. The RVE obtained are depicted in Fig. 8.

The RVE are first loaded under bi-axial conditions. The obtained stressstrain curves are plotted in Fig. 9 (a). It is observed that the homogenised stiffness increases with inclusion volume fraction. In particular, the linear elastic parts of the curves are correctly estimated as there is very good agreement between the semi-analytical and numerical results. Both models then predict larger and more sudden load drops with increasing inclusion volume fraction. The semi-analytical and the numerical models especially show a good match to the results for low to moderate inclusion volume fractions (40% and below).

The same RVE (φ = 30%) but expanded in a homothetic manner is used to observe the effect of inclusion size. The results for both models are represented in Fig. 9 (b). The linear elastic part and the beginning of debonding region of the curve seem not be affected by the variation of the inclusion size (as the initial stiffness of the interface, the first slope of the CZM, was chosen very large compared the stiffnesses of the phases). However, the RVE with the largest particles show a more sudden load drop. The same observations were made in [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF]. The last phase of the loading (completely debonded particles) is reached much faster when using large particles compared with small ones. This strong particle size effect is also described in [START_REF] Gentieu | Size effect in particle debonding: Comparisons between finite fracture mechanics and cohesive zone model[END_REF], which states that large particles tend to debond prior to smaller ones. The size effect is especially generated by the CZM in both the analytical and the numerical models as it introduces a length parameter in the problem. Indeed, the critical displacement u c , characteristic of the CZM, is homogeneous to a length. It is also to be noted that the size of the inclusion intervenes in the calculation of the H The results for the inclusion volume fraction and size effects under uniaxial loading are represented in Fig. 10 (a) and (b) respectively. Similar to the bi-axial loading, good agreements are found between the semi-analytical and numerical results. Once again, the fit of the two models is the best at low to moderate inclusion volume fractions. Also under this loading condition, large particles debond prior to smaller ones.

Conclusion

A new semi-analytical homogenisation model based on a Mori-Tanaka homogenisation scheme, containing inclusions with imperfect interfaces, was developed through this study. To describe the progressive debonding of the reinforcements, the compliance of the inclusion interfaces evolves according to a bilinear traction-separation law, similar to that of a CZM.

The model is applicable to any type of inclusion geometry as long as its modified Eshelby tensor can be determined. This improved version of the homogenisation models presented in [16] and [START_REF] Othmani | Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law[END_REF] led to better predictions of the influence of the imperfect interface behaviour on the resulting homogenised composite. Moreover, unlike the approaches of [16] and [START_REF] Othmani | Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law[END_REF], this new model is able to simulate the debonding phase within an analytical homogenisation scheme. The results obtained with the present model were compared to that of numerical FE simulations based on 2D RVE under plane strain assumptions. The debonding was simulated using CZM. The same CZM parameters were attributed to both models so that no calibration of the semi-analytical model was required. Bi-axial and uniaxial loadings were tested. Given the debonding of the inclusions was considered to be hydrostatic, excellent agreements were found between the semi-analytical and numerical results under hydrostatic loading. Under bi-axial loading, the stress seemed to be slightly underestimated by the semi-analytical homogenisation model compared to that of the numerical one.

The effects of inclusion volume fraction and inclusion size were also tested. Similar trends were observed for both models:

• the volume fraction of particles stiffens the composite during the elastic loading but a higher load drop is observed as the particles start to debond. The effect of particle volume fraction on the debonding onset was not found to be a first order phenomenon;

• An important size effect is visible when debonding occured for large particles (large particles are debonded prior to smaller ones in accordance with the results of the literature [START_REF] García | Fiber-size effects on the onset of fibermatrix debonding under transverse tension: A comparison between cohesive zone and finite fracture mechanics models[END_REF] and [START_REF] Gentieu | Size effect in particle debonding: Comparisons between finite fracture mechanics and cohesive zone model[END_REF]).

The proposed model could also be applied to a material containing inclusions of multiple sizes by considering the inclusions of different sizes as different phases. Accordingly, more slope changes would be observed when a greater number of inclusion populations are introduced. A continuous distribution of particles would then be required in order to obtain a smooth evolution of the homogenised stress-strain curves.

Tables

E [GP a] ν [ ] σ max [M P a] G c [J.m -2 ]

≈

  tensor for the determination of the modified Eshelby tensor S ≈ EshM .

Figures α β Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 β12345 Figures

Figure 6 :Figure 7 :

 67 Figure 6: (a) Homogenised stress-strain curves from the numerical and semi-analytical models under bi-axial loading; (b) Associated stress-displacement curve obtained at the interface of the inclusion.

Figure 8 :

 8 Figure 8: Microstructures obtained from the Dropping and Rolling algorithm [22].

Figure 9 :Figure 10 :

 910 Figure 9: Homogenisedtress-strain curves from the numerical (dotted lines) and semianalytical models under bi-axial loading, (a) effect of inclusion volume fraction, (b) effect of inclusion size.

Table 1 :

 1 Mechanical properties of the materials

	Particle	450	0.2	×	×
	Matrix	3	0.3	×	×
	Interface	×	×	50	100

Acknowledgements

DGA of France, supporting the first author through Allocation de thèse 2015, is gratefully acknowledged.

Data availability statement

The data that support the findings of this study are available within the article and from the corresponding author, A. Catapano, upon reasonable request.


Charlotte Spay

Garance Meyer

Guillaume Lio

Gianni Pezzoli

Benedicte Ballanger

Roberto Cilia

Philippe Boulinguez
email: philippe.boulinguez@univ-lyon1.fr

Bénédicte Ballanger

Resting state oscillations suggest a motor component of Parkinson's Impulse Control Disorders

Keywords: Parkinson's disease, impulsivity, electroencephalography, resting state, beta, gamma

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

Introduction

Impulse control disorders (ICDs) are debilitating neuropsychiatric disorders that include hypersexuality, pathological gambling, compulsive eating and compulsive shopping, and concern up to 46% of patients with Parkinson's disease (PD) [START_REF] Avanzi | Prevalence of pathological gambling in patients with Parkinson's disease[END_REF][START_REF] Grosset | Problematic gambling on dopamine agonists: Not such a rarity[END_REF][START_REF] Voon | Prevalence of repetitive and rewardseeking behaviors in Parkinson disease[END_REF][START_REF] Weintraub | Impulse Control Disorders in Parkinson Disease: A Cross-Sectional Study of 3090 Patients[END_REF][START_REF] Vela | The high prevalence of impulse control behaviors in patients with early-onset Parkinson's disease: A cross-sectional multicenter study[END_REF][START_REF] Antonini | ICARUS study: prevalence and clinical features of impulse control disorders in Parkinson's disease[END_REF][START_REF] Corvol | Longitudinal analysis of impulse control disorders in Parkinson disease[END_REF]. ICDs in PD are associated mainly with cognitive impulsivity, also termed choice or decisional impulsivity.

Cognitive impulsivity reflects dysfunctions in various aspects of decision making and motivation involving the reward system [START_REF] Czernecki | Motivation, reward, and Parkinson's disease: influence of dopatherapy[END_REF][START_REF] Abler | At-risk for pathological gambling: imaging neural reward processing under chronic dopamine agonists[END_REF]Cilia and van Eimeren 2011;[START_REF] Aarts | Aberrant reward processing in Parkinson's disease is associated with dopamine cell loss[END_REF][START_REF] Antonelli | Dopamine-agonists and impulsivity in Parkinson's disease: impulsive choices vs. impulsive actions[END_REF][START_REF] Piray | Impulse control disorders in Parkinson's disease are associated with dysfunction in stimulus valuation but not action valuation[END_REF][START_REF] Santangelo | The relationship between Impulse Control Disorders and cognitive dysfunctions in Parkinson's Disease: A meta-analysis[END_REF][START_REF] Dawson | Neurocognitive correlates of medication-induced addictive behaviours in Parkinson's disease: A systematic review[END_REF][START_REF] Meyer | Functional imaging studies of Impulse Control Disorders in Parkinson's disease need a stronger neurocognitive footing[END_REF]. These dysfunctions can lead to irrational preference for immediate small rewards rather than delayed large rewards (temporal discounting), anticipated decision without enough accumulation of available evidence (reflection impulsivity), and/or biased evaluation of probabilistic gains and losses (probabilistic discounting). ICDs are considered as a side-effect of dopamine replacement therapy [START_REF] Dodd | Pathological Gambling Caused by Drugs Used to Treat Parkinson Disease[END_REF][START_REF] Weintraub | Impulse Control Disorders in Parkinson Disease: A Cross-Sectional Study of 3090 Patients[END_REF][START_REF] Leroi | Dopaminergic influences on executive function and impulsive behaviour in impulse control disorders in Parkinson's disease[END_REF][START_REF] Antonelli | Dopamine-agonists and impulsivity in Parkinson's disease: impulsive choices vs. impulsive actions[END_REF][START_REF] Lopez | Impulse Control Disorders and Related Complications of Parkinson's Disease Therapy[END_REF][START_REF] Voon | Impulse control disorders and levodopa-induced dyskinesias in Parkinson's disease: an update[END_REF]). Thus, their management mainly involves discontinuing or decreasing dopamine agonists [START_REF] Samuel | Management of impulse control disorders in Parkinson's disease: Controversies and future approaches[END_REF][START_REF] Cossu | The rise and fall of impulse control behavior disorders[END_REF]. But this is not a satisfactory therapeutic option since this can cause a worsening of motor symptoms or a dopamine agonist withdrawal syndrome [START_REF] Evans | Punding in Parkinson's disease: its relation to the dopamine dysregulation syndrome[END_REF][START_REF] Rabinak | Dopamine agonist withdrawal syndrome in Parkinson disease[END_REF][START_REF] Pondal | Clinical features of dopamine agonist withdrawal syndrome in a movement disorders clinic[END_REF][START_REF] Samuel | Management of impulse control disorders in Parkinson's disease: Controversies and future approaches[END_REF], and even does not guarantee symptoms remission at follow-up [START_REF] Ávila | Impulsive and compulsive behaviors in Parkinson's disease: a one-year follow-up study[END_REF][START_REF] Cilia | Tryptophan hydroxylase type 2 variants modulate severity and outcome of addictive behaviors in Parkinson's disease[END_REF]. A better understanding of these disorders is mandatory to develop more efficient therapeutic strategies [START_REF] Grall-Bronnec | Dopamine Agonists and Impulse Control Disorders: A Complex Association[END_REF].

Actually, the cognitive and neural bases of ICDs in PD have not been completely elucidated to date. Results of neuroimaging studies are largely inconsistent (see [START_REF] Meyer | Functional imaging studies of Impulse Control Disorders in Parkinson's disease need a stronger neurocognitive footing[END_REF] for recent review). They show numerous possible abnormalities in the mesocorticolimbic circuits consistent with various dysfunctions of object and action valuation caused by biased probability estimation or reward processing, indeed. But abnormalities are also frequently observed in many other brain regions [START_REF] Cilia | Functional abnormalities underlying pathological gambling in Parkinson disease[END_REF](Cilia et al. , 2011;;[START_REF] Frosini | Parkinson's disease and pathological gambling: results from a functional MRI study[END_REF][START_REF] Rao | Decreased ventral striatal activity with impulse control disorders in Parkinson's disease[END_REF][START_REF] Van Eimeren | Drug-induced deactivation of inhibitory networks predicts pathological gambling in PD[END_REF][START_REF] Voon | Dopamine agonists and risk: impulse control disorders in Parkinson's disease[END_REF][START_REF] Politis | Neural response to visual sexual cues in dopamine treatment-linked hypersexuality in Parkinson's disease[END_REF][START_REF] Carriere | Impaired corticostriatal connectivity in impulse control disorders in Parkinson disease[END_REF][START_REF] Imperiale | Brain structural and functional signatures of impulsive-compulsive behaviours in Parkinson's disease[END_REF][START_REF] Tessitore | Resting-state brain networks in patients with Parkinson's disease and impulse control disorders[END_REF][START_REF] Petersen | Ventral striatal network connectivity reflects reward learning and behavior in patients with Parkinson's disease[END_REF]). In addition, these studies have limitations. First, influential papers focused only on decision-making by means of tasks probing specifically risk-taking and motivation, or by means of region of interest approaches centered on the decisional circuit [START_REF] Frosini | Parkinson's disease and pathological gambling: results from a functional MRI study[END_REF][START_REF] Rao | Decreased ventral striatal activity with impulse control disorders in Parkinson's disease[END_REF][START_REF] Van Eimeren | Drug-induced deactivation of inhibitory networks predicts pathological gambling in PD[END_REF][START_REF] Kassubek | Neural reward processing under dopamine agonists: imaging[END_REF][START_REF] Voon | Dopamine agonists and risk: impulse control disorders in Parkinson's disease[END_REF][START_REF] Politis | Neural response to visual sexual cues in dopamine treatment-linked hypersexuality in Parkinson's disease[END_REF]). These studies did not test the possibility of response inhibition dysfunctions (motor impulsivity). Yet, studies of impulsivity in psychiatric conditions [START_REF] Dalley | Fractionating impulsivity: neuropsychiatric implications[END_REF] as well as neuropsychological and behavioral investigations in PD [START_REF] Nombela | Multiple modes of impulsivity in Parkinson's disease[END_REF][START_REF] Voon | Models of Impulsivity with a Focus on Waiting Impulsivity: Translational Potential for Neuropsychiatric Disorders[END_REF] suggest that multiple modes of impulsivity are likely to develop in ICDs. Second, neuroimaging studies mostly used fMRI or SPECT/PET, although blood flow changes are only indirect measures of neural activity. Critically, these studies cannot disentangle concurrent excitatory and inhibitory mechanisms [START_REF] Logothetis | What we can do and what we cannot do with fMRI[END_REF], which is crucial for discriminating the neural events that contribute to the imbalance between excitatory and inhibitory activity in impulsivity.

Assessing resting state brain activity is essential for identifying clinically relevant brain dysfunctions [START_REF] Eidelberg | Metabolic brain networks in neurodegenerative disorders: a functional imaging approach[END_REF]. This allows probing simultaneously the functional integrity of a variety of networks which specific dynamics can only be assessed separately in various functional imaging studies. Here, we analyzed resting state oscillations with high-density electroencephalography (EEG) in PD patients with and without ICDs. Indeed, EEG provides better cortical functional discrimination power than blood flow-derived measures often privileged for resting state studies since frequency-specific activity provides markers of particular cognitive functions, even at rest (e.g., [START_REF] Siegel | Spectral fingerprints of large-scale neuronal interactions[END_REF][START_REF] Cohen | Where Does EEG Come From and What Does It Mean?[END_REF]. This discrimination power is essential for separating multiple neural activities that might be confounded otherwise. This is especially true if spectral analyses are performed directly at the source level after blind source separation -BSS-, and might for instance be helpful for disentangling excitatory from inhibitory activity through analyses of alpha activity [START_REF] Klimesch | EEG alpha oscillations: the inhibition-timing hypothesis[END_REF][START_REF] Lorincz | Temporal framing of thalamic relay-mode firing by phasic inhibition during the alpha rhythm[END_REF][START_REF] Jensen | Shaping functional architecture by oscillatory alpha activity: gating by inhibition[END_REF][START_REF] Mathewson | Pulsed out of awareness: EEG alpha oscillations represent a pulsed-inhibition of ongoing cortical processing[END_REF]. In order to investigate all cortical networks potentially involved in ICDs' impulsivity, we performed whole-brain analyses without a priori about the anatomofunctional origins of the disorder.

Although we are here testing more specifically the neglected motor hypothesis, it is worth mentioning that the cognitive and motor hypotheses make distinct predictions not only about the anatomy of the circuits displaying cortical activity changes at rest, but also about the spectral signatures of these changes. The cognitive account of impulsivity predicts differences in the mesocorticolimbic system, the orbitofrontal and the lateral prefrontal cortices (e.g., [START_REF] Aracil-Bolaños | Molecular imaging and neural networks in impulse control disorders in Parkinson's disease[END_REF][START_REF] Marques | Impulse control disorders in Parkinson's disease[END_REF][START_REF] Vriend | The neurobiology of impulse control disorders in Parkinson's disease: from neurotransmitters to neural networks[END_REF], mainly expressed in the delta and theta bands [START_REF] Van Wingerden | Theta-band phase locking of orbitofrontal neurons during reward expectancy[END_REF][START_REF] Knyazev | EEG delta oscillations as a correlate of basic homeostatic and motivational processes[END_REF][START_REF] Nacher | Coherent delta-band oscillations between cortical areas correlate with decision making[END_REF][START_REF] Cavanagh | Cortical delta activity reflects reward prediction error and related behavioral adjustments, but at different times[END_REF][START_REF] Pinner | Frontal theta accounts for individual differences in the cost of conflict on decision making[END_REF][START_REF] Fatahi | Functional connectivity between anterior cingulate cortex and orbitofrontal cortex during value-based decision making[END_REF]. The motor account of impulsivity predicts differences in a medial network composed of the medial prefrontal cortex (mPFC), the dorsal anterior cingulate (dACC), the inferior frontal gyrus (iFG), the precuneus and the supplementary motor area (SMA) [START_REF] Aron | From reactive to proactive and selective control: developing a richer model for stopping inappropriate responses[END_REF][START_REF] Criaud | Have we been asking the right questions when assessing response inhibition in go/no-go tasks with fMRI? A meta-analysis and critical review[END_REF][START_REF] Manza | A dual but asymmetric role of the dorsal anterior cingulate cortex in response inhibition and switching from a non-salient to salient action[END_REF][START_REF] Criaud | Testing the physiological plausibility of conflicting psychological models of response inhibition: a forward inference fMRI study[END_REF], mainly expressed in the alpha and beta bands (Supplementary Table 1) [START_REF] Zhang | Response preparation and inhibition: the role of the cortical sensorimotor beta rhythm[END_REF][START_REF] Criaud | Have we been asking the right questions when assessing response inhibition in go/no-go tasks with fMRI? A meta-analysis and critical review[END_REF][START_REF] Huster | Electroencephalography of response inhibition tasks: functional networks and cognitive contributions[END_REF][START_REF] Kilavik | The ups and downs of β oscillations in sensorimotor cortex[END_REF][START_REF] Hwang | Cortical neurodynamics of inhibitory control[END_REF][START_REF] Fonken | Frontal and motor cortex contributions to response inhibition: evidence from electrocorticography[END_REF][START_REF] Liebrand | Temporal Dynamics of Proactive and Reactive Motor Inhibition[END_REF]. Importantly, we will perform whole brain and large spectrum analyses in order not to miss the potential contribution of unpredicted brain regions/frequency bands.

Methods

Participants

Twenty-seven PD patients with current ICDs (ICDs+) and 22 PD without ICDs (ICDs-) were enrolled at the Parkinson Institute in February 2016. The presence of any ICDs was assessed with the Questionnaire for Impulsive-Compulsive Disorders in Parkinson Disease (QUIP score ≥ 1 for ICDs+, i.e., one ICD or more [START_REF] Weintraub | Validation of the questionnaire for impulsive-compulsive disorders in Parkinson's disease[END_REF]). Considering that ICDs+ patients may underestimate, or even lie about and voluntarily deny any behavioral disturbance [START_REF] Cilia | Multiple compulsive behaviors in multiple system atrophy: the importance of predisposition to addiction[END_REF], caregivers were independently interviewed. For ICDs-, the absence of any behavioral disturbance during the whole disease course (i.e., QUIP score = 0) was stated and confirmed by the caregiver.

The sample size was determined on the basis of the results of previous studies using EEG to assess inhibitory dysfunction in PD (e.g. [START_REF] Spay | Clonidine modulates the activity of the subthalamic-supplementary motor loop: evidence from a pharmacological study combining DBS and EEG recordings in Parkinsonian patients[END_REF]. The estimated sample size n was calculated as the solution of:

= , / ,
² where d = delta/standard deviation, α = alpha (the probability of a type I error), β = 1 -power (the probability of a type II error) and tv, p is a Student t quantile with v degrees of freedom and probability p. Based on former EEG data assessing power modulations associated with inhibitory dysfunction in PD patients [START_REF] Spay | Clonidine modulates the activity of the subthalamic-supplementary motor loop: evidence from a pharmacological study combining DBS and EEG recordings in Parkinsonian patients[END_REF], the expected difference (delta) in normalized spectral power was set at 0.2, with a standard deviation of 0.2. Setting the alpha risk at 5% and the expected power at 95%, the minimum number of patients per group to highlight the expected difference (n) is 18. This sample size was also determined in agreement with former specific recommendations regarding EEG group blind source separation analyses [START_REF] Lio | How Does Sensor-Space Group Blind Source Separation Face Inter-individual Neuroanatomical Variability? Insights from a Simulation Study Based on the PALS-B12 Atlas[END_REF]. Indeed, optimal performance of source separation and subsequent source localization does not require the inclusion of large samples of subjects (n<20) when applying second order statistics (SOS)-based algorithms that use source spectral diversity. These algorithms can identify and gather sources that have similar functional properties despite variable location and orientation due to inter-individual neuroanatomical variability. The main demographic and clinical characteristics of the patients are displayed in Table 1. All patients gave written informed consent before the study, and the protocol was approved by the local Ethical Committee of the Parkinson Institute, Milan (Italy).

Design and procedure

Resting state activity was recorded over two continuous periods of 10min. Subjects

Data analyses

A flowchart of methods pipeline is presented in Fig. 1 to summarize data analyses strategy.

Preprocessing.

First, the quality of each electrode signal was checked, and spherical interpolation was used when necessary. Data were down-sampled at 2048 Hz, filtered (High-pass 0.5-1Hz; Low-pass 46-48Hz) and set to average reference. First, artifacts (movement-induced perturbations including especially eye movements, ballistocardiac noise, electrode displacements and other electrical noises)

were rejected by means of independent component analysis (ICA) / blind source separation (BSS) (UWSOBI, 300 times delays; [START_REF] Yeredor | Blind separation of Gaussian sources via second-order statistics with asymptotically optimal weighting[END_REF] and the EEGLAB toolbox [START_REF] Delorme | EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF]. Second, data were partitioned into epochs of 2 seconds with an overlap of 50% and an automatic rejection procedure for outlier epochs was applied. The Frobenius norm between the covariance matrix of each epoch and the covariance matrix of the mean dataset was calculated. For each dataset, the 5% of the epochs deemed as outliers according to this metric have been rejected.

Group Blind Source Separation (gBSS).

gBSS provides an efficient solution to the issue of multi-subject analysis by concatenating data from all subjects. A unique mixing matrix is therefore obtained for the whole group of subjects, as well as a unique demixing matrix after separation. The main advantage of the method is that gBSS reveals the independent components (ICs) that are consistently expressed across subjects [START_REF] Eichele | EEGIFT: group independent component analysis for event-related EEG data[END_REF][START_REF] Huster | Group-level component analyses of EEG: validation and evaluation[END_REF][START_REF] Huster | A Tutorial Review on Multi-subject Decomposition of EEG[END_REF][START_REF] Lio | How Does Sensor-Space Group Blind Source Separation Face Inter-individual Neuroanatomical Variability? Insights from a Simulation Study Based on the PALS-B12 Atlas[END_REF]. A potential benefit of this method is a better sensitivity for the identification of substantial sources that might be hidden by the most energetic ones [START_REF] Sutherland | Reliable detection of bilateral activation in human primary somatosensory cortex by unilateral median nerve stimulation[END_REF]. We employed UWSOBI, an algorithm relying on the approximate joint diagonalization of lagged-covariance matrices (SOS). We have recently demonstrated that this class of algorithms is especially adapted to gBSS because it is insensitive to inter-subjects differences in neuroanatomy and does not require deleterious dimension reduction before separation [START_REF] Lio | How Does Sensor-Space Group Blind Source Separation Face Inter-individual Neuroanatomical Variability? Insights from a Simulation Study Based on the PALS-B12 Atlas[END_REF]. This method is especially adapted here because it separates the sources on the basis of time-frequency information, not on the basis of data distribution characteristics as classical higher order statistics data-based algorithms do (e.g., FASTICA or INFOMAX, see [START_REF] Lio | Greater robustness of second order statistics than higher order statistics algorithms to distortions of the mixing matrix in blind source separation of human EEG: Implications for singlesubject and group analyses[END_REF]. This is more convenient when searching spectral signatures of cognitive processes, but this is also more convenient because SOS-based algorithms are more robust to the distortions of the mixing matrix that are inevitable in gBSS (Lio andBoulinguez 2013, 2018). Basically, after a set of matrices of SOS between electrodes has been estimated and made symmetric -time delayed covariance is not symmetric-, a linear transformation that enhances spectral diversity between sources is searched -the mixing matrix is calculated by approximate joint diagonalization of the whole set-. Importantly, the output of the separation provides components that are statistically independent regarding the parameters used for separation (whole spectrum signature). This does not mean that independent components cannot be functionally connected and part of the same neural network, i.e., cannot provide similar or interrelated activation patterns in response to specific events or conditions within specific frequency bands.

For each epoch, 300 lagged-covariance matrices (time delays: 0/2048s to 300/2048s) were calculated, averaged across dataset epochs, and finally across subjects. In the end, a total of 600 lagged-covariance matrices have been approximately joint-diagonalized by means of the UWEDGE algorithm [START_REF] Tichavsky | Fast Approximate Joint Diagonalization Incorporating Weight Matrices[END_REF], leading to the detection of 134 ICs. Thanks to this averaging procedure, inter-epochs and inter-subject variability was reduced while sources with spectral modifications between ICDs+ and ICDs-were highlighted [START_REF] Ramoser | Optimal spatial filtering of single trial EEG during imagined hand movement[END_REF][START_REF] Congedo | On the blind source separation of human electroencephalogram by approximate joint diagonalization of second order statistics[END_REF]. Brain regions explaining the group effect (ICDs+/ICDs-) were revealed by sorting components by percent of explained variance. Components explaining at least 1% of variance were retained for further analyses (n=10).

Independent components source localization.

To identify the neuronal generators of the electrical patterns recorded on the scalp, standardized low resolution electromagnetic tomography was performed with the sLoreta software [START_REF] Rd | Standardized low-resolution brain electromagnetic tomography (sLORETA): technical details[END_REF]. The cortical distribution of the electric neuronal activity generating each IC on the basis of its topography was estimated (solving the inverse problem). sLoreta uses the current density estimate obtained from the minimum norm estimate, and standardizes it by means of its variance. Here, standardized current source density was calculated at each of the 6239 voxels (resolution: 5mm) of a head model built on the basis of the boundary element method -BEM-and the MNI152 template [START_REF] Mazziotta | A probabilistic atlas and reference system for the human brain: International Consortium for Brain Mapping (ICBM)[END_REF]). The 3D solution space was constrained to gray matter, and sLoreta solutions were calculated with an amount of Tikhonov regularization optimized for an estimated Signal/Noise Ratio of 100. The output is a map of current density power (arbitrary units)

whose maximum value corresponds to the exact generating dipole location in noise-free conditions.

2.3.4 Blind spectral analysis.

We performed analysis of resting state spectral activity without a priori knowledge about anatomical sources or frequency bands. The ICs obtained with group BSS were used as a filter to go back to subject level for spectral analysis (deconcatenation). For each dataset of each PD patient, and each one of the 10 ICs accounting at least for 1% of explained variance (group effect), relative power was estimated. The power spectral density was calculated with the Welch's method for each epoch (with segments of 2048 samples, windowed with a Hamming window, and an overlap of 2000 samples between segments), and averaged across epochs for each dataset. This procedure generated absolute spectra with 1 Hz resolution. Relative spectra were then computed in the 1-45 Hz frequency range with respect to the total power of the component:

Powerrel (f) = Powerabs / ∑ Powerabs (f) 2.3.5 Statistical analyses.
Two analyses were performed in order to test our predictions twice: 1) by comparing ICDs+ and ICDs-, assuming discrete differences between groups, and 2) by looking for correlations with a clinical score reflecting ICD severity (QUIP) within the group of PD patients with ICDs, assuming gradual dysfunctions.

Group regression analysis:

We performed blind analyses and compared mean relative power differences between ICDs+ and ICDs-for all frequency bands and each component. In order to control for clinical characteristics, we performed a correlation matrix including all the dependent 45 Hz f=1Hz variables reported in Table 1. Since most of these variables were significantly correlated to each other (Supplementary Table 2), we selected the two most important variables (best model fitting F)

for the subsequent group regression analyses. In other words, we controlled only for motor performance (UPDRS-III) and medication (total LEDD) to avoid collinearity issues.

For each frequency band and each IC, a multiple regression analysis was applied based on one factor of interest (ICDs: ICDs+ vs. ICDs-) and two factors of no-interest (the UPDRS-III score and the LEDD), providing one F-statistic for the quality of the model fitting, and one t-statistic for the modeled factor:

Powerrel(IC,Freq.) = b1. ICDs + b2. UPDRS III + b3. LEDD + b4
Where b1, b2, b3 represent the parameters to be estimated for the first, second and third factors, respectively, and b4 represents the estimated intercept. Multiple testing correction was performed when appropriate by means of false discovery rate (FDR) estimation.

Results

Results (controlled for motor performance -UPDRS-III score-and medication -LEDD-) are summarized in Fig. 2.

Power changes accounting for ICDs (group effect)

Sources localized in the dACC and the mPFC revealed power modulations of resting state activity in PD patients with ICDs+ compared with ICDs-(F=5.78; p<0.05). The first source (IC 11/BA32) showed increase in low beta (13-14 Hz) and decrease in gamma (33 Hz) mean relative power in ICDs+ compared to ICDs-(p<0.001). The second source (IC 13/BA9), showed increase in low beta (p<0.001). The third source (IC 16/BA32) showed decrease in gamma (32-44 Hz) mean relative power in ICDs+ compared to ICDs-(p<10 -4 ).

Power changes accounting for ICDs severity (QUIP score)

Only one source (IC 8/BA6) localized in the supplementary motor area (SMA proper) revealed power changes predicting QUIP scores (F=11.26; p<0.01). Within this source, a positive correlation was found between beta band power (18-28 Hz) and ICDs severity.

Discussion

The hypothesis of a contribution of motor impulsivity to ICDs in PD [START_REF] Nombela | Multiple modes of impulsivity in Parkinson's disease[END_REF][START_REF] Palermo | Impulse control disorder and response-inhibition alterations in Parkinson's disease. A rare case of totally absent functionality of the medial-prefrontal cortex and review of literature[END_REF]) has mostly been ignored in neuroimaging studies [START_REF] Frosini | Parkinson's disease and pathological gambling: results from a functional MRI study[END_REF][START_REF] Rao | Decreased ventral striatal activity with impulse control disorders in Parkinson's disease[END_REF][START_REF] Politis | Neural response to visual sexual cues in dopamine treatment-linked hypersexuality in Parkinson's disease[END_REF][START_REF] Aracil-Bolaños | Molecular imaging and neural networks in impulse control disorders in Parkinson's disease[END_REF][START_REF] Voon | Impulse control disorders and levodopa-induced dyskinesias in Parkinson's disease: an update[END_REF][START_REF] Marques | Impulse control disorders in Parkinson's disease[END_REF] or rejected in behavioral studies (e.g., [START_REF] Rossi | Decision-making in Parkinson's disease patients with and without pathological gambling[END_REF][START_REF] Bentivoglio | Neuropsychological features of patients with Parkinson's disease and impulse control disorders[END_REF][START_REF] Leroi | Dopaminergic influences on executive function and impulsive behaviour in impulse control disorders in Parkinson's disease[END_REF][START_REF] Yoo | Patterns of Neuropsychological Profile and Cortical Thinning in Parkinson's Disease with Punding[END_REF][START_REF] Pineau | Executive functioning and risk-taking behavior in Parkinson's disease patients with impulse control disorders[END_REF][START_REF] Ricciardi | Acting without being in control: Exploring volition in Parkinson's disease with impulsive compulsive behaviours[END_REF]) up to date. Here, we used a resting state design and EEG to probe the functional integrity of all the cortical networks that might be dysfunctional in ICDs in the hope of finding a proof of concept for the contribution of motor impulsivity to PD-ICDs. First, by contrast with blood flow imaging studies, EEG provides a direct measure of neural activity and a fair functional discrimination power. Indeed, cognitive processes have spectral signatures [START_REF] Siegel | Spectral fingerprints of large-scale neuronal interactions[END_REF][START_REF] Keitel | Individual Human Brain Areas Can Be Identified from Their Characteristic Spectral Activation Fingerprints[END_REF]. Second, as it is challenging to test specifically all mechanisms involved in motor impulsivity (e.g., [START_REF] Criaud | Testing the physiological plausibility of conflicting psychological models of response inhibition: a forward inference fMRI study[END_REF], and as brain activity during tasks (i.e., the functional architecture of cortical networks) is partly already reflected in intrinsic activity at rest [START_REF] Raichle | The restless brain: how intrinsic activity organizes brain function[END_REF], this is an ideal starting point to probe resting state activity for the analysis of general brain function [START_REF] Keitel | Individual Human Brain Areas Can Be Identified from Their Characteristic Spectral Activation Fingerprints[END_REF]. As illustrated in Supplementary Table 1, our original approach exploits the observation that the motor and the decisional hypotheses of impulsivity make distinct predictions regarding the cortical and spectral origins of ICDs.

Power modulation in brain regions supporting global motor inhibition

We clearly identified power modulations accounting for ICDs in a medial ensemble of areas including the mPFC, the dACC and the SMA (Fig. 2). These areas have repeatedly been found to form a functional network supporting global motor inhibition in task-based imaging studies [START_REF] Jaffard | Proactive inhibitory control of movement assessed by event-related fMRI[END_REF][START_REF] Criaud | Testing the physiological plausibility of conflicting psychological models of response inhibition: a forward inference fMRI study[END_REF] and to be disrupted in PD-ICDs (Cilia et al. 2011). It is especially involved in proactive inhibition, an executive function that inhibits movement-triggering processes in anticipation of stimulation to prevent automatic and potentially inappropriate responses (Ballanger et al. 2009;[START_REF] Boulinguez | The paradoxical effect of warning on reaction time: demonstrating proactive response inhibition with event-related potentials[END_REF][START_REF] Criaud | Proactive inhibitory control of response as the default state of executive control[END_REF][START_REF] Criaud | Slowness in Movement Initiation is Associated with Proactive Inhibitory Network Dysfunction in Parkinson's Disease[END_REF]. Because this locking state is the default state of executive control [START_REF] Criaud | Proactive inhibitory control of response as the default state of executive control[END_REF], it was found to be tonically active at rest [START_REF] Jaffard | Proactive inhibitory control of movement assessed by event-related fMRI[END_REF]. The observation that the network of brains sources whose activity at rest predicts ICDs coincides with the proactive inhibitory network is thus consistent with the hypothesis according to which dysfunctions of motor inhibition account for ICDs [START_REF] Rubia | Mapping motor inhibition: conjunctive brain activations across different versions of go/no-go and stop tasks[END_REF][START_REF] Jaffard | Proactive inhibitory control of movement assessed by event-related fMRI[END_REF][START_REF] Brevers | Impulsive action but not impulsive choice determines problem gambling severity[END_REF][START_REF] Criaud | Have we been asking the right questions when assessing response inhibition in go/no-go tasks with fMRI? A meta-analysis and critical review[END_REF][START_REF] Albares | The dorsal medial frontal cortex mediates automatic motor inhibition in uncertain contexts: evidence from combined fMRI and EEG studies[END_REF][START_REF] Manza | A dual but asymmetric role of the dorsal anterior cingulate cortex in response inhibition and switching from a non-salient to salient action[END_REF].

Power modulation in frequency bands associated with motor processes and executive control

Although in some instances the ACC and the mPFC have been associated with decision making [START_REF] Paus | Primate anterior cingulate cortex: where motor control, drive and cognition interface[END_REF][START_REF] Botvinick | Conflict monitoring and decision making: reconciling two perspectives on anterior cingulate function[END_REF][START_REF] Gasquoine | Localization of function in anterior cingulate cortex: from psychosurgery to functional neuroimaging[END_REF][START_REF] Shenhav | Dorsal anterior cingulate cortex and the value of control[END_REF][START_REF] O'doherty | Learning, Reward, and Decision Making[END_REF], the present spectral data are more consistent with the predictions of the motor hypothesis (Supplementary Table 1). Indeed, the oscillatory activity associated with reward processing in the context of decision making in these regions involves theta and delta frequency bands [START_REF] Wacker | The role of the nucleus accumbens and rostral anterior cingulate cortex in anhedonia: integration of resting EEG, fMRI, and volumetric techniques[END_REF][START_REF] Park | Risk of punishment influences discrete and coordinated encoding of reward-guided actions by prefrontal cortex and VTA neurons[END_REF]. Here, raw differences between groups of patients with and without ICDs or gradual differences indexing ICDs severity were found in beta and low gamma frequency bands (Fig. 1). The beta rhythm has been extensively studied in the sensorimotor system, and was broadly linked to motor processes and executive control [START_REF] Kilavik | The ups and downs of β oscillations in sensorimotor cortex[END_REF][START_REF] Zhang | Response preparation and inhibition: the role of the cortical sensorimotor beta rhythm[END_REF]. Consistent with the inhibitory interpretation, beta activity over the motor cortex during movement preparation is suppressed prior to and during movement [START_REF] Baker | Oscillatory interactions between sensorimotor cortex and the periphery[END_REF], and increases when movement is voluntarily suppressed [START_REF] Zhang | Response preparation and inhibition: the role of the cortical sensorimotor beta rhythm[END_REF]. Beta oscillations are also considered as a marker of inhibitory cortical transmission mediated by GABA [START_REF] Feshchenko | Comparison of the EEG effects of midazolam, thiopental, and propofol: the role of underlying oscillatory systems[END_REF]. It is of interest to note that a broad increase in beta rhythm is a hallmark of PD [START_REF] Oswal | Synchronized neural oscillations and the pathophysiology of Parkinson's disease[END_REF][START_REF] Stein | β oscillations in the cortico-basal ganglia loop during parkinsonism[END_REF]). In addition, some studies have also associated low gamma waves to executive and motor control [START_REF] Engel | Beta-band oscillations--signalling the status quo?[END_REF][START_REF] Karch | Increased gamma oscillations during voluntary selection processes in adult patients with attention deficit/hyperactivity disorder[END_REF][START_REF] Gaetz | Evidence for a motor gamma-band network governing response interference[END_REF][START_REF] Stein | β oscillations in the cortico-basal ganglia loop during parkinsonism[END_REF][START_REF] Iijima | Event-Related Synchronization and Desynchronization of High-Frequency Electroencephalographic Activity during a Visual Go/No-Go Paradigm[END_REF][START_REF] Fonken | Frontal and motor cortex contributions to response inhibition: evidence from electrocorticography[END_REF]) and to cortical inhibition [START_REF] Jensen | An oscillatory mechanism for prioritizing salient unattended stimuli[END_REF][START_REF] Jensen | Temporal coding organized by coupled alpha and gamma oscillations prioritize visual processing[END_REF][START_REF] Bonnefond | The role of gamma and alpha oscillations for blocking out distraction[END_REF][START_REF] Cheng | Sensory gating, inhibition control and gamma oscillations in the human somatosensory cortex[END_REF]. It is therefore likely that the power changes observed here in the dACC, mPFC and SMA are related to the control of motor inhibition.

Taken together, these anatomical and spectral data contrast with the dominant idea that ICDs in PD do not involve motor impulsivity [START_REF] Voon | Impulse control disorders and levodopa-induced dyskinesias in Parkinson's disease: an update[END_REF][START_REF] Weintraub | Impulse Control and Related Disorders in Parkinson's Disease[END_REF]. They rather suggest that inhibitory dysfunction might contribute to some of the motor and non-motor symptoms observed in various movement and psychiatric disorders (Jahanshahi et al. 2015a(Jahanshahi et al. , 2015b;;[START_REF] Jahanshahi | Inhibitory dysfunction contributes to some of the motor and non-motor symptoms of movement disorders and psychiatric disorders[END_REF]. Interestingly, our observation that ICDs severity is related to motor impulsivity is reminiscent of a study on pathological gambling in non-PD patients [START_REF] Brevers | Impulsive action but not impulsive choice determines problem gambling severity[END_REF] according to which both cognitive and motor inhibition underlie the presence of ICD, but only motor impulsivity determines its severity. Motor impulsivity is indeed acknowledged in ICDs in the general population [START_REF] Chowdhury | Pathological Gambling and Motor Impulsivity: A Systematic Review with Meta-Analysis[END_REF].

Limitations and open issues

It is important to emphasize, however, that our positive results do not contradict the role of cognitive impulsivity in PD-ICDs. Indeed, although particularly useful to detect problems with selfinitiation and task-set maintenance in Parkinson's disease [START_REF] Ko | Parkinson's disease: increased motor network activity in the absence of movement[END_REF][START_REF] Tinaz | Deficits in task-set maintenance and execution networks in Parkinson's disease[END_REF], resting state oscillations can just inform incompletely about the functional integrity of neural networks. The fact that intrinsic activity at rest reflects only to some extent brain activity during tasks [START_REF] Raichle | The restless brain: how intrinsic activity organizes brain function[END_REF] is certainly critical when considering reward-related cognitive mechanisms whose dysfunctions expected for instance in the orbitofrontal or lateral prefrontal cortex have been mainly associated with neural activity induced by cues or signals [START_REF] Frosini | Parkinson's disease and pathological gambling: results from a functional MRI study[END_REF][START_REF] Voon | Mechanisms underlying dopamine-mediated reward bias in compulsive behaviors[END_REF][START_REF] Politis | Neural response to visual sexual cues in dopamine treatment-linked hypersexuality in Parkinson's disease[END_REF]. In other words, our data can provide only some clues, but a comprehensive map of dysfunctional cortical networks cannot be expected from resting state studies. In addition, EEG does not provide easy access to subcortical activity, and is therefore blind to substantial activity of the mesocorticolimbic system. Again, the lack of evidence of involvement of the reward system in the present data does not rule out the major hypothesis of mesocorticolimbic dysfunctions. Further, another issue related to patient's demographics must be acknowledged for critical appraisal: Group differences were observed for several clinical variables (Table 1). Most were expected because they are either inherent in disease management or a common comorbidity of ICDs like, respectively, medication status (e.g., [START_REF] Samuel | Management of impulse control disorders in Parkinson's disease: Controversies and future approaches[END_REF][START_REF] Cossu | The rise and fall of impulse control behavior disorders[END_REF]Mamikonyan et al., 2008) and depression [START_REF] Antonini | ICARUS study: prevalence and clinical features of impulse control disorders in Parkinson's disease[END_REF]Martini et al., 2018). But the fact that patients with ICDs may have shorter disease duration (p=.09) represents a potential bias.

Nevertheless, our results allow relaunching the debate on the neglected hypothesis of motor inhibition dysfunctions in PD-ICDs [START_REF] Antonelli | Impulsivity and Parkinson's disease: more than just disinhibition[END_REF][START_REF] Kehagia | Targeting impulsivity in Parkinson's disease using atomoxetine[END_REF][START_REF] Nombela | Multiple modes of impulsivity in Parkinson's disease[END_REF][START_REF] Ye | Improving response inhibition in Parkinson's disease with atomoxetine[END_REF]. Further studies are needed to provide direct causal evidence that, among the numerous neural mechanisms involved in response inhibition [START_REF] Chambers | Insights into the neural basis of response inhibition from cognitive and clinical neuroscience[END_REF][START_REF] Verbruggen | Models of response inhibition in the stop-signal and stop-change paradigms[END_REF][START_REF] Aron | From reactive to proactive and selective control: developing a richer model for stopping inappropriate responses[END_REF][START_REF] Bari | Inhibition and impulsivity: behavioral and neural basis of response control[END_REF][START_REF] Criaud | Have we been asking the right questions when assessing response inhibition in go/no-go tasks with fMRI? A meta-analysis and critical review[END_REF][START_REF] Li | Response Inhibition[END_REF][START_REF] Criaud | Testing the physiological plausibility of conflicting psychological models of response inhibition: a forward inference fMRI study[END_REF], some might be dysfunctional in PD-ICDs. In this respect it is noteworthy that the valence of beta power changes observed in the present study are difficult to interpret precisely.

Indeed, beta activity is known to be involved in different forms of top-down proactive control. It would directly drive, for instance, local inhibition in targeted motor structures (e.g., [START_REF] Hwang | Cortical neurodynamics of inhibitory control[END_REF]. It could also drive, however, the executive setting of local, automatic, self-inhibitory networks that gate information processing locally in the motor system 1 [START_REF] Albares | The dorsal medial frontal cortex mediates automatic motor inhibition in uncertain contexts: evidence from combined fMRI and EEG studies[END_REF]. The first mechanism predicts decreased beta power as motor impulsivity increases (see also [START_REF] Lee | Resting-state EEG activity related to impulsivity in gambling disorder[END_REF].

The second mechanism predicts increased beta power as motor impulsivity increases (see also [START_REF] Threadgill | Resting beta activation and trait motivation: Neurophysiological markers of motivated motor-action preparation[END_REF]. In addition, it is not possible to determine from these data if brain activity changes are a direct expression of ICDs-related dysfunctions or the manifestation of compensatory mechanisms. Another issue here is the fact that we found no difference in alpha power, a direct physiological marker of active inhibition [START_REF] Klimesch | EEG alpha oscillations: the inhibition-timing hypothesis[END_REF][START_REF] Jensen | Shaping functional architecture by oscillatory alpha activity: gating by inhibition[END_REF][START_REF] Haegens | α-Oscillations in the monkey sensorimotor network influence discrimination performance by rhythmical inhibition of neuronal spiking[END_REF][START_REF] Mathewson | Pulsed out of awareness: EEG alpha oscillations represent a pulsed-inhibition of ongoing cortical processing[END_REF][START_REF] Klimesch | α-band oscillations, attention, and controlled access to stored information[END_REF][START_REF] Hindriks | Thalamo-cortical mechanisms underlying changes in amplitude and frequency of human alpha oscillations[END_REF].

This does not mean that no difference in the activity of local inhibitory populations can account for dysfunctions of motor inhibition in ICDs. We just did not find significant modulations of α oscillations at rest. These issues require further investigations using specific behavioral tasks coupled with appropriate event-related brain imaging protocols (e.g., [START_REF] Verbruggen | Models of response inhibition in the stop-signal and stop-change paradigms[END_REF][START_REF] Hu | Neural processes of preparatory control for stop signal inhibition[END_REF][START_REF] Swann | Roles for the pre-supplementary motor area and the right inferior frontal gyrus in stopping action: electrophysiological responses and functional and structural connectivity[END_REF][START_REF] Huster | Electroencephalography of response inhibition tasks: functional networks and cognitive contributions[END_REF][START_REF] Zandbelt | Expectations and violations: delineating the neural network of proactive inhibitory control[END_REF][START_REF] Hwang | Cortical neurodynamics of inhibitory control[END_REF][START_REF] Lavallee | When holding your horses meets the deer in the headlights: time-frequency characteristics of global and selective stopping under conditions of proactive and reactive control[END_REF][START_REF] Albares | Interaction of noradrenergic pharmacological manipulation and subthalamic stimulation on movement initiation control in Parkinson's disease[END_REF][START_REF] Zhang | Independent component analysis of functional networks for response inhibition: Inter-subject variation in stop signal reaction time[END_REF][START_REF] Wessel | Stop-related subthalamic beta activity indexes global motor suppression in Parkinson's disease[END_REF][START_REF] Bartoli | Topography and timing of activity in right inferior frontal cortex and anterior insula for stopping movement[END_REF][START_REF] Wagner | Establishing a Right Frontal Beta Signature for Stopping Action in Scalp EEG: Implications for Testing Inhibitory Control in Other Task Contexts[END_REF].

Conclusions and perspectives

Our data clearly bring to the forefront the hypothesis that PD-ICDs might partly be due to motor dysfunctions leading to action impulsivity, and not only to cognitive dysfunctions leading to choice impulsivity. Potential implications for PD therapy are substantial. Indeed, recent work on the neural and neurochemical bases of response inhibition, a central function controlling the initiation of any response whatever a complex decision has to be made or not, has questioned the implication of the dopaminergic (DA) system (e.g., [START_REF] Obeso | Levodopa medication does not influence motor inhibition or conflict resolution in a conditional stop-signal task in Parkinson's disease[END_REF][START_REF] Favre | Deep brain stimulation of the subthalamic nucleus, but not dopaminergic medication, improves proactive inhibitory control of movement initiation in Parkinson's disease[END_REF][START_REF] Michely | Dopaminergic modulation of motor network dynamics in Parkinson's disease[END_REF], and has identified the key role of the noradrenergic (NA) system [START_REF] Chamberlain | Neurochemical modulation of response inhibition and probabilistic learning in humans[END_REF][START_REF] Chamberlain | Atomoxetine modulates right inferior frontal activation during inhibitory control: a pharmacological functional magnetic resonance imaging study[END_REF][START_REF] Chamberlain | The neuropsychiatry of impulsivity[END_REF][START_REF] Robbins | The Neuropsychopharmacology of Fronto-Executive Function: Monoaminergic Modulation[END_REF][START_REF] Chamberlain | Noradrenergic modulation of cognition: therapeutic implications[END_REF]. In particular, proactive response inhibition, the basic executive function supported by the circuit identified in the present results, has been linked to the NA system and movement initiation dysfunctions in PD [START_REF] Albares | Interaction of noradrenergic pharmacological manipulation and subthalamic stimulation on movement initiation control in Parkinson's disease[END_REF][START_REF] Criaud | Slowness in Movement Initiation is Associated with Proactive Inhibitory Network Dysfunction in Parkinson's Disease[END_REF][START_REF] Spay | Clonidine modulates the activity of the subthalamic-supplementary motor loop: evidence from a pharmacological study combining DBS and EEG recordings in Parkinsonian patients[END_REF]. Other clues point to the involvement of the serotoninergic (5HT) system [START_REF] Eagle | The neuropsychopharmacology of action inhibition: cross-species translation of the stop-signal and go/no-go tasks[END_REF][START_REF] Pattij | The neuropharmacology of impulsive behaviour[END_REF][START_REF] Bari | Dissociable effects of noradrenaline, dopamine, and serotonin uptake blockade on stop task performance in rats[END_REF][START_REF] Paterson | Impulsive action and impulsive choice are mediated by distinct neuropharmacological substrates in rat[END_REF][START_REF] Ye | Predicting beneficial effects of atomoxetine and citalopram on response inhibition in Parkinson's disease with clinical and neuroimaging measures[END_REF]. Dysfunction of specific inhibitory processes in PD patients with ICDs might thus be treated with non-dopaminergic pharmacological agents as suggested by recent experimental data [START_REF] Kehagia | Targeting impulsivity in Parkinson's disease using atomoxetine[END_REF][START_REF] Ye | Improving response inhibition in Parkinson's disease with atomoxetine[END_REF][START_REF] Rae | Atomoxetine restores the response inhibition network in Parkinson's disease[END_REF]). This approach is successful in psychiatric conditions with impulsivity symptoms like Attention Deficit Hyperactivity Disorder [START_REF] Pattij | The neuropharmacology of impulsive behaviour[END_REF][START_REF] Dalley | Fractionating impulsivity: neuropsychiatric implications[END_REF], but is still to be tested in clinical trials for PD-ICDs [START_REF] Tanwani | A systematic review of treatments for Impulse Control Disorders and related behaviours in Parkinson's disease[END_REF]. Nonetheless, further studies are warranted to identify all possible sources of dysfunction leading to ICDs. This should help defining subtypes of ICDs in PD differently associated with the DA, NA and 5HT systems, as inspired by former work in psychiatric conditions [START_REF] Campo | The roles of dopamine and noradrenaline in the pathophysiology and treatment of attention-deficit/hyperactivity disorder[END_REF][START_REF] Dalley | Fractionating impulsivity: neuropsychiatric implications[END_REF].

Footnote

1. This is a paradoxical mechanism by which voluntary control of action may be achieved. The ability to provide controlled responses in unpredictable environments would require the automatic activation of the self-inhibitory circuitry within the SMA (i.e., automatic inhibition of automatic responses). Conversely, enabling automatic behavior when the environment becomes predictable would require top-down control to deactivate anticipatorily and temporarily the default and automatic inhibitory set. For this inhibitory mechanism, more control (more β) would mean more automatic (more impulsive) behavior [START_REF] Albares | The dorsal medial frontal cortex mediates automatic motor inhibition in uncertain contexts: evidence from combined fMRI and EEG studies[END_REF].

Figure legends

  Inclusion criteria were: age between 40 and 70 years old, with idiopathic PD, benefiting from a stable antiparkinsonian drug therapy for at least 2 months. Exclusion criteria were: dementia, cognitive impairment (MMSE < 26), other neurologic or psychiatric disease, pharmacological treatment with cerebral or psychic impact, substance abuse according to the criteria DSM-IV-TR (except tobacco smoking). Patients remained on their usual medication during the study. Clinical and neuropsychological assessment included the Mini Mental State Examination (MMSE), the Frontal Assessment Battery (FAB), the Minnesota Impulsive Disorders Interview (MIDI) and the QUIP to assess ICDs, the Barratt Impulsiveness Scale (BIS-11), the Beck Depression Inventory (BDI), the UPDRS-III (Unified Parkinson's Disease Rating Scale part III) and the evaluation of Hoehn & Yahr (H&Y) stage. Data from 10 subjects had to be discarded following technical issues during data collection due to difficulties inherent to bedside acquisition (excessive unfilterable noise, defective electrodes). In the end, two groups of 22 ICDs+ and 17 ICDs-were included in the analysis. Groups were matched for age, disease duration, levodopa equivalent daily dose (LEDD) and UPDRS-III.

  comfortably sat and were instructed to stay quietly with the eyes closed. The EEG signal was recorded at a sampling rate of 4096Hz with the BioSemi™ ActiveTwo Mk2 system from 128 electrodes (BioSemi™ ABC system standard locations, electrode offsets <20 mV). Six supplementary electrodes were used in temporal (Biosemi spherical coordinates: Phi -103.5 Theta -18 -36; Phi 103.5 Theta 18 36) and eye (Phi 103.5-103.5 Theta 81-81) regions. The CMS active electrode and the DRL passive electrode served to create a feedback loop steering the average potential of the participant in the Common Mode voltage.

2. 3

 3 .5.2 Oscillatory modulations predicting ICDs severity (QUIP score, ICDs+ only): For each IC and each frequency band, a multiple regression analysis was applied based on one factor of interest (the QUIPA -ICDs score) and two factors of no-interest (the UPDRS-III score and the LEDD): Powerrel(IC,Freq.) = b1. QUIPA + b2. UPDRS III + b3. LEDD + b4Where b1, b2, b3 represent the parameters to be estimated for the first, second and third factors, respectively, and b4 represents the estimated intercept. The second and the third factors are constants which represent a modulation of the source spectral activity associated with the UPDRS-III motor score and the LEDD that are non-clinically relevant for the ICDs severity (QUIP score). Modeling these effects is essential because motor performance and medication-induced spectral modulations that do not account for ICDs severity are likely. The multiple regression provides an F-statistic for the quality of the model fitting, and one t-statistic for the modeled factor.

Fig. 1 :

 1 Fig. 1: Overview of data processing and analysis pipeline.

Fig. 2 :

 2 Fig. 2: Resting state cortical oscillatory activity (spectral power) predicting impulse control disorders. (A) Group effect, controlled for UPDRS-III and LEDD. The left side of the figure displays the frequency bands in which significant differences are observed for each one of the identified sources. The right side of the figure shows the location (current density map obtained with sLoreta) of the corresponding source. (B) Power changes correlating with ICDs severity (as indexed by QUIP scores), controlled for UPDRS-III and LEDD. The left side of the figure displays the frequency bands in which significant differences are observed. The right side of the figure shows the location (current density map obtained with sLoreta) of the unique source revealing power changes predicting ICD severity.

Acknowledgments

This work was supported by a grant PALSE 2016 to CS and a grant ANR to PB. The authors are thankful to Francesca Natuzzi, Francesco Turco and Chiara Siri (Parkinson Institute in Milan) who helped to collect neuropsychological data, and are also thankful to Mauro Schiavella and Matteo Pelagatti from the University of Milan-Bicocca who substantially contributed to the build-up of local capacity.

Declaration of interest

None of the authors have potential conflicts of interest to be disclosed.

Authors' Roles

Financial Disclosures

None.