Pierre Inizan

HOMOGENEOUS FRACTIONAL EMBEDDINGS

Fractional equations appear in the description of the dynamics of various physical systems. For Lagrangian systems, the embedding theory developped by Cresson ["Fractional embedding of differential operators and Lagrangian systems", J. Math. Phys. 48, 033504 (2007)] provides a univocal way to obtain such equations, stemming from a least action principle. However, no matter how equations are obtained, the dimension of the fractional derivative differs from the classical one and may induce problems of temporal homogeneity in fractional objects. In this paper, we show that it is necessary to introduce an extrinsic constant of time. Then, we use it to construct two equivalent fractional embeddings which retain homogeneity. The notion of fractional constant is also discussed through this formalism. Finally, an illustration is given with natural Lagrangian systems, and the case of the harmonic oscillator is entirely treated.

Introduction

For about twenty years, the fractional calculus has known significant development, according to its successes in various domains, such as chaotic dynamics, viscoelasticity, acoustics, electricity or polymer chemistry [START_REF] Bagley | A theoritical basis for the application of fractional calculus in viscoelasticity[END_REF][START_REF] Bagley | On the fractional calculus model of viscoelasticity behavior[END_REF][START_REF] Carpinteri | Fractals and Fractional Calculus in Continuum Mechanics[END_REF][START_REF] Hilfer | Applications of Fractional Calculus in Physics[END_REF][START_REF] Samko | Fractional integrals ans derivatives: theory and applications[END_REF]. In this context, fractional equations are used for modeling physical phenomena. No more considered as formal objects, but as physically significant, one may expect them to be homogeneous, a fundamental principle of the physical equations. In other words, the objects of a same equation must have the same dimension, expressible in terms of fundamental units, such as length L, mass M or time T . For example, the dimension of a force F is

[F ] = M L 2 T -2 .
However, in many fractional equations we can find in litterature, homogeneity is not respected. Let firm up details of our topic: from a strictly mathematical point of view, this notion of homogeneity is meaningless. The problem arises only if the equations considered are susceptible to model a physical system. But in that case, this question is fundamental. However, the problem is often hidden in modelisations because the constants are seen as parameters to fit.

How to obtain fractional equations in time ? The usual approach consists in subsistuting in the classical equation considered the derivative d/dt (homogeneous to T -1 ) by a new one, d α /dt α (homogeneous to T -α ), with 0 < α < 1. Clearly, this transformation does not conserve the temporal homogeneity of the equation. This remark would apply to fractional derivatives in space, but in this article, only the temporal ones will be considered.

The problem of the passage from the classical case to the fractional one has already been discussed in [START_REF] Cresson | Fractional embedding of differential operators and Lagrangian systems[END_REF]. Shifting the derivative leads to fractional equations, but, from a physical point of view, is it the right procedure ? Do the fractional equations obtained remain physically relevant ? Answers are given in the case of Lagrangian systems. Extending the least action principle, Cresson [START_REF] Cresson | Fractional embedding of differential operators and Lagrangian systems[END_REF] provides an univocal method, the fractional embedding, which leads to fractional equations. The respect of this other fundamental principle makes the equations susceptible to conserve a physical meaning.

This article is in keeping with this philosphy. In section 2, it is shown that an extrinsic characteristic time has to be introduced. After having presented the notion of embedding in section 3, we show in section 4 that this time constant leads to two equivalent methods of fractional embedding, which additionally conserve homogeneity. The notion of fractional constant is also adressed. Finally, an illustration is given in section 5 with the case of natural Lagrangian systems. A complete resolution is given for the harmonic oscillator.

Extrinsic characteristic time

In this paper, the notation d α /dt α will concern an unspecified fractional derivative, except in section 5.2. Moreover, the difficulties inherent to fractional derivatives (for example, d α /dt α • d α /dt α = d 2α /dt 2α in general) won't be addressed here.

As it has already been said, the fractional aspect of equations occurs with derivatives generalized to noninteger orders. If this sustitution is done in one term of an equation, the other ones have also to be modified so as to preserve homogeneity. As a first example, we consider the damped oscillator equation,

d 2 dt 2 x(t) + λ d dt x(t) + ω 2 x(t) = 0. (1)
The constants λ and ω are homogeneous to T -1 . Replacing d/dt by d α /dt α gives

d 2α dt 2α x(t) + λ d α dt α x(t) + ω 2 x(t) =
d 2α dt 2α x(t) + λ α d α dt α x(t) + ω 2α x(t) = 0. ( 2
)
Things are getting trickier with the diffusion equation,

∂ ∂t u(x, t) -D ∂ 2 ∂x 2 u(x, t) = 0, with [D] = L 2 T -1 . (3)
If the operator ∂/∂t is directly remplaced by ∂ α /∂t α , the only way to preserve the temporal homogeneity is to shift D into D α . We obtain

∂ α ∂t α u(x, t) -D α ∂ 2 ∂x 2 u(x, t) = 0.
Unfortunately, this equation becomes now nonhomogeneous in space: ∂ α /∂t α has no spatial dimension, whereas the spatial dimension of D α ∂ 2 /∂x 2 is L 2 (1-α) . The only way to preserve the spatial homogeneity is to change ∂/∂x by ∂ α /∂x α , which leads to

∂ α ∂t α u(x, t) -D α ∂ 2α ∂x 2α u(x, t) = 0. ( 4
)
Regarding the numerous studies dealing with that equation (for example, [START_REF] Meerschaert | Limit theorems for continuous-time random walks with infinite mean waiting times[END_REF][START_REF] Zaslavsky | Hamiltonian Chaos & Fractional Dynamics[END_REF]), one of the most used in fractional calculus, this case seems too restrictive. According to [13, chap. 16], anomalous transport cannot be described by (4) (because the transport exponent remains classical).

As a last example, we consider the free fall equation,

d 2 dt 2 x(t) + g = 0. ( 5
)
The dimension of the gravity acceleration g is

[g] = L T -2 . If d 2 /dt 2 is changed directly into d 2α /dt 2α
, it is impossible to obtain an fractional and homogeneous equation.

A first conclusion could be drawn: a necessary condition for an equation to be physically relevant in the fractional case is the possibility of obtaining a homogeneous formulation. In this case, the damped oscillator and the diffusion fractional equations fill this condition, unlike the free fall equation.

However, the constants which appear the equations may be built from other constants. For example, the gravity acceleration g is in fact defined by

g = G M T R -2
T , where G is the gravity constant, M T et R T the mass and the radius of the Earth. So, equation ( 5) can provide a fractional and homogeneous equation,

d 2α dt 2α x(t) + g α R 1-α T = 0.
Therefore, the initial equation itself is not suffisant to conclude if the fractional equation derived is physically relevant or not. This first conclusion is consequently invalid.

Moreover, if no restrictions are added, there may be several ways to obtain fractional equations, which is unsatisfactory. Indeed, (1) can notably turn into

d 2α dt 2α x(t) + λ α d α dt α x(t) + ω 2 λ 2α-2 x(t) = 0, or into d 2α dt 2α x(t) + λ 2α-β d β dt β x(t) + ω 2α x(t) = 0, 0 < β < 1.
Both of these equations differ from [START_REF] Bagley | On the fractional calculus model of viscoelasticity behavior[END_REF].

Facing this problem of underdetermination, equations have to be constrained so as to provide a unique fractional and homogeneous equation. As it was mentioned above, the fractional embedding plays this part in the case of Lagrangian systems. It implies, among others, that there is a single fractional exponent in fractional equations describing such systems. Furthermore, it does not modify the constants of the initial equation. Once the Lagrangian is given, the structure is totally determined and the fractional aspect appears only through the switch of d/dt into d α /dt α . Consequently, a suitable fractional operator should not modify the homogeneity of the initial equation. Two possibilities are conceivable. The first one is the temporal nondimensionalization of the system: the classical and fractional temporal operators become dimensionless, and the homogeneity of the equation is preserved. The second method consists in choosing a fractional operator which dimension is the same as the classical one, i.e. T -1 . Both of them require the introduction of a time constant τ . The examples above show that τ may not be determined directly by the equation (see [START_REF] Hilfer | Foundations of fractional dynamics[END_REF]), or that several constants could suit for τ (see ( 1)). Consequently, the time constant τ has to be postulated, and defined a priori, i.e., independently of the classical equation. The introduction of such a constant enables to solve the problem of homogeneity, while remaining compatible with the fractional embedding.

General embeddings

We adapt here the embedding notion presented in [START_REF] Cresson | Fractional embedding of differential operators and Lagrangian systems[END_REF].

3.1. Embedding of differential operators. Let f = (f 1 , . . . , f p ) and g = (g 1 , . . . , g p ) be two p-uplets of C ∞ functions R k+2 -→ R. Let a, b ∈ R with a < b. We denote by O(f, g) the differential operator defined by O(f, g) : C ∞ ([a, b]) -→ C ∞ ([a, b]) x -→ p i=0 f i • d dt i • g i x(•), . . . , d dt k x(•), • , (6)
where, for any function u : R k+2 -→ R, any functions x 0 , . . . ,

x k : [a, b] -→ R, u (x 0 (•), . . . , x k (•), •) : [a, b] -→ R t -→ u(x 0 (t), . . . , x k (t), t),
and, for any functions f and g, (f

• g)(t) = f (t) • g(t)
. Now we extend this class of operators with an operator which generalizes d/dt.

Definition 1. Let D : C ∞ ([a, b]) -→ C ∞ ([a, b]) be a differential operator. With the previous notations, the D-embedding of O(f, g), denoted E(O(f, g), D), is defined by E(O(f, g), D) : C ∞ ([a, b]) -→ C ∞ ([a, b]) x -→ p i=0 f i • D i • g i x(•), . . . , D k x(•), • .
We define the ordinary differential equation associated with O(f, g) by

O(f, g)(x) = 0, x ∈ C ∞ ([a, b]). ( 7
) Definition 2. Let D : C ∞ ([a, b]) -→ C ∞ ([a, b]
) be a differential operator. With the previous notations, the D-embedding of equation ( 7) is defined by

E(O(f, g), D)(x) = 0, x ∈ C ∞ ([a, b]).

3.2.

Embedding of Lagrangian systems. We consider a Lagrangian system, with Lagrangian L(x, v, t), t ∈ [a, b], and

D : C ∞ ([a, b]) -→ C ∞ ([a, b]) a differential operator.
The Lagrangian L can naturally lead to a differential operator of the form ( 6)

O(1, L) : C ∞ ([a, b]) -→ C ∞ ([a, b]) x -→ L x(•), d dt x(•), • .
Now we identify L and O(1, L).

The D-embedding of L, E(L, D), will be denoted L(D) or shortened into L,

L(D) : C ∞ ([a, b]) -→ C ∞ ([a, b]) x -→ L(x(•), D x(•), •).
In Lagrangian mechanics, the action and its minima play a central role. Let us define an action suitable for the embedding formalism.

Definition 3. Let g : C ∞ ([a, b]) -→ C ∞ ([a, b]
) be a mapping. The action of g, denoted A(g), is defined by

A(g) : C ∞ ([a, b]) -→ R x -→ b a g(x)(t) dt.
For example, with the identification L ≡ O(1, L), the action of L is given by

A(L)(x) = b a L x(t), d dt x(t), t dt.
Concerning the D-embedding of L, the associated action is

A( L(D))(x) = b a L (x(t), Dx(t), t) dt.
The extremum of the associated action to a Lagrangian system provides the equation of motion of the system trough the following theorem.

Theorem 1. The action A(L) is extremal in x if and only if x satisfies the Euler-Lagrange equation, given by

∀t ∈ [a, b], ∂ 1 L x(t), d dt x(t), t - d dt ∂ 2 L x(t), d dt x(t), t = 0. (8)
Equation ( 8) is denoted EL(L). For the embedded Lagrangian, a connected result states the following.

Theorem 2. Under some appropriate variations of the action (see [START_REF] Inizan | Compatibility between Hamiltonian formalisms[END_REF]), A( L(D)) is extremal in x if x satisfies the causal Euler-Lagrange equation, given by

∀t ∈ [a, b], ∂ 1 L(x(t), Dx(t), t) -D ∂ 2 L(x(t), Dx(t), t) = 0. ( 9
)
Equation ( 9) is denoted EL( L(D)). This leads to the following result.

Theorem 3. In the causal case (i.e. with the appropriate variations of the action), we have the commutative scheme,

EL(E(L, D)) ≡ E(EL(L), D).

This embedding procedure is qualified as coherent.

Proof. Equation ( 8) can be written in the form [START_REF] Inizan | Compatibility between Hamiltonian formalisms[END_REF], with f = (1, 1) and

g = (∂ 1 L, -∂ 2 L).
Its D-embedding is exactly (9).

Homogeneous fractional embeddings

Let us now explain the two methods presented in the Introduction for the Lagrangian sytems. We show that they are equivalent. Then we present what could be a third method, also equivalent, which is farther from the philosophy of the embedding theory, but which makes fractional constants to appear.

Let L(x, v, t) be a Lagrangian, with t ∈ [a, b], and τ the time constant introduced in the previous part. The temporal dimensions of the variables x, v and t are respectively T 0 , T -1 and T . 4.1. Temporal nondimensionalization. Nondimensionalization is a widespread method used in physics, particularly in fluid mechanics, to simplify equations and to exhib relevant parameters. Concerning fractional dynamics, the constant τ introduced above seems physically relevant. In the following part, it is used to nondimensionalize equations according to the temporal dimension. None of the variables of L n has temporal dimension, which justify the denomination. We choose here D = d α /dt α . As the new temporal evolution variable u has no dimension, the classical derivative according to this new time, d/du, is also dimensionless, such as d α /du α . Therefore, the substitution d/du → d α /du α does not break homogeneity anymore.

For example, if d α /dt α is taken as the Caputo derivative, defined by

s 0 D α s f (s) = 1 Γ(1 -α) s s 0 (s -v) -α f ′ (v) dv,
we see that if t 0 and t have the dimension of a time,

d α /dt α = t 0 D α t is homogeneous to T -α , whereas if u 0 and u are dimensionless, d α /du α = u 0 D α u is dimensionless. The d α /du α -embedding of L n , Ln (d α /du α ), is Ln d α du α : C ∞ ([ã, b]) -→ C ∞ ([ã, b]) x -→ L n x(•), d α du α x(•), • .
For sake of lisibility, Ln (d α /du α ) will be shortened into Ln . It verifies

∀x ∈ C ∞ ([ã, b]), ∀u ∈ [ã, b], Ln (x)(u) = L x(u), τ -1 d α du α x(u), τ u .
The associated action to Ln is

A( Ln )(x) = b ã L n x(u), d α du α x(u), u du
and leads to the following result.

Theorem 4. The Euler-Lagrange equation associated with the d α /du α -embedding of the nondimensionalized Lagrangian L n is given, in the causal case, by

∀u ∈ [ã, b], ∂ 1 L n x(u), d α du α x(u), u - d α du α ∂ 2 L n x(u), d α du α x(u), u = 0, ( 10
)
with x ∈ C ∞ ([ã, b]).
This is the equation of motion of the Lagrangian system which Lagrangian is L n , and which dynamics evolution is fractional, according to the operator d α /du α . The solution x of (10) depends on the nondimensionalized evolution variable u. As d α /du α has no dimension, equation ( 10) is clearly homogeneous.

This method eludes the question of the dimension of the fractional operator, and, by modifying the Lagrangian variables (but not the Lagrangian itself), it leads to an homogeneous fractional embedding. However, it occults the real dynamics, based on a real (i.e., dimensionalized) time. The following method will lift this veil. 4.2. Homogeneous fractional derivative. The time constant τ can be used to nondimensionalize the temporal operator, but it also enables to build a fractional operator which dimension remains T -1 : τ α-1 d α /dt α . Therefore, this operator preserves the homogeneity of the fractional embedding. Let us detail the procedure.

We consider the same initial Lagrangian L(x, v, t), with t ∈ [a, b]. Contrary to the previous method, there is no need to define a new Lagrangian nor a new evolution variable. Now the differential operator D becomes τ α-1 d α /dt α and acts on functions which depend on the real time t.

The τ α-1 d α /dt α -embedding of L, L (τ α-1 d α /dt α ) will be denoted Lh .

Lh : C ∞ ([a, b]) -→ C ∞ ([a, b]) x -→ L x(•), τ α-1 d α dt α x(•), • , The associated action A( Lh ) verifies ∀x ∈ C ∞ ([a, b]), A( Lh )(x) = b a L x(t), τ α-1 d α dt α x(t), t dt.
The equation of motion is now given by Theorem 5. The Euler-Lagrange equation associated with the (τ α-1 d α /dt α )-embedding of L is given, in the causal case, by

∀t ∈ [a, b], ∂ 1 L x(t), τ α-1 d α dt α x(t), t -τ α-1 d α dt α ∂ 2 L x(t), τ α-1 d α dt α x(t), t = 0. ( 11
)
In this method, the fractional Euler-Lagrange equation and its solutions x depend of the real time t, and all of the fractional objects (variables, operator, Lagrangian, equation) hold the same dimension as their classical predecessors.

4.3.

Equivalence between the two methods. Both of these two methods constitute homogeneous fractional embeddings. Fortunately, if a quite natural condition is respected, they turn out to be equivalent.

We suppose that the fractional derivative d α /dt α verifies the following condition:

∀x ∈ C ∞ ([a, b]), τ α d α dv α x(v)| v=t = d α du α x(u)| u=t/τ , where x : u → x(uτ ). ( 12
)
This property is notably verified by the usual fractional derivatives. For example, for the Caputo derivative, we have

ãD α u x(u) = 1 Γ(1 -α) u ã (u -v) -α x′ (v) dv = 1 Γ(1 -α) uτ ãτ (u - s τ ) -α x′ ( s τ ) ds τ , s = vτ = τ α-1 Γ(1 -α) uτ a (uτ -s) -α (τ x ′ (s)) ds = τ α Γ(1 -α) t a (t -s) -α x ′ (s) ds, t = uτ = τ α a D α t x(t).
The two methods are related through the following result.

Theorem 6. If the fractional operator d α /dt α respects condition [START_REF] Tarasov | Dynamics with low-level fractionality[END_REF], then the two methods are equivalent: x : t → x(t) is solution of (11) if and only if x : u → x(uτ ) is solution of (10).

Proof. Partial derivatives of L and L n verify

∂ 1 L n (x, y, u) = ∂ 1 L x, y τ , uτ , (13)
∂ 2 L n (x, y, u) = 1 τ ∂ 2 L x, y τ , uτ . (14)
From ( 12), ( 13) and ( 14), we infer

∂ 1 L n x(u), d α du α x(u), u = ∂ 1 L x(t), τ α-1 d α dt α x(t), t , ∂ 2 L n x(u), d α du α x(u), u = 1 τ ∂ 2 L x(t), τ α-1 d α dt α x(t), t ,
which proves that ( 10) and ( 11) are similar.

Therefore, if condition ( 12) is verified, nondimensionalized solution u ∈ [a/τ, b/τ ] → x(u) of ( 10) can be redimensionalized into t ∈ [a, b] → x(t/τ ), solution of the dimensionalized and equivalent equation [START_REF] Stanislavsky | Hamiltonian formalism of fractional systems[END_REF]. 4.4. Fractional constants. In various litterature dealing with fractional equations using the operator d α /dt α (homogeneous to T -α ), homogeneity is sometimes preserved by introducing fractional constants (see [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF] for example). For a wide class of Lagrangians, the d α /dt α -embedding of a modified Lagrangian leads to a homogeneous fractional equation involving fractional constants.

We consider the same Lagrangian system as above. The temporal dimension of L is here denoted T n 0 (n 0 = -2 in most of the cases). We suppose that L is equal to its Laurent series in the variable v,

L(x, v, t) = i∈Z a i f i (x, t) v i , (15)
where [a i ] = T i+n 0 . The functions f i have hence no temporal dimension. Using the fractional operator τ α-1 d α /dt α , we obtain for any function

x ∈ C ∞ ([a, b]) L x(t), τ α-1 d α dt α x(t), t = i∈Z a i f i (x, t) τ α-1 d α dt α x(t) i = i∈Z a i f i (x, t)τ i(α-1) d α dt α x(t) i = i∈Z āi f i (x, t) d α dt α x(t) i , with āi = a i τ i(α-1) .
The dimension of āi is [ā i ] = T iα+n 0 . With this fractional dimension, those coefficients can be considered as fractional constants.

We can form a new Lagrangian with those constants.

Definition 5. For a Lagrangian L(x, v, t) = i∈Z a i f i (x, t) v i , the fractional Lagrangian, denoted L f , is defined by

L f (x, v, t) = i∈Z āi f i (x, t) v i . The d α /dt α -embedding of L f , denoted Lf , verifies Lf : C ∞ ([a, b]) -→ C ∞ ([a, b]) x -→ i∈Z āi f i (x(•), •) d α dt α x(•) i . Hence we have L τ α-1 d α dt α = Lf d α dt α , (16)
which leads to Theorem 7. The Euler-Lagrange equation associated with the d α /dt α -embedding of L f is given, in the causal case, by

∀t ∈ [a, b], i∈Z āi ∂ 1 f i (x, t) d α dt α x(t) i -i āi d α dt α f i (x, t) d α dt α x(t) i-1 = 0, (17)
with x ∈ C ∞ ([a, b]).
With relation ( 16), we infer the following result.

Theorem 8. If condition (15) is verified, the (τ α-1 d α /dt α )-embedding of L f and the d α /dt α -embedding of L f are equivalent, EL E L, τ α-1 d α dt α ≡ EL E L f , d α dt α , (18)
E EL(L), τ α-1 d α dt α ≡ E EL(L f ), d α dt α . (19)
In the causal case, we have besides (18) ≡ (19).

One verifies that equation ( 17) is homogeneous: inhomogenity of the operator d α /dt α is counterbalanced by the fractional constants āi . However, the homogeneity problem is here only shifted: L f (x(t), d/dt x(t), t) and EL(L f ) are now inhomogeneous. Furthermore, the physical relevance of L f is discutable, and this Lagrangian may appear as an ad hoc construction to obtain homogeneous fractional equations. In this sense, this third method does not fit with an underlying idea of the embedding theory. For any Lagrangian system, its Lagrangian is independant of the dynamics: it is only the choice of the evolution operator, i.e. the temporal derivative, which conditions the dynamics of the system.

Finally, it seems that the second method is the most relevant homogeneous fractional embedding.

Natural Lagrangian systems

5.1. General case. An illustration of those methods is given here with the most widespread type of Lagrangians: the natural ones. Definition 6. We consider an autonomous Langrangian system with Lagrangian L(x, v), evolving on a temporal interval [a, b]. L is a natural Lagrangian if it can be written as

L(x, v) = 1 2 mv 2 -V (x), ( 20
)
where m is the mass of the system, 1 2 mv 2 is the kinetic energy, and V (x) the potential energy. The temporal dimension of L is T -2 .

We introduce a time constant τ , and we apply now the three embeddings to those systems. The new Lagrangian is

L n (x, y) = L x, y τ = 1 2τ 2 my 2 -V (x),
and its embedding verifies

∀x ∈ C ∞ ([ã, b]), ∀u ∈ [ã, b], Ln (x)(u) = 1 2τ 2 m d α du α x(u) 2 -V (x(u)) .

The associated action is

A( Ln )(x) = b ã L n x(u), d α du α x(u) du = b ã 1 2τ 2 m d α du α x(u) 2 -V (x(u)) du,
∀x ∈ C ∞ ([a, b), ∀t ∈ [a, b], Lh (x)(t) = 1 2 mτ 2(α-1) d α dt α x(t) 2 -V (x(t)) ,
the action is given by

A( Lh )(x) = b a L x(t), τ α-1 d α dt α x(t), t dt = b a 1 2 mτ 2α-2 d α dt α x(t) 2 -V (x(t)) dt,
and the causal Euler-Lagrange equation is

∀t ∈ [a, b], mτ 2(α-1) d 2α dt 2α x(t) + V ′ (x(t)) = 0. (21)
5.1.3. With fractional constants. Let x 0 be a position such as V (x 0 ) = 0. The potential energy can hence be written as V (x) = a 0 f 0 (x, t), with a 0 = V (x 0 ), which temporal dimension is T -2 , and f 0 (x, t) = V (x)/V (x 0 ), which is dimensionless.

The Lagrangian (20) can therefore be expressed in the form (15), with

• a i = 0 for i < 0,

• a 0 = V (x 0 ), f 0 (x, t) = V (x) V (x 0 ) , • a 1 = 0, • a 2 = m, f 2 (x, t) = 1 2 , • a j = 0 for j > 2.
The fractional constants are 1) , and lead to the new fractional Lagrangian

• ā0 = a 0 = V (x 0 ), • ā2 = a 2 τ 2(α-1) = m τ 2(α-
L f (x, v) = 1 2 m τ 2(α-1) v 2 -V (x).
The associated action is defined by

A( Lf )(x) = b a 1 2 mτ 2(α-1) d α dt α x(t) 2 -V (x(t)) dt.
Its extremum provides the following Euler-Lagrange equation:

∀t ∈ [a, b], mτ 2(α-1) d 2α dt 2α x(t) + V ′ (x(t)) = 0, which is similar to (21). Now, let us solve those equations in the case of the harmonic oscillator. 5.2. Harmonic oscillator. This system has already been studied in the fractional case (see [START_REF] Stanislavsky | Hamiltonian formalism of fractional systems[END_REF][START_REF] Zaslavsky | Chaotic and pseudochaotic attractors of perturbed fractional oscillator[END_REF][START_REF] Tarasov | Dynamics with low-level fractionality[END_REF]).

The harmonic oscillator is a natural Lagrangian system which Lagrangian L is given by:

L(x, v) = 1 2 mv 2 - 1 2 kx 2 , (22) with
[x] = L, [v] = L T -1 , [m] = M et [k] = M T -2 .
As above, we introduce a time constant τ (which is a priori not linked with m/k). We arbitrarily choose here the Caputo derivative for the fractional operator d α /dt α . 5.2.1. Temporal nondimensionalization. The nondimensionalized Lagrangian L n is:

L n (x, y) = 1 2τ 2 my 2 - 1 2 kx 2 .
Its embedding, Ln (x)(u) = 1 2τ 2 m (d α /du α x(u)) 2 -1 2 kx(u) 2 leads to the following Euler-Lagrange equation:

∀u ∈ [ã, b], m τ 2 d 2α du 2α x(u) + k x(u) = 0.
With the usual notation ω = k/m, we obtain

∀u ∈ [ã, b], d 2α du 2α x(u) + (ωτ ) 2 x(u) = 0.
The (nondimensionalized) solution, denoted x n , is given by

∀u ∈ [ã, b], x n (u) = x n (ã)E 2α -(ωτ ) 2 (u -ã) 2α ,
where, for λ > 0, E λ is the Mittag-Leffler function defined on C by

∀z ∈ C, E λ (z) = ∞ k=0 z k Γ(λk + 1)
.

The dimensionalized solution, denoted

x d , is hence ∀t ∈ [a, b], x d (t) = x n ( t τ ) = x d (a)E 2α -ω 2 τ 2(1-α) (t -a) 2α .
One verifies that the argument of the Mittag-Leffler function is dimensionless.

5.2.2.

Homogeneous fractional derivative. The embedded Lagrangian Lh , given by

Lh = 1 2 mτ 2(α-1) d α dt α x(t) 2 -k x(t) 2 , provides the equation ∀t ∈ [a, b], mτ 2(α-1) d 2α dt 2α x(t) + k x(t) = 0. (23)
Introducing ω, we obtain

∀t ∈ [a, b], d 2α dt 2α x(t) + ω 2 τ 2(1-α) x(t) = 0,
which admits the solution x h defined by

x h (t) = x h (a)E 2α -ω 2 τ 2(1-α) (t -a) 2α .
One verifies that x n = x h .

5.2.3.

With fractional constants. The Lagrangian (22) can be written as (15), with

• a 0 = k, f 0 (x, t) = 1 2 x 2 , • a 1 = 0, • a 2 = m, f 2 (x, t) = 1 2 , • a i = 0 for i ≥ 3.
The fractional constants are given by 1) . The associated fractional Lagrangian, L f , is

• ā0 = k, • ā2 = a 2 τ 2(α-1) = m τ 2(α-
L f (x, v) = 1 2 mτ 2(α-1) v 2 - 1 2 k x 2 .
We obtain the Euler-Lagrange equation, ∀t ∈ [a, b], mτ 2(α-1) d 2α dt 2α x(t) + k x(t) = 0, which is similar to equation (23).

5.2.4.

Comparison with the inhomogeneous fractional embedding. If we directly apply the fractional embedding presented in [START_REF] Cresson | Fractional embedding of differential operators and Lagrangian systems[END_REF] to the Lagrangian (22), we obtain

d 2α dt 2α x(t) + ω 2 x(t) = 0. ( 24
)
The temporal dimension of d 2α /dt 2α x(t) is T -2α , whereas for ω 2 x(t), it is T -2 . The solution x i of (24) is

x i (t) = x i (a)E 2α -ω 2 (t -a) 2α .

The argument of the Mittag-Leffler function is T 2(α-1) , whereas it should be dimensionless.

Conclusion

With the first two embeddings presented above (temporal nondimensionalization and homogeneous fractional derivative), it becomes possible to preserve homogeneity of embedded equations. Both of them lead to the same fractional equation. Therefore, from any Lagrangian system, it is possible to construct a fractional equation which respects two fundamental physical principles: homogeneity and least action principle.

To do so, a time constant τ has to be introduced. Similarly to the fractional exponent α, this parameter is not defined by the equation itself. Contrary to the nondimensionalization of classical equations, fractional solutions depend on τ (see the example of the harmonic oscillator). This time constant is hence more than a calculus intermediary: it conditions the dynamcis of the system. Its physical relevance seems as much important as the constants of the initial equation. To sum up, a fractional equation should be characterized by two parameters: the fractional exponent α and the time constant τ . So as to understand the physical meaning of this characteristic time, it may be useful to set the context in which such equations arise. In [START_REF] Hilfer | Foundations of fractional dynamics[END_REF][START_REF] Zaslavsky | Hamiltonian Chaos & Fractional Dynamics[END_REF], the fractional aspect of the dynamics appears only on long time scales. At a microscopic level, the dynamics is classical and evolves according to a microscopic time, but when one zooms out at a macroscopic level, ruled by a new macroscopic time, the fractional aspect may emerge. It might be through the relation between these two time scales that the time constant τ may appear.

  0. This equation is non-homogeneous: the three terms have different temporal dimensions (T -α , T -(1+α) and T -2 ). A homogeneous version could be

Definition 4 .

 4 We note ã = a/τ and b = b/τ . The temporal nondimensionalized Lagrangian, denoted L n , evolving on the interval [ã, b], is defined by ∀u ∈ [ã, b], L n (x, y, u) = L(x, y τ , τ u).

5. 1 . 1 .

 11 Temporal nondimensionalization. We denote ã = a/τ and b = b/τ .

2 d

 2 and the Euler-Lagrange equation, in the causal case, turns into ∀u ∈ [ã, b], m τ 2α du 2α x(u) + V ′ (x(u)) = 0. 5.1.2. Homogeneous fractional derivative. The initial Lagrangian remains inchanged. Its embedding is now

Acknowledgements

The author thanks his Ph.D. director, J. Cresson, for his support and for fruitful discussions, W. Thuillot, director of the IMCCE and J.E. Arlot, in charge of the GAP section, for their hospitality.


M Azevedo

Da Silva

A Fournier

M.-C Boutron-Ruault

B Balkau

F Bonnet

H Nabi

G Fagherazzi
email: guy.fagherazzi@lih.lu

Increased risk of type 2 diabetes in antidepressant users: evidence from a 6-year longitudinal study in the E3N cohort

Keywords:

What's new?

 Although meta-analyses have concluded that the use of antidepressants is associated with an increased risk of type 2 diabetes, this association remains debated. Two longitudinal studies found no association between the use of antidepressants and change in glucose levels over time.

 This prospective study found that current antidepressant users were at a higher risk of developing type 2 diabetes during follow-up than non-users, regardless of the type of antidepressant and even after adjusting for symptoms of severe depression.

 The negative effects of using antidepressants on the risk of type 2 diabetes might have important clinical implications for the management of people in need of chronic antidepressant medications.

Introduction

Preventing type 2 diabetes is a public health priority worldwide, as this disease is associated with major complications such as retinopathy, neuropathy, nephropathy, cardiovascular disease, depression and poor quality of life [START_REF] Fosse-Edorh | The burden of diabetes-related complications in France in 2013. Summary and perspectives[END_REF]. In addition, significant healthcare costs are associated with its management [START_REF] Da Rocha Fernandes | IDF Diabetes Atlas estimates of 2014 global health expenditures on diabetes[END_REF].

Prominent risk factors for type 2 diabetes include a family history of diabetes, eating behaviours, overweight, high waist circumference, physical inactivity, advanced age, high blood pressure, ethnicity and education level [START_REF] Who | diabetes prevention program[END_REF].

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The association between the use of antidepressants and the risk of developing type 2 diabetes in the general population has been controversial. Findings from secondary analyses of the Diabetes Prevention Programme randomized controlled trial supported a positive association between their use and the risk of developing type 2 diabetes [4]. Subsequent observational studies yielded conflicting results. Some of these reported antidepressant users to be at greater risk of developing type 2 diabetes compared to nonusers [START_REF] Pan | Use of antidepressant medication and risk of type 2 diabetes: results from three cohorts of US adults[END_REF][START_REF] Kivimäki | Antidepressant medication use, weight gain, and risk of type 2 diabetes: a population-based study[END_REF], while others did not find an association [START_REF] Bhattacharya | Use of antidepressants and statins and short-term risk of new-onset diabetes among high risk adults[END_REF][START_REF] Chang | Effect of Psychotropic Drugs on Development of Diabetes Mellitus in Patients With Alzheimer's Disease[END_REF].

Three meta-analyses [START_REF] Bhattacharjee | Antidepressant use and newonset diabetes: a systematic review and meta-analysis[END_REF][START_REF] Yoon | Antidepressant use and diabetes mellitus risk: a meta-analysis[END_REF][START_REF] Salvi | The risk of new-onset diabetes in antidepressant users -A systematic review and meta-analysis[END_REF] and a systematic review [START_REF] Barnard | Antidepressant medication as a risk factor for type 2 diabetes and impaired glucose regulation: systematic review[END_REF] concluded that the use of antidepressants was associated with an increased risk of type 2 diabetes. However, two longitudinal studies found no association between the use of antidepressants and change in glucose levels over time [START_REF] Kivimäki | Antidepressant medication use and risk of hyperglycemia and diabetes mellitus: a noncausal association?[END_REF][START_REF] Da Silva | Antidepressant medication use and trajectories of fasting plasma glucose, glycated haemoglobin, β-cell function and insulin sensitivity: a 9-year longitudinal study of the D.E.S.I.R. cohort[END_REF]. The association between antidepressant use and type 2 diabetes therefore remains unclear and requires additional studies that can take into account the evolution over time of the use of different classes of antidepressants.

We used longitudinal data from the French E3N cohort study (Étude Épidémiologique de Femmes de la Mutuelle Générale de l'Éducation Nationale) to prospectively examine the association between the use of antidepressants and the risk of developing type 2 diabetes; information on exposure to antidepressants was updated during follow-up, and data on type of antidepressant used were available.

Methods

Study population

The French E3N prospective cohort study was established in 1990 with the initial objective being to study risk factors for cancers in women [START_REF] Clavel-Chapelon | Cohort Profile: The French E3N Cohort Study[END_REF]. This goal has since been expanded to include other chronic diseases. The E3N study is the French component of the European Prospective Investigation into Cancer and Nutrition (EPIC) and is part of EPIC-InterAct, a case-cohort study on type 2 diabetes nested within EPIC [START_REF] Kengne | Non-invasive risk scores for prediction of type 2 diabetes (EPIC-InterAct): a validation of existing models[END_REF]. The E3N cohort included 98 995 women, aged 40-65 years, living in France, who were insured by a health insurance plan that mostly covers teachers. Data are obtained from mailed

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This article is protected by copyright. All rights reserved questionnaires sent every 2-3 years. The health insurance plan has provided all outpatient reimbursements for health expenditure since 1 January 2004. The follow-up questionnaire response rate has remained stable at approximately 80%, and the total loss to follow-up since 1990 has been 3%. The present analyses included 63 999 women who responded to the questionnaire in 2005, were free of drug-treated type 2 diabetes in 2005 and were still affiliated with the health insurance plan (Fig. 1). This questionnaire in 2005 was the baseline for the present study, as it was the first questionnaire available after the health expenditure reimbursement database was initiated in 2004, and the first time that information on depressive symptoms was available. The latest available data were from 1 January 2011, representing the end of follow-up.

Measurements

Antidepressant use

From the drug reimbursement files, antidepressant drugs were identifiable through the Anatomical Therapeutic Chemical (ATC) classification code N06A ('antidepressants'). The different classes of antidepressants were established using ATC codes N06AA for imipramine type, N06AB for selective serotonin reuptake inhibitors, N06AF and N06AG for monoamine oxidase inhibitors and N06AX for other antidepressants (e.g. serotonin and noradrenaline reuptake inhibitors, tetracyclics, and selective serotonin reuptake enhancers). The drug reimbursement files also included the French identifier code for presentations (CIP) and the date of purchase of the drug.

All antidepressant reimbursements during follow-up were extracted. To identify episodes of use versus non-use of antidepressants and assuming that the date of purchase corresponded with the 'start date' of use, we estimated the 'end date' using a three-step approach.

We first obtained the dose (mg) per container (packet of tablets, packet of ampoules, bottle) from the CIP for each antidepressant. In the second step, usual daily doses for adults of the different antidepressants were estimated based on Thesorimed® (https://theso.prod-un.thesorimed.org/index), a database of drugs

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This article is protected by copyright. All rights reserved marketed in France, developed and made available by the French Health Insurance System. For each drug, the period of use in days was calculated by dividing the dose per container by the usual daily dosages. Finally, in the third step, with the recommended number of treatment days per container, it was possible to calculate an estimated 'end date' of use per purchase. If fewer than 90 days separated the end date of a purchase and the date of the next purchase, the period of use was considered as nondiscontinued. Otherwise, the time between the end date of the first purchase and the start date of the next was considered a period of non-use.

Throughout the follow-up, a three-category variable indicating the use of antidepressants was created: 1, none; 2, past; and 3, current. A six-category variable was also created: 1, participants not using antidepressants; 2, past use of any antidepressant; 3, current use of imipramine-type antidepressants only; 4, current use of 'selective serotonin reuptake inhibitors' only; 5, current use of 'other' class of antidepressants; and 6, current use of a combination of antidepressants ('mixed' group). For this work, the category 'current use of monoamine oxidase inhibitors only' was not included as there were only few monoamine oxidase inhibitor reimbursements in the E3N cohort during follow-up. This can be explained by the low prescription rate for this drug, given its recognized adverse side effects.

Ascertainment of type 2 diabetes cases

Type 2 diabetes cases were identified through the drug reimbursement database; women reimbursed at least twice for glucose-lowering medications within a sliding period of 1 year were classified as having type 2 diabetes, with the date of diagnosis defined as the date of first reimbursement [START_REF] Mancini | Micronutrient dietary patterns associated with type 2 diabetes mellitus among women of the E3N-EPIC (Etude Epidémiologique auprès de femmes de l'Education Nationale) cohort study[END_REF].

Statistical analysis

Differences in baseline characteristics of participants by antidepressant medication use over the period 2005-2011 were assessed using chi-squared tests or analysis of variance, as appropriate.

To examine relationships between antidepressant use and incident type 2 diabetes, we used Cox proportional hazard regression models with age as the time scale [START_REF] Thiébaut | Choice of time-scale in Cox's model analysis of epidemiologic cohort data: a simulation study[END_REF], treating antidepressant use as a

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This article is protected by copyright. All rights reserved time-varying exposure to allow for participants starting and stopping and also changing treatments during follow-up. Hazard ratios (HRs) and 95% CIs for incident type 2 diabetes were computed from 2005 to the date of incident type 2 diabetes, death, or end of follow-up in 2011, whichever occurred first. The proportional hazards assumptions were confirmed by Schoenfeld tests [START_REF] Schoenfeld | Partial residuals for the proportional hazards regression model[END_REF].

Statistical analyses were controlled for potentially confounding variables, selected based on the literature and formal bivariate tests (considered relevant at P ≤ 0.1). In model 1, HRs were adjusted for sociodemographic factors that included marital status (single, married or living with a partner, widowed, divorced or separated), education level (<12, 12-15, ≥15 years) and occupational status (working, nonworking). In model 2, HRs were additionally adjusted for smoking (non-smoker, former smoker, smoker), alcohol consumption (none, moderate, heavy), physical activity (quartiles of MET-h/week, where MET is metabolic equivalent of task), BMI (kg/m 2 ), hip circumference (cm) and waist circumference (cm). In model 3, HRs were additionally adjusted for severe depressive symptoms [identified using the Centre for Epidemiological Studies Depression Scale (CES-D) [START_REF] Radloff | The CES-D Scale: A self-report depression scale for research in the general population[END_REF], with a score threshold of ≥23 chosen to define severe depressive symptoms [START_REF] Fuhrer | La version française de l'échelle CES-D (Center for Epidemiologic Studies-Depression Scale). Description et traduction de l'échelle d'autoévaluation. [The French version of the CES-D (Center for Epidemiologic Studies-Depression Scale)[END_REF]], family history of diabetes (no, yes), hypertension (no, yes), the number of ambulatory medical visits in the past 12 months (grouped by tertile of number of ambulatory medical visits). In the final model 4, HRs were additionally adjusted for baseline medication use other than antidepressants and included menopausal hormone therapy use (none, past, current), antipsychotic medication use (ATC N05A; none, past, current), anxiolytic medication use (ATC N05B; none, past, current), hypnotic medication use or sedative medication use (ATC N05C; none, past, current). In these four models, confounding variables were ascertained at baseline (date of response to the 2005 questionnaire). Missing data on confounding variables were replaced according to the three hierarchical steps: 1) replacement by the data from the previous self-administered questionnaire where the variable concerned was measured; 2) replacement by mode (for qualitative variables) or median (for quantitative variables) in the case of a missing data count of ≤5%; and 3) creation of a 'missing' category.

Analyses used STATA software (version 16; Stata Corp, College Station, TX, USA).

Results

The baseline characteristics of the 63 999 women, mean (SD) age 64 (6) years, are presented in Table 1, according to the use of antidepressants over the follow-up period. Based on data from reimbursement files, a total of 16 779 (26%) women used antidepressants at least once during follow-up. Antidepressant users were slightly older, more often widowed, divorced or separated, had a lower level of education, were more likely to be smokers and to not drink alcohol, and had a lower physical activity level, a higher BMI, and higher waist and hip circumferences when compared to non-users. They were also more likely to have severe symptoms of depression, hypertension, were more often on menopausal hormone treatment and more frequently used other psychotropic drugs. Finally, they had a higher number of visits in ambulatory medicine.

The 63 999 women had a median follow-up of 6.4 years, which contributed 377 140 person-years, and 1124 women were diagnosed with type 2 diabetes (incidence rate 2.98 per 1000 person-years). When antidepressant medication use was considered as a six-category variable, only 63 996 women were included because of the exclusion of three women using monoamine oxidase inhibitors.

Table 2 shows the association of antidepressant medication use with the risk of incident type 2 diabetes.

In model 1 adjusted for sociodemographic characteristics, current antidepressant users were at higher risk of developing type 2 diabetes [HR 1.82 (95% CI 1.54, 2.14)] than non-users. After further adjusting for all covariates (model 4), the strength of the association was reduced [hazard ratio 1.34 (95% CI 1.12,

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This article is protected by copyright. All rights reserved 1.61)] in current antidepressant users. There were no differences in the risk of incident type 2 diabetes among past-users [HR 0.97 (95% CI 0.79, 1.18)] compared to non-users.

When the type of antidepressant used was considered, after adjusting for all confounding factors, model 4, women using imipramine-type antidepressants [HR 1.66 (95% CI 1.12, 2.46)), selective serotonin reuptake inhibitors [HR 1.25 (95% CI 0.99, 1.57)], other antidepressants [HR 1.35 (95% CI 1.00, 1.84)] or mixed antidepressants [HR 1.82 (95% CI 0.85, 3.86)] had an increased risk of developing type 2 diabetes compared to non-users. There was no heterogeneity among these four pharmacological classes of current antidepressant (P homogeneity = 0.516), after adjusting for all covariates.

In a sensitivity analysis in which the updated information on BMI from the 2008 questionnaire was also used in each model (i.e. BMI was introduced as a time-varying covariate rather than being fixed at baseline), our risk estimates were only marginally modified (data not shown).

Discussion

This study, conducted in nearly 64 000 women from the E3N cohort, sought to examine the association between antidepressant use and the risk of developing type 2 diabetes, over 6 years of follow-up. After adjusting for a wide range of potential confounding factors, including severe depressive symptoms, current antidepressant users were still at a higher risk of developing type 2 diabetes during follow-up, compared to non-users. When the type of antidepressant was considered, there was no significant difference between the hazard ratios according to the type of antidepressant, after controlling for confounders.

Our results are consistent with the majority of registry-based observational studies [START_REF] Pan | Use of antidepressant medication and risk of type 2 diabetes: results from three cohorts of US adults[END_REF][START_REF] Kivimäki | Antidepressant medication use, weight gain, and risk of type 2 diabetes: a population-based study[END_REF][START_REF] Andersohn | Long-term use of antidepressants for depressive disorders and the risk of diabetes mellitus[END_REF][START_REF] Ma | Elevated depressive symptoms, antidepressant use, and diabetes in a large multiethnic national sample of postmenopausal women[END_REF][START_REF] Khoza | Use of antidepressant agents and the risk of type 2 diabetes[END_REF] and three meta-analyses [START_REF] Bhattacharjee | Antidepressant use and newonset diabetes: a systematic review and meta-analysis[END_REF][START_REF] Yoon | Antidepressant use and diabetes mellitus risk: a meta-analysis[END_REF][START_REF] Salvi | The risk of new-onset diabetes in antidepressant users -A systematic review and meta-analysis[END_REF] that found an association between antidepressant medication use and the risk of type 2 diabetes. However, these meta-analyses pointed out the large heterogeneity among studies as well as a lack of robustness in the results, particularly when certain studies were excluded from the models in sensitivity analyses. In a Finnish occupational case-control study nested in a cohort [START_REF] Kivimäki | Antidepressant medication use, weight gain, and risk of type 2 diabetes: a population-based study[END_REF], antidepressant

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This article is protected by copyright. All rights reserved use was associated with an increased risk of type 2 diabetes in participants without evidence of severe depression [odds ratio 1.68 (95% CI 1.27, 2.21)], as well as in participants with severe depression [odds ratio 2.76 (95% CI 1.93, 3.94)]. In a US study [START_REF] Pan | Use of antidepressant medication and risk of type 2 diabetes: results from three cohorts of US adults[END_REF] pooling data from the Health Professional Study (1990Study ( -2006)), the Nurses' Health Study (1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008) and the Nurses' Health Study II (1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005), the authors found an association between the use of antidepressants and the risk of type 2 diabetes [HR 1.17 (95% CI 1.09, 1.25)]. In a case-control study nested in the UK General Practice Research cohort of people with depression [START_REF] Andersohn | Long-term use of antidepressants for depressive disorders and the risk of diabetes mellitus[END_REF], the recent long-term (>24 months) use of antidepressants at moderate to high daily doses was associated with an increased risk of type 2 diabetes [incidence rate ratio 1.84 (95% CI 1.35, 2.52)] compared to non-users in the past 2 years. Finally, in the Women's Health Initiative including postmenopausal women [START_REF] Ma | Elevated depressive symptoms, antidepressant use, and diabetes in a large multiethnic national sample of postmenopausal women[END_REF], an increased risk of type 2 diabetes was observed in users of antidepressants [HR 1.18 (95% CI 1.10, 1.28)] compared to non-users. The three meta-analyses conducted in 2013 [START_REF] Bhattacharjee | Antidepressant use and newonset diabetes: a systematic review and meta-analysis[END_REF][START_REF] Yoon | Antidepressant use and diabetes mellitus risk: a meta-analysis[END_REF][START_REF] Salvi | The risk of new-onset diabetes in antidepressant users -A systematic review and meta-analysis[END_REF] and the systematic review [START_REF] Barnard | Antidepressant medication as a risk factor for type 2 diabetes and impaired glucose regulation: systematic review[END_REF] concluded that the use of antidepressants was associated with an increased risk of type 2 diabetes.

The mechanisms underlying the association between the use of antidepressants and type 2 diabetes remain to be explored. The literature suggests that weight gain induced by psychotropic drugs is usually preceded by an increased appetite, especially for foods high in fat and sugars [START_REF] Kazes | Eating behaviour and depression before and after antidepressant treatment: a prospective, naturalistic study[END_REF]. An increased dietary intake was also highlighted in non-depressed individuals treated with imipramine antidepressants [START_REF] Garland | Weight gain with antidepressants and lithium[END_REF]. Therefore, it is possible that the functioning of the central nervous system regulating appetite and food intake is altered by antidepressant use. Indeed, fluoxetine (selective serotonin reuptake inhibitor) has been shown to activate a decrease in leptin concentration in rodents [START_REF] Dryden | Decreased plasma leptin levels in lean and obese Zucker rats after treatment with the serotonin reuptake inhibitor fluoxetine[END_REF]. Leptin is a signalling protein participating in the regulation of the hypothalamic satiety centre and fat reserves of the body [START_REF] Ahima | Revisiting leptin's role in obesity and weight loss[END_REF] and a low plasma leptin concentration stimulates food intake and vice versa [START_REF] Friedman | Leptin and the regulation of body weight in mammals[END_REF]. Moreover, hyperglycaemia was associated with the use of antidepressants with a high affinity for the histamine 1 receptor (H1-R; mainly imipramine-type antidepressants, such as amitriptyline, imipramine, trimipramine, clomipramine, and some other types, such as mirtazapine), a receptor associated with weight gain and metabolic abnormalities [START_REF] Salvi | High H1-affinity antidepressants and risk of metabolic syndrome in bipolar disorder[END_REF]. However,

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This article is protected by copyright. All rights reserved whether adjusting our models for BMI measured at baseline or BMI updated during follow-up (so as to take into account variations in weight during follow-up), we consistently found higher risks of developing type 2 diabetes in current antidepressant users.

Strengths of the present study include the fact that it comprised a large number of women with a 6-year follow-up, taking into account different pharmacological classes of antidepressants and a wide range of potential confounders. In contrast to a number of previous studies, we were able to account for changes in antidepressant use during follow-up. Since antidepressant reimbursement data have been prospectively recorded prior to the onset of type 2 diabetes, memory bias, if it exists, is likely to be non-differential.

Finally, it was possible to take into account a wide range of potential confounding factors, in particular, the existence of severe depression symptoms; thus, risk estimates should relate to the use of antidepressants rather than severe depression itself. Since people treated with antidepressants are more likely to use the healthcare system than untreated people, and therefore may be more likely to be diagnosed with another health problem such as diabetes, our analyses were adjusted for the frequency of ambulatory medical visits in the year preceding baseline, in order to limit detection bias.

Nevertheless, the results of this study should be interpreted in the light of some limitations. First, the majority of potential confounding variables were self-reported and may therefore be subject to reporting bias. Second, some biases may persist in the comparisons between antidepressant types if the choice of a particular antidepressant class was influenced by risk factors for type 2 diabetes, however, a wide range of potential confounders were taken into account in the analysis to reduce differences between groups.

Third, residual confounding may still be present due to variables that have not been collected or not recorded with enough detail to completely remove confounding from the analysis. Although analyses were adjusted for severe symptoms of depression, because the CES-D scale is not a diagnostic tool, it was not possible to capture all the relevant components of depression. Fourth, misclassification of antidepressant use may have occurred because some women may not have taken all or even part of their prescription. These misclassification errors may have underestimated the associations reported in the

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This article is protected by copyright. All rights reserved study. Fifth, cases of type 2 diabetes which were not pharmacologically treated were classified as noncases. This may have lowered the magnitude of the association between antidepressant use and type 2 diabetes. However, we think this would have an inconsequential influence on the results [START_REF] Fagherazzi | Associations Between Migraine and Type 2 Diabetes in Women[END_REF]. Finally, as the study population included only women, mostly employed in education, the results cannot be directly extrapolated to men or to the general population.

In conclusion, findings from this cohort study in French women suggest that the use of antidepressants is associated with the risk of developing type 2 diabetes. The negative effects of using antidepressants on the risk of type 2 diabetes might have important clinical implications for the management of people in need of chronic antidepressant medications. Nonetheless, the nature of the mechanisms underlying this association remains unclear and requires further study.

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Table 1

 1 Baseline characteristics of women (2005) by antidepressant use over follow-up (2005-

	2011) in the E3N study
			Antidepressant use over follow-up (2005-2011)
		Total
	Characteristic at baseline (2005)	N = 63 999	None	At least once	P
			N = 47 220	N = 16 779
	Mean (SD) age, years	64.3 (6.3)	64.2 (6.2)	64.6 (6.6)	<0.001
	Marital status, n (%)				<0.001
	Single	3965 (6)	2958 (6)	1007 (6)
	Married or living with a partner	45 721 (71)	34 188 (73)	11 533 (69)
	widowed	6706 (11)	4827 (10)	1879 (11)
	Divorced or separated	7607 (12)	5247 (11)	2360 (14)
	Education level, n (%)				<0.001
	<12 years	6129 (10)	4381 (9)	1748 (10)
	12-14 years	34 726 (54)	25 513 (54)	9213 (55)
	≥15 years	23 144 (36)	17 326 (37)	5818 (35)
	Occupational status, n (%)				0.070
	Working	50 823 (79)	37 580 (80)	13 243 (79)
	Non-working	13 176 (21)	9640 (20)	3536 (21)
	Smoking status, n (%)				<0.001
	Non-smoker	34 054 (53)	25 408 (54)	8646 (51)
	Former smoker	25 540 (40)	18 852 (40)	6688 (40)
	Smoker	4405 (7)	2960 (6)	1445 (9)
	Alcohol consumption, n (%)				<0.001
	None (0 g per day)	8851 (14)	6112 (13)	2739 (16)
	Moderate (1-20 g per day)	41 446 (65)	31 005 (66)	10 441 (62)
	Heavy (>20 g per day)	13 702 (21)	10 103 (21)	3599 (22)

This article is protected by copyright. All rights reserved CES-D, Centre for Epidemiological Studies Depression Scale; MET, metabolic equivalent of task.

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Table 2

 2 Association between antidepressant medication use and the risk of type 2 diabetes onset in the E3N study Analyses with antidepressant use as a three-category variable are based on 63 999 women, comprising 377 140 person-years, 1124 type 2 diabetes onset. Analyses with antidepressant use as a six -category variable are based on 63 996 women, comprising 377 027 person-years, 1124 type 2 diabetes onset. Antidepressant medication use treated as a time-varying exposure; covariates considered at baseline (2005). Model 1: adjusted for age (time scale), marital status, education level and occupational status. † Model 2: model 1 additionally adjusted for smoking, alcohol consumption, physical activity, BMI, hip circumference and waist circumference. ‡ Model 3: model 2 additionally adjusted for severe depressive symptoms, family history of diabetes, hypertension and the number of ambulatory medicine visits in the past 12 months. § Model 4: model 3 additionally adjusted for menopausal hormone therapy use, antipsychotic medication use, anxiolytic medication use, hypnotic or sedative medication use. ¶ Test for homogeneity among different pharmacological classes of current antidepressant.

			Incidence		Model 1*			Model 2 †			Model 3 ‡			Model 4 §
		Number of
			rate per
		diabetes
			1000
		cases/N		HR	95% CI	P	HR	95% CI	P	HR	95% CI	P	HR	95% CI	P
			person-
		person-years
			years
	Antidepressant use
	None	834/304 415	2.73		Reference			Reference			Reference			Reference
	Past	120/39 791	3.02	1.08 (0.89, 1.31)	0.440	1.04 (0.86, 1.26)	0.688	0.99 (0.81, 1.20)	0.894 0.97 (0.79, 1.18)	0.760
	Current	170/32934	5.16	1.82 (1.54, 2.14)	<0.001 1.48 (1.25, 1.75)	<0.001 1.38 (1.16, 1.64) <0.001 1.34 (1.12, 1.61)	0.002
	Antidepressant use					0.543 ¶			0.458 ¶			0.514 ¶		0.516 ¶
	None	834/304 415	2.74		Reference			Reference			Reference			Reference
	Past (any type)	120/39 791	3.02	1.08 (0.89, 1.31)	0.440	1.04 (0.86, 1.26)	0.692	0.99 (0.81, 1.20)	0.892 0.97 (0.80, 1.18)	0.768
	Current imipramine
		27/4691	5.76	2.00 (1.36, 2.93)	<0.001 1.80 (1.23, 2.65)	0.003	1.70 (1.16, 2.51)	0.007 1.66 (1.12, 2.46)	0.012
	type
	Current selective
		88/18 539	4.75	1.69 (1.36, 2.11)	<0.001 1.35 (1.09, 1.69)	0.007	1.28 (1.02, 1.60)	0.033 1.25 (0.99, 1.57)	0.061
	serotonin reuptake

This article is protected by copyright. All rights reserved *

Acknowledgements

The authors are indebted to all participants for their continued participation and to all members of the E3N study group.

Funding sources M.A.D.S. was the recipient of a doctoral fellowship from the Ministère de l'Enseignement Supérieur et de la Recherche, Paris, France. The E3N cohort is being conducted with the financial support of the Mutuelle Générale de l'Education Nationale, the European Community, the French League against Cancer, Gustave Roussy, and the French Institute of Health and Medical Research. The validation of potential type 2 diabetes cases was supported by the European Union (Integrated Project LSHM-CT-2006-037197 in the 6th European Community Framework Programme). G.F. was supported by a grant for the E4N study from the Agence Nationale de Recherche (ANR-10-COHO-0006 grant), and an IDEX grant (NUTRIPERSO project) from Paris Saclay University.

Accepted Article

This article is protected by copyright. All rights reserved

Accepted Article

This article is protected by copyright. All rights reserved

Availability of data and materials

The datasets generated and/or analysed for the present study are not publicly available due to ethical reasons, permission by the participants to use their data, according to the signed informed consent, and from the MGEN, their insurance company. Data use is restricted to the team in charge of the cohort. However, data use can be extended to collaborators with a specific research agreement in collaboration with the corresponding author, on reasonable request.

Accepted Article

This article is protected by copyright. All rights reserved 4. Rubin RR, Ma Y, Marrero DG, Peyrot M, Barrett-Connor EL, Kahn SE et al. Elevated depression symptoms, antidepressant medicine use, and risk of developing diabetes during the

Ethics

All participants signed an informed consent form at study entry, and the study protocol was approved by the French National Commission for Computed Data and Individual Freedom. E3N was approved by this French Commission for Data Protection and Privacy.

Competing interests

None declared.