Amandine Aftalion

Peter Mason

Juncheng Wei

Vortex-peak interaction and lattice shape in rotating two-component Bose-Einstein condensates

Keywords:

When a two-component Bose-Einstein condensate is placed into rotation, a lattice of vortices and cores appear. The geometry of this lattice (triangular or square) varies according to the rotational value and the intercomponent coupling strengths. In this paper, assuming a Thomas-Fermi regime, we derive a point energy which allows us to determine for which values of the parameters, the lattice goes from triangular to square. It turns out that the separating curve in the phase diagram agrees fully with the complete numerical simulations of the Gross-Pitaevskii equations. We also derive a formula for the critical velocity of appearance of the first vortex and prove that the first vortex always appears first in the component with largest support in the case of two disks, and give a criterion in the case of disk and annulus.

INTRODUCTION

When a two-component condensate is set into rotation, topological defects of both order parameters are created, which lead to more exotic defects than in a single component condensate. Experiments on two component condensates have shown how the condensates can exhibit either triangular or square vortex lattices [1]. According to the values of the interaction strengths, the defect patterns can vary a lot, as illustrated in the numerical simulations [2,3]. One specific feature is the appearance of coreless vortices: the existence of a vortex in component-1 corresponds to a peak in component-2 and vice-versa. The interaction between vortices and peaks leads to changes in the geometry of the vortex lattice. We are interested in determining the equations governing this vortex peak behaviour in the Thomas-Fermi regime and estimating the interaction energy between the lattices of the two components. Indeed, for a single condensate, the vortex lattice is triangular, while for a two component, the vortex-peak interaction can lead to a square lattice. In [4], an asymptotic interaction between two half-quantized vortices is derived for two-component homogeneous condensates. In this paper, we want to take into account the nonhomogeneity of the condensate due to the trapping potential and estimate the vortex-peak energy according to the parameters of the system. We derive an energy depending on the location of vortices and peaks and determine for which values of the experimental parameters, the lattice goes from triangular to square. These critical values agree well with the ones found from the numerical computations of the full Gross-Pitaevskii equations of [3]. We note that in the rapid rotation regime, using the lowest Landau level approximation, several papers [5,6] (see [2,7] for a review) have analyzed the transition between triangular to square lat-tices. We point out that [8] found a point energy with an interaction term e -|pi-qj | 2 , which is different from ours and from [4]. We first review relevant results for a single condensate (see [9]), before moving to the derivation of homogeneous equations and the computation of the interaction term in two-component condensates.

For a single component condensate, the wave function minimizes the energy

E g,Ω (ψ) = 1 2 |∇ψ-iΩ×rψ| 2 + 1 2 (V (r)-Ω 2 r 2 )|ψ| 2 + g 2 |ψ| 4
(1) under |ψ| 2 = 1, where Ω = Ωe z is the rotation, V (r) is the trapping potential and in most cases V (r) = r 2 . We will denote by ∇ Ω the operator ∇ -iΩ × r. For g large, at Ω = 0, the ground state η of E g,0 approaches the inverted parabola

1 2g (λ -r 2 )
in the disk of radius R 2 = λ = 2 g/π, and goes to 0 outside the disk. If the problem is rescaled on a disk of size 1, then the analysis of the vortex cores leads to a vortex of size 1/ √ g and, close to the core, the wave function behaves like f (r)e iθ where f is the solution tending to 1 at infinity of

f ′′ + f ′ r - f r 2 + f (1 -f 2 ) = 0. ( 2
)
This is the equation of a vortex core in a uniform system.

In the case of a single condensate, from the equation of the vortex core, one can estimate the energy of vortices, the critical velocity for the nucleation of the first vortex and the interaction energy between vortices [9][START_REF] Aftalion | Vortices in Bose Einstein condensates[END_REF][11][12] which is

- i =j log|p i -p j | + i |p i | 2 (3)
where p i are the location of the vortex cores. Numerically, the minimization of (3) yields an almost triangular lattice for a large number of vortex points. The aim of this paper is to describe the equivalent of ( 2)-(3) in the case of two-component condensates. We define g i to be the intra-component coupling strength for component i and g 12 to be the inter-component coupling strength. For simplicity, we assume equal masses for the atoms in each component and equal trapping potentials, but a general case could be handled. The ground state of a two-component condensate is given by the infimum of

E g1,g2,g12,Ω (ψ 1 , ψ 2 ) = E g1,Ω (ψ 1 ) + E g2,Ω (ψ 2 ) +g 12 |ψ 1 | 2 |ψ 2 | 2 (4) under |ψ 1 | 2 = N 1 , |ψ 2 | 2 = N 2 . We set g 1 = α 1 g, g 2 = α 2 g, g 12 = α 0 g where g is large, so that ε = 1/ √ g is small. We change wave functions to ψ 1 (x, y) = √ εu 1 (x √ ε, y √ ε), ψ 2 (x, y) = √ εu 2 (x √ ε, y √ ε). Calling α = (α 0 , α 1 , α 2 )
, the energy we want to minimize is

E α,Ω (u 1 , u 2 ) = ε 2 2 |∇u 1 | 2 + 1 2 r 2 |u 1 | 2 + α 1 2 |u 1 | 4 -εΩ × r(iu 1 , ∇u 1 ) + ε 2 2 |∇u 2 | 2 + 1 2 r 2 |u 2 | 2 + α 2 2 |u 2 | 4 -εΩ × r(iu 2 , ∇u 2 ) + α 0 |u 1 | 2 |u 2 | 2 (5)
where (iu, ∇u) = iu∇ū -iū∇u. For Ω = 0, the ground state is real valued and we denote it by (η 1 , η 2 ). It is a solution of

-ε 2 ∆η 1 + r 2 η 1 + 2α 1 η 3 1 + 2α 0 η 2 2 η 1 = µ 1 η 1 (6) -ε 2 ∆η 2 + r 2 η 2 + 2α 2 η 3 2 + 2α 0 η 2 1 η 2 = µ 2 η 2 . ( 7
)
The shape of the ground state varies according to α and when α 2 0 -α 1 α 2 ≤ 0, can be either two disks or a disk and an annulus, as we will see below.

REDUCTION TO THE CORE EQUATIONS

We recall that (η 1 , η 2 ) is the ground state for Ω = 0 and we consider (u 1 , u 2 ) a ground state of E α,Ω . We call (f 1 , f 2 ) such that u 1 = η 1 f 1 and u 2 = η 2 f 2 . We expect η i to include the slow varying profile and f i to include the vortex or peak contribution, so that f i is 1 almost everywhere except close to the vortex and peak cores. We want to write the energy of (u 1 , u 2 ) as the energy of (η 1 , η 2 ) plus a rest, which is the energy that we are going to study. This follows a trick introduced in [START_REF] Lassoued | GinzburgLandau type energy with discontinuous constraint[END_REF], and used for single Bose Einstein condensates in [9,[START_REF] Aftalion | Vortices in Bose Einstein condensates[END_REF]. We multiply (6) by η 1 (|f 1 | 2 -1) and ( 7) by η 2 (|f 2 | 2 -1), and integrate and add the two equations, which yields the identity

ε 2 2 |∇η 1 | 2 (|f 1 | 2 -1) + ε 2 η 1 f 1 ∇η 1 • ∇f 1 + 1 2 r 2 η 2 1 (|f 1 | 2 -1) + α 1 η 4 1 (|f 1 | 2 -1) + α 0 η 2 2 η 2 1 (|f 1 | 2 -1) + ε 2 2 |∇η 2 | 2 (|f 2 | 2 -1) + ε 2 η 2 f 2 ∇η 2 • ∇f 2 + 1 2 r 2 η 2 2 (|f 2 | 2 -1)+α 2 η 4 2 (|f 2 | 2 -1)+α 0 η 2 2 η 2 1 (|f 2 | 2 -1) = 0. ( 8
)
Note that the Lagrange multiplier term has disappeared because u i and η i are normalized similarly. We replace (u 1 , u 2 ) by (f 1 η 1 , f 2 η 2 ) into the energy (5), use the identity (8) and find

E α,Ω (u 1 , u 2 ) = E α,0 (η 1 , η 2 ) + F α,Ω (f 1 , f 2 )
where

F α,Ω (f 1 , f 2 ) = ε 2 2 η 2 1 |∇f 1 | 2 -εη 2 1 Ω × r(if 1 , ∇f 1 ) + 1 2 α 1 η 4 1 (|f 1 | 2 -1) 2 + α 0 η 2 1 η 2 2 (1 -|f 1 | 2 )(1 -|f 2 | 2 ) + ε 2 2 η 2 2 |∇f 2 | 2 -εη 2 2 Ω × r(if 2 , ∇f 2 ) + 1 2 α 2 η 4 2 (|f 2 | 2 -1) 2 . ( 9
)
This splitting of energy does not assume anything about the scales of energy: it is an exact identity. We point out that as soon as α 1 α 2 -α 2 0 ≥ 0, then the quadratic form in the energy F α,Ω is positive and minimizing

E α,Ω in (u 1 , u 2 ) amounts to minimizing F α,Ω in (f 1 , f 2 ).
Now we assume that we scale everything close to a point p where η 2 1 = ρ 1 , η 2 2 = ρ 2 , and f 1 , f 2 can be written as functions of p + |r -p|/ε. Then, in the new variable r = |r -p|/ε, the functions f 1 , f 2 are a ground state of

F α,Ω (f 1 , f 2 ) = 1 2 ρ 1 |∇f 1 | 2 -ερ 1 Ω × r(if 1 , ∇f 1 ) + 1 2 α 1 ρ 2 1 (|f 1 | 2 -1) 2 + α 0 ρ 1 ρ 2 (1 -|f 1 | 2 )(1 -|f 2 | 2 ) + 1 2 ρ 2 |∇f 2 | 2 -ερ 2 Ω × r(if 2 , ∇f 2 ) + 1 2 α 2 ρ 2 2 (|f 2 | 2 -1) 2 (10)
and solve the system

-ρ 1 ∆f 1 -iεΩ × rρ 1 ∇f 1 + 2α 1 ρ 2 1 (|f 1 | 2 -1)f 1 + 2α 0 ρ 1 ρ 2 f 1 (|f 2 | 2 -1) = λ1 f 1 -ρ 2 ∆f 2 -iεΩ × rρ 2 ∇f 2 + 2α 2 ρ 2 2 (|f 2 | 2 -1)f 2 + 2α 0 ρ 1 ρ 2 f 2 (|f 1 | 2 -1) = λ2 f 2 . (11)
This is exactly the system studied in [4,[START_REF] Lin | Half-Skyrmions and Spike-Vortex Solu-tions of Two-Component Nonlinear Schrödinger Systems[END_REF] for a homogeneous condensate. The splitting of energy has allowed us to reach a homogeneous system. Assuming a vortex in component-1 and a spike in component-2, we have f 1 = v 1 (r)e iθ and f 2 = v 2 (r). We expect that v 1 , v 2 tend to 1 at infinity so that λ1 = εΩ and λ2 = 0. This yields the following system 12)

- (rv ′ 1 ) ′ r + v 1 r 2 + 2α 1 ρ 1 (v 2 1 -1)v 1 + 2α 0 ρ 2 v 1 (v 2 2 -1) = 0(
- (rv ′ 2 ) ′ r + 2α 2 ρ 2 (v 2 2 -1)v 2 + 2α 0 ρ 1 v 2 (v 2 1 -1) = 0.( 13
)
From this system, asymptotic expansions can be obtained for v 1 and v 2 at infinity:

v 1 (r) -1 ∼ -γ 1 /r 2 and v 2 (r) - 1 ∼ γ 2 /
r 2 for some constants γ 1 and γ 2 . Equations ( 12)-( 13) at infinity imply that α 2 ρ 2 γ 2 = α 0 ρ 1 γ 1 and 1 -

4α 1 ρ 1 γ 1 + 4α 0 ρ 2 γ 2 = 0, thus γ 1 = 1 4ρ 1 α 1 Γ 12 and γ 2 = α 0 4ρ 2 α 1 α 2 Γ 12 (14)
where

Γ 12 = 1 - α 2 0 α 1 α 2 . ( 15
)
In particular,

α 0 γ 1 γ 1 ρ 1 ρ 2 = 1 -Γ 12 16α 1 Γ 2 12 . (16)
In order to fully analyze the system ( 12)- [START_REF] Lassoued | GinzburgLandau type energy with discontinuous constraint[END_REF], we need to have information on ρ 1 , ρ 2 , that is the ground states η 1 , η 2 : in particular, we need to know whether the supports of η 1 , η 2 are disks or annuli, and where they reach their maximum.

THOMAS-FERMI PROFILE OF THE GROUND STATE

We recall some properties of the solutions of ( 6)-( 7) obtained in [3]. The following non-dimensional parameters are introduced

Γ 1 = 1 - α 0 α 1 (17) Γ 2 = 1 - α 0 α 2 . ( 18
)
To begin, assume that both components are circular with radii R 1 and R 2 and with R 1 < R 2 . When α 1 α 2 -α 2 0 ≥ 0, that is Γ 12 ≥ 0, and ε is small, both components are in the Thomas-Fermi (TF) regime, and the density profiles for r < R 1 are

|η 1 | 2 = 1 2α 1 Γ 12 µ 1 - α 0 α 2 µ 2 -r 2 Γ 2 (19) |η 2 | 2 = 1 2α 2 Γ 12 µ 2 - α 0 α 1 µ 1 -r 2 Γ 1 (20) and for R 1 < r < R 2 are |η 2 | 2 = µ 2 -r 2 2α 2 (21)
with |η 1 | 2 = 0. The chemical potentials µ 1 and µ 2 , and the radii, R 1 and R 2 , are to be found. In addition we have the normalisation condition

|η k | 2 d = N k , (22)
where, for generality,

N 1 = N 2 . We denote αk = N k α k and α0 = √ N 1 N 2 α 0 and get R 1 = 4 α1 Γ 12 πΓ 2 1/4 , (23)
R 2 = 4(α 2 + α1 (1 -Γ 1 )) π 1/4 , (24)
µ 1 = 4 α1 Γ 12 Γ 2 π 1/2 +(1 -Γ 2 ) 4 π [α 2 + α1 (1 -Γ 1 )] 1/2 , (25)
µ 2 = 4 π [α 2 + α1 (1 -Γ 1 )] 1/2 . ( 26
)
We find from ( 19) and ( 14) that

ρ 1 = η 2 1 (0) = Γ 2 R 2 1 2α 1 Γ 12 = Γ 2 N 1 πα 1 Γ 12 (27)
and

γ 1 = π 16N 1 α 1 Γ 2 Γ 12 while ρ 2 = η 2 2 (0) = 1 α 2 ( 1 π (N 2 α 2 + N 1 α 0 )) 1/2 -α 0 ( N 1 Γ 2 πα 1 Γ 12 ) 1/2
(28) and γ 2 follows from [START_REF] Lin | Half-Skyrmions and Spike-Vortex Solu-tions of Two-Component Nonlinear Schrödinger Systems[END_REF].

Equations ( 23)-( 24) are valid provided Γ 12 /Γ 2 > 0 (to ensure that R 1 and µ 1 are real) and α1 Γ 1 < α2 Γ 2 (to ensure that R 2 > R 1 ). If instead the initial assumption on the size of the radii was taken to be R 1 > R 2 , then the appropriate expressions would also be given by Eq.'s (23)-( 24), however with the indices 1 and 2 alternated. In this case, the conditions would be α1 Γ 1 > α2 Γ 2 and Γ 12 /Γ 1 > 0.

Returning now to R 2 > R 1 , one must ensure that |η 2 | 2 > 0 for all r < R 1 to have a disk rather than an annulus. Suppose that there is a point at the origin, where |η 2 | 2 = 0. Then, from Eq. (20),

α 0 = ᾱ0 = α 1 µ 2 µ 1 = N 1 α 1 2(N 1 + N 2 ) + 1 2
α 2 1 N 2 1 (N 1 + N 2 ) 2 + 4N 2 α 1 α 2 N 1 + N 2 . ( 29
)
The existence of some Γ 12 at which the density in component-2 hits zero at the origin is the indication of a spatial separation of the components. Notice that this critical value for Γ 12 is independent of Ω. In the spatial separation regime, component-1 is circular while component-2 is annular, provided R 2 > R 1 . It is not possible for an annulus to develop in component-1 if R 2 > R 1 ; this can be seen by writing down the TF density expressions for an annular component-1 and a circular component-2 in which the chemical potentials become multi-valued. Similarly, an annulus can only develop in component-

1 if R 1 > R 2 .
Thus under the assumption R 2 > R 1 , an annulus can only develop in component-2 and the condition to have two disks is thus α 0 < ᾱ0 .

VORTEX INTERACTION

Let us call ρ T F,1 , ρ T F,2 Thomas-Fermi limits of 1 | 2 and |η 2 | 2 given by ( 19)-( 20)-( 21) and ( 23), ( 24), (25), (??), (26) in the case of two disks. Then

ρ T F,1 = Γ 2 2α 1 Γ 12 (R 2 1 -r 2 ) (30) ρ T F,2 = Γ 1 2α 2 Γ 12 (R 2 1 -r 2 )+ 1 2α 2 (R 2 2 -R 2 1 ) if r < R 1 (31) and 1 2α 2 (R 2 2 -r 2 ) if r > R 1 . (32)
We want to estimate the various terms in the energy F α,Ω as in [9,[START_REF] Aftalion | Vortices in Bose Einstein condensates[END_REF] and we are going to show that, if p i are the vortices for component-1 and q j are the vortices for component-2, then they minimize the point energy

-πε 2 i =j ρ 1 log |p i -p j | -πε 2 i =j ρ 2 log |q i -q j | + π -ε 2 | log ε| Γ 2 2α 1 Γ 12 + εΩρ 1 i |p i | 2 + π -ε 2 | log ε| Γ 1 2α 2 Γ 12 + εΩρ 2 i |q i | 2 + π 1 -Γ 12 16Γ 2 12 1 α 1 + 1 α 2 ε 4 | log ε| i =j 1 |p i -q j | 2 . (33)

Estimate of the kinetic energy term

Let us call p i the vortices in component-1, and q j in component-2. Then the kinetic energy term (1/2) η 2 1 ε 2 |∇f 1 | 2 provides a leading order term due to the kinetic energy of the phase (which behaves locally like 1/r outside a disk of radius ε around each vortex), which is

πε 2 i ρ T F,1 (p i )| log ε| -πε 2 i =j ρ T F,1 (p i ) log |p i -p j | (34)
with a similar term for component-2, where p i is replaced by q j .

Rotation term

We call X 1 (r) the primitive of -rρ T F,1 (r) which vanishes at R 1 and X 2 (r) the primitive of -rρ T F,2 (r) which vanishes at R 2 . Then for r < R 1

X 1 (r) = Γ 2 8α 1 Γ 12 (R 2 1 -r 2 ) 2 , (35)
X 2 (r) = Γ 1 8α 2 Γ 12 (R 2 1 -r 2 ) 2 + 1 8α 2 (R 2 2 -R 2 1 )(R 2 2 +R 2 1 -2r 2 ). (36) Thus the rotation term -εΩ η 2 1 × r(if 1 , ∇f 1 ) is well approximated by -εΩ ∇X 1 ×(if 1 , ∇f 1
). An integration by parts around each vortex yields

-2πεΩ i X 1 (p i ) ( 37
)
with a similar contribution for component-2.

First vortices

The leading order approximation of the kinetic and rotation energy yields (assuming vortices at points p i for component-1 and q j for component-2):

πε 2 i ρ T F,1 (p i )| log ε| + πε 2 j ρ T F,2 (q j )| log ε| -2πεΩ i X 1 (p i ) -2πεΩ j X 2 (q j ). (38)
The energy is minimized by a configuration such that ρ T F,1 /X 1 or ρ T F,2 /X 2 reaches its minimum. We find that

X 1 (r) ρ T F,1 (r) = 1 4 (R 2 1 -r 2 ) (39) X 2 (r) ρ T F,2 (r) = 1 4 (R 2 1 -r 2 ) + 1 4 (R 2 2 -R 2 1 )(R 2 2 -r 2 ) Γ1 Γ12 (R 2 1 -r 2 ) + (R 2 2 -R 2 1 )
(40) This implies that above a critical value Ω c , vortices become energetically favorable in the system and Ω c is given from (38) by

Ω c = 1 2 ε| log ε| min i,r ρ T F,i X i .
For a harmonic potential, and in the case of two disks, the minimum of

ρT F,i
Xi occurs at the origin. Furthermore, since

ρ T F,2 X 2 = ρ T F,1 X 1 -D(r) (42)
where

D(r) = 4(R 2 2 -R 2 1 )(R 2 2 -r 2 ) (R 2 1 -r 2 ) × 1 Γ1 Γ12 (R 2 1 -r 2 ) 2 + (R 2 2 -R 2 1 )(R 2 1 + R 2 2 -2r 2 ) , (43)
and given the signs of the parameters, we see that D(0) > 0 so that we always have

ρ T F,1 /X 1 > ρ T F,2 /X 2 .
The first vortex is thus preferred in component-2 (i.e. the component with larger support) and occurs at the origin with the critical velocity given by

Ω c = √ π α2 Γ 12 ε| log ε| Γ 12 α2 + α1 (1 -Γ 1 ) -(1 -Γ 1 ) α1 Γ 2 Γ 12 .
(44)

Note that this expression gives Ω c = 0 when α 0 = ᾱ0 (provided α 1 = α 2 otherwise Ω c reduces to a non-zero constant). For some computations in the rest of the paper, we can assume

N 1 α 1 = N 2 α 2 so that R 1 = R 2 and
we have a lattice of peaks and vortices close to the origin. Since min i,r ρ T F,i /X i = 4/R 2 1 , we have

Ω c = ε| log ε| πΓ 1 α 1 Γ 12 . ( 45
)
We have plotted Ω c -Γ 12 curves for two cases in Fig. 1, the first with distinct intracomponent coupling strengths and the second with equal intracomponent coupling strengths (where Eq. ( 44) reduces to Eq. ( 45)) and compared then to the numerical results of [3] (these parameter sets correspond to sets 'ES1' and 'ES3' respectively from [3]). We find good agreement between the two results.

Energy expansion

We now have to go further into the energy expansion to estimate the interaction energy. We assume that the vortices appear close to the origin. Then (34) and (37) can be expanded around the origin, using (30)-( 32) and ( 35)-( 36), which yields

-πε 2 i =j ρ 1 log |p i -p j | -πε 2 i =j ρ 2 log |q i -q j | + πε -ε| log ε| Γ 2 2α 1 Γ 12 + Ωρ 1 i |p i | 2 + πε -ε| log ε| Γ 1 2α 2 Γ 12 + Ωρ 2 i |q i | 2 (46)
Interaction energy

We find from ( 9) that the interaction energy is

α 0 ρ 1 ρ 2 (1 -|v 1 | 2 )(1 -|v 2 | 2 ).
Near a vortex-peak, this reduces to

α 0 ρ 1 ρ 2 γ 1 γ 2 1 r 2 (1,0) 1 r 2 (0,1)
where we take the notations of [4]: r (1,0) is the local distance to the vortex in component-1, and r (0,1) is the distance to the next peak in component-1, or equivalently to the vortex in component-2. From (16), we find that the coefficient in front of the integral is equal to π

1-Γ12 16α1Γ 2 12 .
The computations in [4] allow us to estimate the integral term and we find for the interaction term

π 1 -Γ 12 16Γ 2 12 1 α 1 ε 4 | log ε| 1 |p i -q j | 2 .
This is for a vortex in v 1 . Of course, if the vortex is in v 2 , it would be different by a factor 1/α 2 .

The interaction energy is thus

π 1 -Γ 12 16Γ 2 12 1 α 1 + 1 α 2 ε 4 | log ε| i =j 1 |p i -q j | 2 . (47)
Together with (46), this leads to (33).

NUMERICAL SIMULATION OF THE RENORMALISED ENERGY

We now want to find the ground state of (33) when the radii of both components are equal. Under this condition we have ρ 1 = ρ 2 , Γ 1 = Γ 2 and α 1 = α 2 . This allows us to perform a rescaling that leaves the renormalised energy only dependent on a single parameter. We write Ω = ωΩ c for Ω c defined in (45). Then

ρ 1 - | log ε| R 2 1 + Ω ε = ρ 1 | log ε| ω πΓ 2 α 1 Γ 12 - 1 R 2 1 ( 48
)
which implies that we can rescale the p i 's and q j 's as p i = γ pi and q j = γ qj with

γ 2 = 1 2| log ε| ω πΓ2 α1Γ12 -1 R 2 1 (49)
so that the new energy is

1 2 πε 2 ρ 1 - i =j log |p i -p j | 2 - i =j log |q i -q j | 2 + i |p i | 2 + i |q i | 2 + c ω i =j 1 |p i -q j | 2 (50)
with

c ω = π(1 -Γ 12 ) 4α 1 Γ 2 12 (2ω -1)ε 2 | log ε| 2 . ( 51
)
We simulate this renormalised energy (50) using a conjugate gradient method varying the parameter c ω and the number of vortex points. For a fixed number of lattice points N , when increasing c ω , the ground state lattice goes from triangular to square at a critical c ts ω . Note that when N gets large, c ts ω no longer depends on N . In Fig. 2 we plot the ground state for two values of c ω which give a triangular (c ω = 0.05) and a square lattice (c ω = 0.3) when N = 46.

For each N , we can calculate the critical c ts ω and compare this with the simulations on the full GP equations, as performed in [3] (they provide the appropriate value of N ). From (51), we thus find the critical value of Ω for which the lattice goes from triangular to square:

Ω ts = 1 2 + 2c ts ω α 1 Γ 2 12 π(1 -Γ 12 )ε 2 | log ε| 2 πΓ 2 g 1 Γ 12 | log ε| (52)
It turns out that when Ω gets close to 1, the condensate expands and one has to include in the TF profile a term (1 -Ω 2 )r 2 instead of just r 2 . The radii R 1 and R 2 vary like (1 -Ω 2 ) 1/4 . This changes c ω from (51) to

c ω = π(1 -Γ 12 ) 4α 1 Γ 2 12 2ω - √ 1 -Ω 2 √ 1 -Ω 2 ε 2 | log ε| 2 (53)
so that if we define

β ts = 1 2 + 2c ts ω g 1 Γ 2 12 π(1 -Γ 12 )| log ε| 2 πΓ 2 g 1 Γ 12 | log ε| (54)
then Ω ts = β ts 1 -(Ω ts ) 2 , which yields

Ω ts = β ts 1 + β 2 ts . (55)
We plot this form of Ω ts as a function of Γ 12 in Fig. 3 where we have taken ε = 0.0358 and α 1 = α 2 = 1 (note that this parameter set corresponds to set 'ES3' in [3]). This provides good agreement with simulations of the full Gross Pitaevskii energy and confirms that our point energy (33) well describes the system.

DISK PLUS ANNULUS

Equation (29) introduces the critical ᾱ0 (or equivalently Γ 12 ) for which the two disk state is no longer a solution. In this section we consider α 0 > ᾱ0 so that the system is given by a disk in component-1 and an annulus in component-2. This requires α 1 to be different to α 2 , because otherwise the conditions α 0 > ᾱ0 and α 2 0 -α 1 α 2 ≤ 0 are not consistent. Defining the inner and outer radii of component-2 to be at r = R 2-and r = R 2+ such that R 2-< R 1 < R 2+ , the appropriate density profiles are

ρ T F,1 (r) = |η 1 | 2 = µ 1 -r 2 2α 1 (56) for 0 < r < R 2-, ρ T F,1 (r) = |η 1 | 2 = 1 2α 1 Γ 12 µ 1 - α 0 α 2 µ 2 -r 2 Γ 2 (57) ρ T F,2 (r) = |η 2 | 2 = 1 2α 2 Γ 12 µ 2 - α 0 α 1 µ 1 -r 2 Γ 1 (58)
for R 2-< r < R 1 , and

ρ T F,2 (r) = |η 2 | 2 = µ 2 -r 2 2α 2 (59) for R 1 < r < R 2+ .
The normalisation condition (22) gives the following expressions for the radii

R 2-= R -- Γ 2 Γ 1 S (60) R 1 = R -- Γ 1 Γ 2 (1 -Γ 2 ) 2 S (61) R 2+ = R + -Γ 1 Γ 2 S (62)
and

µ 1 = R (63) µ 2 = R + -Γ 1 Γ 2 S (64)
for the chemical potentials. Here we have introduced the It follows that

min ρ T F,1 X 1 =      4 α2Γ2R 2 1 +α0Γ1R 2 2 - α2Γ2R 4 1 +α0Γ1R 4 2 - for 0 < r < R 2- 4 (R 2 1 -R 2 2 - ) for R 2-< r < R 1 .
(76) Using the expressions above for the radii, we see that

4 α 2 Γ 2 R 2 1 + α 0 Γ 1 R 2 2- α 2 Γ 2 R 4 1 + α 0 Γ 1 R 4 2- = 2 α1 π α 2 (α 1 α2 + α1 α 2 ) (77)
which we note is independent of α 0 . If we compare the values of Eq. ( 76), we see that they are equal when α 0 = ᾱ0 (which we can consider a degenerate case) and otherwise min(ρ T F,1 /X 1 ) over the whole space is simply given by the expression in Eq. (77). This implies that the vortex in component-1 is preferred at the origin rather than in the interface region of the two components.

To complete the analysis we must also find the min(ρ T F,2 /X 2 ) over the whole space to determine whether vortices appear first in component 1 or com-ponent 2. It is not possible to find an analytic expression for this minimum and so we must resort to plotting the function and comparing this to min(ρ T F,1 /X 1 ). In Fig. 4, we plot the functions ρ T F,1 /X 1 and ρ T F,2 /X 2 in two different cases where min(ρ T F,1 /X 1 ) < min(ρ T F,2 /X 2 ) and min(ρ T F,1 /X 1 ) > min(ρ T F,2 /X 2 ). We also plot the critical Ω defined in Eq. (41) as a function of Γ 12 . We see that until some critical value of Γ 12 , the vortex first nucleates at the origin in component-1, while above in some small region in Γ 12 before reaching Γ12 , the vortex first nucleates in the interface region in the second component. Then, increasing Ω, leads to a vortex lattice close to this point of nucleation. Recall that the giant vortex is always present.

Conclusion

From the energy of a rotating two component condensate, we have derived a reduced energy (33) governing the location of peaks and vortices. We have found that the ground state of this reduced energy yields a square lattice of vortices in regimes consistent with the ones found numerically for the full Gross Pitaevskii energy. We have analyzed in detail the geometry of the ground state (two disks or disk and annulus) and derived formula of the critical velocity for the appearance of the first vortex. We can determine in which component the first vortex appears and the shape of the lattice.

FIG. 1 :

 1 FIG. 1: The critical velocity for creation of the first vortex plotted analytically from Eq. (44) [solid line] and numerically [dotted line] as a function of Γ12 for two parameter sets: (a) ε = 0.0352, α1 = 0.97, α2 = 1.03 and (b) ε = 0.0358, α1 = α2 = 1.

FIG. 2 :FIG. 3 :

 23 FIG. 2: The ground state of the renormalised energy (50) calculated for (a) cω = 0.05 and (b) cω = 0.3 with the number of points in each component equal to N = 46. Component-1 is shown in the left columns and component-2 in the right columns.

FIG. 4 :

 4 FIG. 4: (a) Functions ρT F,1/X1 (solid lines) and ρT F,2/X2 (dashed lines) given by Eq's (72)-(75) for Γ12 = 0.0202 (black lines) and Γ12 = 0.015 (blue lines). (b) The critical velocity for creation of the first vortex plotted from Eq. (41) [black line] and numerically [blue line] as a function of Γ12. The solid part corresponds to a vortex in component 1 at the center and the dashed part in component 2 in the interface region. The parameters are ε = 0.0352, α1 = 0.97 and α2 = 1.03. Note that Γ12 = 0.02

Acknowledgments

The authors acknowledge support from the French ministry Grant ANR-BLAN-0238, VoLQuan.

A number of conditions can immediately be found from these expressions. In order for R 2-< R 1 , it must be that Γ 12 > 0 [The inequalities R 2+ > R 1 and R 2+ > R 2-are automatically valid]. Secondly, all the expressions (except that for µ 1 ) require Γ 1 Γ 2 < 0. This is equivalent to α 2 > α 1 (under the assumption that the annulus develops in component-2). Thus the range of Γ 12 for which component-1 is circular and component-2 is annular is given by 0 < Γ 12 < Γ12 , where Γ12 = Γ 12 evaluated at α 0 = ᾱ0 given by (29).

When α 0 = ᾱ0 we have shown in Eq. (44) that the critical velocity for creation of the first vortex (that occurs at the origin in component-2) is identically zero. This is precisely the point at which the annulus develops (ρ T F,2 (r = 0) = 0 with R 2-= 0). In the region α 0 > ᾱ0 , the first appearance of a topological defect is the development of a giant vortex in component-2 and this can be analyzed fully. We need to introduce

Then the critical velocity for nucleation of the giant vortex is determined by Ω gv = εΛ 2 and the circulation of this giant vortex is the integer part of Ω/(εΛ 2 ). Next, we want to determine vortices in the bulk and apply (38) to the case of the disk plus annulus. Thus we need to define X 2 (r) to take into account the giant vortex.

We define X 1 (r) the primitive of -rρ T F,1 (r) which vanishes at R 1 giving

for 0 < r < R 2-, and is the same as Eq. ( 35) in the region R 2-< r < R 1 . In order to account for the development of vortices beyond the giant vortex, we must recall that additional to the term -εΩ rρ T F,2 (if 2 , ∇f 2 ), there will be a kinetic energy term coupling the giant vortex and the vortex cores, namely

where d is the degree of the giant vortex, of order Ω/(εΛ 2 ). Therefore, we define X 2 (r) to be the primi-tive of -rρ T F,2 (r) + (1/Λ 2 )ρ T F,2 (r)/r which vanishes at both R 2-= 0 and R 2+ = 0,

where Λ 2 is given by (67). Completing the integrals gives

for R 2-< r < R 1 and

for R 1 < r < R 2+ . As before we write down the ratios ρ T F,i /X i :

for R 2-< r < R 1 and

(75)


Ph.D Dorit Ron
email: dorit.ron@ucsf.edu

Noribogaine, but not 18-MC, Exhibits Similar Actions as Ibogaine on GDNF Expression and Ethanol Self-Administration

Keywords: Ethanol self-administration, GDNF, ibogaine, 18-Methoxycoronaridine, noribogaine

. Here, we determined whether noribogaine and/or 18-MC, like ibogaine, increase GDNF expression, and whether their site of action to reduce alcohol consumption is the VTA. We used SH-SY5Y cells as a cell culture model and found that noribogaine, like ibogaine, but not 18-MC, induces a robust increase in GDNF mRNA levels.

Next, we tested the effect of intra-VTA infusion of noribogaine and 18-MC on rat operant alcohol self-administration and found that noribogaine, but not 18-MC, in the VTA decreases responding for ethanol. Together, our results suggest that noribogaine and 18-MC have different mechanisms and sites of action.

Introduction

Alcohol and drug addiction are chronic relapsing diseases characterized by compulsive drug use, and are major societal issues. However, available medications to treat drug addiction in general and alcoholism in particular are very limited. Additionally, the two leading FDA-approved medications to treat alcoholism, Naltrexone and Acamprosate, have limited efficacy and success rates [START_REF] Garbutt | The state of pharmacotherapy for the treatment of alcohol dependence[END_REF][START_REF] Johnson | Update on neuropharmacological treatments for alcoholism: scientific basis and clinical findings[END_REF].

One drug that initially showed great promise as a possible treatment for multiple types of addiction is ibogaine [START_REF] Maciulaitis | Ibogaine, an antiaddictive drug: pharmacology and time to go further in development. A narrative review[END_REF]. Ibogaine is a naturally occurring indole alkaloid claimed to reverse addiction to opiates, stimulants, alcohol, and nicotine [START_REF] Alper | The ibogaine medical subculture[END_REF][START_REF] Mash | Medication development of ibogaine as a pharmacotherapy for drug dependence[END_REF][START_REF] Popik | 100 years of ibogaine: neurochemical and pharmacological actions of a putative anti-addictive drug[END_REF]Vastag, 2002). Ibogaine is extracted from the root bark of the West African shrub Tabernanthe iboga and is used by indigenous people in low doses to keep hunters awake and motionless, and in higher doses for religious rituals because of its psychostimulant and hallucinogenic properties [START_REF] Alper | The ibogaine medical subculture[END_REF][START_REF] Maciulaitis | Ibogaine, an antiaddictive drug: pharmacology and time to go further in development. A narrative review[END_REF]. The initial discovery that ibogaine eliminates signs and symptoms of opioid withdrawal and diminishes craving for heroin was made in the 1960s by a group of heroin addicts [START_REF] Maciulaitis | Ibogaine, an antiaddictive drug: pharmacology and time to go further in development. A narrative review[END_REF][START_REF] Mash | Medication development of ibogaine as a pharmacotherapy for drug dependence[END_REF]. Since that time, several human anecdotal reports and studies have shown that ibogaine possesses many attractive properties as an anti-addiction agent. First, the compound crosses the blood brain barrier and can be given to addicts orally [START_REF] Mash | Medication development of ibogaine as a pharmacotherapy for drug dependence[END_REF].

Second, a single dose of ibogaine is claimed to abolish drug craving for up to six months, and repeated ibogaine treatment (a series of four oral administrations) was reported to be effective in blocking craving and relapse for up to three years in cocaine and opiate addicts [START_REF] Alper | Ibogaine in acute opioid withdrawal. An open label case series[END_REF][START_REF] Mash | Medication development of ibogaine as a pharmacotherapy for drug dependence[END_REF]Sheppard, 1994). Since compliance is a significant problem in medication development to treat addiction, the fact that ibogaine has very long lasting effects (months to years) is very desirable.

Studies in animal models recapitulate human observations. Ibogaine reduces selfadministration and withdrawal symptoms for alcohol and various drugs of abuse in mice, rats and monkeys [START_REF] Glick | 18-Methoxycoronaridine (18-MC) and ibogaine: comparison of antiaddictive efficacy, toxicity, and mechanisms of action[END_REF]. For example, single and repeated intraperitoneal (i.p.) injections of a moderate dose of ibogaine (40 mg/kg) can reduce self-administration of cocaine or morphine in rats and cocaine in mice for several days [START_REF] Cappendijk | Inhibitory effects of ibogaine on cocaine selfadministration in rats[END_REF][START_REF] Glick | Effects of iboga alkaloids on morphine and cocaine self-administration in rats: relationship to tremorigenic effects and to effects on dopamine release in nucleus accumbens and striatum[END_REF][START_REF] Sershen | Ibogaine reduces preference for cocaine consumption in C57BL/6By mice[END_REF]. Ibogaine decreases heroin selfadministration and naloxone/naltrexone-precipitated withdrawal in morphine-dependent rats [START_REF] Cappendijk | The inhibitory effect of norharman on morphine withdrawal syndrome in rats: comparison with ibogaine[END_REF][START_REF] Glick | Effects of ibogaine on acute signs of morphine withdrawal in rats: independence from tremor[END_REF]; but see [START_REF] Sharpe | Ibogaine fails to reduce naloxone-precipitated withdrawal in the morphine-dependent rat[END_REF]. Finally, ibogaine reduces ethanol intake in several strains of alcohol-preferring rats in a dosedependent manner, using concentrations ranging from 10 to 60 mg/kg, with no changes in blood alcohol concentration or food intake [START_REF] Rezvani | Attenuation of alcohol intake by ibogaine in three strains of alcohol-preferring rats[END_REF]. We replicated and extended the findings of Rezvani and colleagues (1995) on different ethanol-drinking behaviors. We found that an acute injection of ibogaine (40 mg/kg, i.p.) in male Long-Evans rats 3 hrs before the test session significantly reduced ethanol intake, but not the concurrent water intake or sucrose intake, using a 2-bottle preference procedure, as well as operant ethanol self-administration [START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF]. We were also interested in testing whether ibogaine would have efficacy in a model of relapse in which rats are allowed to self-administer alcohol in operant chambers following 3 weeks of extinction training.

Ibogaine blocked the excessive intake produced by such forced abstinence period [START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF], suggesting a potent action of ibogaine to prevent relapse. In summary, human anecdotal reports, as well as several preclinical studies in rodents, suggest that ibogaine is a long-lasting and selective anti-addiction medication for alcohol, opioids, psychostimulants, and nicotine.

Unfortunately, despite its attractive properties, ibogaine is not used in the US to treat addiction because of severe side effects, which include hallucination, bradycardia, whole-body tremors and ataxia [START_REF] Alper | The ibogaine medical subculture[END_REF][START_REF] Glick | 18-Methoxycoronaridine (18-MC) and ibogaine: comparison of antiaddictive efficacy, toxicity, and mechanisms of action[END_REF][START_REF] Maas | Fatalities after taking ibogaine in addiction treatment could be related to sudden cardiac death caused by autonomic dysfunction[END_REF][START_REF] Maciulaitis | Ibogaine, an antiaddictive drug: pharmacology and time to go further in development. A narrative review[END_REF][START_REF] Popik | 100 years of ibogaine: neurochemical and pharmacological actions of a putative anti-addictive drug[END_REF]. In addition, while a moderate dose of ibogaine (40 mg/kg) that is effective to reduce self-administration of several drugs of abuse is not neurotoxic in rodents, [START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF][START_REF] Molinari | Ibogaine neurotoxicity: a re-evaluation[END_REF] cerebellar Purkinje cell death has been reported in rats after administration of a higher concentration (100 mg/kg, i.p.) (O' Hearn andMolliver, 1993, 1997). Moreover, although ibogaine is not addictive, the FDA classified the drug in the schedule I category of narcotics because of its stimulant and hallucinogenic properties.

In an attempt to separate the desirable anti-addictive properties of the drug from the undesirable side effects, we set out to identify the molecular mechanism mediating ibogaine's effects on voluntary ethanol consumption. We found that systemic administration of ibogaine induced a long-lasting increase in the expression of the glialderived neurotrophic factor (GDNF) in the dopaminergic ventral tegmental area (VTA) of rodents and that the effect of ibogaine to reduce ethanol self-administration is localized in the VTA [START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF]. Importantly, when the GDNF pathway was inhibited in the VTA, ibogaine was significantly less effective in reducing ethanol self-administration [START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF]. More recently, we observed that upregulation of tyrosine hydroxylase levels resulting from long-term exposure of ethanol in a dopaminergic-like cell line is blocked by ibogaine in a mechanism that requires GDNF [START_REF] He | Glial cell line-derived neurotrophic factor reverses ethanolmediated increases in tyrosine hydroxylase immunoreactivity via altering the activity of heat shock protein 90[END_REF]. Finally, similarly to ibogaine, GDNF administered into the VTA resulted in a long-lasting inhibition of ethanol-seeking and -drinking behaviors in models of moderate and excessive drinking of ethanol, as well as in a model of relapse (Carnicella et al., 2009b;[START_REF] Carnicella | GDNF is a fast-acting potent inhibitor of alcohol consumption and relapse[END_REF]Carnicella and Ron, 2009;[START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF]. Together, our results suggest that the upregulation of the GDNF pathway in the VTA mediates, at least in part, the desirable actions of ibogaine to reduce voluntary ethanol consumption and relapse.

Ibogaine is metabolized by cytochrom P4502D6 into a major metabolite noribogaine, (12-hydroxyibogamine) [START_REF] Obach | Cytochrome P4502D6 catalyzes the Odemethylation of the psychoactive alkaloid ibogaine to 12-hydroxyibogamine[END_REF]. Interestingly, while ibogaine levels in the plasma rapidly decline, noribogaine levels remain high for at least 24 hrs after oral administration of ibogaine in humans [START_REF] Mash | Medication development of ibogaine as a pharmacotherapy for drug dependence[END_REF][START_REF] Mash | Ibogaine: complex pharmacokinetics, concerns for safety, and preliminary efficacy measures[END_REF][START_REF] Mash | Identification of a primary metabolite of ibogaine that targets serotonin transporters and elevates serotonin[END_REF]. Similarly, it has been shown in rats that peak blood levels of noribogaine were reported to exceed those of ibogaine, and a small, but significant portion of the administered noribogaine was still detected in the bloodstream 24 hrs after an acute i.p. administration [START_REF] Baumann | In vivo neurobiological effects of ibogaine and its O-desmethyl metabolite, 12-hydroxyibogamine (noribogaine), in rats[END_REF]. Behavioral studies show that noribogaine acts as an active metabolite [START_REF] Baumann | In vivo neurobiological effects of ibogaine and its O-desmethyl metabolite, 12-hydroxyibogamine (noribogaine), in rats[END_REF] and possesses a similar anti-addictive profile as ibogaine (Glick et al., 1996b). For example, systemic administration of noribogaine induces a long-lasting decrease of morphine and cocaine self-administration (Glick et al., 1996b). Importantly, noribogaine does not produce tremors and ataxia [START_REF] Baumann | In vivo neurobiological effects of ibogaine and its O-desmethyl metabolite, 12-hydroxyibogamine (noribogaine), in rats[END_REF]Glick et al., 1996b), suggesting that it is not, or less, neurotoxic than its parent compound. Moreover, noribogaine was recently found to be more than two times less toxic than its parent compound in mice [START_REF] Kubiliene | Acute toxicity of ibogaine and noribogaine[END_REF].

18-MC is a synthetic derivative of ibogaine (Glick et al., 1996a). 18-MC shares ibogaine's desirable properties but, interestingly, the derivative lacks some of the adverse effects of ibogaine [START_REF] Glick | 18-Methoxycoronaridine (18-MC) and ibogaine: comparison of antiaddictive efficacy, toxicity, and mechanisms of action[END_REF]. Like ibogaine, 18-MC administration results in a long-lasting decrease in ethanol, morphine, cocaine, methamphetamine and nicotine selfadministration, and attenuation of opioid withdrawal symptoms (reviewed in [START_REF] Glick | 18-Methoxycoronaridine (18-MC) and ibogaine: comparison of antiaddictive efficacy, toxicity, and mechanisms of action[END_REF]. However, unlike ibogaine, 18-MC does not induce bradycardia, whole body tremor, and cerebellar toxicity in rats, even at high doses [START_REF] Glick | 18-Methoxycoronaridine (18-MC) and ibogaine: comparison of antiaddictive efficacy, toxicity, and mechanisms of action[END_REF]. Moreover, 18-MC is probably not hallucinogenic as it does not affect serotonin levels [START_REF] Glick | 18-Methoxycoronaridine (18-MC) and ibogaine: comparison of antiaddictive efficacy, toxicity, and mechanisms of action[END_REF].

Because of similarities in structure and anti-addictive actions of ibogaine, noribogaine, and 18-MC, we hypothesized that like ibogaine, the ibogaine derivatives upregulate GDNF expression and that their mechanism of action to reduce ethanol selfadministration is localized in the VTA. Therefore, we set out to determine whether noribogaine and 18-MC increase GDNF expression in SH-SY5Y cells and whether the ibogaine derivatives affect operant ethanol self-administration in rats when infused into the VTA.

Methods

Reagents

Reverse Transcription -Polymerase Chain Reaction (RT-PCR)

Total RNAs were isolated using Trizol reagent and reversely transcribed using a Reverse Transcription System kit at 42°C for 30 min. GDNF was analyzed by PCR with temperature cycling parameters consisting of initial denaturation at 94°C for 2 min followed by 35 cycles of denaturation at 94°C for 30 s, annealing at 52°C for 30 s, extension at 72°C for 2 min, and a final incubation at 72°C for 7 min. PCR for actin, an internal control, was run with the same temperature cycling parameters for 30 cycles. The primers used in RT-PCR are as follows: human GDNF, upstream 5'-TGC CAG AGG ATT ATC CTG ATC AGT TCG ATG-3' and downstream 5'-TTG TCG TAC GTT GTC TCA GCT GCA TCG CAA-3'; human actin, upstream 5'-TCA TGA AGT GTG ACG TTG ACA TC-3' and downstream 5'-AGA AGC ATT TGC GGT GGA CGA TG-3'. PCR products were separated on 1.8% agarose gels in Tris/acetic acid/EDTA buffer with 0.25 µg/ml ethidium bromide, photographed by Eagle Eye II (Stratagen, La Jolla, CA) and quantified by NIH Image 1.61.

Operant ethanol self-administration after a history of high voluntary ethanol consumption

High levels of voluntary ethanol consumption were obtained in an intermittent access two-bottle choice drinking paradigm to 20% ethanol as previously described (Carnicella et al., 2009b). Briefly, animals were given 24-h concurrent access to one bottle of 20% (v/v) ethanol in tap water and one bottle of water on Monday, Wednesday, and Friday, with 24-or 48-h ethanol deprivation periods in between the ethanol-drinking sessions. After achieving a stable baseline of ethanol consumption of ~ 6 g/kg/24 hrs and preference for the 20% ethanol solution over water (see Figure 2A-C), rats were trained to self-administer a 20% ethanol solution in operant self-administration chambers. The chambers contained two levers: an active lever, for which presses result in delivery of 0.1 ml of the ethanol solution, and an inactive lever, for which responses are counted but no programmed events occur. After 2 nights in the chambers to allow acquisition of a leverpress response for ethanol under a fixed ratio 1 (FR1), operant sessions were conducted 5 days per week, with the schedule requirement increased to FR2 and the length of session shortened from 60 to 30 min over the first two weeks. Because the level of presses on the inactive lever was extremely low after acquisition of the self-administration procedure (< 10 presses), and the activity on this lever was not affected by any of the experimental treatments, this measure was excluded from the figures and the analysis for better clarity.

After 1 month of training, surgery to implant cannulae was conducted.

Surgery and intra-VTA infusions

Rats were anesthetized continuously with isoflurane (Baxter Health Care Corporation, Deerfield, IL). Bilateral guide cannulae (C235G-1.5, 26 ga, Plastics One, Roanoke, VA) were aimed dorsal to the VTA (5.6 mm posterior to bregma, 0.75 mm mediolateral, 8.0 mm ventral to the skull surface), according to [START_REF] Paxinos | The rat brain in stereotaxic coordinates[END_REF]. The coordinates were chosen according to previous studies [START_REF] Carnicella | GDNF is a fast-acting potent inhibitor of alcohol consumption and relapse[END_REF][START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF]. After 3 to 5 days of recovery, rats returned to self-administration training and intra-VTA infusions began when operant responding returned to a stable baseline. Noribogaine and 18-MC were dissolved in PBS containing 2% DMSO. Three hours before the beginning of the session, 0.8 μl of noribogaine (1, 10, 100 μM; equivalent to 3, 30, 300 ng/μl), 18-MC (1, 10, 100 μM; equivalent to 3.7, 37, 370 ng/μl) or vehicle were infused over 2 min to gently-restrained rats via injection cannula extending 0.5 mm beyond the guide cannula tip. The time point of infusion and concentration were chosen according to [START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF], in which we found that intra-VTA infusion of ibogaine 3 hrs before the beginning of the session decreased ethanol self-administration. The experiment was performed using a within-subject, Latin Square Design, with an interval of at least one week between two infusions, allowing lever responding for ethanol to return to baseline (e.g., at least 3 sessions with similar operant performance as the pre-infusion period) between treatments. This repeated procedure of infusions does not produce significant physical damage in the VTA [START_REF] Carnicella | GDNF is a fast-acting potent inhibitor of alcohol consumption and relapse[END_REF] and does not affect VTA function. Indeed, baseline operant performance stayed the same throughout the experiment without any adverse effects in response to repeated infusion of the vehicle. Moreover, the efficacy of the agents infused into the VTA to reduce operant ethanol self-administration remained similar throughout the experiment. It should be noted that responding for ethanol in the vehicle group decreased somewhat over the 4 days of self-administration. This effect is due to the fact that rats have access to ethanol five days a week. Therefore, a small deprivation effect is detected at the beginning of the week (after two days of ethanol withdrawal) that disappears over the self-administration session. Importantly, we also observe this effect without manipulations, indicating that it is independent of the intra-brain region infusion."

Histology

Locations of cannulae were verified in 60-μm coronal sections of paraformaldehyde-fixed tissue stained with thionine. Only data from subjects with cannulae located in the region of interest were included in the analysis (see Figure 2D).

Statistical analysis

Data were analyzed using a one-way or two-way ANOVA with repeated measures, followed by Student-Newman-Keuls test or the method of contrasts [START_REF] Keppel | Design and analysis: A researcher's handbook[END_REF]) when indicated.

Results

Noribogaine, but not 18-MC, is a potent inducer of GDNF expression

First, we determined whether exposure of the dopaminergic-like SH-SY5Y cell line to noribogaine or 18-MC results in an increase in the expression of GDNF. We, therefore, treated cells for 3 hrs with different concentrations of noribogaine or 18-MC and compared the level of GDNF expression to those obtained upon incubation of cells to ibogaine. As shown in Figure 1A andB, noribogaine induced a dose-dependent increase in GDNF expression similar to the one observed upon exposure of cells to ibogaine (F(3, 12) = 14.64, p < 0.001 and F(3, 20) = 10.39, p < 0.001, for ibogaine and noribogaine, respectively). In contrast, incubation of SH-SY5Y cells with 18-MC produced a very small, albeit significant, increase in GDNF expression, even at high concentrations (Figure 1C, F(4, 34) = 4.02, p < 0.01). Noribogaine and 18-MC were also tested at other time points (0.5 to 6 hrs). Noribogaine increased GDNF expression at 1 hr and 6 hrs, however no significant effects were observed when cells were treated with 18-MC (data not shown), indicating that the absence of effect of 18-MC on GDNF expression is not due to an issue of the time window of action. These results, therefore, suggest that the ibogaine metabolite, noribogaine, but not its synthetic derivative, 18-MC, is a potent inducer of GDNF expression.

Intra-VTA infusion of noribogaine, but not 18-MC, reduces operant ethanol selfadministration in rats

Next, we tested whether infusion of noribogaine or 18-MC into the VTA, the site of action of both ibogaine and GDNF (Carnicella et al., 2009b;[START_REF] Carnicella | GDNF is a fast-acting potent inhibitor of alcohol consumption and relapse[END_REF][START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF], decreases operant ethanol self-administration in rats with a history of high levels of ethanol consumption. Specifically, intermittent access to a 20% ethanol solution in a two-bottle choice drinking paradigm led to an escalation of ethanol consumption (Figure 2A), a progressive decrease in the drinking of the water solution (Figure 2B), and consequently to an increase in preference for the 20% ethanol solution over water (Figure 2C). Rats were then trained to self-administer a 20% ethanol solution in operant selfadministration chambers and noribogaine, 18-MC or vehicle was infused into the VTA (Figure 2D) after acquisition of a stable baseline of responding. As shown in Figure 3A and B, noribogaine, but not 18-MC, significantly reduced operant responding for ethanol 3 hrs after the intra-VTA infusion (F(3, 36) = 6.72, p < 0.001 and F(3, 36) = 1.23, p = 0.31, for noribogaine and 18-MC, respectively), leading to a substantial decrease in ethanol intake only for the rats that received noribogaine (F(3, 36) = 7.66, p < 0.001 and F(3, 36) = 1.85, p = 0.16, for noribogaine and 18-MC, respectively; Figure 3C andD).

To ensure that the absence of behavioral effects in response to intra-VTA administration of 18-MC was not due to the use of too low of a dose, the dose of 10 μg/μl of 18-MC, which has been shown to efficiently reduce morphine and methamphetamine self-administration when infused into the medial habenula or the interpeduncular nucleus [START_REF] Glick | 18-Methoxycoronaridine acts in the medial habenula and/or interpeduncular nucleus to decrease morphine selfadministration in rats[END_REF][START_REF] Glick | Brain regions mediating alpha3beta4 nicotinic antagonist effects of 18-MC on methamphetamine and sucrose selfadministration[END_REF], was also tested in the VTA and did not affect operant ethanol self-administration (data not shown).

Interestingly, as shown in Figure 4A andC Session interaction, F(3, 36) = 1.87, p = 0.15, for ethanol deliveries and ethanol intake, respectively). Taken together, these data suggest that the VTA is an important site of action of noribogaine, but not 18-MC, to reduce operant ethanol self-administration.

Discussion

Here we show that noribogaine, like ibogaine, but not 18-MC, potently increases GDNF expression in a dopaminergic-like cell culture model. In addition, we found that administration of noribogaine, but not 18-MC, into the VTA results in a significant and long-lasting decrease in operant self-administration of ethanol.

Different factors can contribute to a decrease in operant performance, including potential suppressant effects of the pharmacological agent, such as sickness, sedation or locomotor deficits. It is, however, unlikely that such factors contribute to the decrease in operant behavior induced by noribogaine as systemic administration of noribogaine does not induce tremor or ataxia [START_REF] Baumann | In vivo neurobiological effects of ibogaine and its O-desmethyl metabolite, 12-hydroxyibogamine (noribogaine), in rats[END_REF]Glick et al., 1996b), and does not alter locomotor activity in rats [START_REF] Baumann | In vivo neurobiological effects of ibogaine and its O-desmethyl metabolite, 12-hydroxyibogamine (noribogaine), in rats[END_REF]. While intra-VTA infusion of noribogaine, and not systemic administration was used in the present study, the present data suggest that noribogaine in the VTA reduces the motivation to self-administer ethanol.

Our results suggest that the ibogaine metabolite noribogaine exhibits similar activities as ibogaine in both increasing the level of GDNF and reducing ethanol operant self-administration after infusion of the drug into the VTA, the site of action of both ibogaine [START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF] and GDNF [START_REF] Carnicella | GDNF is a fast-acting potent inhibitor of alcohol consumption and relapse[END_REF]Carnicella and Ron, 2009).

Interestingly, noribogaine is as potent as ibogaine in increasing GDNF expression in SH-SY5Y cells and in reducing operant ethanol self-administration [START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF]. Also, like ibogaine, the decrease in ethanol self-administration induced by noribogaine is longlasting, and although in this study we performed a 72-hr (4 sessions) time course for noribogaine and previously we tested only a 48-hr (3 sessions) time course for ibogaine [START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF], the results for both drugs are likely to be similar.

Ibogaine is metabolized to noribogaine [START_REF] Obach | Cytochrome P4502D6 catalyzes the Odemethylation of the psychoactive alkaloid ibogaine to 12-hydroxyibogamine[END_REF] and the levels of noribogaine were reported to remain high for a prolonged period of time following oral administration of ibogaine in humans [START_REF] Mash | Medication development of ibogaine as a pharmacotherapy for drug dependence[END_REF][START_REF] Mash | Ibogaine: complex pharmacokinetics, concerns for safety, and preliminary efficacy measures[END_REF][START_REF] Mash | Identification of a primary metabolite of ibogaine that targets serotonin transporters and elevates serotonin[END_REF]. In rats, however, in which metabolism of drugs in the liver and in the brain is generally much faster (Smith, 1991;Woodland et al., 2008), only a small fraction of the noribogaine injected systemically is detected 24 hrs after the administration [START_REF] Baumann | In vivo neurobiological effects of ibogaine and its O-desmethyl metabolite, 12-hydroxyibogamine (noribogaine), in rats[END_REF]. Therefore, it seems unlikely that the decrease in operant ethanol selfadministration observed 24 hrs or more after the infusion results from a persistent level of noribogaine in the brain. However, this possibility cannot be excluded. We previously showed that ibogaine blocks or reverses biochemical and behavioral adaptations obtained in response to exposure of SH-SY5Y cells and rodents to ethanol via GDNF [START_REF] He | Glial cell line-derived neurotrophic factor mediates the desirable actions of the anti-addiction drug ibogaine against alcohol consumption[END_REF][START_REF] He | Glial cell line-derived neurotrophic factor reverses ethanolmediated increases in tyrosine hydroxylase immunoreactivity via altering the activity of heat shock protein 90[END_REF]. We also obtained evidence to suggest that the long-lasting increase in GDNF expression induced by ibogaine is due to the induction of a positive autoregulatory loop in which GDNF increases its own expression [START_REF] He | Autoregulation of glial cell line-derived neurotrophic factor expression: implications for the long-lasting actions of the anti-addiction drug, Ibogaine[END_REF]. As GDNF in the VTA has been shown to be a potent inhibitor of ethanol-drinking andseeking behaviors (Carnicella et al., 2009b;[START_REF] Carnicella | GDNF is a fast-acting potent inhibitor of alcohol consumption and relapse[END_REF], these data strongly suggest that the long-term action of noribogaine on operant ethanol self-administration, as with ibogaine, is mediated by GDNF via the activation of the same positive feedback loop.

In contrast, we did not observe a significant increase in GDNF expression upon treatment of cells with the synthetic derivative of ibogaine, 18-MC. In line with these results, infusion of 18-MC into the VTA did not lead to a reduction of operant selfadministration of ethanol, even at high concentrations. Interestingly, systemic administration of 18-MC in selectively bred alcohol-preferring rats was found to dosedependently decrease ethanol intake and preference in a two-bottle choice paradigm [START_REF] Rezvani | Attenuation of alcohol consumption by a novel nontoxic ibogaine analogue (18-methoxycoronaridine) in alcohol-preferring rats[END_REF]. These results thus suggest that the site of action of 18-MC is not the VTA, and that this synthetic derivative of ibogaine does not mediate its actions on ethanol consumption via the upregulation of the GDNF pathway. 18-MC has been reported to act as an antagonist of nicotinic α3β4 receptors (Glick et al., 2002b;[START_REF] Maisonneuve | Anti-addictive actions of an iboga alkaloid congener: a novel mechanism for a novel treatment[END_REF]. Indeed, low dose combinations of 18-MC with other pharmacological agents that antagonize the nicotinic α3β4 acetylcholine receptors (mecamylamine, dextromethorphan or burproprion), decrease morphine, methamphetamine, and nicotine self-administration at doses that were ineffective when administered alone (Glick et al., 2002a;Glick et al., 2002b). Moreover, 18-MC reduces morphine and methamphetamine self-administration after infusion into the medial habenula or interpeduncular nucleus [START_REF] Glick | 18-Methoxycoronaridine acts in the medial habenula and/or interpeduncular nucleus to decrease morphine selfadministration in rats[END_REF][START_REF] Glick | Brain regions mediating alpha3beta4 nicotinic antagonist effects of 18-MC on methamphetamine and sucrose selfadministration[END_REF], two diencephalon structures with a high level of nicotinic α3β4 receptors [START_REF] Perry | Measuring nicotinic receptors with characteristics of alpha4beta2, alpha3beta2 and alpha3beta4 subtypes in rat tissues by autoradiography[END_REF][START_REF] Quick | Alpha3beta4 subunit-containing nicotinic receptors dominate function in rat medial habenula neurons[END_REF] that are known to influence the activity of the mesocorticolimbic system (e.g., [START_REF] Jhou | The rostromedial tegmental nucleus (RMTg), a GABAergic afferent to midbrain dopamine neurons, encodes aversive stimuli and inhibits motor responses[END_REF][START_REF] Nishikawa | Evidence for, and nature of, the tonic inhibitory influence of habenulointerpeduncular pathways upon cerebral dopaminergic transmission in the rat[END_REF]. In contrast, and in line with the present data, 18-MC is not effective in decreasing morphine and methamphetamine selfadministration when infused into the VTA [START_REF] Glick | 18-Methoxycoronaridine acts in the medial habenula and/or interpeduncular nucleus to decrease morphine selfadministration in rats[END_REF][START_REF] Glick | Brain regions mediating alpha3beta4 nicotinic antagonist effects of 18-MC on methamphetamine and sucrose selfadministration[END_REF], in which the expression of the nicotinic α3β4 is low [START_REF] Klink | Molecular and physiological diversity of nicotinic acetylcholine receptors in the midbrain dopaminergic nuclei[END_REF][START_REF] Perry | Measuring nicotinic receptors with characteristics of alpha4beta2, alpha3beta2 and alpha3beta4 subtypes in rat tissues by autoradiography[END_REF].

Therefore, while the two ibogaine congeners noribogaine and 18-MC reduce drug selfadministration when administered systemically, our data suggest that their site and mechanism of action are different. Like 18-MC, ibogaine is a potent antagonist of the nicotinic α3β4 receptors (Glick et al., 2002b). However, ibogaine also has a plethora of mechanisms of action in the brain, and it is proposed that this variety of effects accounts for its anti-addictive properties [START_REF] Glick | Mechanisms of anti-addictive actions of ibogaine[END_REF]Sweetnam et al., 1995). It is, therefore, likely that both the effects of ibogaine on GDNF expression in the VTA, as well as its antagonist activity at the nicotinic α3β4 receptors, are likely to contribute significantly to its anti-addictive activities.

As described in the introduction, ibogaine's side effects preclude its use as a medication. Noribogaine, as suggested by the present study and others (e.g., Glick et al., 1996b), shares the same desirable long-lasting actions and anti-addictive mechanisms as ibogaine. Importantly, however, noribogaine is safer than its parent compound and, for example, has no tremorigenic effects [START_REF] Baumann | In vivo neurobiological effects of ibogaine and its O-desmethyl metabolite, 12-hydroxyibogamine (noribogaine), in rats[END_REF]Glick et al., 1996b). As such, noribogaine may be considered to have a greater therapeutic profile than ibogaine.

The present data also suggest that the effects of noribogaine on GDNF expression greatly contribute to its potential anti-addictive profile. We recently showed that another molecule, cabergoline, reduces ethanol consumption and relapse in rodent preclinical models by the upregulation of the GDNF pathway in the VTA (Carnicella et al., 2009a).

Together, these results put forward the use of GDNF inducers as a valuable strategy to combat alcohol use and abuse disorders.

In summary, the present data strongly suggest that activation of the GDNF pathway in the VTA is a mechanism of action of noribogaine, but not 18-MC, to reduce ethanol taking. Taken with other studies, it emphasizes an important role of the GDNF pathway within the VTA to modulate ethanol-drinking behaviors. Understanding how ibogaine derivatives decrease drug intake and delineating the different signaling events mediating the anti-addictive properties of ibogaine derivatives may lead to the development of a new generation of drugs to treat addiction.

  Noribogaine hydrochloride was a generous gift from the Addiction Research Institute (Austin, Texas), and 18-MC hydrochloride was obtained from Albany Molecular Research, Inc. (Albany, NY). Ibogaine was purchased from Sigma (St. Louis, MO). Growth medium Dulbecco's modified Eagle's medium (DMEM) and Trizol reagent were purchased from Invitrogen (Carlsbad, CA). The Reverse Transcription System kit was purchased from Promega (Madison, WI). Animals Male Long-Evans rats (280-300 g at the beginning at the experiment) obtained from Harlan (Indianapolis, IN) were used. Rats were housed at a 12-hour light/dark cycle, with lights on 7:00 a.m. and food and water available ad libitum. All animal procedures in this report were approved by the Gallo Center Institutional Animal Care and Use Committee and were conducted in agreement with the Guide for the Care and Use of Laboratory Animals, National Research Council, 1996. Cell culture and drug treatment SH-SY5Y human neuroblastoma cells were cultured in the growth medium DMEM containing 10% fetal bovine serum (FBS) plus 1x MEM non-essential amino acid solution. Cells were incubated in a low-serum medium containing 1% FBS for 2 days before experiments. Cells were then treated for 3 hrs with different concentrations of ibogaine (5, 10 or 50 µM), noribogaine (5, 10 or 50 µM) or 18-MC (10, 20, 50 or 100 µM) dissolved in the medium. This time point was chosen since we previously observed that 3 hrs incubation of SH-SY5Y cells with ibogaine or cabergoline was sufficient time to upregulate GDNF pathway (Carnicella et al., 2009b; He et al., 2005).

  , noribogaine at a concentration of 10 µM produced a decrease in operant ethanol self-administration that was long-lasting and persisted for more than 48 hrs (Treatment, F(1, 36) = 10.63, p < 0.01, Session, F(3, 36) = 2.46, p = 0.08, Treatment x Session interaction, F(3, 36) = 3.26, p < 0.05 and Treatment, F(1, 36) = 11.26, p < 0.01, Session, F(3, 36) = 2.18, p = 0.11, Treatment x Session interaction, F(3, 36) = 3.13, p < 0.05, for ethanol deliveries and ethanol intake, respectively). Similar results were obtained with the 100 µM concentration of noribogaine (data not shown). In contrast, and consistent with the absence of effect 3 hrs post-infusion, no effect of 18-MC on level of response and intake was observed at any of the test sessions(Figure 4B and D; Treatment, F(1, 36) = 0.07, p = 0.80, Session, F(3, 36) = 2.67, p = 0.06, Treatment x Session interaction, F(3, 36) = 2.26, p = 0.10 and Treatment, F(1, 36) = 0.01, p = 0.92, Session, F(3, 36) = 2.32, p = 0.09, Treatment x

Figure 1 .

 1 Figure 1. Ibogaine and noribogaine, but not 18-MC, dose-dependently increase

Figure 2 .

 2 Figure 2. Escalation of ethanol intake and preference in an intermittent access two-

Figure 3 .

 3 Figure 3. Intra-VTA infusion of noribogaine, but not 18-MC, decreases operant

Figure 4 .

 4 Figure 4. Intra-VTA infusion of noribogaine, but not 18-MC, induces a long-lasting

Acknowledgments

The authors thank the Addiction Research Institute for their generous gift of noribogaine. This work was supported by the State of California for Medical Research on Alcohol and Substance Abuse through the University of California, San Francisco (D.R.) and by the Department of the Army, Grant # W81XWH-07-1-0079 (D.R.) for which the U.S. Army Medical Research Acquisition Activity, 820 Chandler Street, Fort Detrick, MD 21702-5014 is the awarding and administering acquisition office. The content of the information represented does not necessarily reflect the position or the policy of the Government, and no official endorsement should be inferred. Sheppard SG (1994) A preliminary investigation of ibogaine: case reports and recommendations for further study. J Subst Abuse Treat 11:379-385. Smith DA (1991) Species differences in metabolism and pharmacokinetics: are we close to an understanding? Drug Metab Rev 23:355-373. Sweetnam PM, Lancaster J, Snowman A, Collins JL, Perschke S, Bauer C, Ferkany J (1995) Receptor binding profile suggests multiple mechanisms of action are responsible for ibogaine's putative anti-addictive activity. Psychopharmacology (Berl) 118:369-376. Vastag B (2002) Addiction treatment strives for legitimacy. JAMA 288:3096, 3099-3101. Woodland C, Huang TT, Gryz E, Bendayan R, Fawcett JP (2008) Expression, activity and regulation of CYP3A in human and rodent brain. Drug Metab Rev 40:149-168.

Authors' contribution

SC and DR designed research. SC, DYH and QVY performed research. SC, DYH and DR assisted with data analysis and interpretation of findings. SC and DR wrote the manuscript and SDG provided material consultation and discussion. All authors critically reviewed content and approved final version for publication.