Nadine Braxmeier-Even

Stefano Olla May

Hydrodynamic Limit for a Hamiltonian system with Boundary Conditions and Conservative Noise

We study the hyperbolic scaling limit for a chain of N coupled anharmonic oscillators. The chain is attached to a point on the left and there is a force (tension) τ acting on the right. In order to provide good ergodic properties to the system, we perturb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that momentum and energy are locally conserved. We prove that in the macroscopic limit the distributions of the elongation, momentum and energy, converge to the solution of the Euler system of equations, in the smooth regime.

Introduction

The aim of this paper is to study the hydrodynamic limit for a non-equilibrium system subject to an exterior time dependent force at the boundary. We consider the most simple mechanical model with non-linear interaction, i.e. a one dimensional chain of N anharmonic oscillators. The left side is attached to a fixed point, while on the right side is acting a force τ (tension). For each value of τ there is a family of equilibrium (Gibbs) measures parametrized by the temperature (and by the tension τ ). It turns out that these Gibbs measures can be written as a product.

We are interested in the macroscopic non-equilibrium behavior of this system as N tends to infinity, after rescaling space and time with N in the same way (hyperbolic scaling). We also consider situations in which the tension τ depends slowly on time, such that it changes in the macroscopic time scale. In this way we can also take the system originally at equilibrium at a certain tension τ0 and push out of equilibrium by changing the exterior tension.

The goal is to prove that the 3 conserved quantities (elongation, momentum and energy) satisfy in the limit an autonomous closed set of hyperbolic equations given by the Euler system.

We approach this problem by using the relative entropy method (cf. [START_REF] Yau | Relative entropy and hydrodynamics of Ginzburg-Landau models[END_REF]) as already done in [START_REF] Olla | Hydrodynamical limit for a Hamiltonian system with weak noise[END_REF] for a system of interacting particles moving in R 3 (gas dynamics).

The relative entropy method permits, in general, to obtain such hydrodynamic limit if the system satisfy certain conditions:

A) The dynamic should be ergodic in the sense that the only conserved quantities that survive the limit as N → ∞ are those we are looking for the macroscopic autonomous behavior (in this case elongation, momentum and energy). More precisely, the only stationary measure for the infinite system, with finite local entropy, are given by the Gibbs measures.

1

B) The macroscopic equations have smooth solutions.

C) Microscopic currents of the conserved quantities should be bounded by the local energy of the system.

We do not know any deterministic Hamiltonian system that satisfy condition A), and this is a major challenging open problem in statistical mechanics. Stochastic perturbation of the dynamics that conserves energy and momentum can give such ergodic property and have been used in [START_REF] Olla | Hydrodynamical limit for a Hamiltonian system with weak noise[END_REF] (cf. also [START_REF] Liverani | Ergodicity in infinite Hamiltonian systems with conservative noise[END_REF][START_REF] Fritz | Reversibility in infinite Hamiltonian systems with conservative noise[END_REF][START_REF] Fritz | Stationary states of random Hamiltonian systems[END_REF]). We use here a simpler stochastic mechanism than in [START_REF] Olla | Hydrodynamical limit for a Hamiltonian system with weak noise[END_REF]: at random independent exponential times we exchange the momentum of nearest neighbor particles, as if they were performing an elastic collision. Under this stochastic dynamics, every stationary measure has the property to be exchangeable in the velocity coordinates, and this is sufficient to characterize it as a convex combination of Gibbs measures (cf. [START_REF] Fritz | Stationary states of random Hamiltonian systems[END_REF] and [START_REF] Bernardin | Non-equilibrium macroscopic dynamics of chains of anharmonic oscillators[END_REF]).

About condition B), it is well known that nonlinear hyperbolic equations in general develop shocks also starting from smooth initial condition. Characterization and uniqueness of weak solutions in presence of shock is a challenging problem in the theory of hyperbolic equations. We expect that a shock will increase the thermodynamic entropy associated to the profiles of the conserved quantities.

The relative entropy method compares the microscopic Gibbs entropy production (associated to the probability distribution of the system at a given time) with the macroscopic (thermodynamic) entropy production. If no shocks are present both entropy productions are small. The presence of the boundary force changes a bit this balance, since one should take into account the (macroscopic) change of entropy due to the work performed by the force. It turns out that the right choice of the boundary conditions in the macroscopic equation does compensate this large entropy production, keeping the time derivative of the relative entropy small. It would be interesting to prove similar cancellation of entropy productions when this is caused by shocks, as it would allow to prove the hydrodynamical limit in these cases, and provide a microscopic derivation of irreversible thermodynamic adiabatic transformations, between thermodynamic equilibrium states that increase the thermodynamic entropy. Recent efforts in this direction use different methods (cf. [START_REF] Fritz | Microscopic Theory of Isothermal Elastodynamics[END_REF]). Similar results on isothermal transformation are mathematically easier (cf. [START_REF] Olla | Microscopic derivation of an isothermal thermodynamic transformation[END_REF]).

About condition C), it created a problem in [START_REF] Olla | Hydrodynamical limit for a Hamiltonian system with weak noise[END_REF]: in the usual gas dynamics the energy current has the convecting term cubic in the velocities, while energy is quadratic. This was fixed in [START_REF] Olla | Hydrodynamical limit for a Hamiltonian system with weak noise[END_REF] by modifying the kinetic energy of the model: if the kinetic energy grows linearly as a function of the velocity, also the energy current will grow linearly. Since we work here in Lagrangian coordinates, our energy current does not have the cubic convecting term. This allows us to work with the usual quadratic kinetic energy. This paper has been partially supported by European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953) and by a grant of the MAThematics Center Heidelberg (MATCH).

The Model and the Main Theorem

We will study a system of N + 1 coupled oscillators in one dimension. Each particle has the same mass that we set equal to 1. The position of atom i (i = 0, . . . , N ) is denoted by qi ∈ R, while its momentum is denoted by pi ∈ R. We assume that particle 0 is attached to a fixed point and it does not move, i.e. (q0, p0) ≡ (0, 0), while on particle N we apply a force τ (t) depending on time. Observe that only the particle 0 is constrained not to move, and that qi can assume also negative values.

Denote by q := (q0, . . . , qN ) and p := (p0, . . . , pN ). The interaction between two particles i and i -1 will be described by the potential energy V (qi -qi-1) of an anharmonic spring relying the particles. We assume V to be a positive smooth function that grows quadratically at infinity, i.e. there exist strictly positive constants C+ and C-such that for any r ∈ R:

V ′ (r) 2 ≤ C+(1 + V (r)), r 2 ≤ C-(1 + V (r)), V (0) = 0. (2.1)
Energy is defined by the following Hamiltonian:

HN (q, p) : = N i=1 p 2 i 2 + V (qi -qi-1) .
Since we focus on a nearest neighbors interaction, we define the distance between particles by ri = qi -qi-1, i = 1, . . . , N.

Consequently the phase space is given by (R 2 ) N . We define the energy of particle i ∈ {1, . . . , N } as

ei := p 2 i 2 + V (ri)
so that HN (r, p) = N i=1 ei, where r := (r1, . . . , rN ). Given a smooth function τ (s) that represents the force applied to particle N at the macroscopic time s, the dynamics of the system is determined by the generator

N G τ (t) N := N L τ (t) N + N γSN . (2.2)
Here the Liouville operator L τ N is given by

L τ N = N i=1 (pi -pi-1) ∂ ∂ri + N-1 i=1 V ′ (ri+1) -V ′ (ri) ∂ ∂pi + τ -V ′ (rN ) ∂ ∂pN , (2.3)
where we used the fact that p0 ≡ 0. Notice that the time scale in the tension is chosen such that it changes smoothly on the macroscopic scale.

The symmetric operator SN is the generator of the stochastic part of the dynamics that exchanges at random times velocities of nearest neighbors particles. For any smooth function f , we define the operator Υi,i+1 by

Υi,i+1 = f r, p i,i+1 -f (r, p) (2.4)
where p i,i+1 ∈ R N is defined from p ∈ R N by exchanging the coordinates pj and pj+1

p i,i+1 j =    pj if j = i, i + 1 pi+1 if j = i pi if j = i + 1 .
Then SN is defined through

SN f (r, p) := N-1 i=1 f r, p i,i+1 -f (r, p) = - 1 2 N-1 i=1 Υ 2 i,i+1 f (r, p) (2.5)
With this choice of the noise, the three balanced quantities, i.e. locally conserved, are given by ri, pi, ei.

We define ζ(r, p) = (r, p, -e(r, p)) T ∈ R 2 × R-, and the Gibbs thermodynamic potential:

Θ(λ) := log R 2 e λ•ζ(r,p) drdp.
(2.6)

By the condition imposed on V , this function is always finite.

For ζ ∈ R 2 × R-we define Φ : R 2 × R-→ R the Legendre transform of Θ(λ): Φ(ζ) := sup η∈R 2 ×R + {η • ζ -Θ(η)} . (2.7)
We denote by λ(ũ) and ũ(λ) := (r, p, -E) T the corresponding convex conjugate variable, that satisfy λ = DΦ(ũ) and ũ = DΘ(λ),

where the operator D is defined by

Df (a) := ∂f ∂a1 , ∂f ∂a2 , ∂f ∂a3 (2.9)
for any C 1 function f : R 3 → R and a := (a1, a2, a3) ∈ R 3 .

On the one particle state space R 2 we define a family of probability measure

ν λ (dr, dp) = e λ•ζ(r,p)-Θ(λ) drdp.
(2.10)

Observe that

Eν λ [ζ(r, p)] = ũ so we can identify ũ = (r, p, -E) T as respectively the average distance, velocity and (negative) energy. We also define the internal energy e = Ep 2 /2. We have the relations

Eν λ (p 2 ) -p 2 = λ -1 3 := β -1 , P (r, e) := Eν λ [V ′ (r)] = λ1 λ3 := τ
that identify β -1 as temperature and τ as tension. This thermodynamic terminology is justified by observing that, for constant τ in the dynamics, and any β > 0, with the choice λ = (βτ, 0, β) the family of product measures given by:

ν N (τ β,0,β) (dr, dp) = N i=1 ν (τ β,0,β) (dri, dpi), β ∈ R +
is stationary for the dynamics. These are the Gibbs measures at an average temperature β -1 , pressure τ and velocity 0. In what follows we need also Gibbs measure with average velocity different from 0, and we will use the following notation:

ν N λ := N i=1 e λ•ζ i -Θ(λ) dridpi := g N λ (r, p)drdp,
where ζ i := (ζi,1, ζi,2, ζi,3) T := (ri, pi, -ei) T .

In a similar way we may introduce the local Gibbs measures: For any continuous profile ũ(x), x ∈ [0, 1], we have correspondingly a profile of parameters λ(x), and we define the inhomogeneous product measure

ν N λ(•) := N i=1 e λ(i/N)•ζ i -Θ(λ(i/N)) dridpi,
that we call Local Gibbs measures. We are interested in the macroscopic behavior of the elongation, momentum and energy of the particles, at time t, as N → ∞. Notice that t is already the macroscopic time, since we have already multiplied the generator by N . Taking advantage of the one-dimensionality of the system, we will use lagrangian coordinates, i.e. our space variables will be given by the lattice coordinates {1/N, . . . , (N -1)/N, 1}. Also observe that at this time scale, the generator of the process is given by N G

τ (t) N .
Consequently, we introduce the (time dependent) empirical measures representing the spatial distribution (on the interval [0, 1]) of these quantities:

η N α (dx, t) := 1 N N i=1 δ x - i N ζi,α(t) dx, for α = 1, 2, 3.
We expect the measures η N α (dx, t), α = 1, 2, 3 to converge, as N → ∞, to measures r(x, t)dx, p(x, t)dx, -E(x, t)dx being absolutely continuous with respect to the Lebesgue measure and with density satisfying the following system of three conservation laws:

           ∂tr -∂xp = 0 ∂tp -∂xP (r, e) = 0 ,    r0(x) = r(x, 0), p0(x) = p(x, 0), E0(x) = E(x, 0) p(0, t) = 0, P (r(1, t), e(1, t)) = τ (t) ∂tE -∂x(pP (r, e)) = 0
(2.11) for bounded, smooth initial data r0, p0, E0 : [0, 1] → R and the force τ (t) depending on time t. Here we denoted by r the specific volume, p the velocity, E the total energy and e := E -1 2 p 2 the internal energy. We also assume that the internal energy is always positive: e(x, 0) > 0.

We need the solutions of the system (2.11) to be C 2 -solutions. To assure this, the following additional compatibility conditions at the space-time edges (x, t) = (0, 0) and (x, t) = (1, 0) have to be satisfied:

lim x→0 p0(x) = p(0, 0) = 0 , lim x→1 P (r0(x), e0(x)) = τ (0)
(2.12)

lim x→0 d dx P (r0(x), e0(x)) = 0 , lim x→1 d dt P (r0(x), e0(x)) = τ ′ (0) (2.13) lim x→0 d 2 (dt) 2 p0(x) = 0 , lim x→1 d 2 (dt) 2 P (r0(x), e0(x)) = τ ′′ (0).
(2.14)

A proof of this can be adapted from Chapters 4.3, 7.5 and 3.5 of [START_REF] Li | Boundary value problems for quasilinear hyperbolic systems[END_REF].

For any test function J : [0, 1] → R with compact support in (0, 1) consider the empirical densities

η N α (t, J) := η N α (dx, t); J = 1 N N i=1 J i N ζα,i(t). (2.15)
Our goal is to show that, starting with an initial distribution such that there exist smooth functions r0, p0 and E0 satisfying

{η N 1 (0, J), η N 2 (0, J), η N 3 (0, J)} → J(x)r0(x)dx, J(x)p0(x)dx, -J(x)E0(x)dx (2.16) in probability as N → ∞, then at time t ∈ [0, T ] we have the same convergence of η N α (t, J), α = 1, 2, 3 to the corresponding profiles r(x, t), p(x, t) and E(x, t) respectively, that satisfy (2.11)-(2.14).
Here is the precise statement of our main result, where we make a stronger assumption on the initial measure: Theorem 2.1 (Main Theorem). For any time t ≤ ts, ts being the time at which the solution u produces the first shock, denote by µ N t the probability measure on the configuration space R 2N at time t, starting from the local Gibbs measure ν N λ(•,0) corresponding to the initial profiles ũ0. Then for any smooth function J :

[0, 1] → R and any δ > 0 lim N→∞ µ N t 1 N N i=1 J( i N )ζ i - 1 0 J(x)ũ(x, t)dx > δ = 0, (2.17)
where u is a C 2 -solution to the system of conservation laws (2.11)-(2.14).

Remark 2.2. As our proof is based on the relative entropy method of [START_REF] Yau | Relative entropy and hydrodynamics of Ginzburg-Landau models[END_REF], it is only valid as long as the solution to (2.11) are C 2 . Since, even for smooth initial data, it may happen that the solution develops shocks, we are forced to restrict our derivation to a time 0 < t < ts, where ts is the time when the solution to the system of conservation laws enters the first shock. Remark 2.3. A proof for the existence of smooth solutions to the initial-boundaryvalue problem (2.11) can be found in chapter 4.3, 7.5 and 3.5 of [START_REF] Li | Boundary value problems for quasilinear hyperbolic systems[END_REF]. Notice that we can rewrite the pressure P as a function of specific volume r and entropy s:

P (r, s) := P (r, e).
Then we can rewrite the initial boundary value problem (2.11), in the smooth regime, in terms of the unknown r, p and s(r, e) as follows:

           ∂tr -∂xp = 0 ∂tp -∂x P (r, s) = 0 ∂ts = 0 ,    r0(x) = r(x, 0), p0(x) = p(x, 0), s0(x) = s(x, 0) p(0, t) = 0, P (r(1, t), s(1, t)) = τ (t) , (2.18)
where we used the thermodynamic relation

P (r, s) = - ∂e(r, s) ∂r .
Hence the specific entropy s does not change in time and for any x ∈ [0, 1] is given through the initial data s(x, 0) := s0(x).

In the non-conservative form, equation (2.18) reads as:

∂t   r p s   -A(r, p, s)∂x   r p s   = 0
where the 3 × 3-matrix A is defined by

A :=   0 1 0 ∂ P ∂r 0 ∂ P ∂s 0 0 0   = S •   c 0 0 0 -c 0 0 0 0   • S -1 with c := c(r, s) = ∂ P ∂r and S := S(r, p, s) =   1 1 -1 c ∂ P ∂s c -c 0 0 0 c   .
With these notations we can rewrite (2.18) in the characteristic form

S -1 • ∂t   r p s   -   c 0 0 0 -c 0 0 0 0   • S -1 • ∂x   r p s   = 0 ⇒ c(∂tr -c∂xr) + (∂tp -c∂xp) + 1 c ∂ P ∂s (∂ts -c∂xs) = 0 c(∂tr + c∂xr) -(∂tp + c∂xp) + 1 c ∂ P ∂s (∂ts + c∂xs) = 0 ∂ts = 0.
In this way we can apply the existence proof for C 2 solutions to (2.11)-(2.14) for short times from [START_REF] Li | Boundary value problems for quasilinear hyperbolic systems[END_REF].

3 The Hydrodynamic Limit

The Relative Entropy

On the phase space (R 2 ) N we now have two time-dependent families of probability measures. One of them is the local Gibbs measure ν N λ(•,t) constructed from the solution of the system of conservation laws (2.11)-(2.14). We denote its density by

g N t = N i=1 e λ(i/N,t)•ζ i -Θ(λ(i/N,t)) (3.1)
On the other hand we have the actual distribution µ N t , whose density f N t (r, p) is a solution, in the sense of distributions, of the Kolmogorov forward equation:

     ∂f N t ∂t (r, p) = N G τ (t),⋆ N f N t (r, p) f N 0 (r, p) = g N 0 (r, p). (3.2)
By G τ, * N = L τ,⋆ N +γSN we denote the adjoint operator of G τ N with respect to the Lebesgue measure, where L τ,⋆ N can be computed as L τ,⋆ N = -L τ N . The relative entropy of f N t with respect to g N t is defined by

HN (t) = f N t log f N t g N t drdp (3.3)
Our main result will follow from: Remark: Recall that the relative entropy H(α|β) of a probability measure α with respect to a probability measure β can be rewritten as

H(α|β) = sup ϕ ϕdα -log e ϕ dβ (3.4)
where the supremum is taken over all bounded measurable functions ϕ. It is easy to see that the relative entropy has the following properties: H(α|β) is positive, convex, and lower semi continuous function of α. It follows that for any measurable function F and any σ > 0:

F dα ≤ 1 σ log e σF dβ + 1 σ H(α|β). (3.5)

Proof of Theorem 2.1:

A useful special case of the entropy inequality can be stated if we set

F := 1 [A] to be the indicator function on a set A. With the choice σ = log 1 + 1 β[A]
, we obtain the inequality

Eα[1 [A] ] = α[A] ≤ 1 σ log β[exp(σ1 [A] )] + 1 σ H(α|β) = 1 σ log (β[A](e σ -1) + 1) + 1 σ H(α|β) ⇒ α[A] ≤ log 2 + H(α|β) log 1 + 1 β[A] (3.6)
Thus, if we define the set A δ to be

A δ := 1 N N i=1 J( i N )ζ i - 1 0 J(x)u(x, t)dx > δ ,
for any test function J : [0, 1] → R with compact support in (0, 1), then with inequality (3.6), to prove that limN→∞ µ N t [A δ ] = 0, it is enough to show that for each δ > 0

log 1 + 1 ν N λ(•,t) ≥ C(δ)N,
for some constant C not depending on N , since from Theorem 3.1 we have that

HN (t) = o(N ). But this is satisfied if ν N λ(•,t) [A δ ] is exponentially small, i.e ν N λ(•,t) [A δ ] ≤ 1 e C(δ)N . (3.7)
This is a result of the large deviation theory which can be adapted from [START_REF] Bernardin | Non-equilibrium macroscopic dynamics of chains of anharmonic oscillators[END_REF][START_REF] Kipnis | Scaling Limits of Interacting Particle Systems[END_REF][START_REF] Varadhan | Large Deviation and Application[END_REF].

Time Evolution of the Relative Entropy

In this Section we will prove Theorem 3.1. Notice that by the choice of the initial distribution HN (0) = 0.

The strategy is to show that for some constant C N , we have that

HN (t) ≤ C t 0 HN (s)ds + t 0 RN (s)ds (3.8)
f N t+h log f N t+h dr dp ≤ f N t log f N t dr dp (3.11)
Then, since g N s is smooth and HN (0) = 0, (3.10) follows.

Before we proceed in the proof, we have to introduce some further notations. For any C 1 function F := (f1, f2, f3) T : R 3 → R Equation (3.14) can be rewritten as

(D 2 Θ)∂tλ = (D 2 Σ)∂xλ. Since D 2 Θ(λ(t, x)) -1 = (D 2 Φ)(ũ(t, x)), it follows that ∂tλ(D 2 Φ) = (D 2 Σ)∂xλ. Since (D 2 Σ) = (D 2 Φ)(D J(ũ))
the following system of partial differential equations is satisfied:

∂tλ(t, x) = (D J) T (ũ)∂xλ(t, x). ( 3
N ζ i = Ji-1,i -Ji,i+1.
Finally let us define vj := (0, pj, -p 2 j /2) T . Hence with the definition of the symmetric operator given by (2.5),

SN (ζ j ) = -2vj + vj+1 + vj-1, j = 2, . . . , N -1 SN (ζ N ) = -vN + vN-1, SN (ζ 1 ) = -v1 + v2. Lemma 3.3. N L τ (t) N log g N t = N i=1 ∂xλ( i N , t) • Ji-1,i + N λ2(1, t)τ (t) + aN (t) (3.17)
where aN (t) is such that

lim N→∞ 1 N t 0 aN (s)f N s dpdrds = 0 Proof. N L τ N log g N t (r, p) = N N i=1 λ( i N , t) • (Ji-1,i -Ji,i+1) = N N i=1 λ( i N , t) -λ( i -1 N , t) • Ji-1,i -λ(0, t) • J0,1 + λ(1, t) • JN.N+1
Taking into account the boundary conditions on λ we have

λ(0, t) • J0,1 = λ2(0, t)V ′ (r1) = 0 (3.18) and λ(1, t) • JN.N+1 = -pN λ1(1, t) -λ2(1, t)τ (t) + λ3(1, t)τ (t)pN = -λ2(1, t)τ (t) (3.19) because τ (t)λ3(1, t) = λ1(1, t). Since λ is a C 2 -function, we obtain (3.17) with |aN (t)| = C N N-1 i=1

Ji

It remains to show, that limN→∞ t 0 a N (s) N dµ N s ds = 0. This will be an easy consequence of Lemma 3.15. Lemma 3.4.

∂t log g N t = N i=1 (D J) T ũ( i N , t) ∂xλ( i N , t) • ζ i -ũ( i N , t) Proof. ∂ ∂t log g N t = ∂ ∂t N i=1 λ( i N , t) • ζ i -Θ λ( i N , t) = N i=1 ∂tλ( i N , t) • ζ i -DΘ λ( i N , t)
By (2.8), DΘ λ( i N , t) = ũ( i N , t), and (3.15) the result follows.

Lemma 3.5. Recall the definition of the symmetric operator given by (2.5).

-lim

N→∞ 1 N t 0 N SN log g N s f N s dpdr ds ≤ lim N→∞ 1 σN t 0 HN (s)ds,
where σ is a constant independent of N with 0 < 2σ < infx,s λ3(x, s).

Observe that smoothness of the solution of (2.11) guarantees infx,s λ3(x, s) > 0.

Proof.

SN log g N s = N-1 i=2 λ( i N , t) • (vi-1 -2vi + vi+1) + λ( 1 N , t) • (-v1 + v2) + λ(1, t) • (vN-1 -vN ) = N-1 i=2 λ( i -1 N , t) -2λ( i N , t) + λ( i + 1 N , t) • vi + λ( 2 N , t) -λ( 1 N , t) • v1 + λ( N -1 N , t) -λ(1, t) • vN
In Lemma 3.15 we will show that the expectation of 1 N i vi is uniformly bounded for all N and hence, since λ is in C 2 , the first term vanishes in the limit as N → ∞.

Recall that by the entropy inequality (3.5), for any σ > 0 we have for k ∈ {1, . . . , N }:

1 N p 2 k f N s dpdr ≤ 1 N σ log e σp 2 k ν N λ(•,s) + 1 N σ H(s)
Since this inequality is true for any σ > 0, the integral on the right hand side of the inequality is bounded as long as σ < infx 1 2 λ3(x, s) and hence the first term vanishes as N → ∞. The expected value of p k can be controlled in a similar way.

So far we have from Lemma 3.3, 3.4 and 3.5

HN (t) ≤ t 0 N i=1 ∂xλ( i N , s) Ji-1,i -(D J) T (ũ( i N , s)) ζ i -ũ( i N , s) f N s dp dr ds - t 0 N τ (s)λ2(1, s)ds + 1 σ t 0 HN (s)ds + t 0 RN (s)ds (3.20)
where RN (t) is such that (3.9) holds. By (3.12) we have

1 0 ∂xλ(x, t) • J(ũ(x, t))dx = ∂ ∂x λ1(x, t)λ2(x, t) λ3(x, t) dx = τ (t)λ2(1, t).
and consequently we can replace -N τ (t)λ2(1, t) by

- N i=1 ∂xλ( i N , t) • J(ũ( i N , t))
with an error uniformly bounded in N . It follows that from (3.20) we have

HN (t) ≤ t 0 N i=1 ∂xλ( i N , s)× Ji-1,i -J ũ( i N , s) -(D J) T (ũ( i N , s)) ζ i -ũ( i N , s) f N s dp dr ds + 1 σ t 0 HN (s)ds + t 0 RN (s)ds. (3.21)
Our next goal is to prove a weak form of local equilibrium. In view of this we introduce microscopic averages over blocks of size 2k + 1: In what follows, for any vector field Yi := (Y1,i, Y2,i, Y3,i)

T : (R 2 ) 3 → R 3 we denote by Y k i := (Y k 1,i , Y k 2,i , Y k 3,i )
T , block averages over blocks of length 2k + 1, where k > 0 is independent of N . For example

ζ k i = (ζ k 1,i , ζ k 2,i , ζ k 3,i ) T := (r k i , p k i , -e k i ) T := 1 2k + 1 |i-l|≤k ζ l . (3.22)
These blocks are microscopically large but on the macroscopic scale they are small, thus N goes to infinity first and then k goes to infinity. We also need to introduce another small parameter ℓ and consider small macroscopic blocs of length ℓN at the boundaries.

For any smooth and bounded function F : [0, 1] → R and any bounded function ψ : R 2 → R, we obtain the following summation by parts formula

1 N N i=1 F ( i N )ψ(ri, pi) = 1 N N-[Nℓ] i=[Nℓ] F ( i N ) 1 2k + 1 |j-i|≤k ψ(rj, pj) + O( k + N ℓ N ). (3.23)
Here we first restricted the sum to configurations over {[N ℓ], . . . , N -[N ℓ]}, for some small ℓ > 0, such that ℓ → 0 after N → ∞ and ℓN >> k. In this way, we avoid touching the boundary when we introduce the block averages. The error we made will vanish in the limit since ℓ → 0.

We also need to do some cut off in order to have only bounded variables: Let C i,b := {ei-1, ei ≤ b}, and define

J b i-1,i := Ji-1,i1C i,b and
ζ b i := ζ i 1C i,b ,
then these functions are bounded. Also denote Jb (ũ) the corresponding expectation with respect to the Gibbs measure of parameters λ(ũ), that converges to J(ũ) as b → ∞. Assumptions (2.1) on the potential assert that by the entropy inequality (3.5) with reference measure dν N λ(•,t) , the error we make by the replacement of Ji-1,i and ζ i by

J b i-1,i and ζ b i respectively is small in N if we can show that 1 N HN (s) → 0 as N → 0: For any σ > 0 small enough N i=1 ∂xλ( i N , s)Ji-1,i1 C c i,b dµ N s ≤ 1 σ N i=1 log e σ∂xλ( i N ,s)J i-1,i 1 C c i,b dν λ(•,t) + HN (s) σ ≤ 1 σ N i=1 log 1 + C c i,b e σ∂xλ( i N ,s)J i-1,i dν λ(•,t) + HN (s) σ = N C(b, σ) σ + HN (s) σ (3.24)
where lim b→∞ C(b, σ) = 0 for any σ > 0.

Using that λ and u are in C 2 and formula (3.23), we arrive at

N i=1 ∂xλ( i N , s) J b i-1,i -Jb ũ( i N , s) -(D J) T ũ( i N , s) ζ b i -ũ( i N , s) = N-[Nℓ] i=[Nℓ] ∂xλ( i N , s) × 1 2k + 1 |l-i|≤k J b l-1,l -Jb ũ( i N , s) -(D J) T ũ( i N , s) ζ b,k i -ũ( i N , s) +O(k + N ℓ).
The following theorem will be proved in Section 3.3.

Theorem 3.6 (The one-block estimate). For any ℓ, b:

lim k→∞ lim N→∞ 1 N N-[Nℓ] i=[Nℓ] t 0 1 2k + 1 |l-i|≤k J b l-1,l -Jb ζ k i f N s dp dr ds = 0. (3.25)
With this Theorem we obtain:

HN (t) N ≤ 1 N N-[Nℓ] i=[Nℓ] t 0 Ω ζ k i , ũ( i N , s) f N s dp dr ds + t 0 R N,k,ℓ,b (s) N ds + t 0 HN (s) N σ ds (3.26) for some σ > 0. R N,k,ℓ,b is such that lim b→∞ lim ℓ→0 lim k→∞ lim N→∞ t 0 R N,k,ℓ,b (s) N ds = 0,
and we used (3.15) to define

Ω(z, ũ) := ∂xλ • J (z) -J (ũ) -∂tλ • (z -ũ) . Hence DzΩ(z, ũ) = (D J) T (z) • ∂xλ -∂tλ (3.27)
is equal to zero if z is a solution of (3.15) and consequently:

Ω(ũ, ũ) = 0, DzΩ(ũ, ũ) = 0.
Applying the entropy inequality (3.5) on the sum in (3.26), we obtain that for some σ > 0 it is bounded above by

1 N σ t 0 log exp    σ N-[Nℓ] i=[Nℓ] Ω(ζ k i , u( i N , s))    g N s dp dr ds+ 1 N σ t 0 HN (s)ds (3.28)
Hence it remains to prove, that the first term of this expression is of order O( 1 N ). This will be done using the following special case of Varadhan's Lemma: Theorem 3.7 (Varadhan's Lemma). Let ν n λ be the product homogeneous measure with marginals ν λ given by (2.10) and with rate function I : R 2 × R-→ R defined by

I(x) := Φ(x) -x • λ + Θ(λ).

Then for any bounded continuous function

F on R 2 × R- lim n→∞ 1 n log e nF (ζ) dν n λ = sup x {F (x) -I(x)}
Proof. A proof of this Theorem can be adapted from [START_REF] Bernardin | Non-equilibrium macroscopic dynamics of chains of anharmonic oscillators[END_REF][START_REF] Kipnis | Scaling Limits of Interacting Particle Systems[END_REF][START_REF] Varadhan | Large Deviation and Application[END_REF] In order to apply this Theorem we arrange the sum in (3.28) as sums over disjoint blocks and then take advantage of the fact that the local Gibbs measures are product measures. Assume without loss of generality that 2k

+ 1 divides N -2[N ℓ], then N-[Nℓ] i=[Nℓ] Ω(ζ k i , ũ( i N , s)) = j∈{-k,...,k} i∈B N -2[N ℓ],k [N ℓ] τj Ω(ζ k i , ũ( i N , s))
where

B N-2[Nℓ],k [Nℓ] := r(2k + 1) + [N ℓ] + k; r ∈ 0, . . . , N-2[Nℓ]-2k 2k+1
. In this way, for any fixed j, the terms in the sum over i ∈ B

N-2[Nℓ],k [Nℓ]
depend on configurations in disjoint blocks. Thus the random variables

τjΩ(ζ k i , ũ( i N , s)) are independent under ν N λ(•,s) .
Using Hölder inequality, the first term in (3.28) is bounded above by

1 N σ t 0 log j∈{-k,...,k} exp      σ i∈B N -2[N ℓ],k [N ℓ] τj Ω(ζ k i , u( i N , s))      g N s dp dr ds ≤ 1 N σ(2k + 1) t 0 j∈{-k,...,k} log exp      σ(2k + 1) i∈B N -2[N ℓ],k [N ℓ] τjΩ(ζ k i , ũ( i N , s))      g N s dp dr ds = 1 N σ(2k + 1) N-[Nℓ] i=[Nℓ] t 0 log exp σ(2k + 1)Ω(ζ k i , ũ( i N , s)) g N s dp dr ds.
Then, since all the functions in this expression are smooth and the family of local Gibbs measures converges weakly, we obtain that

lim k→∞ lim N→∞ 1 (2k + 1)N σ N-[Nℓ] i=[Nℓ] t 0 log exp σ(2k + 1)Ω(ζ k i , ũ( i N , s)) g N s dp dr ds = lim k→∞ 1 σ(2k + 1) t 0 1 0 log exp (2k + 1)σΩ(ζ k i , ũ(x, s)) dν λ(x,s) dxds.
So now for each x ∈ [0, 1], the distribution of the particles in a box of size k is given by the invariant Gibbs measure with average u(x, s). such that we can apply Theorem 3.7 on this product measure to obtain that the last expression is equal to To conclude Theorem 3.1 it thus remains to show that this is equal to zero. Since I and Ω are both convex, and both functions and their derivatives are vanishing at z = ũ, it follows from assumption (2.1) on the potential that σΩ(z, ũ) ≤ I(z) for σ small enough. Hence there exists a σ such that the last expression is equal to zero. This concludes the proof of Theorem 3. 3.3 The one block estimate (Theorem 3.6)

We define the space-time average of the distribution

f N,ℓ,k t = 1 t t 0 1 [N (1 -2ℓ)] N(1-ℓ) i=Nℓ f N,k s,i (r -k , p -k , . . . , r k , p k ) ds (3.30)
where we defined the projections

f N,k s,i (r -k , p-k . . . , rk , pk ) = f N s (r1, p1,. . .,r i-k-1 , p i-k-1 , r-k , p-k ,. . .,

Tightness

We have the following Lemma 3.8 asserts that for each fixed k there exists a limit point µ ℓ,k t of the sequence (μ N,ℓ,k t ) N≥1 . On the other hand, since the sequence (µ ℓ,k t ) k≥1 forms a consistent family of measures, by Kolmogorov's Theorem, for k → ∞, there exists a unique probability measure µ on the configuration space {(ri, pi) i∈Z ∈ (R 2 ) ∞ }, such that the restriction of µ on {(rj, pj) j∈{-k,...,+k} ∈ (R 2 ) 2k+1 } is µ ℓ,k t .

Proof of the one-block-estimate

Let us define the formal generator G of the infinite dynamics by

G := L + γS, (3.33)
with the antisymmetric part The proof of Theorem 3.10 is contained in [START_REF] Fritz | Stationary states of random Hamiltonian systems[END_REF], see also [START_REF] Bernardin | Non-equilibrium macroscopic dynamics of chains of anharmonic oscillators[END_REF] for more details. The idea of the proof is the following: by proposition 3.9 one can prove that µ is separately stationary for L and S. This implies that the distribution of momenta conditioned on position µ(dp|r) is exchangeable. This is the only point where we need the noise in the dynamics.

L := j∈Z pj ∂ ∂rj - ∂ ∂rj+1 + V ′ (rj+1) -V ′ (rj) ∂ ∂pj (3.

Proof of Theorem 3.6:

Recall we need to prove (3.25). By Lemma 3.8 and (ii), (iii) of Proposition 3.9, it is enough to show that for each b and ℓ lim sup

k→∞ sup µ∈G 1 2k + 1 k l=-k+1 J b l-1,l -Jb 1 2k + 1 k l=-k+1 ζ l dµ = 0
where G is the set of Gibbs measures. But this is just the law of large numbers and holds in the limit as k → ∞. With (3.32), the integral is equal to

1 t t 0 1 [N (1 -2ℓ)] N(1-ℓ) j=Nℓ GF f N,k s,j |l|≤k dr l dp l ds. (3.38)
Define the space average

F := 1 [N (1 -2ℓ)] N(1-ℓ) j=Nℓ τjF,
and observe that G τ (t) N F = G F , then we can rewrite (3.38) as

1 N t t 0 (N G τ (s) N F ) f N s dr dp ds = 1 tN t 0 F ∂sf N s dr dpds = 1 tN F f N t dr dp - F f N 0 dr dp .
This expression converges to 0 when N → ∞, since F is a bounded function.

Entropy density

For some integer n ≥ 1, define by Λ n a box of length 2n + 1 and by Λ n i a box of length 2n + 1 and centered at i. Furthermore, let

ν ∞ (τ β,0,β) (dr, dp) := i∈Z ν (τ β,0,β) (dri, dpi) and H Λ k (µ|ν ∞ (τ β,0,β) ) := H(µ| Λ k |ν k (τ β,0,β)
). We obtain the following Lemma: Lemma 3.13. The limit point µ has finite entropy density, that means there exists a constant C > 0 such that for all subsets Λ k

H Λ k (µ|ν ∞ (τ β,0,β) ) ≤ C|Λ k |.
Proof. By convexity of the relative entropy, we have Relative entropy is superadditive in the following sense (see for example [START_REF] Bernardin | Non-equilibrium macroscopic dynamics of chains of anharmonic oscillators[END_REF]): let (Λi) i∈I⊂N be a family of disjoint subsets of Z. Then

H μN,ℓ,k t ν k (τ β,0,β) ≤ 1 [N (1 -2ℓ)] N(1-ℓ) j=Nℓ H 1
H i∈I Λ i μN t ν N (τ β,0,β) ≥ i∈I HΛ i μN t ν N (τ β,0,β) .
The sum in (3.39) can be rearranged in 2k + 1 sums of sums over disjoint blocks, then applying the superadditivity (3.39) is bounded by

(2k + 1) [N (1 -2ℓ)]
H μN t ν N (τ β,0,β) .

We will prove in Lemma 3.14, that there exists a finite constant C independent of N , such that H μN )N is tight. Since by Lemmata 3.11 and Theorem 3.12 each limit point µ of (μ N,ℓ,k t )N is translation invariant and stationary, we can conclude the proof by the lower semi continuity of the relative entropy.

To complete the proof of Lemma 3.13 it remains to show (3.41).

Energy Bound

We prove here a deterministic bound on the total energy inside the system, independent of the realizations of the noise of the dynamics.

Theorem 3 . 1 (

 31 Relative entropy). Under the same assumptions as in Theorem 2.1, for any time t ∈ [0, T ], T < ts,

9 )+

 9 Then it follows by Gronwall's inequality that limN→∞ H N (t) N = 0 which concludes the proof of Theorem 3.1. We first prove the following inequality: ∂s log g N s dr dp (3.10)Proof. By convexity of the function φ(f ) = f log f , since Lebesgue measure is stationary for the dynamics generated by G τ (t)

3

 3 we define DF (a) := ((Df1)(a), (Df2)(a), (Df3)(a)) T , with Dfi(a), i = 1, 2, 3 defined by (2.9). Recall that ũ = (r, p, -E) T , e := E -1 2 p 2 and let us denote by J(ũ) := (p, P (r, e), -pP (r, e)) T = p, λ1(r, e) λ3(r, e) , -p λ1(r, e) λ3(r, e) T (3.12) the flux of (2.11). Then the equation (2.11) can be rewritten as ∂tũ = D J(ũ)∂xũ relation (2.8), λ is solution of the symmetric system ∂t[DΘ(λ)] = ∂x[DΣ(λ)], (3.14) where Σ(λ) = λ • J(DΘ(λ)).

  .15) Let us define the microscopic fluxes: Ji-1,i := (-pi-1, -V ′ (ri), pi-1V ′ (ri)) T i = 1, . . . , N -1, JN,N+1 := (-pN , -τ (t), pN τ (t)) T . (3.16) By the definition of the Liouville operator given by (2.3), L τ (t)

  , ũ(x, s)) -I(z)}dx.(3.29)

0 R 0 R 0 R

 000 N,k,ℓ,b (s)ds, for some uniform constant C, it follows by Gronwall inequality that HN (t) ≤ HN (0)e Ct + t N,k,ℓ,b (s)e C(t-s) ds ≤ e Ct HN (0) + t N,k,ℓ,b (s)ds .

Lemma 3 . 8 ()

 38 Tightness). For each k fixed , the sequence (μ N,ℓ,k t N≥1 of probability measures is tight.Proof. From the definition of μN,ℓ,k dr -k dp -k . . . dr k dp k l dp l ds ≤ C by lemma 3.15, and this implies the tightness.

Proposition 3 . 9 .

 39 p j,j+1 -f (r, p)(3.35) In section 3.3.3 we will prove the following Proposition: Any limit point µ of μN,ℓ,k t , for N → ∞ and then k → ∞, satisfies the following properties:(i) it has finite entropy density: there exists a constant C > 0 such that for all subsetsΛ ⊂ Z H µ|Λ ν |Λ| (τ β,0,β) ≤ C|Λ|,(ii) it is translation invariant: For any local function F and any j ∈ Z, F dµ = (τjF ) dµ where τj denotes the spatial shift by j on the configurations. (iii) it is stationary with respect to the operator G: For any smooth bounded local function F (GF )dµ = 0. With this Proposition, we can apply the ergodic theorem from [3]: Theorem 3.10 (Ergodicity). Any limit point µ of μN,ℓ,k t (dr, dp) is a convex combination of Gibbs measures i.e µ(dr, dp) = i∈Z g λ (ri, pi)dridpi.

3. 3 . 3 9 Lemma 3 . 11 .Lemma 3 . 12 .

 339311312 Proof of Proposition 3.Any limit probability µ of μN,ℓ,k t , for N → ∞ and then k → ∞, is translation invariant. Proof. Let F be a bounded, local function depending on configurations only through -m, . . . , m for some m ≥ 0. Then there exists for each z ∈ Z an integer k such that |m + z| ≤ k. Since ( f N,ℓ,k t )N is tight, it suffices to prove that for each z lim k→∞ lim N→∞ (F -τzF ) f N,ℓ,k t |l|≤k dr l dp l = 0 (3.36) that follows easily from the definition (3.32). Any limit measure µ is stationary in time with respect to the generator G = L + γS, that means for any bounded smooth local function F (r, p) GF dµ = 0. (3.37) Proof. We have to show that for some k ≥ m lim k→∞ lim N→∞ GF f N,ℓ,k t |l|≤k dr l dp l = 0.

t t 0 f

 0 N,k s,j dr -k dp -k . . . dr k dp k ds ν k (τ β,0,β)

  sequence (μ N,ℓ,k t

Lemma 3 .τ

 3 14. If H f N 0 drdp | ν N (τ β,0,β) ≤ C1N for some uniform constant C1 > 0, then for any N ∈ N there exists a constant C2 > 0 such thatH μN t | ν N (τ β,0,β) ≤ C2N,where μN t is defined by(3.40).Proof. Recall from Lemma 3.2, that the relative entropy with respect to Lebesgue measure is nonincreasing in time since the Lebesgue measure is stationary with respect to the generator G τ (t) N(for any t). Therefore, as in the proof of Lemma 3.2 we can writeH f N t dr dp g N (τ β,0,β) dr dp -H f N 0 dr dp g N (τ β,0,β) dr dp ≤ -log g N (τ β,0,β) f N t dr dp + log g N (τ β,0,β) f N 0 dr dp.The last line is then equal to = -(s) f N s pN dr dp ds.Since the last line is equal to the expectation of β N j=1 (ej(t) -ej(0)), by lemma 3.15 it is bounded by CN for some constant C.Hence, by convexity of H(•|•), H μN t | ν N (τ β,0,β) ≤ (C1 + C)N.

Lemma 3 .

 3 15. If the initial configuration satisfyN j=1 ej(0) ≤ CN then there exists a constant C(t) independent of N such that N j=1 ej(t) ≤ C(t)N (3.42) Proof. Define FN (t) = N j=1 ej (t) -τ (t)qN (t) = N j=1 (ej(t) -τ (t)rj(t))Computing the time evolution of this function we haveFN (t) = FN (0) -) = τ (t)qN (t) -τ(0)qN (0) + that implies (3.42).

  r k , pk , r i+k+1 , p i+k+1 ,. . .,rN , pN )

				dr l dp l .
				|i-l|>k
					(3.31)
	and we denote	dμ N,ℓ,k t	:= f N,ℓ,k t	dr l dp l	(3.32)
			|l|≤k


Saman Noorzadeh

Bertrand Rivet

Christian Jutten

BEYOND 2D FOR BRAIN-COMPUTER INTERFACES: TWO 3D EXTENSIONS OF THE P300-SPELLER

Keywords: 3D BCI, P300-Speller Interface, Virtual Keyboard, 3D Stereoscopic Images

This paper, investigates the use of a 3D setting for Brain-Computer Interface (BCI) by implementing the 3D interface for the P300-Speller device. The 3D configurations were implemented using two different approaches which are called Natural 3D and Parallel 2D. The theoretical analysis concerning these two approaches are provided considering the modifications in speed, accuracy, and capacity. The experimental results on subjects who tested the 3D interfaces are then provided to validate the theoretical analysis.

INTRODUCTION

BCI has been commonly the most suitable application to allow communication, or control of external devices only based on brain brain activities [START_REF] Hochberg | Neuronal ensemble control of prosthetic devices by a human with tetraplegia[END_REF][START_REF] Kennedy | Restoration of neural output from a paralyzed patient by a direct brain connection[END_REF]. These activities are recorded non-invasively as the EEG signal [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. P300-Speller is a BCI device which works based on Event-Related Potentials (ERPs) appearing in EEG signal, and has been first developed by Farwell and Donchin [START_REF] Farwell | Talking off the top of your head: toward a mental prosthesis utilizing event-related brain potentials[END_REF] with an interface containing a matrix of symbols, and sequential random flashes on rows and columns as depicted in Figure 1.

If a user keeps a mental count of the number of the stimulus flashes of a specific symbol, the intensifications of the row and the column containing that symbol elicits a positive wave about 300 ms after the stimulus in his/her EEG signal, which is called P300 evoked potentials. Most of the researches on the P300-speller have focused on enhancing the signals by removing the artifacts [START_REF] Serby | An improved P300-based braincomputer interface[END_REF][START_REF] Kaper | BCI 2003-Data Set IIb: Support Vector the P300 Speller Paradigm[END_REF][START_REF] Rivet | Optimal linear spatial filters for evoked potentials: an asymptotic performance analysis[END_REF],

or on classification of P300 from non-P300 evoked potentials [START_REF] Dean | A comparison of classification techniques for the P300 Speller[END_REF]. More recently, the flashing paradigm on rows and columns has been modified, for example motion of rows and columns is used as the stimulus instead of flashes [START_REF] Martens | Overlap and refractory effects in a brain-computer interface speller based on the visual p300 event-related potential[END_REF]. Similarly, others used variations on motion, colors or flash patterns [START_REF] Hong | N200-speller using motion-onset visual response[END_REF][START_REF] Takano | Visual stimuli for the p300 brain-computer interface: A comparison of white/gray and green/blue flicker matrices[END_REF][START_REF] Salvaris | Visual modifications on the p300 speller bci paradigm[END_REF]. Another paradigm, called checkerboard paradigm, is proposed in [START_REF] Townsend | A novel p300-based brain-computer interface stimulus presentation paradigm: Moving beyond rows and columns[END_REF] in which the standard matrix of symbols is virtually superimposed on a checkerboard to avoid the wrong detection of a character and its adjacent one. For almost the same reason, [START_REF] Cecotti | One step beyond rows and columns flashes in the p300 speller: a theoretical description[END_REF] has also proposed a new flashing paradigm, which will be used in our 3D virtual keyboards.

We consider here the use of a 3D virtual keyboard instead of the classical 2D one. Firstly, we intend to investigate the application of 3D settings in BCI devices by verifying its performance on the P300-Speller. This can also increase the user's acceptability of the device, which is a factor that has not been taken into much consider. Secondly, considering the virtual keyboard in 3 dimensions causes changes in flashing paradigm and can increase the device's speed and capacity.

This paper is organized as follow: The methods that are used to implement a 3D virtual keyboard are presented in Section 2. In Section 3 the theoretical analysis of the proposed 3D extensions are presented. The mentioned theories are then evaluated according to the results obtained through the experiments in Section 4. Finally, Section 5 sums up the techniques which were presented in previous sections.

3D EXTENSIONS OF P300 BASED BCI

Here, we consider modifying the interface of the P300-Speller by expanding the 2D keyboard to 3 dimensions. Two approaches of flashing strategies are implemented and compared in this paper: "natural 3D"(section 2.1) and "parallel 2D" (section 2.2).

Natural 3D

The natural generalization of the classical 2D interface is to reorganize the symbols in a three dimensional matrix. This is implemented according to 3D stereoscopic methods [START_REF] Howard | Binocular Vision and Stereopsis[END_REF][START_REF] Wang | Disparity manipulation for stereo images and video[END_REF][START_REF] Lambooij | Visual discomfort in stereoscopic displays: a review[END_REF][START_REF] Vlad | State-of-the-art report on stereoscopic 3D issues with focus on quality[END_REF] in 3 depth layers, and plane flashes are used as stimuli. This configuration is reffered to as natural 3D in which each (a) Flash on a row (b) Flash on a column(c) Flash on a depth symbol is the intersection of a row, a column, and a depth plane (Figure 2).

Parallel 2D

Another flashing approach is considered by treating the different depths of the 3D keyboard as separate 2D keyboards. Using such a strategy (denoted as parallel 2D), the flashes on each layer are the classical row and column flashes. The depth is coded by a delay between the stimuli onsets on the different layers. Figure 3 illustrates this statement with two layers by showing the stimuli timelines: the dotted line corresponds to one of the depth levels and the continuous line corresponds to the other one. As one can see the interlacing flashes lead to independent or parallel layers. It has to be considered that the number of 2D keyboards that can function in parallel is limited, first because of the limited frame rate of the imaging device and secondly, the delay between layers should not be too small for ensuring a good detection. In this way we can avoid the wrong detection of depth to a large amount. The flashing paradigm of [START_REF] Cecotti | One step beyond rows and columns flashes in the p300 speller: a theoretical description[END_REF] is also implemented on the parallel 2D keyboard to avoid the wrong detections between adjacent symbols.

THEORETICAL ANALYSIS AND COMPARISON OF 3D INTERFACES

In this section, theoretical analysis of the proposed 3D interfaces is provided by comparing their accuracy (Section 3.1), their speed (Section 3.2) and their capacity (Section 3.3).

Accuracy

Let's consider the accuracy of the classical 2D interface as a reference. To detect the symbol that the user intends to spell, a permutation of flashes on all rows and columns (so that each row and column is flashed once and only once) is needed and is referred to as repetition. Usually one repetition is not enough for an acceptable accuracy; therefor, there are several repetitions with different permutations. Now, let p(R = r, C = c) denotes the probability that the row r and the column c are jointly detected. Considering the classical 2D interface, the probability of detecting the row can be assumed to be independent from that of the column:

p(R = r, C = c) = p(R = r)p(C = c). ( 1
)
As a consequence, assuming that the detection probability of a correct row and a column is equal and is denoted P ∈ [0, 1], the accuracy of the classical 2D interface is expressed as

Acc 2D = p(R = r t , C = c t ) = P 2 , ( 2
)
where r t (resp. c t ) is the true row (resp. column) containing the selected symbol. In natural 3D the probability of detecting the symbol in row plane r, column plane c, and depth plane d, considering the same assumptions of equation ( 1), is given by:

Acc natural3D = p(R = r t , C = c t , D = d t ) = P 3 . ( 3
)
In parallel 2D, assuming that the depth is coded by the time delay between layers, the probability of detecting a symbol as the intersection of row r and column c in the layer d is independent of d. So the accuracy is:

Acc parallel2D = p(R = r t , C = c t , D = d t ) = P 2 . ( 4
)
According to equations 2, 3, and 4, the detection rates of classical and parallel 2D were estimated to be equal, and it is higher than the estimated accuracy of natural 3D.

Number of Flashes: Speed of the Interface

Considering S as the total number of symbols on the interface, a minimum of N 2D flashes are required in each repetition, to cover all rows and columns in classical 2D:

N 2D = 2 × S 1 2
where ⌈•⌉ is the ceiling function. For the proposed row-column-depth stimulation of natural 3D interface this number would be

N natural3D = 3 × S 1 3
On each layer of the parallel row-column stimulation (parallel 2D) with d layers, the minimum number of flashes would be: 4 compares the minimum number of flashes. As one can see, with more than nine symbols into the interface (S = 9), the classical 2D interface needs more flashes than the natural 3D; and for less than 85 symbols, the parallel 2D interface needs more flashes than the natural 3D. On the contrary, the parallel 2D always requires less flashes than the classical 2D one.

N parallel2D =     2 × S d 1 2     Figure

capacity

A more relevant performance measure to quantize the information transfer rate (ITR) [START_REF] Kronegg | Analysis of bit-rate definitions for braincomputer interfaces[END_REF] is the capacity, which is defined as equation ( 5) for a S-symbol interface.

C = log 2 (S)+Acc log 2 (Acc)+(1-Acc) log 2 1 -Acc S -1 , (5)
Where Acc is the accuracy of the interface. The capacity C is expressed in bits per repetition. Due to the different number of flashes, the duration of a repetition in each approach (classical 2D, natural 3D or parallel 2D) is different from the other. So the comparison is more relevant considering the bit rate B, in bits per second, defined as

B = C × F (6)
or the relative bit rate B ′ , expressed in bits per flash (because of a same duration for every flash), defined as

B ′ = C N (7)
As shown in Figure 5, the bit rates of the natural 3D and parallel 2D interfaces are always larger than the bit rate of the classical 2D interface for a given accuracy. For parallel 2D, this is due to the smaller number of flashes with the same accuracy. This improvement factor is given by N 2D /N parallel2D . However, since the relationships between the accuracy and the marginal probability of good detection are nonlinear, the bit rate of the natural 3D interface is only larger than the bit rate of the classical 2D interface for large marginal probability P . This threshold decreases as the number of symbols increases.

RESULTS AND ANALYSIS

The preliminary experiments with almost a small number of subjects just intend to prove the functionality of 3D settings in BCIs and validate the theoretical results that have been explained. Two comparisons are presented: the classification accuracy and the capacity of the two 3D extensions.

Data, Feature Extraction, and Classification

The data is collected from 16 volunteers between 22 and 34 years old, with normal stereoscopic vision and neurological state. 8 participants have been subjected to the natural 3D interface, and the other 8 ones to the Parallel 2D. They all have done an experiment on classical 2D before 3D, so that the results can be compared. For each approach there is a train and a test session. The interfaces are implemented using OpenGL, in 5 repetitions with ISI equal to 133 milliseconds, and a duty cycle of 50% for the flashes. The interfaces characteristics are summarized in table 1. The EEG is recorded The signal is sampled at a rate of 1200 Hz, and filtered by a bandpass filter in the frequency band between 1 and 12 Hz with a fourth order Butterworth filter. To enhance the signals, the spatial filters are estimated [START_REF] Rivet | P300 speller braincomputer interface: Enhancement of P300 evoked potential by spatial filters[END_REF][START_REF] Rivet | xDAWN algorithm to enhance evoked potentials: application to brain-computer interface[END_REF][START_REF] Rivet | Optimal linear spatial filters for evoked potentials: an asymptotic performance analysis[END_REF] from the training data. The temporally and spatially filtered signal is then categorized into two target and non-target epochs, and is used as the two-class data to train the classifier: Bayesian Linear Discriminant Analysis (BLDA) [START_REF] Hoffmann | An efficient P300-based brain-computer interface for disabled subjects[END_REF]. K-fold cross-validation is adopted: the database is divided into 40 symbols used to the training and 10 symbols for the tests. This partition is per-formed randomly 2000 times for each testing configuration.

Interface # Symbols row× column× depth Classical 2D 36 2 × 2 Natural 3D 27 3 × 3 × 3 Parallel 2D 32 4 × 4 × 2

Accuracy

The classification accuracies of the two 3D extensions are in Figure 6. First of all, one notices that the classification accuracy obviously increases with the number of repetitions. Then, as expected in the theoretical section (Section 3.1), it is worth noting that the classification accuracy of the classical 2D is larger than the classification of the natural 3D extension, while they are the same between the classical 2D and the parallel 2D interfaces. It should be noted that the results concerning classical 2D in different figures are the same in average. And the small difference is due to the fact that they are tested on different subjects.

To verify the theoretical assumptions on the marginal probabilities, P, Figure 7 is depicted. Figure 7.a proves the equal marginal probability hypothesis for row and column of the natural 3D and they are similar to the marginal probability of good detection with the classical 2D interface; however, this is not true for marginal probability of good detection of depth, which is smaller than that of row Regarding Figure 7.b, for parallel 2D the two assumptions regarding the marginal probabilities are well validated: first, these probabilities are similar on rows and columns, and also similar to that of classical 2D. Second, the marginal probability of detecting the depth is equal to one after at least 3 repetitions. This proves that the depth is well coded by the flashing paradigm.

Capacity

Figure 8 compares the capacity of 3D approaches with the classical 2D. For a better understanding, the figures show directly the difference of new interfaces' capacities with the classical 2D's. It is depicted that the capacity of the parallel 2D interface is higher than that of classical 2D which is confirmed by the theoretical analysis (Figure 5). The variance of this result can be interpreted as the visual effect while looking at 3D images [START_REF] Shibata | Stereoscopic 3-D display with optical correction for the reduction of the discrepancy between accommodation and convergence[END_REF]. However, Figure 8.a shows that the classical 2D has higher capacity comparing the natural 3D. As it is already proved that the natural 3D has less accuracy comparing classical 2D (Figure 6.a), it is natural that the capacity does not follow Figure 6.d which assumes the same accuracy for all approaches.

CONCLUSION

The paper presented two 3D extensions of the classical 2D interface represented in [START_REF] Farwell | Talking off the top of your head: toward a mental prosthesis utilizing event-related brain potentials[END_REF]. The methods are called Natural 3D and Parallel 2D. Experiments on 16 subjects showed that the proposed interfaces have several advantages over the classical 2D interface. One of these advantages is the ergonomics of the device. according to a survey from the subjects, we found that 75% of the subjects preferred 3D keyboards to 2D because of its comfort and higher ergonomic features.

The next advantage is the speed which is proven to be higher in both proposed interfaces comparing classical 2D, since they need smaller number of flashes.

As the modification of the P300-Speller is done just on the interface part, we do not expect any accuracy improvement in the theoretical part, and also this is confirmed in the experiments. This accordance of theoretical results with the experimental ones is missing only for the marginal probability of detecting the depth which was not equal to detection of other dimensions in natural 3D.

Finally, with capacity of the faces, parallel 2D approach performance than the other ones.

The validation of theoretical results along with the increase of speed and accuracy mostly in the parallel 2D interface, can prove the functionality of the 3D settings not only in the p300-Speller device, but also in other BCI technologies which require more ergonomic features, as in virtual worlds [START_REF] Bayliss | Use of the evoked potential P3 component for control in a virtual apartment[END_REF].

Figure 1 :

 1 Figure 1: The classical 2D interface; the flashes of (a) and (b) indicate the letter 'T'.

Figure 2 :

 2 Figure 2: Flashes corresponding to the planes of Natural 3D interface with 3 rows, 3 columns, and 3 depths.

Figure 3 :

 3 Figure 3: Parallel flashes on 2 depth levels in Parallel 2D interface.

Figure 4 :

 4 Figure 4: Comparison of the interfaces.Figure 4.a: relative speed F of the interfaces defined as N × T where T is the ISI.

Figure 4 .

 4 Figure 4: Comparison of the interfaces.Figure 4.a: relative speed F of the interfaces defined as N × T where T is the ISI.

Figure 5 :

 5 Figure 5: Relative bit rate per flash B ′ of equation (7) against marginal probability of good detection P (Figures 5.a, 5.b) and against the accuracy (Figures 5.c, 5.d). In Figure 5.d, B ′ natural3D = B ′ parallel2D since N natural3D = N parallel2D .

Figure 6 :

 6 Figure 6: Accuracy against the number of repetitions. The median accuracies are plotted, and the error bars extend from the 10% quantile to the 90% quantile.

Figure 7 :

 7 Figure 7: Marginal probability (P ) against the number of repetitions.

Figure 8 :

 8 Figure 8: Comparison of relative bitrates. B ′ M ax is the maximum bitrate assuming a perfect accuracy (i.e. Acc = 100%).

Table 1 :

 1 Implemented Interfaces via 16 active electrodes with g.USBamp device from g.tec.


Hubert Cecotti

Bertrand Rivet

PERFORMANCE ESTIMATION OF A COOPERATIVE BRAIN-COMPUTER INTERFACE BASED ON THE DETECTION OF STEADY-STATE VISUAL EVOKED POTENTIALS

Keywords: Brain-Computer Interface, Steady-State Visual Evoked Potentials, Cooperative BCI

To be better suited for the expectations of healthy people, new brain-computer interface (BCI) paradigms should be proposed. To tackle this problem, we investigate the emerging field of cooperative BCIs, which involves several users in a single BCI system. Because combining trials over time improves performance, combining trials across subjects can significantly improve performance compared to a single user. However, cooperative BCIs can only be used in particular settings, and new paradigms must be proposed to efficiently use this approach. To show the advantages of a cooperative BCI based on steady-state visual evoked potentials (SSVEP), we evaluate and compare the performance of combining decisions across subjects, and over time. By considering a reliable brain response such as SSVEP responses, this study represents a first evolution for the combination, and the choice of the combination method for creating a cooperative SSVEP based BCI. The results suggest that six people would be enough to obtain a perfect accuracy within one second of EEG signal.

INTRODUCTION

Pioneer works such as the P300 speller [START_REF] Farwell | Talking off the top of your head: toward a mental prosthesis utilizing event-related brain potentials[END_REF] have opened the research field of brain-computer interface (BCI). In the last decades, BCIs have always been used as a way to interact with different devices in relation to the subject's will. The goal is typically to send commands to a computer, e.g., spelling words [START_REF] Cecotti | Spelling with non-invasive brain-computer interfaces -current and future trends[END_REF], and in assistive technologies [START_REF] Millán | Combining braincomputer interfaces and assistive technologies: state-of-the-art and challenges[END_REF], where it is possible to move a wheelchair or robot. Due to the low signal-tonoise ratio (SNR) in the electroencephalogram (EEG) signal, many strategies have been employed to increase the SNR by enhancing the signal through filtering or by combining several trials over time [START_REF] Townsend | A novel P300-based brain-computer interface stimulus presentation paradigm: Moving beyond rows and columns[END_REF]. Whereas combining EEG signals over time has been used since the first P300 speller, where the stimuli appearing on the keys of the virtual keyboard are repeated several times, combining EEG signals across subjects has not been efficiently used in BCI until recently for decreasing reaction time [START_REF] Yuan | Study on an online collaborative BCI to accelerate response to visual targets[END_REF] and for 2-D pointer control with two users [START_REF] Poli | Some steps towards realtime control of a space-craft simulator via a brain-computer interface[END_REF]. In this type of BCI, several users are implicated for enabling a command. In this study, we propose to evaluate the performance and the number of subjects that are needed for a BCI combining several users, and based on the detection of steady state visual evoked potentials, which is a robust brain response [START_REF] Lyskov | Steady-state visual evoked potential to computer monitor flicker[END_REF][START_REF] Vialatte | Steady-state visually evoked potentials: focus on essential paradigms and future perspectives[END_REF][START_REF] Wang | Measuring steadystate visual evoked potentials from non-hair-bearing areas[END_REF].

Combination strategies

The literature of multi-classifier systems offers a large choice of methods for combining the signals and/or the decisions from the different subjects [START_REF] Ludmila | Combining Pattern Classifiers: Methods and Algorithms[END_REF][START_REF] Fumera | Performance analysis and comparison of linear combiners for classifier fusion[END_REF]. Increasing the number of subjects can have a different impact on the performance. By adding more subjects, it is possible to add subjects who have had bad performances: while their single performance does not allow them to use a BCI for a single user, their brain features can still contribute to the global performance. The choice of the combination strategy may depend on how possible it is to find subjects that can complement each other, particularly for adaptive combination methods. In this study, we limit the applications to strategies for binary classification.

Let O i denote the output value of a binary classifier for subject i at a particular trial. O i represents the confidence value of assigning the input to one of the two classes. Six classical combination scores over N sub subjects are considered: the sum of the classifier outputs

(O sum = N sub i=1 O i ), the weighted sum (O w-sum = w 0 + N sub i=1 w i • O i ), the product (O prod = N sub i=1 O i ), the maximum score (O max = max 1≤i≤N sub O i ), the minimum score (O min = min 1≤i≤N sub O i )
, the majority voting (more than half of the outputs agree for the decision), and the consensus vote (the decision should be the same for all the outputs).

The maximum score (O max ) and the minimum score (O min ) are related to the notions of OR and AND in logic, respectively. The maximum score bases its decision with the most relevant subject. On the contrary, the minimum score bases its decision with the less relevant subject (with the lowest individual score): all the subjects have to be relevant to provide a high combined score. In addition, the sum (O sum ) and weighted sum (O w-sum ) scores can be seen as smooth or weighted OR, whereas the product score (O prod ) can be considered as a smooth or weighted AND.

Performance evaluation

To show the relevance and the type of performance that can be expected by combining the decision from several individuals, we report the Area Under the ROC curve (AUC) [START_REF] Fawcett | An introduction to ROC analysis[END_REF] and the ITR [START_REF] Shannon | The mathematical theory of communication[END_REF] in bits per minute (bpm) defined by IT R = 60 T • ϑ where

ϑ = log 2 (N out ) + P log 2 (P ) + (1 -P )log 2 ( 1-P Nout-1 ) (1
) and P being the probability of the good detection, i.e. the accuracy, N out being the number of possible different outputs, and T being the time in seconds of recorded EEG signal that is required to take the decision among the N out outputs.

EXPERIMENTAL PROTOCOL

SSVEP responses are not time locked to an event; an SSVEP response is evoked when the subject pays attention to a flickering visual stimulus. The efficiency of the classification of an SSVEP response depends on the time that is required to detect a response. A challenge is to reduce the time segment for the SSVEP response detection because it allows to decrease the required time for enabling a command. To qualify the effect of combining brain responses from several subjects, we evaluated the effect of combining several subjects for improving SSVEP response detection with a time segment of 1s.

The experiment consisted of a visual task where 10 healthy subjects (age=27.2 ± 2.4 years old, two females) were paying attention to a series of different flickering lights (6.66Hz, 7.50Hz, 8.57Hz, 10.00Hz and 12.00Hz) on a computer screen (diagonal size=15.4 inches, vertical refresh rate=60Hz, luminance= 180.0cd/m 2 , with an estimated contrast of 280 : 1). Subjects were sitting in a comfortable chair at about 60cm from the computer screen, in a non shielded room. Each stimulus had a luminance of about 0.46cd. The experiments were carried out sequentially. The order of the flickering boxes on which the subject had to pay attention was identical across subjects. In this study, we focused on the detection of a single flickering light that was at 7.50Hz. The EEG was recorded with a g.USBamp EEG amplifier from g.tec with a sampling rate of 128Hz. The electrodes were placed on AF Z for ground, C Z for the reference and P O 3 , P O 4 , P Z , O 9 , O 10 , and O Z for the input electrodes.

SIGNAL PROCESSING AND CLASSIFICATION

An analog bandpass filter between 2 and 30Hz, and a notch filter around 50Hz (main frequency in Europe) were applied directly inside the amplifier during the EEG acquisition. For the classification, we consider the responses corresponding to the following frequencies: 7.50Hz for the target class, and 6.66Hz, 8.57Hz, 10.00Hz and 12.00Hz, for the non-target class. The frequency corresponding to the target class also corresponds to the frequency of the flickering light. The frequencies for the non-target classes are used as frequencies that should not be detected when the subject is looking at the flickering light corresponding to the target. The following method has been used for the detection of the SSVEP responses [START_REF] Cecotti | Effect of the visual signal structure on steady-state visual evoked potentials detection[END_REF][START_REF] Friman | Multiple channel detection of steady-state visual evoked potentials for brain-computer interfaces[END_REF]. We consider a visual stimulation flickering at f Hz. The signal y i (t) as the voltage between the electrode i and a reference electrode at a time t is considered as:

y i (t) = N h k=1 a i,k sin(2πkft + Φ i,k ) + b i,t (2)
where N h is the number of considered harmonics. The signal is divided into two parts: the SSVEP response and the remaining EEG activity, which is considered as noise. The first part corresponds to the evoked SSVEP response signal, which is composed of a number of sinusoids with frequencies in relation to the stimulus frequency and a number of N h harmonic frequencies. Each sinusoid is defined by its amplitude and phase: a i,k and Φ i,k . b i,t corresponds to the background EEG activity. The detection of an SSVEP response on an EEG signal requires a time segment of N t samples of EEG signal, with a sampling frequency of F s Hz:

y i = X f a i + B i (3)
where y i = [y i (1), . . . , y i (N t )] T contains the EEG signal for the i th electrode in one time segment. The SSVEP model of the frequency f, X f , is contained in a matrix N t ×2N h defined by

X f (t, 2k -1) = sin(2πkft) (4) X f (t, 2k) = cos(2πkft) (5)
with 1 ≤ k ≤ N h . The vector a i of size 2N h contains the amplitudes. For N y electrodes, the signal is defined as:

Y = X f A f + B (6)
where Y = [y 1 , . . . , y Ny ] contained the sampled EEG signals from all the electrodes. A f contains all the amplitudes for all the expected sinusoids for every electrode signal related the the expected frequency to detect. Spatial filters are used to enhance the SSVEP response in the signal. A spatial filter is represented by a linear combination of the signals measured by different electrodes. We denote by s, a linear combination of y i , the EEG after a spatial filter:

s = Ny i=1 w i y i = Y w (7)
where w i is the weight for the i th electrode. Several components can be created by using several sets of weights w.

We note N s as the number of channels. We first estimate the background activity by removing the potential SSVEP components from the signal. It is achieved by projecting the signal onto the orthogonal complement of the SSVEP model matrix (X).

Yf = Y -X f (X T f X f ) -1 X T f Y (8)
Spatial filters Ŵf that maximize the Signal-to-Noise Ratio are obtained though determining the generalized Rayleigh quotient that maximizes the following expression:

Ŵf = argmax W T r(W T Y T Y W ) T r(W T Yf T Yf W ) (9)
We denote by Ŷf = X T f Y Ŵf the signal after spatial filtering. The power of the expected frequencies and their harmonics are calculated for the N s components. For each frequency, the evaluation of the SSVEP response is defined by:

R(f) = 1 N s • N h Ns i=1 N h k=1 Ŷf (i, 2k -1) 2 + Ŷf (i, 2k) 2 (10)
In the experiment, N s = 6 and N h = 3, the detection of an SSVEP response is performed by selecting the frequency with the maximum associated value R(f). The database contained 713 trials for each subject (238 for target: when the subjects paid attention to the flickering light at 7.50Hz, 475 for non-target: when the subjects paid attention to other flickering lights).

RESULTS

The performances for combining classification results across subjects or across trials are estimated by the AUC, the classification accuracy and the ITR. They are depicted in Figure 1, which represents the mean AUC, accuracy and ITR across all the possible combinations of n subjects (Fig. 1(a), 1(c), 1(e)), and across all the possible combinations of n trials from the same subject (Fig. 1(b),1(d),1(f)). For single-event detection, the mean AUC across subjects is 0.920 ± 0.079. The results show that combining trials across subjects can improve the AUC. With the sum combined score (O sum ), the AUC increases from about 0.92 with a single subject to a perfect detection with eight subjects. Yet, the AUC achieved with the product and minimum scores do not monotonically increase with the number of subjects: there is a maximum AUC with about four subjects. These combined scores require that all the individual scores are relatively high to achieve a good detection while increasing the number of subjects reduces this probability.

Performance curves present the same behavior depending on the number of combined scores between the evolution of the AUC as a function of the number of subjects (Fig. 1(a)), and as a function of the number of trials (Fig. 1(b)). For the combination of trials from the same subject and the combination of trials from different subjects, the accuracy starts at about 85.41 ± 7.6% with a single-trial, and increases sublinearly until around 99.7 ± 0.27% with ten trials and the methods O sum and O w-sum . Figure 1(e) shows the theoretical ITR as a function of the number of subjects that are involved in the decision with N out = 5. The best performance is obtained with the sum method with an ITR of 137.93±1.36 bpm. The ITR was estimated based on the choice between five commands and a time of 1s for the selection of a command (T = 1s). The same way that the AUC increases with the number of involved subjects, the ITR also increases for most of the combination methods. The increase of the AUC is directly reflected on the ITR (Fig. 1(e)), while it is not the case when combining trials over time (Fig. 1(f)). The evolution of the performance indicates a continuous increase of performance with the addition of new subjects. With only two subjects, it is possible to obtain a performance that is reliable enough for single-event detection.

DISCUSSION AND CONCLUSION

Cooperative BCIs involve the creation of new paradigms to be used online. SSVEP-BCIs are known as being fast and reliable compared to other BCIs [START_REF] Gao | A BCI based environmental controller for the motiondisabled[END_REF][START_REF] Bin | An online multi-channel SSVEP-based braincomputer interface using a canonical correlation analysis method[END_REF][START_REF] Cecotti | A self-paced and calibration-less SSVEP based brain-computer interface speller[END_REF]. To further improve the performance, we have shown that it was possible to significantly increase the ITR with simple combination methods by considering the detection of SSVEP responses occurring on a group of six subjects. With cooperative BCIs, it would be more efficient to have several users with relatively cheap systems with few sensors, than to have a single user with a very expensive system, e.g., with 32 channels. A cooperative BCI system should ideally require six persons to provide optimal results in the case of SSVEP based BCI. Therefore, each subject should wear an EEG cap and have a personal amplifier. By considering the EEG signals from different users, one issue could be the synchronization the different EEG signals coming from each user. However, with a system based on the detection of SSVEP response based only on the frequency of the stimuli, a jitter would not be an obstacle if the phase is not used [START_REF] Garcia | Detection of high-frequency steady state visual evoked potentials using phase rectified reconstruction[END_REF].

We have shown that combining trials across subjects can be advantageously used in particular BCI applications. Cooperative BCIs represent a fundamental new approach compared to current BCI because the user does not interact independently, but he interacts with other people who share the same goal. Moreover, the performance does not depend on a single user, but on the group of users. This promising approach could be successfully implemented in cooperative video games or in other applications involving several people.

Fig. 1 .

 1 Fig.1. SSVEP response detection results based on the AUC, the accuracy (ACC), and ITR. Figures1(a) and 1(b): AUC for single-event detection as a function of the number of subjects and AUC for a single subject as a function of the number of trials, respectively. Figures1(c) and 1(d): Accuracy for single-event detection as a function of the number of subjects and accuracy for a single subject as a function of the number of trials, respectively. Figures1(e) and 1(f): theoretical ITR as a function of the number of subjects and the number of trials, respectively.