Benoît Perthame
email: benoit.perthame@sorbonne-universite.fr

Delphine Salort
email: delphine.salort@sorbonne-universite.fr

DERIVATION OF A VOLTAGE DENSITY EQUATION FROM A VOLTAGE-CONDUCTANCE KINETIC MODEL FOR NETWORKS OF INTEGRATE-AND-FIRE NEURONS *

Keywords: Integrate-and-fire neurons, Voltage-conductance Vlasov equation, Neural networks, Slow-fast dynamics, Asymptotic analysis, Fokker-Planck kinetic equation, AMS subject classifications. 35B65, 35Q84, 62M45, 82C32, 92B20

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Derivation of a voltage density equation from a voltage-conductance kinetic model for networks of integrate-and-fire neurons

Benoît Perthame, Delphine Salort

To cite this version:

Benoît Perthame, Delphine Salort. Derivation of a voltage density equation from a voltageconductance kinetic model for networks of integrate-and-fire neurons. Communications in Mathematical Sciences, 2019, 17 [START_REF] María | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF], pp.1193-1211. [START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF].4310/CMS.2019.v17.n5.a2. hal-02431772

Introduction

The voltage-conductance density systems for neural networks are nonlinear (2+1) dimensional kinetic Fokker-Plank equation which are established in [START_REF] Cai | An effective kinetic representation of fluctuationdriven neuronal networks with application to simple and complex cells in visual cortex[END_REF]25]. Based on neuro-physical concepts adapted in particular to the visual cortex, they describe the probability density p ε (v,g,t) to find neurons at time t with a membrane potential v ∈ (V R ,V E ) and a conductance g > 0. Here, V E denotes the excitatory reversal potential and the reset potential satisfies V R ≤ V L with V L the leak potential. There are several variants of the equation depending on the physical interpretation of variable, see, e.g., [START_REF] Fourcaud | Dynamics of the firing probability of noisy integrate-and-fire neurons[END_REF][START_REF] Gerstner | Spiking neuron models: Single neurons, populations, plasticity[END_REF]. The mathematical structure of these systems is rather complex and has attracted the interest of mathematicians [START_REF] Bossy | Clarification and complement to "Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons[END_REF][START_REF] Cáceres | A numerical solver for a nonlinear fokker-planck equation representation of neuronal network dynamics[END_REF][START_REF] Perthame | On a voltage-conductance kinetic system for integrate & fire neural networks[END_REF]. In particular difficulties related to boundary conditions and partial parabolicity make the system rather uneasy to handle and connect it to present interest about hypoellipticity in kinetic equations, see [13,[START_REF] Herda | Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit[END_REF]27] and the references therein. A rather striking finding in [START_REF] Cáceres | A numerical solver for a nonlinear fokker-planck equation representation of neuronal network dynamics[END_REF] is the numerical observation of periodic solutions describing spontaneous activity of the network, a phenomena which is common to other neural assembly models (see [START_REF] Mischler | Relaxation in time elapsed neuron network models in the weak connectivity regime[END_REF][START_REF] Khashayar Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Khashayar Pakdaman | Relaxation and self-sustained foscillations in the time elapsed neuron network model[END_REF][START_REF] Pham | Activity in sparsely connected excitatory neural networks: effect of connectivity[END_REF]).

Here, we deal with the probability density p ε (v,g,t) to find a neuron with potential v, conductance g at time t, which is assumed to be driven by the equation

∂ ∂t p ε + ∂ ∂v g L (V L -v) + g(V E -v) p ε + 1 ε ∂ ∂g [(G eq (v,bN ε (t)) -g)p ε ] -a ε ∂ 2 ∂g 2 p ε + φ F (v)p ε = 0, t ≥ 0, V R < v < V E , g ≥ 0, (1.1)
with the no-flux boundary conditions (G eq (v,bN ε (t)) -g)p ε (v,g,t) -a ∂ ∂g p ε (v,g,t) = 0 for g = 0,

[g L (V L -V R ) + g(V E -V R )]p ε (V R ,g,t) = N ε (g,t) and p ε (V E ,g,t) = 0, (

and with an initial data that satisfies p 0 (v,g) ≥ 0,

V E V R ∞ 0 p 0 (v,g)dv dg = 1. (1.3)
A neuron spikes with a rate φ F (v), typically very large for v larger than a firing potential, and we assume that its membrane potential is instantaneously set at the reset potential V R . For the sake of analytical tractability, we have assumed that the firing potential is randomly distributed, with the firing rate φ F (v), while it is often prescribed at a deterministic value V F of the firing potential. Therefore, at each time, the firing rate (activity of the network) for neurons with conductance g and respectively the total firing rate of the network are defined as

N ε (g,t) := V E V R φ F (v)p ε (v,g,t)dv ≥ 0, N ε (t) := +∞ 0 N ε (g,t)dg. (1.4)
Those definitions and boundary conditions, when integrating the equation (1.1), imply the conservation property

V E V R ∞ 0 p ε (v,g,t)dv dg = 1, (1.5)
which is in accordance with the interpretation that the solution is the probability density of neurons with potential v and conductance g at time t.

For the ease of use, we summarize the parameter interpretation, according to [START_REF] Cai | An effective kinetic representation of fluctuationdriven neuronal networks with application to simple and complex cells in visual cortex[END_REF]25],

• V E is the excitatory reversal potential, • Firing occurs with a rate φ F (v) ≥ 0, φ F (v) > 0, • Reset is at V R , • V L is the leak potential, V R < V L < V E , • g L > 0 denotes the leak conductance, • G eq (v,
•) ≥ 0 is the conductance equilibrium (when ignoring noise),

• a represents the intensity of the synaptic noise, • ε > 0 denotes the time decay constant of the excitatory conductance, • b ≥ 0 denotes the synaptic strength of network excitatory coupling.

Concerning parameter range, in [START_REF] Cai | Kinetic theory for neuronal network dynamics[END_REF], see Eq. (3.2a), the authors choose V R = V L .

Here we consider the more general case V R ≤ V L . Also, to establish (1.1), [START_REF] Cai | An effective kinetic representation of fluctuationdriven neuronal networks with application to simple and complex cells in visual cortex[END_REF]25,[START_REF] Aaditya | Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train[END_REF] assume that the value ε is small enough. Following [START_REF] Cai | An effective kinetic representation of fluctuationdriven neuronal networks with application to simple and complex cells in visual cortex[END_REF][START_REF] Cai | Kinetic theory for neuronal network dynamics[END_REF], this motivates to consider the limit ε → 0 of Equation (1.1), which formally leads to a reduction of dimension with a (1+1) dimensional equation easier to tackle. More precisely, we are going to show that it can be described by the following voltage only neural network model for integrate&fire neurons (see [START_REF] María | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF][START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF][11][12]14] and references therein)

for t ≥ 0, V R < v < V E ,          ∂ ∂t n(v,t) + ∂ ∂v G(v,bN (t)) V (bN (t)) -v n(v,t) + φ F (v)n(v,t) = 0, N (t) = V E V R φ F (v)n(v,t)dv, G(V R ,bN (t)) V (bN (t)) -V R n(V R ,t) = N (t) and n(V E ,t) = 0. (1.6)
Our purpose is then, on the one hand, to derive this system in the slow-fast limit ε → 0, to explain how noise comes in the expression of the drift, together with the property

V L < V (bN (t)) ≤ V E , G(v,bN (t)) > 0, (1.7)
which explains the possibility to state the boundary conditions in (1.6). On the other hand, we aim at studying the qualitative properties of Equation (1.6), in order to formally obtain, with this more simplified equation, the main properties of Equation (1.1). The situation at hand is similar to the derivation of macroscopic (fluid) models from kinetic (Boltzmann) equation in the phase space (position, velocity). This explains our terminology 'kinetic equation' for the voltage-conductance model.

Assumptions on φ and G and initial data. In the rest of the paper, we make the following assumptions. We assume that the firing rate satisfies

φ F , φ F ∈ L ∞ , φ F ≥ 0. (1.8)
We also assume that G(v,A) ≥ 0 and that there is a smooth increasing function Ḡ ≥ 0 such that, sup

V R ≤v≤V E [ G(v,A) + | ∂G(v,A) ∂v | ] ≤ Ḡ(A), ∀A ≥ 0. (1.9)
For example, in [START_REF] Cai | An effective kinetic representation of fluctuationdriven neuronal networks with application to simple and complex cells in visual cortex[END_REF]25], we find the choice G(v,bN ) = bN . Finally, for the initial data, we assume, for ε ≤ 1 and k ≥ 0

V E V R +∞ g=0 g k p 0 (v,g)dvdg ≤ Q (k),0 < ∞. (1.10)
The microscopic model.

For the sake of completeness, we present the microscopic dynamics which generates the voltage-conductance kinetic equation in the limit of a large number of neurons. More on this model and its limit can be found in [START_REF] Bossy | Clarification and complement to "Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons[END_REF][START_REF] Cáceres | A numerical solver for a nonlinear fokker-planck equation representation of neuronal network dynamics[END_REF][START_REF] Aaditya | Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train[END_REF]. The following dynamical system governs the temporal evolution of the membrane potential V i (t) and the excitatory conductance G i (t) ≥ 0 of the i-th neuron in a pool of N excitatory neurons,

τ dV i dt = -g L (V i -V L ) -G i (V i -V E ), σ E dG I (t) dt = -G i (t) + µ f E δ(t -t µ i ) + S EE N E N j=1 µ p Eµ j δ(t -t µ j ).
With the rate φ F (V i ), the i-th neuron will fire at some random times t µ i and the potential is reset to V R , then it spikes and sends instantaneously a current described by its strength f E (self excitation) and S EE the strength of excitatory coupling with N E a normalizing constant. Noise (as a Wigner process) or random jumps can be added to also include the second derivative in g.

Outline.

The rest of the paper is organized as follows. Section 2 is devoted to the slow-fast limit study of Equation (1.1) to Equation (1.6). To this, we preliminary explain how to obtain the formal derivation. We then give uniforms estimates on the moments of the solution and of the firing rate with respect to ε, leading to prove rigorously the weak convergence of Equation (1.1) to Equation (1.6) when ε goes to 0. We finish this section by the construction of super-solutions leading to uniform L ∞ bounds for the solutions of (1.1) with respect to ε and for the solutions of Equation (1.6). In Section 3 we give the main qualitative properties of Equation (1.6) with a slightly simplified drift. To this, we first study the steady states of this equation as well as the asymptotic long time convergence to those steady states when the strength of connectivity parameter b is small enough. The method of proof is based on entropy methods. We finally consider the case with stronger interconnections where instabilities is proved for the linearized problem and we perform numerical simulations in order to illustrate the emergence of oscillatory solutions. We finish the paper with conclusions and perspectives.

From conductance-voltage to voltage only model

Departing with a formal derivation, we first explain, how, Equation (1.1) leads to Equation (1.6) when ε goes to 0. We then prove uniform estimates on the total firing rate and on the moments on the solution with respect to ε. This leads us to prove rigorously weak convergence of solutions of Equation (1.1) to Equation (1.6) (see Theorem 2.1). Finally, we prove uniform estimates in L ∞ when V R < V L by the construction of supersolutions in Theorem 2.2.

Formal derivation

The relation between Equations (1.6) and (1.1) can be simply observed setting

n ε (v,t) = ∞ 0 p ε (v,g,t)dg.
Then, integration in g of (1.1), using the no-flux boundary condition in g, gives

         ∂ ∂t n ε + ∂ ∂v G ε (v,t) V ε (t) -v n ε + φ F (v)n ε = 0, t ≥ 0, V R < v < V E , N ε (t) = V E V R φ F (v)n ε (v,t)dv, G ε (V R ,t) V ε (t) -V R n ε (V R ,t) = N (t) and n ε (V E ,t) = 0, (2.1)
with the bulk conductance and voltage

         g L ≤ G ε (v,t) = ∞ 0 g L + g p ε (v,g,t) n ε (v,t) dg, V L < V ε (v,t) = 1 Gε(v,t) ∞ 0 g L V L + gV E p ε (v,g,t) n ε (v,t) dg ≤ V E . (2.2)
In order to close this formula, we need to identify the first moment in g of the distribution p ε (v,g,t) in terms of n ε .

To do so, we consider the limit p of p ε and it formally solves

∂ ∂g [(G eq (v,bN (t)) -g)p] -a ∂ 2 ∂g 2 p = 0,
With the no-flux boundary condition, we find

(G eq (v,bN (t)) -g)p -a ∂p ∂g = 0, thefore p = n(v,t)P(v,g,bN (t)),
with n solution of Equation (1.6) and

P(v,g,A) = 1 Z(v,a,A) exp -(Geq (v,A)-g) 2 2a , Z(v,a,A) = ∞ 0 exp -(Geq (v,A)-g) 2 2a
dg. (2.3) This expression allows us to compute the limiting flux in Equation (1.6) as stated Theorem 2.1 below.

Notice that Z(v,a,A) is controlled from above and below as

1 2 √ 2πa = ∞ Geq(v,A) exp- (G eq (v,A) -g) 2 2a dg ≤ Z(v,a,A) ≤ √ 2πa.
The effect of noise at the synaptic level is not a diffusion in v, as it is assumed in the standard voltage-based population model for Integrate and Fire neurons, see for instance [START_REF] Brunel | Fast global oscillations in networks of integrate-and-fire neurons with long firing rates[END_REF]. Noise rises a change of the excitatory bulk conductance G ε (v,t). It can be compared to the noiseless case a → 0 which gives

G ε (v,0,bN ε (t)) = G eq (v,bN ε (t)).
In mathematical terms, the analysis of the limit a → 0, i.e., the case without noise, yields difficulties similar to those of friction dominant particle flows, see [19] for instance.

Estimates on the firing rate/moments on the solution and rigorous limit

We now prove rigorously the above formal derivation. The result is stated in the following Theorem Theorem 2.1 (Slow-fast limit result). We assume (1.8)-(1.10). Then, for all k ≥ 1, there exists a constant C(k) > 0 and a constant C > 0 such that, for all ε > 0 and t ≥ 0, the following estimates hold

+∞ 0 g k p ε (v,g,t)dvdg ≤ C(k) and N ε L ∞ (R + ) + N ε L ∞ (R + ) ≤ C. (2.4)
Hence, in the weak topology of bounded measures,

p ε n(v,t)P(v,g,bN (t)) > 0, ∞ 0 P(v,g,bN (t))dg = 1,
where the smooth function P(v,g,bN (t)) is determined by (2.3) and where n(v,t) satisfies Equation (1.6) with the initial data n 0 (v) = +∞ 0 p 0 (v,g)dg and

G(v,A) = g L + +∞ 0 gP(v,g,A)dg > g L , V L < V (v,A) = g L V L + V E +∞ 0 gP(v,g,A)dg G(v,A) < V E .
This theorem states a general weak convergence result. It can be strengthen, with the expense on stronger assumptions as stated afterwards, see Theorem 2.2 in the next subsection.

Proof. Proof of Theorem 2.1.

Moments estimates.

Let us first prove the first inequality of (2.4). We set

Q (k) ε (t) := V E V R +∞ 0 g k p ε (v,g,t)dvdg.
Multiplying Equation (1.1) by g k and integrating, we compute for k ≥ 2

1 k d dt Q (k) ε (t) = 1 ε V E V R +∞ 0 g k-1 [G eq (v,bN ε (t)) -g]p ε dvdg - a ε V E V R +∞ 0 g k-1 ∂p ε ∂g dvdg.
Therefore we find

dQ (k) ε (t) dt + k ε Q (k) ε (t) = V E V R +∞ 0 k ε g k-1 G eq (v,bN ε (t)) + k(k -1) a ε g k-2 p ε dvdg.
As p ε is a density probability, then

N ε L ∞ (R + ) ≤ φ F L ∞ .
Using assumption (1.9), we deduce that there exists a constant C such that

V E V R +∞ 0 g k-1 G eq (v,bN ε (t))p ε dvdg ≤ C V E V R +∞ 0 g k-1 p ε dvdg.
Using again that p ε is a density probability, and splitting the integral in g in two parts : from 0 to µ and from µ to +∞, we deduce that for all µ > 0 large enough, there exists a constant C(µ) such that

V E V R +∞ 0 (g k-1 + g k-2 )p ε dvdg ≤ C(µ) + 1 µ Q (k) ε (t).
Consequently, we have for all µ > 0 large enough

d dt Q (k) ε (t) + k ε Q (k) ε (t) ≤ 1 ε C(k,µ) + 1 µ Q (k) ε (t) .
Taking µ large enough and using Gronwall inequality, we deduce the first part of estimate (2.4).

Uniform estimates on the firing rate.

The bound on N ε (t) is easy to obtain. Because the total mass of p ε is 1, we conclude from its definition that

N ε (t) ≤ φ F L ∞ .
Next, we prove the Lipschitz bound. We multiply equation (1.1) by φ F and integrate in (v,g), we find

d dt N ε (t) ≤ φ F (v) g L (V L -v) + g(V E -v) p ε (v,g,t)dgdv ≤ φ F (v) L ∞ V E gp ε (v,g,t)dgdv
and we conclude using that the moments are uniformly bounded with respect to ε, thanks to (2.4).

Weak convergence.

Let us first deal with the term G eq (v,bN ε (t)). With the second part of estimate (2.4), we deduce using the Ascoli theorem that, up to a subsequence, there exists a Lipschitz function N such that for all T > 0,

lim ε→0 N ε -N L ∞ (0,T ) = 0.
As G is assumed to be regular, we deduce that, up to a subsequence, for all T > 0,

lim ε→0 G(v,bN ε ) -G(v,bN ) L ∞ ((V R ,V E )×(0,T )) = 0.
Let us now study the convergence of p ε . As p ε is a density measure and as its moments is uniformly bounded with respect to ε, we deduce that up to a subsequence, there exists a bounded measure function p with total mass 1 and finite moments such that p ε p gp ε gp.

Integrating Equation (2.1) with respect to the variable g, we find that

n(v,t) := +∞ 0 p(v,g,t)dv
is solution of the equation

∂ t n(v,t) + ∂ v g L (V L -v) n + (V E -v) +∞ 0 gpdg + φ(v) n(v,t) = δ v=V R N (t).
Combining this with the equality

∂ ∂g [(G eq (v,bN (t)) -g)p] -a ∂ 2 ∂g 2 p = 0,
and following the subsection 2.1, we conclude the proof of Theorem 2.1.

L ∞ bound on p ε and n

We now wish to prove uniform bounds on p ε (v,g,t) for the full problem (1.1). For this, we need a kind of non-characteristic condition between the transport in v and the boundary flux at V R . Hence, to avoid technicalities, we assume that V R < V L for Equation (1.1). For the limit Equation (1.6), the above condition can be relaxed with V R ≤ V L . The following theorem holds Theorem 2.2 (L ∞ bound for p ε and for n). We assume (1.8)-(1.10), V R < V L , that p ε (t = 0) ∈ L ∞ with sufficient (Gaussian) decay at g = ∞ uniformly in ε. Then, for all T > 0, there exists a constant C(T ) independent of ε small enough such that the solution of (2.5) satisfies

sup 0≤t≤T sup g≥0 N ε (g,t) ≤ C(T ), sup 0≤t≤T sup V R ≤v≤V E sup g≥0 p ε (v,g,t) ≤ C(T ).
Assume that V R ≤ V L and (1.8)-(1.10). Then, there is a constant C(T ) such that, the solutions of (1.6) satisfy

sup 0≤t≤T sup V R ≤v≤V E n(v,t) ≤ C(T ) n 0 L ∞ .
Proof. Proof of Theorem 2.2. We begin with the estimates on p ε and then treat those for n.

Estimate for p ε . We consider N ε (t) as a given data in the term G eq and we build a super solution pε (v,g,t) of Equation (1.1) for p ε (v,g,t), that is a solution of the following Problem for t ≥ 0, g ≥ 0 and

V R < v < V E ,              ∂ ∂t pε + ∂ ∂v g L (V L -v) + g(V E -v) pε + 1 ε ∂ ∂g [(G eq (v,bN ε (t)) -g)p ε ] -a ε ∂ 2 ∂g 2 pε ≥ 0, a ∂ ∂g pε -(G eq (v,bN ε (t)) -g)p ε = 0, at g = 0,, p ε (V E ,g,t) = 0, [g L (V L -V R ) + g(V E -V R )]p ε (V R ,g,t) ≥ N ε (g,t) := φ F (v)p ε (v,g,t)dv.
(2.5) We construct it under the form

pε (v,g,t) = Be α(V R -v) M (v,g,t)e εg 2 e µt , with M (v,g,t) = exp -(Geq (v,bNε(t))-g) 2 2a
, with B , α and µ three constants which are large enough.

Firstly, the constant B is just used to satisfy the initial condition and we ignore it in the end of the proof.

Secondly, we fix α so that the boundary condition at V L is fulfilled. That simply means (recall that we assume

V R < V L ) [g L (V L -V R ) + g(V E -V R )]M (V R ,g,t) ≥ φ F (v)e α(V R -v) M (v,g,t)dv
which is possible, because large values of g are favorable, for α large enough.

Thirdly, because M (v,g,t) satisfies (G eq -g)M -a∂ g M = 0, we deduce that a∂ g pε -(G eq -g)p ε = 2ap ε εg ≥ 0, and hence, the zero flux boundary condition is satisfied at g = 0.

Finally, building on the above calculation, we also find successively that

∂ g ((G eq -g)p ε -a∂ g pε ) = -2aεBe µt ∂ g (gM ε e εg 2
),

∂ g ((G eq -g)p ε -a∂ g pε ) = (-2aε -a(2gε) 2 + εg(g -G eq ))p ε .
We deduce that there exists a constant C independent of ε small enough, such that for all v ∈ (V R ,V E ) for all g ≥ 0

ε -1 ∂ g ((G eq -g)p ε -a∂ g pε ) ≥ [ g 2 2 -C(1 + g)]p ε . (2.6)
On the other hand, we have

∂ t pε = (µ + z ε (v,g,t))p ε with z ε (v,g,t) = b a ∂G eq ∂A (g -G eq ) d dt N ε (t).
Using that N ε is uniformly Lipschitz with respect to ε, see (2.4), we deduce that there exists a constant C independent of ε such that for all v ∈ (V R ,V E ) for all g ≥ 0

|z ε | ≤ C(1 + g).
We then deduce that

∂ t pε ≥ (µ -C(1 + g))p ε . (2.7)
Simple computations show that there exists a constant C independent of ε such that for all v ∈ (V R ,V E ) for all g ≥ 0

∂ v ((g L (V L -v) + g(V E -v))p ε ) ≥ -C(1 + α)(1 + g)p ε . (2.8)
Combining (2.6), (2.7), (2.8) and taking µ large enough, we deduce that

∂ t pε + ∂ v [(g L (V L -v) + g(V E -v))p ε ] + 1 ε ∂ g [(G eq -g)p ε -a∂ g pε ] ≥ 0.
This proves the first part of Theorem 2.2.

Estimate for n.

Due to (1.7), there exists a constant b > 0, such that

G(V R ,A) V (V R ,A) -V R ≥ b > 0. (2.9)
Moreover, as n is a probability density, we have

N L ∞ ≤ φ F L ∞ .
To show that we can build a super-solution, we can reduced to construct n ≥ 0 such that

∂ t n + ∂ v G(v,bN (t)) V (bN (t)) -v n ≥ 0 and G(V R ,bN (t)) V (bN (t)) -V R n ≥ φ F L ∞ .
Choosing n(v,t) := B n 0 L ∞ e At with A and B large enough and using assumption (2.9), we deduce that n is a supersolution which ends the proof of Theorem 2.2.

Qualitative study of the associatedpopulation model for I&F neurons

This section is devoted to the understanding of the qualitative and asymptotic behavior of Equation (1.6) with respect to the strength of interconnections of the network. To simplify the presentation, we consider a slightly simplified model of Equation (1.6) as follows.

∂n ∂t + ∂ ∂v [(V 0 (N ) -v)n] + φ F (v)n = 0, 0 ≤ v ≤ V E , V 0 (N (t))n(v = 0,t) = N , n(V E ,t) = 0, N (t) = V E 0 φ F (v)n(v,t)dv. (3.1)
Here, firstly, following [START_REF] Cai | An effective kinetic representation of fluctuationdriven neuronal networks with application to simple and complex cells in visual cortex[END_REF]25], we choose V L = V R = 0. Secondly, we use a simpler drift V 0 independent of the variable v.

Steady states

The associated steady equation is given by

∂ ∂v [(V 0 ( N ) -v)n] + φ F (v)n = 0, 0 ≤ v ≤ V E , V 0 ( N )n(v = 0) = N , p(V E ) = 0, N = V E 0 φ F (v)n(v)dv. (3.2)
We assume that the smooth function V 0 (•) satisfies

0 < V 0 (N ) < V E , ∀N ≥ 0. (3.3)
One observes that, to avoid concentration as a Dirac mass at v = V 0 (N ), it is useful to also assume

φ F (V 0 (N )) > 0 ∀N ≥ 0. (3.4)
Indeed, we may write Equation (3.2) as

(V 0 ( N ) -v) ∂ n ∂v + (φ F (v) -1)n = 0. (3.5)
Therefore, near v = V 0 ( N ), the solution behaves as

∂ ln(n) ∂v = -φ F (V0( N ))-1 V0( N )-v
and thus, for some constant

q s n ≈ q s (V 0 ( N ) -v) α , α = φ F (V 0 ( N )) -1 > -1,
which means that Condition (3.4) implies that n has an integrable singularity.

More precisely we have the Theorem 3.1. With the assumptions (1.8), (3.3), (3.4), there is at least one steady state solution n which satisfies

V E 0 n = 1.
Proof. It is possible to give an expression of the stationary solution n. Using (3.5), we can write

∂ ln(n) ∂v = 1-φ F (v) V0( N )-v , for v < V 0 ( N ), 0, for v > V 0 ( N ).
Therefore we define

F (v, N ) = v 0 1 -φ F (w) V 0 ( N ) -w dw,
and we conclude that (and recall this n is integrable near V 0 ( N ) as we saw it above)

n(v) = N V0( N ) e F (v, N ) , for v < V 0 ( N ), 0, for v > V 0 ( N ).
(3.6) Indeed, from (3.5), we may also infer that n ≡ 0 for v > V 0 ( N ) and the sign of ∂ n ∂v is the sign of 1 -φ F (v). Next, we choose the value N so as to enforce the constraint V0( N ) 0 n = 1. For N = 0, the corresponding solution n vanishes, and for N → ∞, we have n → ∞. By continuity, we may achieve the constraint. These considerations explain the numerical solutions depicted in the Figure 3.1. These are obtained with Cases of uniqueness of the steady state. We may complete the existence result in Theorem 3.1 with uniqueness cases Theorem 3.2. With the assumptions (3.3), (3.4), the steady state is unique in the two following cases.

V R = 0, V E = 1, φ F (v) = A1 I {v>.5} , V 0 (N ) = .8 + .2 N 1 + N , ( 3
1. φ F > 1 on [0,V E ] and 0 < N V 0 (N ) < V 0 (N ), ∀N > 0. (3.8)
2. φ F ≥ 0 and V 0 < 0 Notice that Condition (3.8) is satisfied, for instance, by V 0 (N ) = aN b+N + c, with a ≥ 0, b > 0 and c ≥ 0, three constants.

Proof. Proof of Theorem 3.2. Case 1. The condition providing the steady states in Theorem 3.1, that is

V0( N ) 0 n = 1, is also written V 0 (N ) N = V0(N ) 0 e F (v,N ) dv,
and we show that the two uniqueness cases correspond to left and right hand sides with opposite monotonicity. The derivative of the left hand side is given by

N V 0 (N ) -V 0 (N ) N 2 < 0.
To treat the right hand side, we observe that, thanks to the assumption φ F (v) > 1, we have e F (N ,V R ) = 0. Therefore its sign is given by that of V 0 (N ) because

∂F (v,N ) ∂N = -V 0 (N ) v 0 1 -φ F (w) (V 0 (N ) -w) 2 dw.
Because the latter expression has the sign opposite to V 0 (N ) we find the result. Case 2. To consider also possible values φ F < 1 and to compute the derivative in N requires additional steps. We write, for v < V 0 (N ),

F (v,N ) = v/V0(N ) 0 1 -φ F (zV 0 (N )) 1 -z dz, and G(w,N ) = w 0 1 -φ F (zV 0 (N )) 1 -z dz.
The formula (3.6) gives,

V0( N ) 0 n(v) = N 1 0 e G w,V0( N ) dw,
and n is a probability measure if

1 N = 1 0 e G w,V0( N ) dw.
Uniqueness follows under the condition that G is increasing in N , which means

-φ F zV 0 (N ) V 0 (N ) > 0,
and, since φ F ≥ 0, this gives the condition V 0 < 0.

Asymptotic stability

Back to the evolution Equation (3.1), the next step is to determine if the solutions converge to a steady state in long time. We first consider the linear case and by a perturbation method conclude that the result still holds, assuming a weak nonlinearity. We use the Doeblin method which is very well adapted to the problem at hand, see [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF]9,[START_REF] María | Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activity[END_REF] for recent presentations and results.

Linear case.

In the linear case (V 0 constant), we may apply the Theorem 2.3 in [9] to equation (3.1). We obtain the following criteria for exponential convergence to a unique stationary state. If, for a time t 0 > 0, we are able to construct non-negative subsolution, uniformly with respect to all initial data in L 1 , then the solution of our equation converges exponentially to a unique stationary state. More precisely, the following result holds Theorem 3.3 ( [9]). Assume that V 0 > 0 is constant and assume that there exist t 0 > 0 and a nonnegtive function ν = 0 such that for all initial data n 0 ∈ L 1 + (0,V E ) with V E 0 n 0 (v)dv = 1, the solution of (3.1) at time t 0 satisfies n(v,t 0 ) ≥ ν(v).

(3.9)

Then, there exists a unique stationary state n of Equation (3.1), there exist α > 0, C > 0 such that for all t ≥ 0 and for all density initial data n

0 ∈ L 1 n(t) -n L 1 ≤ Ce -αt n 0 -n L 1 .
Notice that Doeblin's method is particular adapted to work with measures. However, in our context, we have control in L 1 and thus we restrict ourselves to this context. Once adapted to our case, we conclude that the following result holds Theorem 3.4. Assume that V 0 is constant, V 0 > 0 and φ F (V 0 ) > 0. Then, there exists t 0 > 0 and a non-negative function ν = 0 such that for all initial data n 0 ∈ L 1 + (0,V E ) with

V E
0 n 0 (v)dv = 1, the solution of (3.1) at time t 0 satisfies estimate (3.9). As a consequence, there exist α > 0, C > 0 such that for all t ≥ 0

n(t) -n L 1 ≤ Ce -αt n 0 -n L 1 ,
where n is the stationary state of Equation (3.1).

Proof. Proof of Theorem 3.4. Equation (3.1) can be written as

∂n ∂t + (V 0 -v) ∂n ∂v + (φ F (v) -1)n = 0.
With the method of characteristics, we obtain that

n(v,t) = 0 for v ≥ [V E -V 0 ]e -t + V 0 .
In the interval

V 0 (1 -e -t ) ≤ v ≤ [V E -V 0 ]e -t + V 0 , we have n(v,t) = e t-t 0 φ F (v+V0(e -t -1)+V0(1-e -s ))ds n(0,ve t + V 0 (1 -e t )), (3.10)
and finally, in the interval

0 ≤ v ≤ V 0 (1 -e -t ),
we have

n(v,t) = 1 V 0 N (t + ln(V 0 -v) -ln(V 0
))e -ln(V0-v)+ln(V0)-ln(V 0 -v)+ln(V 0 ) 0 φ F (V0(1-e -s ))ds .

(3.11) Using formula (3.10), we deduce that

N (t) ≥ e t(1-φ F L ∞ ) [V E -V0]e -t +V0 V0(1-e -t ) φ F (v)n 0 ve t + V 0 (1 -e t ) dv.
Now, we use that φ F (V 0 ) > 0 and that φ F is regular. This implies that there exist a constant C > 0 and t 1 > 0 such that for

V 0 (1 -e -t1 ) ≤ v ≤ [V E -V 0 ]e -t1 + V 0 , we have φ F (v) ≥ C > 0.
Hence, for t ≥ t 1 , we have

N (t) ≥ Ce t(1-φ F L ∞ ) [V E -V0]e -t +V0 V0(1-e -t )
n 0 ve t + V 0 (1 -e t ) dv.

Now, using that

V E 0 n 0 (v)dv = 1, we obtain the following estimate, independent of the initial data, that is

N (t) ≥ Ce -φ F L ∞ t ∀t ≥ t 1 .
(3.12)

Using now formula (3.11), we deduce that for t 0 = 2t 1 , and for

v ∈ [0,V 0 (1 -e -t 0 2 )] with 0 ≤ -ln(V 0 -v) + ln(V 0 ) ≤ t 0 2 , we have n(v,t 0 ) ≥ Ce -φ F L ∞ t0 .
This implies, that there exists a time t 0 := 2t 1 and a nonnegative function ν with ν > 0, such that, for all initial data in L 1 , (3.9) condition is satisfied. This ends the proof of Theorem 3.4.

Nonlinear case.

We now consider the nonlinear Equation (3.1), which means that V 0 is not necessary a constant. The following theorem holds Theorem 3.5. Assume (3.3), (3.4) and that there is a constant D > 0, small enough, such that

N V 0 (N ) L ∞ + V 0 L ∞ φ F L ∞ ≤ D.
Then, there exists a unique stationary state n * of Equation (3.1) and there exists constants α > 0 and C > 0 such that for all initial data n 0 ∈ L 1 (0,V E ),

V E 0 n 0 (v)dv = 1, the solution of Equation (3.1) satisfies n(t) -n * L 1 ≤ Ce -αt n0 -n * L 1 .
Proof. Proof of Theorem 3.5. As the assumptions of Theorem 3.1 hold, there exists a stationary state of Equation (3.1), we denote it by (n * ,N * ).For a density n 0 as above, we call S t (n 0 ) the solution of the linear equation

∂n ∂t + ∂ ∂v [(V 0 (N * ) -v)n] + φ F (v)n = 0, 0 ≤ v ≤ V E , V 0 (N * )n(v = 0,t) = N * , n(V E ,t) = 0, n(v,0) = n 0 . (3.13)
Then, using Duhamel's principle, the solution of Equation (3.1) with initial data n 0 can be written as

n(v,t) = S t (n 0 ) + t 0 S t-τ (V 0 (N * ) -V 0 (N (τ )) n(τ,v)dτ + δ v=0 t 0 S t-τ N (τ ) V0(N (τ )) -N * V0(N * ) dτ.
From this, inserting absolute values, integrating in v and using that the total mass of n is 1, we obtain

n(t)-n * L 1 ≤ S t (n 0 )-n * L 1 + t 0 S t-τ V 0 (N * ) -V 0 (N (τ )) + N (τ ) V 0 (N (τ )) - N * V 0 (N * ) dτ
and we now use Theorem 3.4 to obtain two constants C > 0 and β > 0 such that

n(t) -n * L 1 ≤ e -βt n 0 -n * L 1 + t 0 e -β(t-τ ) V 0 (N * ) -V 0 (N (τ )) + N (τ ) V 0 (N (τ )) - N * V 0 (N * ) dτ.

Conclusion and perspectives

Neural networks are by nature highly complex systems and mean field models are a way to circumvent the modeling. In [START_REF] Cai | An effective kinetic representation of fluctuationdriven neuronal networks with application to simple and complex cells in visual cortex[END_REF]25], the authors propose a kinetic mean field equation of neural assemblies to make a fine description of the dynamic of neural networks with respect to the membrane potential and the conductance of the neurons. The patterns that emerge from those models are very rich, and exhibit oscillations, bifurcations... However, this model is very difficult to tackle both from a numerical viewpoint [START_REF] Cáceres | A numerical solver for a nonlinear fokker-planck equation representation of neuronal network dynamics[END_REF], where efficient numerical schemes are complex to implement and from a theoretical viewpoint [START_REF] Perthame | On a voltage-conductance kinetic system for integrate & fire neural networks[END_REF], where classical methods failed due to the particular boundary conditions and the degenerate diffusion involved in the equation. To overcome those difficulties, in [START_REF] Cai | An effective kinetic representation of fluctuationdriven neuronal networks with application to simple and complex cells in visual cortex[END_REF]25] The first one concerns a more precise theoretical study on the mechanisms beyond the oscillations in the Integrate and Fire model presented in this equation. Indeed, to our knowledge, there do not exists theoretical results of existence of periodic solutions for this kind of PDE. However, Equation (3.1) shares, in its structure, some similarity with the time elapsed model, where we can built explicit periodic solutions (see [START_REF] Khashayar Pakdaman | Relaxation and self-sustained foscillations in the time elapsed neuron network model[END_REF]). Indeed, as for the time elapsed model, in some regimes, the total firing rate satisfy a delay type equation, which may be tractable. As an example, considering for k ∈ N big enough (let us assume to simplify V R = 0, V E = 1) and the particular function φ F (v) = kI v≥α ,α > 0, 0 < α < 1, using the method of characteristics, we obtain the following formula on the total firing rate of Equation (3.1)

N (t) = k 1 - t ϕ(t)
N (s)ds with ϕ(t) ≤ t such that α = e -(t-ϕ(t)) t-ϕ(t) 0 V 0 (N (s + ϕ(t)))e s ds. This system is however more complex than the one obtained in [START_REF] Khashayar Pakdaman | Relaxation and self-sustained foscillations in the time elapsed neuron network model[END_REF], due to its specific coupling, but may be exploitable.

Coming back to the conductance and voltage-conductance kinetic model developed in [START_REF] Cai | An effective kinetic representation of fluctuationdriven neuronal networks with application to simple and complex cells in visual cortex[END_REF]25], a second important question, concerns the theoretical study of this equation which was initiated in [START_REF] Perthame | On a voltage-conductance kinetic system for integrate & fire neural networks[END_REF]. Indeed, beside the question of well-posedness which is rather difficult, typical questions as the asymptotic convergence to a stationary state for very weak interconnections, is not yet understood.

The standard I&F model for networks is closely related to (1.6) with two main differences. On the one hand firing is taken pointwise, rather than the nonlocal definition of firing in (1.6), which is our choice of a simplification. On the other hand noise is taken into account differently, directly by a diffusion term in the standard I&F model, rather than by dispersion of the drift terms. It is written

∂ ∂t n + ∂ ∂v h(bN (t),v)n -a ∂ 2 n ∂v 2 = N (t)δ(v -V R ), t ≥ 0, v < V E , n(V E ,t) = 0, N (t) = -a ∂n ∂v | v=V E .
Another question is to derive this model from the voltage-conductance equation in a diffusive limit rather than an hyperbolic limit.

. 7 )

 7 and A = .5 for the figure on the left, A = 5. for the figure on the right. The value V 0 (N ) can be identified because the solution vanishes for v > V 0 (N ).

Fig. 3 . 1 .

 31 Fig. 3.1. Numerical solutions of the steady states in (3.2) with the date in (3.7) and two choices of the term φ F (v). Abscissae are v. Left: firing rate A = .5; Right: Firing rate A = 5.

Fig. 3 . 2 .

 32 Fig. 3.2. Numerical solutions of the evolution equation (3.1) V 0 (N ) = .95, that is the linear equation. This shows that the solution relaxes to the steady state depicted in Fig. 3.1 with damped oscillations.

Fig. 3 . 3 .

 33 Fig. 3.3. Numerical solutions of the evolution equation (3.1) with data in (3.15). Abscissae are v. Left: firing rate V 0 (N ) = .75 + .15 N .1+N ; Right: Zoom on two oscillations.

  , the authors propose a reduction of dimension of this equation by closure moments. In this article, we give an alternative with a new kinetic model where the singular boundary condition is replaced by an integral absorption in order to obtain more precise theoretical results and to simulate the different types of dynamics that can emerge with a relatively simple algorithm on the limiting voltage-based population equation for I&F neurons. Several open questions about the present voltage only I&F model remain open.

Acknowledgments BP has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 740623). B.P. and D.S. acknowledge partial funding from the ANR blanche project Kibord ANR-13-BS01-0004 funded by the French Ministry of Research.

To conclude, it remains to estimate

To do so, we notice that

With the weak nonlinearity assumption

we conclude Theorem 3.5 using the Gronwall lemma and choosing

Oscillatory states

As observed in [START_REF] Cáceres | A numerical solver for a nonlinear fokker-planck equation representation of neuronal network dynamics[END_REF], the voltage-conductance model (1.1) can produce periodic solutions which represent the spontaneous activity of the network. This desirable property also holds for other neural network models; see [6] for the Leaky Integrate&Fire population model and [START_REF] Khashayar Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Khashayar Pakdaman | Relaxation and self-sustained foscillations in the time elapsed neuron network model[END_REF] for the time elapsed model.

Linear instability.

While in Subsection 3.2 we have studied the nonlinear asymptotic stability of the steady state, one can also find conditions for its instability.

The linearization of Equation (3.2) around a steady state n(v) is to find (r(v),µ) satisfying

We look for a solution with exponential behavior in time r(v)e λt which gives

Because n(v) = 0 for v > V0 , we also have r(v) = 0 for v > V0 and it remains to solve the problem for v ∈ (0, V0 ). Notice that the condition

0 rdv = 0 follows by integration when λ = 0. Proposition 3.1. We assume that φ F (v) = φ 1 1 I {v>v1} and φ 1 > 1. We also assume that

Then there is a solution of the problem (3.14) with λ > 0. In other words the steady state n is unstable. Proof. To solve this problem, we notice that q = r n satisfies

and thus, using (3.2),

where φ F (v) ranges for some average of φ F on (0,v). At this stage we recall the singularity of n(v)

Then, eliminating µ, the parameter λ is chosen so as to satisfy

a relation that we write, with obvious notations, as

Now we assume that φ F > 1 is constant on its support. On the one hand, as λ → φ F , the term n(v) ( V0-v) λ dv tends to +∞ and λ -1 → φ F -1 for v ≈ V0 . Therefore,

On the other hand, as λ → 0,

Since n = N V0( V0-v1) ( V0 -v), Proposition 3.1 follows by the mean value theorem. Numerical illustration. Numerical evidence, indicates that periodic solutions occur for the simple limiting model (1.6). We use the range of parameters produced in the previous subsection, when the firing rate φ F is large enough and the nonlinear voltage V 0 (N ) is stiff enough. The computed solution is depicted in the Figure 3.3 for the evolution equation (3.1).

Here the choice of parameters and nonlinearities (which are far form the regimes where uniqueness of a steady state has been established) is given by the expressions φ F (v) = 5 1 I {v>.7} , V 0 = .75 + .22min(N ,1). (3.15)


Michael Berhanu

Eric Falcon

Guillaume Michel

C Gissinger

S Fauve

Capillary wave turbulence experiments in microgravity

Keywords: 47, 35, Pq -Capillary waves 47, 27, -i -Turbulent flows 81, 70, Ha -Testing in microgravity environments

published or not. The documents may come

Wave turbulence describes the statistical behavior of a random set of dispersive waves in nonlinear interaction. This phenomenon occurs at very different scales in a great variety of systems [START_REF] Nazarenko | Wave Turbulence[END_REF][START_REF] Newell | [END_REF]. The case of surface waves at the interface between a gas and a liquid constitutes an example of prime interest, which has deserved numerous experimental investigations [3,4]. With specifically the hypotheses of weak nonlinearity and negligible dissipation, wave turbulence theory [START_REF] Nazarenko | Wave Turbulence[END_REF][START_REF] Newell | [END_REF]4,[START_REF] Zakharov | Kolmogorov spectra of turbulence[END_REF] predicts, in stationary state, turbulent self-similar regimes, in which energy (or another conserved quantity) is transferred from an injection scale towards a dissipation scale. These regimes are characterized by power law spectra both in spatial and temporal Fourier spaces, for scales belonging to the inertial range, where both the dissipation and the forcing are negligible. In contrast to hydrodynamic turbulence for which the Kolmogorov spectrum has not been deduced yet from the Navier-Stokes equation, energy spectra can be analytically computed in the framework of weak wave turbulence. Since the hypotheses used in this derivation are drastic, the relevance of weak wave turbulence theory to describe experimental and natural systems remains questionable. In particular, wave turbulence theory assumes a scaleinvariant dispersion relation of the form ω ∝ |k| α . That is not verified for surface waves on Earth, as a result of the competition of two restoring forces. Gravity dominates at large scales, whereas capillarity is the main restoring mechanism for wavelengths smaller typically than 15 mm for water. Experimentally, forcing low frequency gravity waves leads in the capillary wave range to power-law spectra of wave elevation whose exponents are in agreement to those predicted for pure capillary waves [START_REF] Zakharov | [END_REF] both for the frequency spectrum f -17/6 [7] and for the wavenumber spectrum k -15/4 [8,9], although the transition between gravity and capillary waves is not described by wave turbulence theory. Moreover, in such experiments, the degree of nonlinearity is not weak [9] and dissipation is significant [10,11] even in the decade over which the predictions of weak wave turbulence are observed (typically between 20 Hz and 200 Hz). Recent direct numerical simulations [12,13] have demonstrated the relevance of weak wave turbulence theory for pure capillary waves in the limit of low dissipation and nonlinearity. Operating in microgravity conditions is required to study turbulence of pure capillary waves in laboratory experiments. Note that using two fluids of the same density leads to a different phenomenology [14]. A single study has been performed in parabolic flights to investigate turbulence of capillary waves inside a spherical container [15]. In this geometry surface waves propagate at the spherical air-water interface in absence of walls causing reflections or enhanced dissipation due to the contact line dynamics [16,[START_REF] Michel | Solid boundaries and surface waves: dissipation, Doppler effect and nonlinear interactions[END_REF]. Self-similar frequency spectra of wave elevation have been reported over two decades in satisfying agreement with the weak wave turbulence theory [15]. Nevertheless, due to the short duration (22 s) of the zero gravity phases in parabolic flights, statistically steady states were not reached: in particular, the behavior at low frequency was not accurately resolved and the transient motion of the fluid possibly interfered with the wave dynamics. The observed power-law spectrum could correspond to a transient regime rather than to a capillary wave turbulence regime. To strengthen these previous observations, we report in this letter experiments in the international space station (ISS), where the duration of the weightlessness is not limited. The low gravity condition is excellent with a root-mean-square (r.m.s.) acceleration due to external vibration less than 20 × 10 -6 g with g = 9.81 m•s -2 , whereas the r.m.s. residual acceleration in parabolic flight is of order 5 × 10 -2 g is less. We use the FLUIDICS (Fluid Dynamics in Space) facility [START_REF] Mignot | 68th international astronautical congress (IAC)[END_REF] designed by Airbus Defence and Space and by the French spatial agency the CNES ("Centre National d' Études Spatiale") primarily to study the sloshing motion of liquid propellants of satellites [START_REF] Dalmon | [END_REF]20]. A specific tank equipped with two fluid level sensors is dedicated to the study of capillary wave turbulence. The experiment was installed in the ISS and operated by ESA (European Space Agency) astronauts (Thomas Pesquet on May 3rd 2017, Paolo Nespoli on October 27th 2017, Norishige Kanai on March 13th 2018 and Alexander Gerst on September 27th 2018).

Experimental setup. -The experimental setup is depicted in Fig. 1. A spherical container of internal radius R = 50 mm and made of polycarbonate is partially filled with 30 % of water (mineral water Luchon c ) and with air at atmospheric pressure. In the ISS, the temperature is regulated around 22 • C. The water density and the water/air surface tension are ρ = 998 kg•m -3 and γ = 72 mN•m -1 . The tank is rotated about the axis O z distant of L = 175.4 mm with an oscillating angle θ(t) in order to generate waves at the air-water interface, θ(t) being either a sine function or a low-frequency noise. Two cameras image the transparent container in the rotating frame (see Fig. 1 (c)). A force sensor (S in Fig. 1 (d)) located approximately at a distance of 74.8 mm from C provides the three components of the force exerted by the tank on the rotating arm, so that the tank acceleration can be evaluated. Two capacitive probes [21] (isolated copper wires of diameter 0.32 mm) measure the thickness of the liquid layer, at two different positions η 1 (t) and η 2 (t). More details on the experimental setup and on the data interfacing can be found in Ref. [START_REF] Mignot | 68th international astronautical congress (IAC)[END_REF]. In weightlessness, the water wets the internal surface of the tank at rest, forming a large air bubble in the center and a homogeneous water shell of thickness h 0 = 5.60 mm. This observation previously reported [15] can be understood by an energetic argument. Let V 0 denote the volume of wa- ter, R h = R -h 0 the inner radius of the uniform shell and R 0 = (3V 0 /4π) 1/3 the radius of a water sphere not in contact with the solid surface, and ϑ = 78 • the static contact angle between water and polycarbonate [START_REF] Cho | Adhesion Aspects of Thin Films[END_REF]. The capillary potential energy of the wetting liquid shell is smaller than the inner liquid sphere, if (R 2 h -R 2 0 ) < cos ϑ R 2 , i.e. if the filling ratio (R 0 /R) 3 > 0.31. Although the experimental value of this ratio 0.3 is slightly smaller than the critical value, we observe at rest a homogeneously wetted sphere (see Fig. 1 (a)). Nevertheless, the capillary forces are not sufficient to prevent some dewetting events when the tank is oscillated. Considering now small oscillations of angular frequency ω of the wetting spherical shell the free-surface deformation η can be decomposed using the spherical harmonics Y m l and the amplitudes C l,m :

η(t, θ, φ) = R h + ∞ l=0 l m=-l C l,m Y m l (θ, φ) e i ωt .
With the assumption of potential flow, the dispersion relation can be derived [START_REF] Lamb | Hydrodynamics[END_REF]:

ω 2 = γ ρ l(l -1)(l + 2) R 3 h R R h 2l+1 -1 1 + l l + 1 R R h 2l+1 (1)
When l ≥ 9 (i.e. f = ω/(2π) > 3.7 Hz), the usual dispersion relation for capillary waves

ω 2 = γ ρ k 3 (2)
with k = l/R h is accurate with an error less than 5 %. Note that this corresponds to a scale-invariant dispersion relation as assumed by the theory of weak wave turbulence.

Forcing characterization. -We consider first a sinusoidal forcing, θ(t) = θ 0 sin(2π F t), applied in most cases during 450 s. In the reference frame of the tank, the tangential and centrifugal accelerations at the center of the sphere read respectively a θ = -L θ 0 (2π F ) 2 sin(2π F t) e θ and a r = -L θ 2 0 (2π F ) 2 cos 2 (2π F t) e r . Therefore, to decrease the relative influence of the centrifugal force, which acts as an effective gravity, we restrict our study to the case of small oscillations, i.e. θ 0 ≤ 0.04 (less than 2.5 • ). For a typical forcing F = 1.75 Hz and θ 0 = 0.04, the liquid is strongly agitated and remains most of the time in contact with the surface of the sphere. However, a significant amount of bubbles of few millimeters are generated by the motion as it can be seen on the images from camera 1 (see Fig. 2). Morever a few dewetting events occur especially close to the tank fastening. The second set of measurements corresponds to a random forcing with θ(t) = θ 0 sin(φ(t)), where φ(t) is a random noise, band-pass filtered in the frequency interval ∆F = [F 1 , F 2 ] with typically F 1 = 1 Hz and F 2 = 2 Hz and θ 0 = 0.04 . The visual observations are very similar.

The actual forcing can be characterized using the angular displacement of the motorized arm and the force sensor, power spectra being reported in Fig. 3. The arm follows satisfyingly the programmed instructions, although we notice significant third and fifth harmonics for the sinusoidal forcing in the spectrum of the tangential force. This spectrum differs from the one from the force sensor, due to the inertia of the water during the motion and we observe a higher content between 10 and 100 Hz. For the random forcing, the power spectrum is not flat in the expected range [0.5, 2] Hz due to the mechanical response of the motor, with a maximum of excitation at about 1.7 Hz. The peak around 9 Hz in the force sensor data could be due to a sloshing eigenmode or to a vibration of the structure. For both sinusoidal and random forcing the spectra of the radial force is smaller by roughly a factor of 500 for f ∈ [1, 30] Hz. Moreover, the average value of the radial centrifugal acceleration at the furthest point of the tank from the rotation axis is g ef f = 0.058 m•s -2 in the sinusoidal case with θ 0 = 0.04 and F = 2 Hz, making this effective gravity negligible for frequencies f 0.29 Hz. At the same location, the r.m.s. tangential acceleration is a θ = 1.4 m•s -2 . In this setup in microgravity, deformation of the air-water interface can therefore be considered to be pure capillary waves, for frequencies larger than 1 Hz instead of about 20 Hz on Earth.

Turbulent dynamics of the air-water interface. -The fluctuations of the free-surface are monitored using the two capacitive probes. Examples of temporal evolutions are displayed in Fig. 4. The repetition of the sequences shows a good reproducibility. The sensor 1 η 1 (t) can be affected by the dewetting. As the position of the sensor 2 corresponds to the point of maximal centrifugal force, dewetting does not occur there and we favor this sensor in our further analysis. The sensor 2 η 2 (t) is sometimes saturated, which means that the sensor is completely immersed in water. However, the shape of the power spectra is not perturbed by these short saturations. is interpreted as a dewetting of the liquid from the tank wall at the location of the sensor 1. Note that sensor 2 is not affected.

For a sinusoidal forcing of frequency F = 2 Hz and amplitude θ 0 = 0.04 corresponding to a forced wavenumber of k ∼ 130 m -1 we observe r.m.s. fluctuations of the free surface of order σ η ∼ 5 mm. This order of magnitude close to the simple relation a t /(2π F ) 2 implies a relatively large value of the wave steepness s ≈ k σ η ∼ 0.6 using the linear dispersion relation Eq. 2. This high value of s which quantifies the importance of nonlinear effects for surface waves shows that these experiments do not lie in the weakly nonlinear regime. However, the analytic solution of Crapper [START_REF] Crapper | [END_REF] shows that the change of the dispersion relation of a plane monochromatic capillary wave does not exceed 10 % at the maximal steepness value s = 4.59 corresponding to the trapping of air bubbles. Therefore, we use in the following the linear dispersion relation Eq. 2, which was experimentally verified in microgravity [15].

The dynamics of the free-surface is characterized by the frequency power spectrum of wave elevation S η (f ). The runs last for 450 s or more and the spectra are computed on a duration of 400 s (typically 20 times longer than for the parabolic flight experiments). Typical spectra for sensors 1 and 2 are plotted in Fig. 5 (a) for a sinusoidal forcing of high amplitude (F = 2 Hz and θ 0 = 0.04). We observe for the sensor 2, an absence of the forcing peak and a decay of the spectrum with frequency following fairly the power law predicted by the weak wave turbulence theory for capillary waves S η (f ) ∼ f -17/6 [START_REF] Zakharov | [END_REF]. Similarly to previous experiments operated in parabolic flights [15], a capillary wave turbulent cascade is reported for a sinusoidal forcing of the tank, which is only observed for few experiments on Earth using a parametric forcing in the capillary wave range [25,26]. In contrast, when capillary waves are generated from nonlinear interactions of gravity waves, a random forcing appears necessary to generate the capillary wave power law spectrum [7][8][9]. In the present experiment, we observe a peak at 4 Hz (the second harmonic of the forcing frequency) superposing on the turbulent cascade. To avoid a possible saturation, the sensitivity of the capacitive probes specially realized for this setup is lower than the ones used in our previous experiments on Earth or in the parabolic flights [15] and the spectrum reaches the electrical noise level typically for f ≥ 20 Hz. We display for information, the power-law f -1 recently reported for turbulent capillary waves and interpreted as a statistical equilibrium of large scales [27]. In addition, we compute the magnitude-squared coherence function C xy (f ), which indicates the correspondence degree between two temporal signals x(t) and y(t) at a given frequency f . C xy between η 1 (t) and the tangential force F t measured with the force sensors in Fig. 5 (b) shows that the local free-surface dynamics results mainly from the motion of the tank. In contrast, C xy between η 2 (t) and F t is lower, except at the peak at 4 Hz, showing that the free-surface dynamics at the position of the sensor 2 is weakly correlated to the forcing and results from the turbulent dynamics of the capillary waves. Similarly, the coherence between the two wave height signals is small.

For runs operated with a random forcing, typically with an angular amplitude θ 0 = 0.04 and a frequency range f ∈ [0.5, 2] Hz, the spectrum displayed in Fig. 6 (a) shows significant differences. The wave amplitude is lower at the maximal motor capacities, because the forcing is less coherent. The capillary wave turbulence power law is again observed for the second sensor without forcing peaks but on a narrower frequency range f ∈ [START_REF] Nazarenko | Wave Turbulence[END_REF]8] Hz. At frequencies lower than the forcing range, the spectrum is also compatible with the power law f -1 expected for a statistical equilibrium of large scales [27]. At the location of the first sensor, we observe a significant peak at 9 Hz also visible on the data of the force sensor, which could be interpreted as a sloshing mode or a vibration of the structure. The coherence function in Fig. 6 (b) between η 1 and F t displays indeed a quite high value around 9 Hz. The coherence between η 2 and F t is low outside frequencies belonging to the forcing range. The coherence between η 1 and η 2 remains also quite low, except for f > 12 Hz, where the spectrum becomes dominated by the electrical noise, which is identical on both sensors.

During the operations of the FLUIDICS experiment, 20 independent sinusoidal forcing runs (F varying from 1.5 to 6 Hz and θ 0 varying from 0.01 to 0.04) and 13 random forcing runs (F c varying from 1 to 3.5 Hz, ∆F varying from 0.5 to 1.75 Hz and θ 0 varying from 0.01 to 0.04) have been carried out, mostly for the highest possible torque of the motor, which is performing small oscillations. To summarize, the results from the different runs the standard deviation of wave elevation σ η is plotted in Fig. 7 (a) as a function of the forcing tangential acceleration a t at the tank center location. For most of the runs, a t is of order 1.2 m•s -2 and 0.9 m•s -2 respectively for sinusoidal and random forcing, but the r.m.s. free-surface deformation remains close to 5 mm for both sensors. Using the linear dispersion rela- tion Eq. 2, the spatial spectrum S η (k) is computed and the wave steepness from the signals of the second sensor can be evaluated as s = k 2 S η (k) dk 1/2 = | ∇η 2 |. Fig. 7 (b) shows that s is a slowly growing function of σ η , with a typical wavenumber of k ∼ s/σ η ≈ 110 m -1 corresponding to a frequency of 1.6 Hz, which corresponds to the beginning of the capillary cascade. This estimation validates this indirect estimation of the wave steepness and shows that different runs of these experiments can be compared equivalently using the wave amplitude or the wave steepness. Moreover, s reaches quite large values, showing that the presented experiments are not performed in a weakly nonlinear regime. For all these runs, the power spectra are fitted in a variable frequency range as power laws S η = K 1 f -α . The results are gathered in Fig. 8 as a function of the wave steepness s. We observe first in panel (a), that for all measurements the exponent α is surprisingly robust, being close to α theo , especially at high s. The frequency interval of each fit is defined as the range of the capillary wave cascade i.e. f ∈ [2.5, 20] Hz for sinusoidal forcing and f ∈ [1.5, 8] Hz for random forcing. To estimate the quality and the relevance of the fits, we compute a fit error ratio as |S η -K 1 f -α |/(K 1 f -α ) in the fit frequency range. For sinusoidal forcing, we report a fit error of order 15 %. Note that at small values of s, the fit error is significantly larger because it corresponds to runs operated at a smaller motor displacement, the corresponding spectra being not well described by a power law. For random forcing, we obtain a fit error of order 11 %. The error is less than for sinusoidal forcing due to the absence of the peak at 4 Hz superposed on the cascade. Wave turbulence theory predicts an elevation power spectrum S η (ω) = 4π/3 C KZ (γ/ρ) 1/6 1/2 ω -17/6 [START_REF] Zakharov | [END_REF]7,9], where is the energy flux and the dimensionless constant C KZ can be analytically computed [28,29] in the weakly nonlinear limit. To estimate the amplitude of the capillary wave turbulent cascade, our power spectra are fitted by S η (f ) = K 2 f -17/6 . Although the present experiments do not meet all the hypotheses of the theory (weak nonlinearity, negligible dissipation,. . . ), an effective flux ef f = (K 2 (2 π) 11/6 (ρ/γ) 1/6 ) 2 can be computed by assuming arbitrarily 4π/3 C KZ = 1, thus providing a quantity of the same order of magnitude of the energy flux. shows that this effective energy flux increases with the wave steepness. The values of ef f are comparable with the estimations from the dissipated power by viscosity, which were obtained in a previous capillary wave turbulence experiment in a non weakly nonlinear regime [9]. From the scaling of the three-wave interaction collision integral, for a given value of the energy flux a critical wave-number can be defined k N L ≈ 2/3 (γ/ρ) -1 [30,31]. For k < k N L the linear time (the wave period) is smaller than the characteristic nonlinear time (the timescale of the wave interaction). Therefore, capillary waves cannot propagate at too large scales, because they disappear due to non-linear interactions on a time smaller than the period. With the gravest experimental value of ef f ≈ 7×10 -5 m 3 •s -3 , k N L ≈ 23 m -1 . According to Eq. 1, the largest possible mode k 2 = 2/R h ≈ 45 m -1 exceeds k N L and corresponds to a frequency of f 2 = 0.22 Hz. Then, a breakdown of the wave turbulence due to a too large value of the energy flux does not occur in these experiments, although the level of nonlinearity is significant.

Discussion. -Using the FLUIDICS experiment, the turbulent fluctuations of an air-water interface have been investigated in weightlessness. Due to the excitation system by a rotating arm, the dynamics of the interface at the position of the first wave height sensor is perturbed by frequent dewetting events that affect the shape of the power spectra. However, this does not affect the signal at the location of the second wave height sensor that displays a surprisingly robust power law in the frequency range [2.5, 20] Hz for sinusoidal forcing and [1.5, 8] Hz for random forcing. The exponents of these power-law spectra are in agreement with the prediction of capillary waves weak turbulence theory. This observation confirms the previous investigation of capillary wave turbulence in microgravity using parabolic flights [15], which reported a power law spectrum of wave elevation in f -3 close to the theoretical prediction in the frequency range [4,400] Hz both for sinusoidal and random forcing. In these previous experiments, the wetting of ethanol on the glass cylindrical container insured indeed a better quality of the freesurface and the signal to noise ratio of the sensors was better. Nevertheless, the longer duration of the experiments in the ISS and the systematic investigation of forcing parameters insure that the power law spectrum is not caused by a transient behavior and can be attributed to capillary wave turbulence. Moreover, in the parabolic flight experiments [15], the calibration of the capacitive probe was not provided and thus the amplitude of the waves in turbulent regimes was unknown. Here, we show that the corresponding runs are obtained in fact for relatively high values of the wave steepness s ∈ [0.4, 0.7], demonstrating that the capillary wave cascade can be observed outside the weakly nonlinear regime as stated in recent experiments [8,9]. For a steepness level of order 0.1, due to the nonlinear broadening of the dispersion relation, the effects of non resonant interactions are not negligible [32]. Moreover, the analytic derivation of the capillary wave turbu-lence spectra and of associated Kolmogorov-Zakharov constants [28,29] may become invalid, as the theory requires a time scale separation between the nonlinear interaction time and the linear time. Here, the value of the exponent of this strongly nonlinear wave turbulent cascade, close to α theo = 17/6, might be explained by dimensional analysis arguments [30] in combination with the dispersion relation once we have supposed that the quadratic nonlinearity and thus three-wave processes dominate the wave dynamics [9]. Under strong agitation, the air-water interface in microgravity therefore presents a turbulent behavior, whose understanding remains incomplete and outside the validity domain of existing theories. Further experiments performed in microgravity may be useful to better characterize the strong capillary wave turbulence evidenced in this work.

Fig. 1 :

 1 Fig. 1: (a) Picture of the experimental spherical tank. The cylinder visible on the top of the picture is the filling orifice and the fastening point. The two circular electronic boards measure the capacity for each fluid height probe. (b) Technical drawing of the experiment without, the acquisition system and the cameras (Airbus Defence and Space). (c) Horizontal cross section through the center of the tank. (d) Schematic view of the experiment (see text for definitions).

Fig. 2 :

 2 Fig. 2: Snapshots of the tank from camera 1, for a sinusoidal forcing with F = 1.75 Hz and θ0 = 0.04. Presence of air bubbles is noted. In the right image (b), a strong transient dewetting is visible on the top right corner.

Fig. 3 :

 3 Fig. 3: Examples of Fourier forcing power spectra. Dashed magenta line, tangential force evaluated from θ(t). Tangential force (continuous blue line) and radial force (dotted green line) are measured by the force sensor at S. (a) Sinusoidal forcing θ0 = 0.04 and F = 2 Hz. (b) Random forcing θ0 = 0.04 and ∆F = [0.5, 2] Hz.

Fig. 4 :

 4 Fig. 4: Examples of temporal evolution of the local wave amplitude measured with the capacitive probes. Blue (bottom), sensor 1, η1(t). Red (top), sensor 2, η2(t), the signal has been vertically shifted 35 mm for better visualization. (a) Sinusoidal forcing θ0 = 0.04 and F = 2 Hz. (b) Random forcing θ0 = 0.04 and ∆F = [0.5, 2] Hz. At t 127.5 s, a sharp decrease of η1(t)is interpreted as a dewetting of the liquid from the tank wall at the location of the sensor 1. Note that sensor 2 is not affected.

Fig. 5 :

 5 Fig. 5: (a) Power spectrum of wave elevation Sη 1 (f ) (Blue dotted line) and Sη 2 (f ) (Red continuous line) for a sinusoidal forcing θ0 = 0.04 and F = 2 Hz. The power-laws f -17/6 and f -1 are also indicated, respectively as a black continuous line and as a black dashed line (b) Magnitude-squared coherence functions Cxy between η1 and Ft (blue dashed line), between η2 and Ft (red dotted continuous line) and between η1 and η2 (magenta continuous line).

Fig. 6 :

 6 Fig. 6: (a) Power spectrum of wave elevation, Sη 1 (f ) (Blue dotted line) and Sη 2 (f ) (Red continuous line) for a random forcing θ0 = 0.04, Fc = 1.25 Hz and ∆F = 1.5 Hz. The power-laws f -17/6 and f -1 are also indicated, respectively as a black continuous line and as a black dashed line. (b) Magnitude-squared coherence functions Cxy between η1 and Ft (blue dashed line), between η2 and Ft (red dotted continuous line) and between η1 and η2 (magenta continuous line)

Fig. 8 Fig. 7 :

 87 Fig. 7: (a) Standard deviation of the free surface as a function of the tangential forcing r.m.s acceleration. • sensor 1 sinusoidal forcing, * sensor 2 sinusoidal forcing, sensor 1 random forcing, × sensor 2 random forcing. (b) For the sensor 2 only, wave steepness s ≈ | ∇η2 | as a function of the standard deviation of η. s is derived from the spectrum Sη 2(f ) and is a slowly growing function of ση. * random forcing, × sensor 2 random forcing.

Fig. 8 :

 8 Fig. 8: For the sensor 2 only, (a) fitted exponent of the power spectrum Sη ∼ f -α as a function of the steepness s. * sinusoidal forcing, × random forcing. The horizontal dashed line indicates the value predicted by the capillary wave turbulence theory α theo = 17/6. (b) Effective energy flux ef f as a function of the steepness s.

We thank CNES (Centre National d' Études Spatiales) for supporting this project. We acknowledge R. Pierre, S. Bonfanti, A. Papadopoulos and N. Bordes for the conception and the realization of the FLUIDICS experiment at Airbus Defence and Space. J. Mignot, L. Oro Marot, A. Llodra-Perez and M. Steckiewicz members of CADMOS at CNES have planned and supervised the operation of the runs. They have been also in charge of the communication with the ISS crew and the transmission of data. E. F. and S. F. thank the funding of ANR-17-CE30-0004 Dysturb.


A Vansteene

J P Jasmin

G De

La Cruz

S Cavadias

C Mariet

G Cote

Prediction of the formation of segmented flow in microsystems for radiochemical liquid-liquid extraction

PREDICTION OF THE FORMATION OF SEGMENTED FLOW IN MICROSYSTEMS FOR RADIOCHEMICAL LIQUID-LIQUID EXTRACTION

One of the most important separation techniques in radiochemical procedures is solvent extraction. It involves concentrated acids and corrosive solvents that require microsystems built with robust materials. In the last decade, a growing interest in its use in microsystems has emerged because such systems allow a good control of both the interface area between aqueous and organic phases, and the contact time of the two phases. After a study of parallel flow [1], our goal is to understand and compare the formation of segmented flow for T, flow focusing, and co-flow junctions, and to predict the optimal design in order to enhance solvent extraction.

OVERVIEW

Limitation of parallel flow:

 Interface area cannot be optimized  Liquid-liquid extraction controlled only by diffusion Not suitable for low kinetic systems.

COMSOL®

The interface area between the two phases is always greater for droplet-based microfluidics than parallel flows (~5 times higher when optimized).

The size of droplets and their generation rate are shown to be key parameters for liquid-liquid extraction optimization. Microsystems design is therefore investigated in order to determine the operating conditions to control the size and frequency of the droplets, and eventually fasten microsystems prototyping for specific liquid-liquid extractions.

Modelling is an essential asset for rapid prototyping of microsystems in expensive materials resistant to acids and corrosive solvents used in radiochemistry. [6] Utada A S et al., Science, 2005, 308: 537-541.



  Very common geometry  Allows high throughput (>100Hz) and monodisperse (<1%) droplet generation [2]  Difficult to form tiny droplets (smaller than channel size)  Allows double emulsions (i.e. w/o/w emulsions)  Access to tiny droplets' volume (i.e. sub-micron)  Polydispersity  Satellite droplets  Very high throughput droplet generation (>10kHz)  Insertion of a small capillary in another one  High dead volume  Satellite droplets Vansteene A. 1 , Jasmin J.P. 1 , De La Cruz G. 2 , Cavadias S. 2 , Mariet C. 1 , Cote G. 3 1) Den -Service d'Etudes Analytiques et de Réactivité des Surfaces (SEARS), CEA, Université Paris-Saclay, F-91191, Gif sur Yvette, France 2) UPMC -Univ Paris 06, 4 Place Jussieu, 75005 Paris, France 3) PSL, Chimie ParisTech -CNRS, Institut de Recherche de Chimie Paris, 75005, Paris, France



  Fig. 4 : CFD simulation of multiphase flow in a T-junctionVideo accessible online at : https://www.comsol.fr/model/download/11070/droplet_breakup.swf

Fig. 1 :Flow

 1 Fig. 1 : Interface area comparison between parallel flows and droplet-based microfluidics

Fig. 3 :

 3 Fig. 3 : Comparison between the interface area generated in a flow-focusing geometry operating in the droplet jetting regime and parallel flow in a similar microsystem [6]