Nicolás Torres
email: nicolas.torres_escorza@sorbonne-universite.fr

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

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

A multiple time renewal equation for neural assemblies with elapsed time model

Keywords: 2010 Mathematics Subject Classification. 35B40, 35F20, 35R09, 92B20 Structured equations, Renewal equation, Mathematical neuroscience, Neural networks, Doeblin theory

We introduce and study an extension of the classical elapsed time equation in the context of neuron populations that are described by the elapsed time since last discharge. In this extension, we incorporate the elapsed time since the penultimate discharge and we obtain a more complex system of integro-differential equations. For this new system, we prove convergence with exponential rate to stationary state by means of Doeblin's theory in the case of weak non-linearities using an appropriate functional setting, inspired by the case of the classical elapsed time equation. Moreover, we present some numerical simulations to observe how different firing rates can give different types of behaviors and to contrast them with theoretical results of both the classical and extended models.

Introduction

In the study and modelling of neural processes, population density models have proved to be a useful approach to understand brain phenomena at different scales. Among these models, we can mention for example the Wilson-Cowan model [START_REF] Hugh | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF] and Amari model [START_REF] Amari | Dynamics of pattern formation in lateral-inhibition type neural fields[END_REF], which describe dynamics of interactions between inhibitory and excitatory neurons, and they have been widely studied in different extensions with many theoretical results and simulations.

Another class of population-based models is the elapsed time model, which has attracted the attention of many researchers. In this model we consider a neural network where neurons are described by their elapsed time since the last discharge as the key variable. After receiving some stimulation, neurons spike and interact with other neurons leading them to spike as well.

The elapsed time model is closely related to the well-known integrate-and-fire model. This system describes the dynamics of the membrane potential and has been studied by several authors such as Carrillo et al. [START_REF] María | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF][START_REF] José | Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience[END_REF], Perthame et al. [START_REF] Perthame | On a voltage-conductance kinetic system for integrate & fire neural networks[END_REF][START_REF] Perthame | Distributed synaptic weights in a lif neural network and learning rules[END_REF][START_REF] Perthame | Derivation of a voltage density equation from a voltageconductance kinetic model for networks of integrate-and-fire neurons[END_REF] and Zhou et al. [START_REF] Liu | Rigorous justification of the Fokker-Planck equations of neural networks based on an iteration perspective[END_REF] in different variants and approaches. Like the integrate-and-fire model [START_REF] Delarue | Particle systems with a singular mean-field selfexcitation. Application to neuronal networks[END_REF][START_REF] Delarue | Global solvability of a networked integrate-and-fire model of McKean-Vlasov type[END_REF], the elapsed time model is related to the limit of stochastic processes at microscopic scale and the connection with Poisson processes was established in the works of Chevalier et al. [START_REF] Chevallier | Microscopic approach of a time elapsed neural model[END_REF][START_REF] Chevallier | Mean-field limit of generalized hawkes processes[END_REF]. Other important works on spiking neurons include Brunel [START_REF] Brunel | Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons[END_REF], Gerstner et al. [START_REF] Gerstner | Spiking neuron models: Single neurons, populations, plasticity[END_REF], Ly et al. in [START_REF] Ly | Spike train statistics and dynamics with synaptic input from any renewal process: a population density approach[END_REF] and Pham et al. [START_REF] Pham | Activity in sparsely connected excitatory neural networks: effect of connectivity[END_REF] and by Schwalger et al. [START_REF] Pietras | Low-dimensional firing-rate dynamics for populations of renewal-type spiking neurons[END_REF][START_REF] Schwalger | Mind the last spike-firing rate models for mesoscopic populations of spiking neurons[END_REF]. The relation between integrate-and-fire and the elapsed time model was studied in Dumont et al. [START_REF] Dumont | Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model[END_REF][START_REF] Dumont | A theoretical connection between the noisy leaky integrate-and-fire and the escape rate models: the non-autonomous case[END_REF].

From a mathematical and analytical point of view, the elapsed-time model has been studied by several authors using different techniques such as Cañizo et al. in [START_REF] José | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF], Kang et al. [START_REF] Kang | Dynamics of time elapsed inhomogeneous neuron network model[END_REF], Mischler et al. in [START_REF] Mischler | Relaxation in time elapsed neuron network models in the weak connectivity regime[END_REF][START_REF] Mischler | Weak and strong connectivity regimes for a general time elapsed neuron network model[END_REF] and the pioneering works of Pakdaman et al. in [START_REF] Khashayar Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Khashayar Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF][START_REF] Khashayar Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF]. Moreover, different extensions of the elapsed time model have been considered by incorporating new elements such as spatial dependence with connectivity kernel in Salort et al. [START_REF] Torres | An elapsed time model for strongly coupled inhibitory and excitatory neural networks[END_REF], a leaky memory variable in Fonte et al. [START_REF] Fonte | Long time behavior of an age and leaky memorystructured neuronal population equation[END_REF] and a stochastic extension that accounts for finite population size in Schmutz et al. [START_REF] Schmutz | On a finite-size neuronal population equation[END_REF] and Schwalger et al. [START_REF] Schwalger | Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size[END_REF].

The aim of the present work is to extend the classical elapsed time model by taking into account the elapsed time since the penultimate discharge in addition to the last one. This extension is modeled through a multiple time renewal equation, whose study is more complex than that of the classical model. We aim to analyze how the incorporation of a second elapsed time changes dynamics of the classical model and what type of new patterns may arise.

From a biophysical point of view, the main motivation is to represent the dependence on the previous history of spikes in neural dynamics. Examples of models incorporating this approach include the works of Gerstner et al. [START_REF] Gerstner | Neuronal dynamics: From single neurons to networks and models of cognition[END_REF] on the spike-response-model with escape noise, where the sum over the spike history is truncated at the second last spike. Among stochastic models, Chevallier et al. [START_REF] Chevallier | Microscopic approach of a time elapsed neural model[END_REF] consider Wold processes, which are point processes for which the next occurrence of a spike depends on the previous occurrence but also on the previous inter-spike interval. In their work, a connection with Wold processes and multiple-time renewal equations is established by deriving the PDE for the corresponding density. They extended the homogeneous Markovian case, which is study via the forward Kolmogorov's equations, to more general non-Markovian systems with a method called Ogata's thinning algorithm. They also included examples of other stochastic models like Hawkes processes.

Furthermore, multiple time renewal equations have been investigated in Fournier et al. [START_REF] Fournier | A nonexpanding transport distance for some structured equations[END_REF] in the linear case, where a non-expanding distance was introduced via a coupling argument. Moreover, these type of equations have multiple potential applications in other domains such as epidemiology, where the evolution of individuals under possible secondary infections has been studied. For an example, see the work of Ferretti et al. [START_REF] Ferretti | Quantifying sars-cov-2 transmission suggests epidemic control with digital contact tracing[END_REF].

The extended model is described as follows. Let n = n(t, s, a) the probability density of finding a neuron at time t, such that the elapsed times since its last and penultimate discharge are respectively s and a. Hence, for all t ≥ 0, the domain of definition of n in the elapsed time variables is

D := {(s, a) ∈ R 2 : 0 ≤ s ≤ a}.
In this setting, neural dynamics are modeled through the following nonlinear renewal system

           ∂ t n + ∂ s n + ∂ a n + p(s, a, X(t))n = 0 t > 0, a > s > 0, n(t, s = 0, a) = N (t, a) with N (t, a) := ∞ a p(a, u, X(t))n(t, a, u) du t > 0, a > 0, X(t) = ∞ 0 N (t, a) da t > 0, n(t = 0, s, a) = n 0 (s, a) a > s > 0.
(

) 1
We assume that the initial data n 0 ∈ L 1 (D) is a probability density so that System (1) formally verifies

n(t, s, a) da ds = n 0 (s, a) da ds = 1, n(t, s, a) ≥ 0 ∀t ≥ 0. (2)
As in the classical elapsed time model the function p : D × R → R is the firing rate of neurons, which depends on the total activity X(t). Furthermore, for the firing rate function p, we assume that there exist σ, p 0 , p ∞ > 0 such that

p 0 1 {a>s≥σ} ≤ p ≤ p ∞ . (3)
Thus, we get

0 ≤ X(t) ≤ p ∞ , ∀t ≥ 0. (4)
We assume for simplicity that p ∈ W 1,∞ (D × R), although most of the theoretical results are also valid for firing rates with simple jump discontinuities and the behavior of solutions does not depend on this regularity assumption as we show in the numerical simulations. Furthermore, we say that the network is inhibitory if p is decreasing with respect to the total activity X and excitatory if p is increasing. If in addition ∂ X p ∞ is small, we say that System (1) is under a weak interconnection regime.

The function N (t, a) represents the flux of discharging neurons, conditioned to the elapsed time since last discharge with value a > 0. The boundary condition of n at s = 0 states that, when a neurone discharges, its last elapsed time resets to the penultimate elapsed time. The function X models the total activity received by the network, and we assume that it is given by the integral of N (t, a) with respect to all the times a > 0.

Finally, we also remark that when p does not depend on a, the probability density

m(t, s) := ∞ s n(t, s, a) da satisfies the equation      ∂ t m + ∂ s m + p(s, X(t))m = 0 t > 0, s > 0, m(t, s = 0) = X(t) = ∞ 0 p(u, X(t))m(t, u) du t > 0, m(t = 0, s) = ∞ s n 0 (s, a) da s > 0. (5)
In other words, the probability with respect to the last elapsed time is a solution of the classical elapsed time equation. In particular, if we consider firing rates of the form p = ϕ(X(t))1 {s>σ} , with ϕ ∈ W 1,∞ (R) strictly positive and σ > 0, we know from Torres et al. [START_REF] Torres | An elapsed time model for strongly coupled inhibitory and excitatory neural networks[END_REF] that periodic solutions and jump discontinuities may occur. Based on this choice of p, we discuss the impact of taking into account the penultimate discharge time regarding to the emergence of periodic solutions of System (1).

Equation [START_REF] Amari | Dynamics of pattern formation in lateral-inhibition type neural fields[END_REF] represents the probability distribution of neurons in a large (infinite) all-to-all network which can be described as follows. For each neuron, labeled by i = 1, ..., N , the last two spike times S i (t), A i (t) are recorded. They follow a Piecewise Deterministic Markov Process where, between the spike times τ i,j , the 'ages' progress as

Ṡi (t) = 1, Ȧi (t) = 1, τ i,j-1 < t < τ i,j , i = 1, ..., N,
and when a discharge occurs, the penultimate time spike is renewed:

A i (τ + i,j ) = 0, S i (τ + i,j ) = A i (τ - i,j ).
The τ i,j are chosen randomly according to a Poisson distribution of parameter p(S i (t), A i (t), X δ (t)), where the network activity is computed as

X δ (t) = 1 N i,j ω δ (t -τ i,j ),
whith ω δ a smoothing kernel of parameter δ which takes into account the transmission delay. In the limit N → ∞ and δ → 0, one expects the neurons become independent, driven by the average discharge rate E[X δ (t)] ≈ p(s, a, X(t)n(t, s, a) ds da.

In the study of age-structured models, the entropy method has been a useful tool for proving convergence to the steady state. The main idea consists in finding a Lyapunov functional H[n] and a dissipation functional D H [n] such that the solutions of the system satisfy

d dt H[n] = -D H [n] ≤ 0.
Thus if we can find a Poincaré inequality of the type λH[n] ≤ D H [n] for some λ > 0, we can deduce the exponential decay of H[n] by using the classical Gronwall inequality, which eventually allows to deduce convergence to the steady state in some convenient norm. This method was developed in the works of [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF][START_REF] Perthame | Transport equations in biology[END_REF] with extensions to measure initial data in [START_REF] Gwiazda | Generalized entropy method for the renewal equation with measure data[END_REF], and it has been applied to different types of models. However, when such inequalities are not available the study of asymptotic behavior becomes more complex.

Another important approach, that we use in this paper, is Doeblin's theory, which was first introduced in the context of Markov chains [START_REF] Doeblin | Sur deux problèmes de m. kolmogoroff concernant les chaînes dénombrables[END_REF][START_REF] Doeblin | Éléments d'une théorie générale des chaînes simples constantes de markoff[END_REF] and later developed in the work of Harris [START_REF] Harris | The existence of stationary measures for certain markov processes proceedings of the third berkeley symposium on mathematical statistics and probability 1954-1955 2 univ[END_REF] which has inspired several works such as Bansaye et al. [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized doeblin's conditions[END_REF], Cañizo et al. [START_REF] José | Spectral gap for the growth-fragmentation equation via harris's theorem[END_REF] and Hairer & Mattingly [START_REF] Hairer | Yet another look at harris' ergodic theorem for markov chains[END_REF]. It ensures that exponential convergence in L 1 to the stationary state follows from the existence of a probability measure ν, a time t 0 , and 0 < α, such that for all initial data which is a probability measure, the solution at time t 0 is bounded from bellow by αν. Several works in PDE, also consider this approach, see for example the works of Gabriel et al. [START_REF] Gabriel | Measure solutions to the conservative renewal equation[END_REF], B. Perthame et al. [START_REF] Perthame | Derivation of a voltage density equation from a voltageconductance kinetic model for networks of integrate-and-fire neurons[END_REF], Dumont et al. [START_REF] Dumont | The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states[END_REF]. In particular, this theory has already been an alternative to the classical entropy methods to prove exponential convergence to the steady state for a large class of time elapsed model since the last discharge (see Cañizo et al. [START_REF] José | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF]).

The article is organized as follows. In Section 2 we prove that System (1) is well-posed in a suitable space for weak non-linearities. Starting with the asymptotic analysis for the linear case, we prove in Section 3 the existence of a stationary state and exponential convergence via Doeblin's theory. For the non-linear problem in the case of weak interconnections, we show in Section 4 the uniqueness of the steady state and in Section 5 we prove the exponential convergence via a perturbation argument.

Finally in Section 6 we present some examples of numerical simulations for different initial data and class of firing rates.

Well-posedness for weak non-linearities

We prove that System (1) is well-posed in the weak interconnection regime. To do so, we start by studying an auxiliary linear problem where the total activity is fixed and then we proceed to prove well-posedness of System (1) via the Picard fixed point contraction theorem.

This requires some function analytic definitions. For a Banach space X we denote C b ([0, ∞), X ) as the Banach space given by

C b ([0, ∞), X ) := {f : [0, ∞) → X | f is continuous and bounded}.
In this setting we look for weak solutions satisfying

n ∈ C b ([0, ∞), L 1 (D)), so that N ∈ C b ([0, ∞), L 1 (0, ∞)) and X ∈ C b [0, ∞).

The linear problem

Given X ∈ C b [0, ∞), we consider the following linear problem

     ∂ t n + ∂ s n + ∂ a n + p(s, a, X(t))n = 0 t > 0, a > s > 0, n(t, s = 0, a) = N (t, a) := ∞ a p(a, u, X(t))n(t, a, u) du t > 0, a > 0, n(t = 0, s, a) = n 0 (s, a) ≥ 0 a > s > 0. (6)
Even though the boundary condition holds only for t > 0, we may define N (t, a) at t = 0 by the same formula

N (t, a) = ∞ 0 p(a, u, X(t))n(t, a, u) du,
for almost all a > 0, ∀t ≥ 0, which implies that n may have discontinuities along the set {(t, s, a) ∈ [0, ∞) × D : t = s} as a consequence of the method of characteristics (7) explained below.

Lemma 1. Assume that n 0 ∈ L 1 (D) is a probability density and p ∈ W 1,∞ (D × R) satisfies (3). Then for a given X ∈ C b [0, ∞), Equation (6) has a unique weak solution n ∈ C b ([0, ∞), L 1 (D)) with N ∈ C b ([0, ∞), L 1 (0, ∞)).
Moreover n is non-negative and verifies the property (2).

In particular this lemma proves the property (2) for the non-linear System (1).

Proof. From the method of characteristics, we start by noticing that a solution of the linear System (6) satisfies the following fixed point equation

n(t, s, a) = Ψ[n](t, s, a) := n 0 (s -t, a -t)e -t 0 p(t +s-t,t +a-t,X(t ))dt 1 {a>s>t} + N (t -s, a -s)e -s 0 p(s ,s +a-s,X(s +t-s))ds 1 {t,a>s} . (7)
Let T > 0 and

X T := {n ∈ C b ([0, T ], L 1 (D)), n(0) = n 0 }, it readily follows that Ψ maps X T → X T .
We prove by the Picard contraction theorem that Ψ has a unique fixed point in X T for T > 0 small enough, i.e., there exists a unique weak solution of ( 6) defined on

[0, T ]. Consider n 1 , n 2 ∈ X T , we compute |Ψ[n 1 ] -Ψ[n 2 ]|(t, s, a) ds da ≤ t 0 ∞ s |N 1 -N 2 |(t -s, a -s) da ds ≤ T sup t∈[0,T ] ∞ 0 |N 1 -N 2 |(t, a) da ≤ T p ∞ sup t∈[0,T ] n 1 (t, s, a) -n 2 (t, s, a) L 1 (D) , (8)
thus for T < 1 p∞ , we have proved that Ψ is a contraction and there exists a unique n ∈ X T such that Ψ[n] = n. Since the choice of T is independent of n 0 , we can reiterate this argument to get a unique solution of ( 6), which is defined for all t ≥ 0.

From Formula (7) we can extend the notion of a weak solution for Equation [START_REF] José | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF] for an initial data n 0 ∈ (M(D), • M 1 ), the space of finite regular measures on D endowed with the total variation norm (see section 3 for the definition). Therefore we can use the same argument to prove existence and uniqueness of a weak solution

n ∈ C b ([0, ∞), M(D)) with N ∈ C b ([0, ∞), M(0, ∞)) and X ∈ C b [0, ∞).
Next, we prove the mass conservation property. For all t ≥ 0, consider S t : M(D) → M(D) the semi-group given by

S t [f ](s, a) = f (s -t, a -t)1 {a>s>t} ,
whose infinitesimal generator is the operator Lf = -∂ s f -∂ a f . From Duhamel's formula, the solution of the fixed point problem (7) also verifies the following equality

n(t, s, a) = S t [n 0 ](s, a) + t 0 S t-τ [δ {s=0} (s, a)N (τ, a)] dτ - t 0 S t-τ [p(s, a, X(τ ))n(τ, s, a)] dτ, (9)
where δ {s=0} (s, a) is the measure along the line {(0, a) : a ≥ 0}. This formula is translated as

n(t, s, x) = n 0 (s -t, a -t)1 {a>s>t} + N (t -s, a -s)1 {t,a>s} - t 0 p(s -t + τ, a -t + τ, X(τ ))n(τ, s -t + τ, a -t + τ )1 {a>s>t-τ } dτ, (10)
and we get the mass conservation property by integrating with respect to (s, a) on the domain D.

Finally, since n 0 is non-negative then Ψ preserves positivity, so by uniqueness of the fixed point, the corresponding solution n is also non-negative.

The non-linear problem

Like the classical elapsed-time model, studying well-posedness for a general firing rate is a difficult problem. Indeed, examples of multiple solutions for a same initial data has been observed in Torres et al. [START_REF] Torres | An elapsed time model for strongly coupled inhibitory and excitatory neural networks[END_REF] and some ill-posed examples have been remarked on Cañizo et al. [START_REF] José | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF].

As in the linear problem, and with the assumptions below, we may define the quantities

N (0, a) = ∞ 0 p(a, u, X(0)) n 0 (a, u) du
with X(0) the solution of

X(0) = ∞ 0 ∞ a p(a, u, X(0)) n 0 (a, u) du da. ( 11
)
But again, the boundary condition is not imposed at t = 0 and in general X(0) = ∞ 0 n 0 (0, a) da. By using the results of the linear problem, we are now ready to prove that System (1) is well-posed in the case of weak interconnection.

Theorem 1 (Well-posedness for weak interconnections). Assume that

n 0 ∈ L 1 (D) is a probability density and that p ∈ W 1,∞ (D × R) satisfies (3). Then for ∂ X p ∞ < 1, (12)
System (1) has a unique solution with n ∈ C b ([0, ∞), L 1 (D)), N ∈ C b ([0, ∞), L 1 (0, ∞)) and X ∈ C b [0, ∞).
Moreover n verifies Condition (2) for all t > 0.

Proof. Consider T > 0. We fix a function X ∈ C b [0, ∞) and define the functions

n ∈ C b ([0, ∞), L 1 (D)) and N ∈ C b ([0, ∞), L 1 (0, ∞))
which are solutions of System (6) by Lemma 1. Furthermore, the solution of this linear equation satisfies [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized doeblin's conditions[END_REF].

So we have a solution of System (1) defined on [0, T ] if X satisfies for all 0 ≤ t ≤ T and x ∈ Ω, the following fixed point condition

X(t) = T [X](t) := ∞ 0 N [X](t, a) da. (13)
We prove that T defines for all T > 0 an operator that maps X T → X T with X T := C b ([0, T ]). First, we observe the following estimate

N (t, a) da ≤ p ∞ , ∀t ∈ [0, T ], (14)
and it is immediate that T [X] ∈ X T .

We now prove that for T small enough, T is a contraction. Let X 1 , X 2 ∈ X T with their respective solutions (n 1 , N 1 ), (n 2 , N 2 ) of System [START_REF] José | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF]. For the difference between N 1 and N 2 we have

|N 1 -N 2 |(t, a) da ≤ |p(a, u, X 1 ) n 1 (t, a, u) -p(a, u, X 2 ) n 2 (t, a, u)| du da ≤ |p(a, u, X 1 ) -p(a, u, X 2 )| n 1 du da + p(a, u, X 2 )|n 1 -n 2 |(t, a, u) du da ≤ ∂ X p ∞ X 1 -X 2 ∞ + p ∞ n 1 -n 2 L 1 (D) . (15)
To estimate the difference between n 1 and n 2 , we combine this inequality with [START_REF] Delarue | Particle systems with a singular mean-field selfexcitation. Application to neuronal networks[END_REF] and get

n 1 -n 2 L 1 (D) ≤ 2T ∂ X p ∞ X 1 -X 2 ∞ + 2T p ∞ n 1 -n 2 L 1 (D) .
Then, for T < 1 2p∞ , we obtain

n 1 -n 2 L 1 (D) ≤ 2T ∂ X p ∞ 1 -2T p ∞ X 1 -X 2 ∞ . (16)
Finally, by using again estimate [START_REF] Dumont | Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model[END_REF], the operator T satisfies

T [X 1 ] -T [X 2 ] ∞ ≤ ∂ X p ∞ 1 + 2T p ∞ 1 -2T p ∞ X 1 -X 2 ∞ . (17)
Therefore, for ∂ X p ∞ < 1 and T small enough, T is a contraction.

From Picard's fixed point we get a unique X ∈ X T such that T [X] = X, and this implies the existence of a unique solution of (1) defined on [0, T ]. Since estimate ( 14) is uniform in T , we can iterate this argument to get a unique solution of (1) defined for all t > 0.

Furthermore, we conclude from this construction that the non-linear System (1) satisfies (2) like the linear System (6).

Asymptotic behavior for the linear case

In order to study the behavior of System (1), we start by studying the case when X ≥ 0 is a fixed constant. Thus we consider the linear problem given by

     ∂ t n + ∂ s n + ∂ a n + p(s, a, X)n = 0 t > 0, a > s > 0, n(t, s = 0, a) = N (t, a) := ∞ a p(a, u, X)n(t, a, u) du t > 0, a > 0, n(t = 0, s, a) = n 0 (s, a) a > s > 0. (18)
To determine the behavior of System (18), we consider (n X , N X ) as the solution of the steady state problem given by

∂ s n + ∂ a n + p(s, a, X)n = 0 a > s > 0, n(s = 0, a) = N (a) := ∞ a p(a, u, X)n(a, u) du a > 0, (19)
In the classical elapsed time model the generalized relative entropy inequality for the linear problem is a well-known property of this class of age-structured models. For a reference on this method, see Perthame et al. [START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF] and for some applications in the non-linear elapsed time model see for instance [START_REF] Khashayar Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Kang | Dynamics of time elapsed inhomogeneous neuron network model[END_REF]. Following these ideas, we can prove this property for the linear System [START_REF] Fonte | Long time behavior of an age and leaky memorystructured neuronal population equation[END_REF].

Proposition 1 (Generalized relative entropy). Assume there exists a steady state solution of the linear System (18) with n X , N X > 0. Then for all convex functions H : [0, ∞) → [0, ∞) with H(0) = 0, the solution n of the linear System (18) satisfies

d dt n X (s, a)H n(t, s, a) n X (s, a) da ds = -D H [n(t, s, a)] ≤ 0 ∀t ≥ 0, D H [n(t, s, a)] = p(s, a, X)n X (s, a)H n(t, s, a) n X (s, a) da ds -N X (a)H N (t, a) N X (a) da, (20)
and in particular the steady state is unique.

Proof. In order to prove the relative entropy property, we follow the arguments in [START_REF] María | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF]. We start by noticing the following identities

∂ s n = n X ∂ s n n X + n n X ∂ s n X , ∂ a n = n X ∂ a n n X + n n X ∂ a n X , (21)
and for simplicity we reformulate Equation ( 18) as follows

     ∂ t n + ∂ s n + ∂ a n + p(s, a, X)n = δ {s=0} (s, a)N (t, a) t > 0, a > s > 0, n(t, s = 0, a) = 0 t > 0, a > 0, n(t = 0, s, a) = n 0 (s, a) a > s > 0, (22)
where δ {s=0} (s, a) is the measure along the line {(0, a) : a ≥ 0}. In the same way, we reformulate the corresponding steady state problem [START_REF] Fournier | A nonexpanding transport distance for some structured equations[END_REF].

∂ s n X + ∂ a n X + p(s, a, X)n X = δ {s=0} (s, a)N X (a) a > s > 0, n X (s = 0, a) = 0 a > 0. ( 23
)
Hence by using the identities (21) along with Equations ( 22) and ( 23), we get the following equation

for n n X ∂ t n n X + ∂ s n n X + ∂ a n n X = δ {s=0} (s, a) N X n X N N X - n n X
and if we multiply this equality by

H n n X , we get ∂ t H n n X + ∂ s H n n X + ∂ a H n n X = δ {s=0} (s, a) N X n X N N X - n n X H n n X .
Therefore, by multiplying the latter equality by n X and using Equation ( 23), we have the correspond-

ing equation for u = n X H n n X ∂ t u + ∂ s u + ∂ a u + p(s, a, X)u = δ {s=0} (s, a)N X N N X - n n X H n n X + H n n X . (24)
Finally, by noticing the following limit

lim s→0 n(t, s, a) n X (s, a) = N (t, a) N X (a)
, for a.e. t, a > 0, we conclude the generalized relative entropy property (20) by integrating Equation ( 24) with respect to (s, a) on the domain D. Moreover, we observe that D H [•] is non-negative by applying Jensen's inequality with the probability measure dµ = p(a, y) n X (a,y) N X (a) dy for each a > 0. In particular when H is strictly convex and D H [n] = 0, we deduce that n n X is constant and subsequently we get n = n X , since both n, n X are probability densities. Therefore, the steady state is unique.

If we consider the entropy method to prove exponential convergence for the linear Equation ( 18) in L 1 (D), we have the following equality for

H(•) = | • | d dt |n -n X | da ds = p(n -n X ) da ds - p|n -n X | da ds ≤ 0.
We observe that for the right-hand side it is not evident to find a L 1 Poincaré inequality involving the entropy in order to conclude exponential convergence, unlike the classical elapsed time model where this approach has proved to be useful (see for example Perthame [START_REF] Perthame | Transport equations in biology[END_REF]).

Furthermore, in Proposition (1) we assumed that n X and N X are strictly positive, which is not necessarily true. Unlike the classical elapsed time model, there exist solutions where n X and N X vanish for some values of (s, a). For example, consider p(s, a, X) = 1 {s>1} which satisfies the bounds (3).

From the method of characteristics we get the following equality n X (s, a) = N X (a -s)e -(s-1)+ , a > s, so that form the boundary condition at s = 0, we observe that N X satisfies

N X (a) = 1 {a>1} e -(a-1) ∞ 0 N X (u) du,
which implies that N X (a) vanishes for a < 1 and subsequently n X (s, a) vanishes for a -s < 1.

Due to the limitations of the entropy method approach we will make use of Doeblin's theory, which will be the key ingredient in proving convergence to steady state. In this context we start by recalling the useful concepts in order to apply Doeblin's theorem. Consider (M(X ), • M 1 ) the space of finite signed measures on the space X with the norm of the total variation

µ M 1 := µ + (X ) + µ -(X ), (25)
where µ = µ + -µ -is the Hahn-Jordan decomposition of the measure µ into its positive and negative parts. For simplicity of the computations, we will treat measures as if they were L 1 functions and we simply write the L 1 -norm instead of M 1 -norm.

We now recall the definition of a Markov semigroup and Doeblin's condition.

Definition 1 (Markov semi-group). Let (X , A) be a measure space and P t : M(X ) → M(X ) be a linear semi-group. We say that P t is a Markov semi-group if P t µ ≥ 0 for all µ ≥ 0 and X dP t µ = X dµ for all µ ∈ M(X ). In other words, (P t ) preserves the subset of probability measures P(X ).

Definition 2 (Doeblin's condition). Let P t : M(X ) → M(X ) be a Markov semi-group. We say that (P t ) satisfies Doeblin's condition if there exist t 0 > 0, α ∈ (0, 1) and ν ∈ P(X ) such that P t0 µ ≥ αν ∀µ ∈ P(X ).

Under this functional setting, we are now ready to state Doeblin's theorem as follows.

Theorem 2 (Doeblin's Theorem). Let P t : M(X ) → M(X ) be a Markov semi-group that satisfies Doeblin's condition. Then the semigroup has a unique equilibrium µ * ∈ P(X ). Moreover, for all µ ∈ M(X ) we have

P t µ -µ µ * M 1 ≤ 1 1 -α e -λt µ -µ µ * M 1 ∀t ≥ 0, with µ = X dµ and λ = -ln(1-α) t0 > 0.
For a proof of Doeblin's Theorem, see for example [START_REF] Gabriel | Measure solutions to the conservative renewal equation[END_REF].

From Lemma 1, the solution of the linear problem ( 18) determines a Markov semi-group acting on L 1 (D). By means of Doeblin's theory, the solutions of linear Equation ( 18) converge exponentially to a unique steady state, as we assert in the following theorem.

Theorem 3. Let n 0 ∈ L 1 (D) be a probability density and assume that p smooth satisfies Assumption (3).

Then for a fixed X > 0, there exists a unique stationary solution n X (s, a) ∈ L 1 (D) of the linear Equation (18) satisfying n X (s, a) da ds = 1. Moreover, the corresponding solution of Equation [START_REF] Fonte | Long time behavior of an age and leaky memorystructured neuronal population equation[END_REF] satisfies

n(t) -n X L 1 s,a ≤ 1 1 -α e -λt n 0 -n X L 1 s,a ∀t ≥ 0, with α = 1 2 p 2 0 σ 2 e -3p∞σ and λ = -log(1-α) 3σ > 0.
In order to obtain the result, we show that after some time the solution of the linear problem is uniformly bounded from below for all probability densities. Thus from Doeblin's theorem we get the exponential convergence to equilibrium.

Lemma 2. Assume (2) and (3). Let n(t, s, a) be a solution of [START_REF] Fonte | Long time behavior of an age and leaky memorystructured neuronal population equation[END_REF], then there exist t 0 > 0, α ∈ (0, 1) and a probability density ν ∈ L 1 such that n(t 0 , s, a) ≥ αν(s, a).

Proof. The main idea of the proof is to control the mass transported along the lines of direction (1, 1). Firstly, we observe the transport of the initial data n 0 . From Assumption (3) and the characteristics Formula (7) the following inequality holds

∞ t ∞ s n(t, s, a) da ds ≥ e -p∞t , ∀t ≥ σ. ( 26
)
Secondly, we see the mass that returns at s = 0. From ( 26) we get for all t ≥ σ

∞ t n(t, s = 0, a) da = ∞ t N (t, a) da ≥ p 0 ∞ t ∞ a n(t, a, u) du da ≥ p 0 e -p∞t . (27)
This means that we reduced by one dimension the problem of finding the uniform lower bound. For t ≥ σ the mass of the region {(s, a) : a > s > t} concentrates in the line {(0, a) : a ≥ t}, as we see in Figure 1.

s a t ≥ σ
Figure 1: First reduction of dimension. For a t ≥ σ, all points in D are transported to the red region, which has a total mass of at least e -p∞t . Then a mass of at least p 0 e -p∞t returns to the green line.

Thirdly, in order to control the point values of n(t, s, a), we regard the values of N (t, a). Observe that from Formula (7) we have

n(t, s, a) ≥ N (t -s, a -s)e -s 0 p(s ,s +a-s,X)ds 1 {t,a>s} ≥ N (t -s, a -s)e -p∞s 1 {t,a>s} , (28)
thus for a ≥ σ and t -a > σ, we obtain by using again Assumption (3) that

N (t, a) ≥ p 0 ∞ a n(t, a, u) du ≥ p 0 e -p∞a ∞ a N (t -a, u -a) du = p 0 e -p∞a ∞ 0 N (t -a, u) du ≥ p 0 e -p∞a ∞ t-a N (t -a, u) du ≥ p 2 0 e -p∞t . (29)
This means we reduced the problem of finding the uniform lower bound by one dimension again, as we see in Figure 2.

Finally, once we have estimated N (t, a) from below, we come back to estimate [START_REF] Luçon | Periodicity and longtime diffusion for mean field systems in R d[END_REF] to conclude that for a -s ≥ σ and t -a > σ we have

n(t, a, s) ≥ N (t -s, a -s)e -p∞s 1 {t,a>s} ≥ p 2 0 e -p∞t 1 {t-a,a-s>σ} , (30)
so that we can choose t = 3σ and conclude that n(3σ, a, s) ≥ p 2 0 e -3p∞σ 1 {2σ>a>s+σ} .

Therefore we get the desired result with t 0 = 3σ, α = 1 2 σ 2 p 2 0 e -3p∞σ ∈ (0, 1) and ν given by whose support is contained in orange region of Figure 3.

ν(s, a) = 2 σ 2 1 {2σ>a>s+σ} ,
From Lemma 2 the hypothesis of Doeblin's theorem are verified and Theorem 3 readily follows.

Concerning the conditioned activity N in System [START_REF] Fournier | A nonexpanding transport distance for some structured equations[END_REF], we conclude from Theorem 3 that for X fixed, there is a unique stationary N X ∈ L 1 (0, ∞) determined by the method of characteristics through the formula

n X (s, a) = N X (a -s) exp - s 0 p(s , a -s + s , X) ds , a > s. (31)
By replacing this expression in the boundary condition at s = 0, we obtain the following integral equation for N X (a)

N X (a) = T X [N X ](a), (32)
with

T X : L 1 (0, ∞) → L 1 (0, ∞) given by T X [N ](a) := ∞ 0 p(a, u + a, X) exp - a 0 p(s , u + s , X) ds N (u) du = - ∂ ∂a ∞ 0 exp - a 0 p(s , u + s , X) ds N (u) du.
Moreover, by integrating Equation ( 31) we get

∞ 0 ∞ 0 N X (a) exp - s 0 p(s , a + s , X) ds da ds = 1. (33)
Therefore we conclude that finding a function N ∈ L 1 (0, ∞) satisfying Equation [START_REF] Mischler | Relaxation in time elapsed neuron network models in the weak connectivity regime[END_REF] and Condition ( 33) is equivalent to finding a steady state n X (s, a) in Equation [START_REF] Fournier | A nonexpanding transport distance for some structured equations[END_REF]. The integral Equation (32) will play an important role in the analysis of the non-linear System (1), thus we prove the following two lemmas on the operator T X that will be useful in the sequel.

Lemma 3. Assume that p Lipschitz satisfies Assumption (3). For each X > 0 the operator T X is compact and it satisfies that dim ker(I -T X ) = 1, which is generated by a non-negative function, and

ran(I -T X ) = f ∈ L 1 (0, ∞) : ∞ 0 f (x) dx = 0 .
Proof. The first step is to prove that T X is a compact operator. This means we have to prove that the set

A = {T X [f ] : ||f || 1 ≤ 1} is relatively compact in L 1 . First observe that T X [f ] 1 ≤ p ∞ for all f with ||f || 1 ≤ 1, so A is bounded. Second, we prove that ∞ r |T X [f ](a)|da → 0 uniformly when r → ∞. Indeed for r > σ we have ∞ r |T X [f ](a)|da ≤ p ∞ ∞ r ∞ 0 |f (u)|e -a 0 p(s ,u+s ,X) ds du da ≤ p ∞ ∞ r ∞ σ |f (u)|e -p0(a-σ) du da ≤ p ∞ e p0σ ∞ σ |f (u)|du ∞ r e -p0a da ≤ p ∞ e p0σ e -p0r p 0 → 0.
Now we prove the equicontinuity property. Observe that

d da T X [f ](a) = ∞ 0 (∂ s p + ∂ a p)(a, u + a, X)e -a 0 p(s ,u+s ,X) ds f (u)du - ∞ 0 p(a, u + a, X) 2 e -a 0 p(s ,u+s ,X) ds f (u)du,

Steady states

Consider n * = n * (s, a) with support in the set {s ≤ a}. We are interested in the stationary solutions of the non-linear System (1) given by

           ∂ s n + ∂ a n + p(s, a, X)n = 0 a > s > 0, n(s = 0, a) = N (a) := ∞ 0 p(a, u, X)n(a, u) du a > 0, X = ∞ 0 N (a) da, n(s, a) da ds = 1, n(s, a) ≥ 0. (34)
We define N X as the respective conditional activity in terms of X. In order to have a steady state of the non-linear Problem (1), we must find X > 0 such that

X = Φ(X) := ∞ 0 N X (a) da. (35)
In the general case this equation has always a solution since the right-hand side is uniformly bounded thanks to estimate (4) and N X (a) depends continuously on X. By using the properties of the operator T X , we prove that under the weak interconnections regime the non-linear System (1) has a unique steady state.

Theorem 4. Assume (2) and that p smooth satisfies Assumption (3). Then for ∂ X p ∞ small enough, System

(1) has a unique steady state (n * , N * , X * ).

Proof. The goal is to prove that Φ is a contraction in order to obtain a unique fixed point. In order to estimate ∂ X N we make use of the implicit function theorem. By differentiating Equation [START_REF] Khashayar Pakdaman | Dynamics of a structured neuron population[END_REF] we get ∂ X (N X )e -s 0 p(s ,a+s ,X) ds da ds = N X e -s 0 p(s ,a+s ,X) ds s 0 ∂ X p(s , u + s , X) ds da ds.

(36) Furthermore, if we differentiate with respect to X the Equation (32), we get

∂ X N X (a) = ∞ 0 -∂ a e -a 0 p(s ,u+s ,X) ds ∂ X N X (u) du + ∞ 0 -∂ a ∂ X e -a 0 p(s ,u+s ,X) ds N X (u) du, i.e. ∂ X N X (a) satisfies the equation (I -T X )[∂ X N ](a) = ∞ 0 p(a, u + a, X)e -a 0 p(s ,u+s ,X) ds N X (u) du - ∞ 0 p(a, u + a, X) a 0 ∂ X p(s , u + s , X
) ds e -a 0 p(s ,u+s ,X) ds N X (u) du.

(37)

By using the implicit function theorem and the Condition [START_REF] Perthame | Transport equations in biology[END_REF] we can define an inverse of I -T X which depends continuously on X. Observe that (I -T X ) -1 is uniformly bounded on X in the operator norm, since X is uniformly bounded. Thus, for the function Φ we get

|Φ (X)| = ∂ X N X (a) da ≤ (I -T X ) -1 ∂ X p ∞ (1 + p ∞ a)e -a 0 p(s ,u+s ,X) ds N X (u)da du ≤ C ∂ X p ∞ (1 + p ∞ a)e -p0a N X (u) da du ≤ C ∂ X p ∞ p ∞ (1 + p ∞ a)e -p0a da ,
so that for ∂ X p ∞ small enough Φ is a contraction and we conclude the result.

Convergence to equilibrium

After studying the linear case, we are now ready to prove convergence to the steady state under the weak interconnection regime, i.e. ∂ X p ∞ small enough, by a perturbation argument.

Theorem 5 (Convergence to equilibrium). Assume that n 0 ∈ L 1 (D) satisfies Assumption (2) and that p Lipschitz satisfies Assumption (3). For ∂ X p ∞ small enough, let (n * , N * , X * ) be the corresponding stationary state of System (1). Then there exist C, λ > 0 such that the solution n of System (1) satisfies

n(t) -n * L 1 s,a ≤ Ce -λt n 0 -n * L 1
s,a , ∀t ≥ 0.

Moreover N (t) -N * L 1 and |X(t) -X * | converge exponentially to 0 when t → ∞.

Proof. Observe that n satisfies the evolution equation

∂ t n = L X [n] := -∂ s n -∂ a n -p(s, a, X(t))n + δ {s=0} (s, a) ∞ 0 p(a, u, X(t))n(t, a, u) du,
where δ {s=0} (s, a) is the measure along the line {(0, a) : a ≥ 0}. We can rewrite the evolution equation as

∂ t n = L X * [n] + (L X [n] -L X * [n]) = L X * [n] + h. (38)
with h(t, s, a) given by h = p(s, a, X * )-p(s, a, X(t) n(t, s, a)+δ {s=0} (s, a)

∞ 0 p(a, u, X(t))-p(a, u, X * ) n(t, a, u) du. [START_REF] Perthame | On a voltage-conductance kinetic system for integrate & fire neural networks[END_REF] Let P t : L 1 (D) → L 1 (D) be the linear semi-group associated to operator L X * . As in the proof of Lemma 1, P t is extended to space (M(D), • M 1 ) in order to be able to evaluate at the measure h. Since P t n * = n * for all t ≥ 0, we get that n satisfies

n -n * = P t (n 0 -n * ) + t 0 P t-τ h(τ, s, a) dτ, (40)
so we need find an estimate for the function h. Observe that we have the following inequalities:

h(t) L 1 s,a ≤ 2 ∂ X p ∞ |X(t) -X * |, |X(t) -X * | ≤ N (t) -N * 1 , N (t) -N * 1 ≤ ∂p ∂X ∞ |X(t) -X * | + p ∞ n(t) -n * L 1
s,a , and since ∂ X p ∞ < 1 we get

h(t) L 1 s,a ≤ 2p∞ ∂ X p ∞ 1-∂ X p ∞ n(t) -n * L 1 s,a , |X(t) -X * | ≤ p∞ 1-∂ X p ∞ n(t) -n * L 1 s,a , N (t) -N * 1 ≤ p ∞ ∂ X p ∞ 1-∂ X p ∞ + 1 n(t) -n * L 1 s,a ,
thus by taking norm in Equality [START_REF] Pham | Activity in sparsely connected excitatory neural networks: effect of connectivity[END_REF] and applying Doeblin's Theorem we obtain

n(t) -n * L 1 s,a ≤ P t (n 0 n * ) L 1 s,a + t 0 P t-τ h(τ ) L 1 s,a dτ ≤ e -λt 1 -α n 0 -n * L 1 s,a + 1 1 -α t 0 e -λ(t-τ ) h(τ ) L 1 s,a dτ ≤ e -λt 1 -α n 0 -n * L 1 s,a + C t 0 e -λ(t-τ ) n(τ ) -n * L 1 s,a dτ, with C := 1 1-α 2p∞ ∂ X p ∞ 1-∂ X p ∞ .
By using Gronwall's inequality with respect to the function e λt n(t) -

n * L 1 s,a we conclude n(t) -n * L 1 s,a ≤ e -(λ-C)t 1 -α n 0 -n * L 1
s,a , so that for ∂ X p ∞ small enough we have C < λ and we deduce the exponential convergence of n(t, •, •), N (t, •) and X(t) when t → ∞.

Numerical simulations

In order to illustrate the theoretical long time results and other possible behaviors of System (1), we present numerical simulations for different firing rates and initial data. The numerical illustrations below are obtained by solving the equation ( 1) with a classical first-order upwind scheme, while the initial activity X(0) is computed by solving numerically the fixed-point equation [START_REF] Delarue | Global solvability of a networked integrate-and-fire model of McKean-Vlasov type[END_REF].

We focus in displaying the discharging flux N (t, a) and the total activity X(t) since these two elements determine the general behavior of system (1).

Example 1: Convergence to equilibrium

For our first example, we consider a firing rate given by p = ϕ(X -e -s -e -a )1 {s>1} , ϕ(u) = 1.5 1 + e -u , which corresponds to an excitatory case, modulated by a sigmoid function ϕ with an absolute refractory period σ = 1. By taking as initial data n 0 (s, a) = e -a , we observe in Figure 4 that the solution (a) Discharging Flux N (t, a). converges to the steady state for both discharging flux N and the total activity X, following the results of Theorem 5. As consequence of the refractory period in the firing rate, we observe in Figure 4a that steady states vanishes for a < 1.

From Example 1.1, we can compute a numerical approximation of the stationary flux N * and the respective total activity X * . In this setting, consider now as initial data the stationary density conditioned to a spike immediately before s = 0, i.e. n 0 (s, a) = e -a δ 0 (s) which corresponds to a synchronization of all neurons at t = 0. The theoretical results on wellposedness in Section 2 allows us to consider probability measures as initial data and we approximate the Dirac mass as δ 0 (s) ≈ 1 ∆s 1 {s∈[0,∆s]} , ∆s small enough.

As in Example 1.1, we observe in Figure 5 that the solution converges to the steady state and it stabilizes at similar times. This numerical result is compatible with Theorem (5) extended for initial data in the space of measures (M(D), • M 1 ). Moreover, as effect of the refractory period and the fact that initial data is supported on the line {s = 0} of the domain D, the total activity vanishes for small times and then it becomes increasing when the time approaches to t = 1.

We continue with some numerical examples beyond the scope of the theory of weak interconnections, in order to explore the possible behaviors of a strongly interconnected regime.

Example 2: Jump discontinuities

We now consider a compactly supported initial data n 0 (s, a) = 2 • 1 {2>a>s+1} , with a firing rate given by p = 1 {s>e -X } + 1 {a-s>e -X } , which corresponds to an excitatory case since p is increasing with respect to X. Like the previous examples, the solution converges to the steady state, but the total activity X shows three jump discontinuities as we see in Figure 6. The multiple jump discontinuities are consequence of the contribution of the term depending on the difference between the two elapsed times. Furthermore, converging solutions with a single jump discontinuity were already observed in Torres et al. [START_REF] Torres | An elapsed time model for strongly coupled inhibitory and excitatory neural networks[END_REF] for the classical elapsed time model. The phenomenon of multiple jumps discontinuities in Figure 6 is an extension for the case of Equation (1). Concerning the strongly inhibitory case, it remains pending to prove uniqueness of the steady state for the non-linear Equation [START_REF] Amari | Dynamics of pattern formation in lateral-inhibition type neural fields[END_REF]. Whilst in the classical elapsed time model this problem is reduced to solving a simple equation, for the model with two elapsed times we have to prove uniqueness for the integral Equation [START_REF] Khashayar Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF], which is not straightforward to deal with. Moreover, we conjecture in the general case that the speed of convergence to a steady state must be exponential like it is expected for the classical elapsed equation.

With respect to the existence of periodic solutions, an interesting problem is to find or construct nontrivial examples relying on dynamics for two elapsed times. The only examples we have found so far are adaptations of solutions of the classical elapsed time equation that were obtained in Torres et al. [START_REF] Torres | An elapsed time model for strongly coupled inhibitory and excitatory neural networks[END_REF] and these types of solutions presents jump discontinuities, making them difficult to analyze. Furthermore, it would be interesting to find some class of firing rates for the System (1) leading to delay differential equation with two discrete delays. Following the ideas in Torres et al. [START_REF] Torres | An elapsed time model for strongly coupled inhibitory and excitatory neural networks[END_REF] for the classical elapsed time model and differential equations with a single delay, we expect to construct specific periodic solutions for the multiple-time renewal equation.

ORCID iDs

Nicolás Torres: https://orcid.org/0000-0001-6059-9754. Benoît Perthame: https://orcid.org/0000-0002-7091-1200. Delphine Salort: https://orcid.org/0000-0003-4289-0286.

Figure 2 :Figure 3 :

 23 Figure 2: Second reduction of dimension. For t ∈ [σ, 2σ] the green lines are transported to the region where s ≥ σ and their mass is of at least p 0 e -2p∞σ . Then the mass of each green line is concentrated in the orange points, whose values are at least p 2 0 e -2p∞σ .

  Total Activity X(t).

Figure 4 : 1 . 1 .

 411 Figure 4: Example 1.1. Case n 0 (s, a) = e -a and p = ϕ(X -e -s -e -a )1 {s>1} .

  (a) Discharging Flux N (t, a).

  Total Activity X(t).

Figure 5 :

 5 Figure 5: Example 1.2. Case n 0 (s, a) = e -a δ 0 (s) and p = ϕ(X -e -s -e -a )1 {s>1} .

  (a) Discharging flux N (t, a).

  Total Activity X(t).

Figure 6 : 2 .

 62 Figure 6: Example 2. Case n 0 (s, a) = 2 • 1 {2>a>s+1} and p = 1 {s>e -X } + 1 {a-s>e -X } .

  Total Activity X(t).

Figure 8 :

 8 Figure 8: Example 3.2. Case n 0 (s, a) = 1 2 e -(a-1) 1 {a>max(s,1)} and p = ϕ(X)1 {s>1} + 1 {a-s>X} .

Acknowledgements

NT has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 754362. 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. DS has received support from ANR ChaMaNe No: ANR-19-CE40-0024.

Therefore by the Kolmogorov-Frechet theorem we conclude that A is relatively compact so the operator T X is.

Furthermore, since n X is the unique steady state of Equation ( 18) that is a probability density, from the linearity we deduce that any other function in ker(I -T X ) is a multiple of N X and thus dim ker(I -T X ) = 1.

Next, we proceed to determine ran(I -T X ). Observe that adjoint operator

and from Fredholm's alternative we get dim ker(I -T * X ) = dim ker(I -T X ) = 1. Since T * X [g] ≡ 0 for any constant function, we deduce that ker(I -T * X ) is the subspace of constant functions. Finally from orthogonality conditions we conclude that ran

A direct consequence of Lemma 3 is the following result Lemma 4. Assume that p is smooth respect to variable X, then N X (a) is also smooth with respect to X.

Proof. Define the

N (a)e -a 0 p(s ,u+s ,X) ds da ds -1 , so that for each X we have F (N X (a), X) = 0. Observe that D N F is given by

)e -a 0 p(s ,u+s ,X) ds da ds

Thus by Lemma 3 this operator is an isomorphism and from the implicit function theorem we conclude that N X (a) depends smoothly on X.

Remark 1. The lower bound condition (3) on the firing rate p is important to verify the existence of a steady state for System (1) and Doeblin's condition. For example, when we consider a fixed X > 0 and p(s, a, X) = 1 {a-s>X} , then there are no steady states of the linear Equation [START_REF] Fonte | Long time behavior of an age and leaky memorystructured neuronal population equation[END_REF], besides the zero solution. Indeed, from Equation [START_REF] Mischler | Relaxation in time elapsed neuron network models in the weak connectivity regime[END_REF] we deduce that the discharging flux N should satisfy

thus N X is an exponential function and by integrating with respect to a we get

Since N is non-negative, we conclude N vanishes on (0, X), implying that N ≡ 0.

Example 3: Periodic solutions and stabilization

Next, we choose initial data n 0 (s, a) = 1 2 e -(a-1) 1 {a>max(s,1)} and the firing rate is given by

which corresponds to an excitatory case since ϕ (u) > 0. Since p does not depend on a, we take advantage by solving the classical elapsed time Equation ( 5) after integrating with respect to a, as we remarked in the introduction.

(a) Discharging Flux N (t, a). For these data, both the discharging flux N and the total activity X are asymptotic to a periodic pattern as we see in Figure 7. Similar examples on periodic solutions were found in Torres et al. [START_REF] Torres | An elapsed time model for strongly coupled inhibitory and excitatory neural networks[END_REF] in the classical elapsed time model for the same type of firing rates. For periodic solutions of Fokker-Planck type models, see [START_REF] Luçon | Periodicity and longtime diffusion for mean field systems in R d[END_REF] and the references therein.

However, when we incorporate the effects of the difference between the elapsed times, the periodic regime changes. For the same initial data and

we observe in Figure 8 that the solution of System (1) converges to the steady state. Like the classical model, we can find examples of continuous solutions of System (1) converging to the steady state under a general strongly interconnected regime.

Perspectives

By means of Doeblin's theory applied to linear Equation (18) than the classical elapsed time model, we managed to understand the dynamics of System (1) for weak non-linearities by adapting the ideas


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Bernard Payrastre

Dr Jacques A Nunès
email: jacques.nunes@inserm.fr

Ph

Cutting edge: Dok-1 and Dok-2 adaptor molecules are

Introduction

Stimulation of membrane receptors such as the TCR on T cells induces the activation of protein tyrosine kinases (PTKs), and subsequently the phosphorylation of substrates, which contribute to the formation of intracytoplasmic multiprotein network. The downstream of tyrosine kinases (Dok) proteins, Dok-1 and Dok-2 are involved in negative T cell activation control [START_REF] Acuto | Tailoring T-cell receptor signals by proximal negative feedback mechanisms[END_REF]. During T cell activation, Dok proteins are tyrosine phosphorylated [START_REF] Yang | The role of Tec proteintyrosine kinase in T cell signaling[END_REF][START_REF] Dong | T cell receptor for antigen induces linker for activation of T cell-dependent activation of a negative signaling complex involving Dok-2, SHIP-1, and Grb-2[END_REF]. Dok-1 and Dok-2 overexpression in T cell lines decreases IL-2 production in activated cells [START_REF] Gérard | Functional interaction of RasGAP-binding proteins Dok-1 and Dok-2 with the Tec protein tyrosine kinase[END_REF][START_REF] Némorin | p62dok negatively regulates CD2 signaling in Jurkat cells[END_REF] and loss or reduction of Dok-1 and Dok-2 expression in primary T cells enhances TCRmediated functions and signaling [START_REF] Dong | T cell receptor for antigen induces linker for activation of T cell-dependent activation of a negative signaling complex involving Dok-2, SHIP-1, and Grb-2[END_REF][START_REF] Yasuda | Dok-1 and Dok-2 are negative regulators of T cell receptor signaling[END_REF]. Moreover, mice lacking both Dok-1 and Dok-2 develop T cell-dependent antibody responses and a spontaneous autoimmunity disease [START_REF] Yasuda | Dok-1 and Dok-2 are negative regulators of T cell receptor signaling[END_REF].

Interestingly, Dok-1 and Dok-2 contain N-terminal pleckstrin homology (PH) domain that has been poorly studied. PH domains are one of the largest and most intensively investigated families of lipid binding domains, the majority bind weakly to phosphoinositides (PIs) with little or no selectivity [START_REF] Lemmon | Pleckstrin homology (PH) domains and phosphoinositides[END_REF]. PIs are constituents of cell membranes playing a critical role in cell signaling pathways. Their metabolism is tightly regulated by several lipid kinases/phosphatases and its disruption gives rise to several pathologies [START_REF] Pendaries | Phosphoinositide signaling disorders in human diseases[END_REF]. These PIs species mediate signals through their binding to proteins containing specific interaction domains including PH, PX, FYVE or ENTH domains [START_REF] Lemmon | Membrane recognition by phospholipid-binding domains[END_REF].

We now report that in T cell lines or primary T cells, the PH domain of Dok proteins is necessary for their phosphorylation and functions. Interestingly, we show that the phosphatidylinositol-5-phosphate (PtdIns5P) is the most relevant ligand for Dok-1 and Dok-2 PH domains.

PtdIns5P is a rare PIs species and is starting to emerge as a potential second messenger [START_REF] Rameh | A new pathway for synthesis of phosphatidylinositol-4,5-bisphosphate[END_REF][START_REF] Coronas | PtdIns5P: a little phosphoinositide with big functions?[END_REF]. Bacterial infection by Shigella flexneri via the virulence factor, IpgD, generates PtdIns5P in the host cells and induces Akt activation [START_REF] Niebuhr | Conversion of PtdIns(4,5)P(2) into PtdIns(5)P by the S.flexneri effector IpgD reorganizes host cell morphology[END_REF][START_REF] Pendaries | PtdIns5P activates the host cell PI3-kinase/Akt pathway during Shigella flexneri infection[END_REF]. Recent evidences report that enhanced tyrosine phosphorylation increases cellular PtdIns5P levels [START_REF] Coronas | Elevated levels of PtdIns5P in NPM-ALK transformed cells: implication of PIKfyve[END_REF][START_REF] Wilcox | Regulation of extranuclear PtdIns5P production by phosphatidylinositol phosphate 4-kinase 2alpha[END_REF]. Altogether, these data argue for an important role of PtdIns5P in cell signaling. Here, we show that PtdIns5P is generated upon TCR triggering and regulates the tyrosine phosphorylation of negative adaptor proteins of T cell activation such as Dok-1 and Dok-2.

Materials and methods

Culture cells and transfection

Jurkat JA16 and Hut-78 T cells were grown in RPMI 1640 medium, Hela cells were cultured in DMEM. Both medium were supplemented with 10 % heat-inactivated FCS, 2 mM Lglutamine, and 1 mM sodium-pyruvate. PBMC from healthy donors (Etablissement Français du Sang, Marseille, France) were isolated on Ficoll-Hypaque (Pharmacia) gradients before purification of the CD4 + T cell subset using the CD4 + T cell isolation kit (Miltenyi Biotec).

Jurkat cells were electroporated at 960 μF and 250 V using a Bio-Rad Gene Pulser or nucleofected according to the manufacturer's instructions (http://www.amaxa.com/). PBMC were transfected using Amaxa Nucleofector technology. Hela cells were transfected using Lipofectamin 2000 ™ (Invitrogen).

Plasmid constructs

The HA-Dok1 and HA-Dok2 constructs were described previously [START_REF] Gérard | Functional interaction of RasGAP-binding proteins Dok-1 and Dok-2 with the Tec protein tyrosine kinase[END_REF]. All the constructs encoding the mutants lacking the PH domain: HA-ΔPHDok1, HA-ΔPHDok2, ΔPHDok1-GFP and ΔPHDok2-GFP, GFP-Dok1 and GFP-Dok2, chimeric PH domain constructs: PHDok1-GFP, PHDok2-GFP, GST-PHDok1 and GST-PHDok2, and finally several other vectors described previously are detailed in supplemental method.

PtdIns binding assays

Protein lipid-blot assays were carried out as described [START_REF] Gozani | The PHD finger of the chromatin-associated protein ING2 functions as a nuclear phosphoinositide receptor[END_REF]. "PIP array" and "PIP strip" were purchased from Echelon biosciences Inc. To reveal binding, GST mAb was used in immunoblotting.

Surface plasmon resonance (SPR) experiments

Detailed protocols for SPR experiments were described previously [START_REF] Mortier | Nuclear speckles and nucleoli targeting by PIP2-PDZ domain interactions[END_REF]. Briefly, the binding of 1 µM protein to lipid layers containing 90% DOPC (1,2-dioleoyl-sn-glycero-3phosphocholine (SIGMA) and 10% PtdIns, PtdIns3P, PtdIns4P or PtdIns5P (Echelon biosciences Inc.) was measured. Protein binding for the 4 lanes were measured at the same time and all proteins were tested on the same fresh lipid layers (at least 6000 RU was coupled). Proteins were removed from the layers by addition of 20µl 25 µM NaOH at a flow rate of 100 µl/min.

Phospholipid extraction and analysis

Hut-78 (10 7 ) were stimulated with CD3 antibody (clone 289) (18) during indicated time.

PtdIns5P was quantified by mass assay as previously described [START_REF] Niebuhr | Conversion of PtdIns(4,5)P(2) into PtdIns(5)P by the S.flexneri effector IpgD reorganizes host cell morphology[END_REF]. The PtdIns(4,5)P 2 generated after lipid kinase assay was extracted and analyzed by HPLC for quantification.

Luciferase assay

Luciferase assay was performed as previously described (4).

T cell proliferation assay

Primary human T cells were transfected with the Amaxa Nucleofector technology, 16 hrs after transfection (transfection efficiency: 40 % GFP + cells), mixed CD4 + T cell population (10 5 ) was plated on 96-wells plate and left non stimulated or stimulated with OKT3 + CD28.2 (0.5µg/ml each), or 1ng/ml PMA + 1µg/ml Ionomycin. Cells were cultured for 4 days and plates were pulsed with 1 µCi [ 3 H]-thymidine/well for the last day of culturing.

Results and discussion

The loss of the PH domain blocks Dok-1 and Dok-2 tyrosine phosphorylation and their inhibitory properties

Tyrosine phosphorylation of Dok proteins is also detectable in Jurkat T cells upon an optimal T cell activation via TCR plus CD28 co-stimulation (Fig. 1A). To investigate the role of the PH domain, we first compared Jurkat cells overexpressing GFP-tagged Dok-1 or Dok-2 proteins full length or deleted for their PH domain. Dok-1 and Dok-2 proteins full length were tyrosine phosphorylated in stimulated cells (Fig. 1B). Interestingly, Dok proteins tyrosine phosphorylation was undetectable when PH domain was deleted. Therefore, PH domain is important for Dok-1 and Dok-2 phosphorylation.

Overexpression of Dok-1 or Dok-2 inhibits IL-2 promoter activity in TCR plus CD28activated Jurkat cells (4). We used this assay to address the importance of the PH domain for the function of Dok proteins. Cells were transfected with HA-tagged Dok-1 and Dok-2 full length or deleted of their PH domain and stimulation was carried out with CD3 and CD28 antibodies for 16 hrs. Luciferase assays revealed that IL-2 promoter activity was inhibited by Dok-1 or Dok-2 overexpression proteins full length (wild type, WT) (Fig. 1C). However, expression of PH-deleted Dok-1 or Dok-2 proteins (ΔPH) resulted in a loss of inhibitory effects on IL-2 promoter activity. To confirm the importance of the PH domain in the regulatory function of Dok proteins, constructs encoding GFP-tagged Dok-1 or Dok-2 full length (WT) or deleted for their PH domain (ΔPH) were transfected in human primary T cells.

The cells were then stimulated by CD3 and CD28 mAbs and proliferation assays were performed (Fig. 1D). We observed an inhibition of T cell proliferation upon overexpression of Dok-1 or Dok-2, but not upon expression of their PH-deleted versions.

Taken together, these results indicate that the PH domain is necessary for Dok phosphorylation and to trigger the inhibitory effects of Dok-1 and Dok-2 in activated T cells.

We then decided to characterize the ligand of these PH domains.

Dok-1 and Dok-2 PH domains bind PtdIns5P

PH domains are commonly described phospholipid binding domains [START_REF] Lemmon | Pleckstrin homology (PH) domains and phosphoinositides[END_REF]. Although little is known about Dok-1 and Dok-2 PH domain binding-specificity, it was reported that the PH domain of Dok-1 recognizes polyphosphoinositides incorporated in multilamellar vesicles [START_REF] Zhao | Phosphoinositide 3-kinase-dependent membrane recruitment of p62(dok) is essential for its negative effect on mitogen-activated protein (MAP) kinase activation[END_REF]. To establish whether both Dok-1 and Dok-2 PH domains have a phospholipid-binding activity, we tested 15 lipids including phosphoinositides in a protein lipid-blot assay using "PIP-Strip" membranes.

After incubation with recombinant GST-PH Dok-1 or GST-PH Dok-2, a signal was clearly detectable with phosphatidylinositol monophosphate species PtdIns3P, PtdIns4P and PtdIns5P (Fig. 2A). By SPR analysis, GST Dok-1 and Dok-2 PH domains interacted mostly with PtdIns4P and PtdIns5P but not with PtdIns3P (Fig. 2B). To quantify the relative affinity of PH Dok-1 and Dok-2 domain for phosphoinositides, we probed a lipid blot with serial dilutions of eight different phosphoinositides with GST-PH Dok-1 or GST-PH Dok-2 proteins ("PIP array" membranes) (Fig. 2C). GST immunoblot analysis revealed that Dok-1 and Dok-2 PH domain bound most tightly to PtdIns5P. Although there is a low binding to other phosphoinositides species, PtdIns5P appears clearly as the preferential ligand for Dok-1 or Dok-2 PH domain. Dok-1 PH domain has been previously described as a potential PtdIns(3,4,5)P 3 binding module [START_REF] Zhao | Phosphoinositide 3-kinase-dependent membrane recruitment of p62(dok) is essential for its negative effect on mitogen-activated protein (MAP) kinase activation[END_REF], however the binding activity to phosphatidylinositol monophosphates was not addressed in this study. Here we observe a slight binding to PtdIns(3,4,5)P 3 (Fig. 2C), but the binding of the PH domains of Dok proteins are stronger to PtdIns5P .Moreover, the PH domain of Dok proteins has not been identified as a PtdIns(3,4,5)P 3 binding module in a recent published large scale screening for PtdIns(3,4,5)P 3 partners [START_REF] Park | Comprehensive identification of PIP3-regulated PH domains from C. elegans to H. sapiens by model prediction and live imaging[END_REF]. We are showing by independent in vitro assays that Dok-1 and Dok-2 PH domain recognize phosphatidylinositol monophosphates and preferentially PtdIns5P. PtdIns5P increase has been described in few reports and this lipid appears as a second messenger involved in cell signaling [START_REF] Coronas | PtdIns5P: a little phosphoinositide with big functions?[END_REF]. Our results raise the possibility of a yet unknown Dok-1 and Dok-2 regulation via PtdIns5P binding, moreover they suggest that PtdIns5P could have an impact on T cell activation.

Consequently, we decided to measure PtdIns5P levels under T cell activation.

TCR triggering induces a PtdIns5P production in T cells

The difficulty to detect PtdIns5P explains why it has only been quite recently biochemically characterized [START_REF] Rameh | A new pathway for synthesis of phosphatidylinositol-4,5-bisphosphate[END_REF]. Few studies report on an increase in PtdIns5P levels [START_REF] Clarke | Inositol lipids are regulated during cell cycle progression in the nuclei of murine erythroleukaemia cells[END_REF][START_REF] Sbrissa | Phosphatidylinositol 5phosphate biosynthesis is linked to PIKfyve and is involved in osmotic response pathway in mammalian cells[END_REF][START_REF] Tronchère | Production of phosphatidylinositol 5-phosphate by the phosphoinositide 3-phosphatase myotubularin in mammalian cells[END_REF], and classically this lipid is present in low amounts in resting cells. As the method to measure PtdIns5P necessitates a large amount of cellular material, we used a T cell line Hut-78, instead of primary T cells. Hut-78 T cell line is not mutated in phosphoinositide biosynthetic enzymes, such as phosphatase and tensin homologue deleted in chromosome 10 (PTEN) or SH2 domain-containing inositol polyphosphate 5'-phosphatase (SHIP) (3). PtdIns5P levels were measured in T cells upon TCR triggering (Fig. 2D). Compared to unstimulated cells, a four time increase in PtdIns5P levels was detected after 2 min of TCR/CD3 stimulation.

PtdIns5P amount then rapidly dropped to basal levels after 10 min. These data show for the first time, that PtdIns5P is detectable in T cells upon TCR triggering and suggest that it might be involved in early T cell signaling. In parallel, we investigated the status of Dok-2 tyrosine phosphorylation (Fig. 2D; upper panel). Dok-2 phosphorylation followed the same kinetic as PtdIns5P increase, suggesting a possible link between Dok-2 tyrosine phosphorylation and PtdIns5P production.

PtdIns5P increase induces Dok-1 and Dok-2 proteins tyrosine phosphorylation

To further establish a link between PtdIns5P production and Dok-1 and Dok-2 regulation, we used the bacterial inositol-4-phosphatase, IpgD [START_REF] Niebuhr | Conversion of PtdIns(4,5)P(2) into PtdIns(5)P by the S.flexneri effector IpgD reorganizes host cell morphology[END_REF]. IpgD is a virulence factor of Shigella flexneri transforming PtdIns(4,5)P 2 into PtdIns5P via its 4-phosphatase activity. IpgD ectopic expression induces high PtdIns5P production in Jurkat T cells [START_REF] Tronchère | Production of phosphatidylinositol 5-phosphate by the phosphoinositide 3-phosphatase myotubularin in mammalian cells[END_REF]. To study the impact of IpgD and thereby of PtdIns5P increase on Dok-1 and Dok-2 tyrosine phosphorylation, Jurkat T cells were transfected with a Myc-tagged IpgD phosphatase (WT) expression construct or its phosphatase-dead C438S mutant (CS) construct. Interestingly, IpgD expression but not its phosphatase-dead version leads to Dok-1 and Dok-2 tyrosine phosphorylation (Fig. 3A). To test for the specific contribution of PtdIns5P increase during ectopic IpgD expression on Dok-1 and Dok-2 phosphorylation, Hela cells expressing HA-tagged Dok-1 were cotransfected with IpgD plus GFP-tagged Inp54 or the type II PtdIns5P 4-kinaseβ (PIP4KIIβ) (Fig. 3D). Inp54 is an inositol-5-phosphatase known to transform PtdIns(4,5)P 2 into PtdIns4P and PIP4KIIβ is a kinase that transforms PtdIns5P into PtdIns(4,5)P 2 [START_REF] Niebuhr | Conversion of PtdIns(4,5)P(2) into PtdIns(5)P by the S.flexneri effector IpgD reorganizes host cell morphology[END_REF]. Inp54 ectopic expression increasing PtdIns4P was unable to induce the tyrosine phosphorylation of Dok-1 (Fig. 3D, HA-Dok-1 plus INP54 GFP conditions). Using an in vitro SPR binding approach both PtdIns4P and PtdIns5P are recognized by Dok PH domains (Fig. 2B), however only PtdIns5P, but not PtdIns4P, is a regulator module for Dok-1 protein (Fig. 3D). When HA-

These data further indicate that PtdIns5P increase induces

Dok

  Dok-1 and Dok-2 phosphorylation, and strongly suggest that Dok-1 and Dok-2 proteins are PTK substrates downstream of PtdIns5P production. To demonstrate a more direct effect of PtdIns5P on Dok-1 and Dok-2 phosphorylation, we tested whether the overexpression of Dok-1 PH domain could inhibit Dok phosphorylation (competitive effect). Jurkat T cells containing detectable levels of tyrosine phosphorylated Dok-1 protein (overexpressing IpgD together with HA-tagged Dok-1), were transfected with plasmids encoding the GFP-PH domain of Dok-1 (PH Dok-1 GFP). The Dok-1 PH domain expression blocks Dok-1 tyrosine phosphorylation induced by IpgD (Fig. 3B). Similar results were reproduced in Hela cells (Fig. 3C) and extended to another PtdIns5P-binding module, the GFP-trimer of PHD (Plant Homeo Domain) motif of ING2 (3X PHD ING2 GFP) (16). Both PtdIns5P partners (PH-Dok-1 or PHD-ING2) were able to block Dok-1 tyrosine phosphorylation induced by PtdIns5P increase.

  -1 plus IpgD were co-expressed in presence of PIP4KIIβ, Dok-1 phosphorylation was lost. In control, co-expression of HA Dok-1 and PIP4KIIβ without IpgD did not lead to Dok-1 phosphorylation. Taken together, our results indicate that PtdIns5P production controls Dok proteins tyrosine phosphorylation in cells. PtdIns5P could be a new unexpected lipid second messenger in T cell activation. It has been reported that the PtdIns5P production induces Akt activation in epithelial cells infected by Shigella flexneri probably via a PTK activation leading to the stimulation of phosphoinositide 3'-kinase (PI3K) (13). As Dok-1 and Dok-2 PH domains are able to bind PtdIns5P and PtdIns5P production induces PTK activation, the PtdIns5P could bring together activated PTKs and their substrates (Dok proteins). PtdIns5P production could support positive cell signals via PI3K signaling and negative cell signals via Dok proteins. PtdIns5P might thereby function as a sensor in setting the threshold for T cell activation and in maintaining T cell homeostasis. To investigate this hypothesis, further studies are required to better understand the molecular mechanisms inducing PtdIns5P increase in eukaryotic cells.

Figure legends Figure 1 .

 legends1 Figure legends

Figure 2 .

 2 Figure 2. The PH domains of Dok-1 and Dok-2 bind PtdIns5P and transient increase in

Figure 3

 3 Figure 3. PtdIns5P production and availability control Dok-1 and Dok-2

Acknowledgments

We would like to thank Sonia Pastor for providing technical assistance in carrying out some expression vectors; Dr. Frédérique Michel, Dr. Jean Luc Teillaud and Dr. Or Gozani for providing some reagents. We are grateful to Dr. Patrice Dubreuil and Damien Ramel for helpful discussions and to Gaëtan Chicanne for his helps with the HPLC analysis.

 [START_REF] Acuto | Tailoring T-cell receptor signals by proximal negative feedback mechanisms[END_REF]This work was supported by grants from Institut National de la Santé et de la Recherche Médicale, the Institut National du Cancer (# PL-06026), the Association pour la Recherche sur le Cancer (# 4202), ANR-blanc (# 0065-01), the Fund for Scientific Research-Flanders (FWO), the Belgian Federation against Cancer, the Concerted Actions Program and the Research Fund of the K.U.Leuven. GG was supported by fellowship from the Inserm / Région Provence-Alpes Côte d'Azur and FRM. AG was supported by fellowship from the Ligue Nationale Contre le Cancer. 2 GG and AG: these authors contributed equally to this