Rachida Chakir

Yvon Maday

A two-grid finite-element/reduced basis scheme for the approximation of the solution of parameter dependent

In the frame of optimization process in industrial framework, where numerical simulation is used at some stage, the same problem, modeled with partial differential equations depending on a parameter has to be solved many times for different sets of parameters. The reduced basis method may be successful in this frame and recent progress have permitted to make the computations reliable thanks to a posteriori estimators and to extend the method to non linear problems thanks to the "magic points" interpolation. However, it may not always be possible to use the code (for example of finite element type that allows for evaluating the elements of the reduced basis) to perform all the "off-line" computations required for an efficient performance of the reduced basis method. We propose here an alternating approach based on a coarse grid finite element the convergence of which is accelerated through the reduced basis and an improved post processing.

Introduction

Let X be a closed subspace of the Sobolev space H 1 (Ω) over a bounded domain Ω ⊂ R d and D a set of parameter. We consider the following problem: given µ ∈ D, find u(µ) ∈ X such that ∀v ∈ X, a(u(µ), v; µ) = (f, v),

where a is a bilinear form, continuous and coercive over X that depends additionally on a parameter µ ∈ D and f ∈ L 2 (Ω).

In order to approximate the solution to this problem, one can use a standard numerical approach, as a finite element method, that provides a very accurate approximation of the solution u(µ) for any fixed value of the parameter µ. This accurate approximation will be called "truth approximation" in what follows.

The computation of the truth approximation of u(µ) for many values of µ can however become very expensive as it has to be repeated for each parameter. The reduced basis method is an alternative that takes its roots upon the low "complexity" of the set of all solutions M D = {u(µ), µ ∈ D} that can e.g. be measured by the Kolmogorov width [START_REF] Kolmogoroff | Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse Anals of Math[END_REF] (see also [START_REF] Pinkus | n-Widths in Approximation Theory[END_REF]). This can for instance be formalized by the fact that for any ǫ > 0, there exist a set of parameters

µ 1 , µ 2 , • • • , µ N , ∈ D, where N = N (ε) is reasonable, such that, ∀µ ∈ D ∃(α i (µ)) ∈ R N , u(µ) - N i=1 α i (µ)u(µ i ) X ≤ ε. (2)
Based on the potential approximation property expressed above, the reduced basis method is in a Galerkin approach to the problem (1) for each new value of µ, within a space X N spanned by N particular truth approximations of u(µ) corresponding to suitably chosen parameters µ.

To keep the interest of this method, i) the parameters µ i have to be adequately chosen, ii) the corresponding solution has to be properly calculated or approximated through an accurate discretization method (as a finite element method, for example) and iii) moreover the construction of the stiffness matrix A(µ) with entries a(u(µ i ), u(µ j ); µ) as to be done for each new value of µ.

All the expensive computations involving in the three previous steps are done off-line which allows to have online computations that scales only like powers of N and do not involve the dimension of the finite element space (see e.g. [START_REF] Maday | Reduced basis method for the rapid and reliable solution of partial differential equations[END_REF]).

Various recent contributions have permitted to extend the range of the reduced basis method, e.g. the a posteriori error estimates for validation and determination of the proper parameters µ i 's [START_REF] Prud | Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods[END_REF], the "magic points" , for generalization to nonlinear problems [START_REF] Ma Grepl | Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations[END_REF].

In an industrial framework, for optimization processes for instance these approaches have a great potential, unfortunately part of the off line computations require to enter in the code that computes the truth approximation which is not possible in case the simulation code has been bought or relies on a long evolution so that only a black box use of the code is possible. Those computations require indeed the use of some component involved in the implementation of discretization method which are not available to the user. As a consequence the reduced basis method cannot be efficiently implemented, an alternative needs to be proposed.

An alternating reduced basis method

Let us assume that the truth approximation is based on a P 1 -finite element code, capable of giving us a good enough approximation of the u(µ) in a finite element space X h such that

∀µ ∈ D, ||u(µ) -u h (µ)|| X ≤ c 1 h < T ol. (3)
Where T ol is a a tolerance chosen in accordance to the final goal we have.

In the standard reduced basis method we first compute the truth approximation u h (µ i ), then form a discrete space

X N h = Span{u h (µ i ), i = 1, .., N } and build a Garlerkin approximation of (1) in X N h : Find u N h (µ) ∈ X N h such that, ∀v ∈ X N h , a(u N h (µ), v; µ) = (f, v).
The implantation of this reduced basis method involves the construction of the stiffness matrix A h (µ) with entries a(u h (µ i ), u h (µ j ); µ).

In an industrial framework, the finite element code is often locked, so we can not decompose the construction of the stiffness matrix A h (µ) into a series of independent part that can be evaluated off line. This prevents us from employing the usual technique to compute quickly each stiffness matrix for a new value of µ, and take away the benefit of the reduced basis method (i.e. having a complexity depending only on N, independently of the dimension of the finite element space). First of all, let us remind that for a stable implementation of the reduced basis technique, it is required to build a better prepared basis than the one composed with the u(µ i ), usually a Gramm-Schmidt method is here advocated. We replace it here by the resolution of an eigenvalue problem: find ξ ∈ X N h and λ ∈ R such that ∀v ∈ X N h , Ω ∇ξ∇v = λ Ω ξv, that provides L 2 (Ω) and H 1 (Ω) orthogonal eigenvectors ξ i,BR (chosen to be normalized in L 2 ). We note, that the ξ i,BR constitute a second basis of the space X N h . Secondly we remark that, the standard reduced basis method aims at evaluating the coefficients intervening in the decomposition of u N h (µ) in the basis of the

ξ i,BR , u N h (µ) = N i=1 β BR i ξ i,BR .
Those can appear as a substitute to the optimal coefficients

β h i (µ) = Ω u h (µ)ξ i,BR of the best approximation of u h (µ) in X N
h . This substitute is still good enough since, from Cea's Lemma we have u(µ) -

u N h (µ) X ≤ c inf v∈X N h u(µ) -v X ,
then by using ( 2) and (3) we derive:

u(µ) -u N h (µ) X ≤ ε + c 2 h. (4)
Our alternative method first presented in [START_REF] Chakir | Une méthode combinée d'éléments finis à deux grilles/bases réduites pour l'approximation des solutions d'une E.D.P. paramétrique[END_REF] and illustrated by numerical results proving the potential interest of this alternative consists in proposing, another surrogate to β h i (µ) defined by β H i (µ) = Ω u H (µ)ξ i,BR . Since, the computation u H (µ), for H >> h and X H ⊂ X h , is less expensive than the one of u h (µ), the use of the industrial code with the parameter H to construct the β H i (µ)is cheap enough. From this computation we derive u Hh

N (µ) = N i=1 β H i (µ)ξ i,BR in X N h .
In what follow we explain in which case this can still be a very good approximation.

Since, 2 . By using the L 2 and H 1 orthogonality of the ξ i,BR basis, we get that u(µ)u Hh N (µ) X ≤ ε + c 3 h + c 4 H 2 where c 4 = c 4 (N ) which is asymptotically similar to (4) when we choose h ∼ H 2 .

|β h i (µ) -β H i (µ)| ≤ u h (µ) -u H (µ) 0,Ω a classical Aubin-Nitsche ar- gument 1 provides the following estimate: u(µ) -u H (µ) 0,Ω ≤ cH u(µ) - u H (µ) X ≤ cH
In the case of an higher order finite element approximation, P k , we can as well use an Aubin-Nitsche argument to get the improved error estimation. First we define Φ i,BR , such that ∀v h ∈ X h , a(v h , Φ i,BR ; µ) = Ω v h ξ i,BR , hence 1 Actually, the convergence results stated here either require that there is no corner or edge type singularities in the solutions -of the primal or dual problem for the Aubin-Nitsche argument -or that we relax somehow the definition of h and H being here a parameter associated with the grid size and the way the global refinement is done for convergence but not the size of the finer elements that should be defined such that the error bound by a conatant times h or H holds.

β h i (µ) -β H i (µ) = Ω (u h (µ) -u H (µ))ξ i,h = a(u h (µ) -u H (µ), Φ i,h ; µ). Since X H ⊂ X h we obtain, from the definition of u h (µ) and u H (µ), that ∀χ H ∈ X H , a(u h (µ) -u H (µ), χ H ; µ) = 0, then that ∀χ H ∈ X H , β h i (µ) -β H i (µ) = a(u h (µ) - u H (µ), Φ i,h -χ H ; µ). Therefore we have |β h i (µ)-β H i (µ)| ≤ c u h (µ)-u H (µ) X Φ h -χ H ; µ) X ≤ cH 2k and then u(µ) -u Hh N (µ) X ≤ ε + c 5 h k + c 6 H 2k (where c 6 = c 6 (N ))
. Finally, we get to the same conclusion as previously by choosing h ∼ H 2 . Here we want to improve even further the accuracy of the approach by proposing a simple post processing of the results.

Post-processing

Let β H (µ j ) be the vector (β H i (µ j )) 1 i N and β h (µ j ) the one corresponding to the (β h i (µ j )) 1 i N . We decide to improve the computation of the β H (µ), by a post-processing that will insure that for each parameters µ (µ j ) j=1,••• ,N that are used in the construction of the reduced basis, the method returns exactly u h (µ j ), indeed contrarily to u h (µ) that we do not want to compute for a large number of values of µ, the truth solutions u h (µ j ), j = 1, • • • , N have been actually computed. In order to define this post-processing, we consider the linear transformation F : R N → R N , that maps β H (µ j ) on to β h (µ j ). The post processing consist in applying it to all the vector β H (µ).

Let T be the matrix associated to the transformation F. For large values of N , the solutions u h (µ j ), j = 1, • • • , N mays become almost linearly dependent which results in a bad conditioning of the matrix T . This loss of stability may result in an important deterioration of the vectors β H (µ). To avoid this problem we propose to map only the first solutions in the previous set and thus construct an alternative matrix denoted k , 1 ≤ k ≤ N verifying:

T k β H (µ p 1 ), • • • , β H (µ p k ), γ k+1 , • • • , γ N t = β h (µ p 1 ), • • • , β h (µ p k ), γ k+1 , • • • , γ N t
where the N vectors γ k are constructed by a Gram -Schmidt method such that

γ 1 = β H (µp 1 ) β H (µp 1 ) 2 , and γ k ∈ Span{β H (µ p 1 ), • • • , β H (µ p k )}.
The set (µ p k ) 1≤k≤N is identical to the one used in the construction of the reduced basis, but it has been arranged differently. Indeed for each iteration k, we choose µ p k among the Nk parameters (µ pq ) ,k≤q≤N , such that max 1≤j≤N T k β H (µ j )β h (µ j ) ∞ is the smallest. We notice that, at the end, the matrix T N and T are similar.

Then we chose the matrix T k with the largest k chosen in such a way that the condition number of the matrix T k is moderate enough.

Numerical results

The problems we consider in this section are in 2 dimensions. From an original coarse triangulation T H 0 , we built successive refined triangulations T H i ,1≤i≤4 by recursively splitting each triangle K in T H i-1 into four triangles with equal is thus even improved by the trivial postprocessing. Note that the mathematical justification of this last ingredient is still missing.

X h = {v ∈ C 0 (Ω), v |T ∈ P 1 (T ), T ∈ T H 4 } ||u(µ h ) -u h (µ h )|| 1,Ω = 3.3 × 10 -2 N k i ||u(µ Hi ) -u hHi N (µ Hi )|| 1,Ω ||u(µ Hi ) -u hHi N (µ Hi )||
= {v ∈ C 0 (Ω), v |T ∈ P 2 (T ), T ∈ T H 3 } u(µ h ) -u h (µ h ) 1,Ω = 3.5 × 10 -3 N k i ||u(µ Hi ) -u hHi N (µ Hi )|| 1,Ω ||u(µ Hi ) -u hHi N (µ Hi )|| 1,Ω ||u(µ Hi ) -u Hi (µ Hi )||

5 ×

 5 (µ h )|| 1,Ω = 3.5 × 10 -3 10 -3 5.6 × 10 -3 5.1 × 10 -2 2 3.5 × 10 -3 3.5 × 10 -3 1.4 × 10 -2 15 ||u(µ h )u BR h (µ h )|| 1,Ω = 3.5 × 10 -3

5 × 6 ×

 56 10 ||u(µ h )u BR h (µ h )|| 1,Ω = 3.5 × 10 -2 h )u BR h (µ h )|| 1,Ω = 3.5 × 10 -2 10 -2 3.6 × 10 -2 7.9 × 10 -2 20 ||u(µ h )u BR h (µ h )|| 1,Ω = 3.5 × 10 -2 10 -27.9 × 10 -2

Table 1 :

 1 Error for the example 1 with

  1,Ω ||u(µ Hi )u Hi (µ Hi )|| 1,Ω

				with post-processing	without post-processing
				||u(µ h ) -u BR h (µ h )|| 1,Ω = 0.19
	5	5	0 1 2	0.15 0.17 0.19	0.13 0.16 0.18	0.49 0.28 0.15
			3	0.19	0.19	7.3 × 10 -2
	10	10	0 1 2	||u(µ h ) -u BR h (µ h )|| 1,Ω =3.6 × 10 -2 3.6 × 10 -2 0.35 3.5 × 10 -2 6.8 × 10 -2 3.5 × 10 -2 3.8 × 10 -2	0.49 0.28 0.15
			3	3.6 × 10 -2	3.5 × 10 -2	7.3 × 10 -2
	15	15 0 7 1	||u(µ h ) -u BR h (µ h )|| 1,Ω =3.4 × 10 -2 3.5 × 10 -2 0.47 3.7 × 10 -2 0.14	0.49 0.28
		15	2 3	3.4 × 10 -2 3.4 × 10 -2	3.4 × 10 -2 3.4 × 10 -2	0.15 7.3 × 10 -2
	20	20	0 1 2	||u(µ h ) -u BR h (µ h )|| 1,Ω =3.3 × 10 -2 5.5 × 10 -2 0.56 3.4 × 10 -2 0.20 3.4 × 10 -2 4.8 × 10 -2	0.49 0.28 0.15
			3	3.4 × 10 -2	3.4 × 10 -2	7.3 × 10 -2

Table 2 :

 2 Error for the example 2 with X h

Table 3 :

 3 Error for the example 2 withX h = {v ∈ C 0 (Ω), v |T ∈ P 1 (T ), T ∈ T H 4 } u(µ h )u h (µ h ) 1,Ω = 3.5 × 10 -2 Hi )u hHi N (µ Hi )|| 1,Ω ||u(µ Hi )u hHi N (µ Hi )|| 1,Ω ||u(µ Hi )u Hi (µ Hi )||1,Ω with post-processing without post-processing 5 ||u(µ h )u BR h (µ h )|| 1,Ω = 3.6 × 10 -2

	N	k ||u(µ 5 i 0 1 2	5.7 × 10 -2 6.2 × 10 -2 6.5 × 10 -2	0.17 9.0 × 10 -2 7.0 × 10 -2	0.45 0.24 0.12
		3	6.6 × 10 -2	6.7 × 10 -2	6.1 × 10 -2

. We get a superspace X H i about four times larger than X H i-1 that satisfies X H 0 ⊂ X Hn . We denote by u hP N (µ), the H 1 projection of u h (µ) on the basis of the ξ i,BR , defined by

It is the best we can expect from the reduced basis, that is one of the ingredient entering in the approximation.

Example 1

We first consider the nonlinear problem: find u ∈ H 1 (Ω) such that

In this example, the set of parameters, µ = (α, η), that we use is varying in [START_REF] Chakir | Une méthode combinée d'éléments finis à deux grilles/bases réduites pour l'approximation des solutions d'une E.D.P. paramétrique[END_REF]37]

The P 1 approximation results's are showed in the table 1. We first remark that we need at least N = elements in the reduced basis to recover the truth error. Second, before post-processing we note that the H 1 -error made with the solution u hH 2 N is close to the one made with u h , for any value of µ in D, despite the fact that T H 2 is eight times less accurate than T H 4 , at least for N ≥ 10. We also note a small deterioration of the evaluation of the solution, when N rises, confirming that the constant c 4 (N ) is growing with . Finally, we note that the post-processing improved even more the approaximation since it allows to recover the truth error even starting from the computations of the coarsest solution u hH 0 N , at least if we use the proper number of reduced elements (10 or 15) , which is a very substantial savings. We note also that the reduction of indices in the post-processing is used, even it is important since, in the case N = 15 the error ||u(µ H 1 )u hH 1 N (µ H 1 )|| 1,Ω with the full matrix is 0.50.

Exemple 2

The second problem is a convection dominated problem :

where v is such as v = (cos µ, sin µ). Here, the varying parameter is the angle of the convection µ

{||u(µ)u h (µ)|| 1,Ω }. The table 3 shows the P 1 approximations result's, while the table 2 shows the P 2 ones. We can make the same conclusion in the previous example : this combined method (reduced bais + two grids)


Khashayar Pakdaman
email: pakdaman@ijm.jussieu.fr

Benoît Perthame
email: benoit.perthame@upmc.fr

Delphine Salort
email: salort.delphine@ijm.jussieu.fr

Dynamics of a structured neuron population

Keywords: Structured population dynamics, neurons populations, nonlinear asymptotic stability, synchronization AMS Class. No. 35F15, 35B10, 92B20

We study the dynamics of assemblies of interacting neurons. For large fully connected networks, the dynamics of the system can be described by a partial differential equation reminiscent of agestructure models used in mathematical ecology, where the "age" of a neuron represents the time elapsed since its last discharge. The nonlinearity arises from the connectivity J of the network.

We prove some mathematical properties of the model that are directly related to qualitative properties. On the one hand we prove that it is well-posed and that it admits stationary states which, depending upon the connectivity, can be unique or not. On the other hand, we study the long time behavior of solutions; both for small and large J, we prove the relaxation to the steady state describing asynchronous firing of the neurons. In the middle range, numerical experiments show that periodic solutions appear expressing re-synchronization of the network and asynchronous firing.

Introduction

Synchronous regular or irregular activities are ubiquitous in nervous systems and can be at the basis of physiological functions such as respiration, cognitive function or pathological conditions such as epilepsy. These synchronous patterns reflect simultaneous and repetitive discharges of large numbers of neurons in specific assemblies at the level of the brainstem, the cortex, etc. Such coherence results from the interplay between the intrinsic neural properties and the coupling principles underlying the network. A number of theoretical investigations have aimed to determine the mechanisms underlying the genesis of synchronous firing in neural assemblies, through both numerical and analytical studies.

One mechanism for obtaining global coherent activity in assemblies is through the interconnection of self oscillating units. Synchrony in such systems strongly depends on the distribution of the frequencies of spontaneous oscillations of the components. This form of oscillation has been the topic of numerous investigations. However, it is also possible to obtain coherent global activities in assemblies that do not contain such pacemaker units. neural assemblies constitute one example where such dynamics have been reported. This paper is mainly concerned with this issue.

Early, experimental motivation for this paper came from recordings that monitored the spontaneous activity of brainstem slices. Simulation of a computational model adjusted to the experiments revealed that the network could sustain regular rhythmic activity in some parameter ranges. Similar observations had been made in [START_REF] Van Ooyen | The emergence of long-lasting transients of activity in simple neural networks[END_REF] for models of long lasting transients observed in the cerebral cortex. Since the earlier studies in [START_REF] Pham | Activity in sparsely connected excitatory neural networks: effect of connectivity[END_REF][START_REF] Pham | Noise-induced coherent oscillations in randomly connected neural networks[END_REF], several other studies obtained such results. That the phenomenon can be observed in many models suggests that the fine details of the model may not be at the core of the mechanism, and that to get the gist of the phenomena, one may focus on a few features of neural dynamics. Indeed, the systematic exploration of the influence of model parameters hints that the key ingredient is the relation between the recovery time scale of the neuron and the input time scales. Furthermore, the numerical simulations show that the phenomenon persists in networks with diverse connectivity. For instance while the first studies in [START_REF] Pham | Activity in sparsely connected excitatory neural networks: effect of connectivity[END_REF][START_REF] Pham | Noise-induced coherent oscillations in randomly connected neural networks[END_REF] where performed in randomly connected networks, the same phenomenon can also be observed in fully connected homogenous networks. These simulations show several points, first that the effect observed in finite size stochastic models appears in the form of oscillations in the averaged model obtained in the limit of infinite sized population. Second that in this version, to the different regimes correspond different asymptotic regimes of the system. In summary, these numerical simulations point to the mathematical analyses of the population models as one way of gaining insight into the mechanisms that control the genesis of various forms of activity in neural populations. This paper is concerned with such mathematical analyses of a continuous model for a random neural assembly. We first introduce the model and show that it exhibits self-sustained oscillations in a certain range of a parameter J measuring the network connectivity. For mathematical analysis, we consider a particularly simple form of interactions, that preserves the main property of the model regarding self-sustained oscillations and for which we can prove existence of solutions. We show that it exhibits relaxation to a steady state when J is small or J is large. Since a population of individual neuron (J = 0) has the property of desynchronization, the interpretation for J small is clear. For J large it means that too many connections also restore this desynchronization of the assembly.

After a presentation of the model in section 2, we analyze the steady states (section 3) and recall a method to prove the resynchronization property (with exponential rate) for disconnected networks (section 4). Existence of solutions to the nonlinear system for the assemblies of neurons is proved afterwards in section 5 and this brings material to show global nonlinear asymptotic stability both for small and large connectivity, this is performed in section 6. We conclude with numerical simulations that illustrate our theoretical results and the existence of self-sustained oscillations in the middle range of the connectivity parameter.

The neuron population model

We consider a population of neurons described by the probability density n(s, t) of finding a neuron in 'state' s at time t where s represents the time elapsed since the last discharge. According to [START_REF] Pham | Activity in sparsely connected excitatory neural networks: effect of connectivity[END_REF][START_REF] Pham | Noise-induced coherent oscillations in randomly connected neural networks[END_REF] a simple model of the dynamics is given by the age-structured equation ∂n(s,t) ∂t

+ ∂n(s,t) ∂s + p(s, X(t)) n(s, t) = 0, N (t) := n(s = 0, t) = +∞ 0 p(s, X(t)) n(s, t)ds, (1)
completed with an initial probability density n 0 (s) that satisfies

0 ≤ n 0 (s) ≤ 1, ∞ 0 n 0 (s)ds = 1. (2)
The coefficient p(s, X) represents the firing rate of neurons in the 'state s' and in an environment X resulting from the global neural activity. It is usually small (or even vanishes) during a rest phase that depends on X, and increases suddenly afterwards. The density of neurons undergoing a discharge at time t is denoted by N (t) and the boundary condition at s = 0 means that the neuron re-enters the cycle at 'age s = 0' after firing. The interactions between the neurons are taken into account through the global neural activity at time t, X(t), that is given by

X(t) = J t 0 α(u)N (t -u)du. (3)
The parameter J ≥ 0 represents the network connectivity (and thus the strength of the interactions), and the delay takes into account an averaged propagation time for the ionic pulse in this network.

Here we assume that the neurons can discharge earlier when the neural activity is high.

Notice that this nonlinear system has a remarkable conservation property

∞ 0 n(s, t)ds = ∞ 0 n 0 (s)ds = 1 ∀t ≥ 0, (4)
which expresses the interpretation of n(t, •) as a probability density. These modeling assumptions can be written on the (nonnegative) coefficients as

∞ 0 α(u)du = 1, α(•) ≥ 0, (5)
∂ ∂s p(s, x) ≥ 0, p(s, x) = 0 for s ∈ (0, s * (x)) p(s, x) > 0 for s > s * (x), p(s, x) ----→ s→∞ 1, (6)
∂ ∂x p(s, x) ≥ 0, p(s, x) ----→ x→∞ 1. (7)
In order to simplify these assumptions and the mathematical analysis, throughout the paper we often use the particular firing rate

p(s, x) = 1 {s>s * (x)} , for some s * ∈ C 1 , d dx s * (x) ≤ 0, s * (0) = s * + , s * (∞) = s * -< s * + < 1. (8)
A discontinuity in this coefficient is not realistic but does not change the behavior of solutions that we describe below; it has the advantage of allowing for simpler explicit expressions and thus it simplifies the proofs. In the range x ≈ 0, the value s * + represents the spontaneous activity due to noise. In the range x ≫ 1 we will use the decay rates D and D defined by lim sup

J→∞ sup x≥ J (1+s * + ) J |s * ′ (x)| 1 + s * (x) 2 := D, (9)
lim sup

J→∞ sup x∈C J J |s * ′ (x)| := D, (10)
where

C(J) = [J(1 -s * + ), J]. (11)
Later on, we will use the condition that the constants D and D are small enough, this means that s * (x) → s * -fast enough as x → ∞. This holds for example if s * (x) = s * -for x large enough or more generally if |s * ′ (x)| ≤ 1/x 2 (then D = 0 and D = 0). The condition D < 1 or D < 1 corresponds to |s * ′ (x)| ≤ A/x with A small enough.

The equation ( 1) is an original nonlinear version of a standard linear equation, the renewal equation. It arises in various biological applications as epidemic spread, cell division cycle or even ecology (see [START_REF] Gyllenberg | A nonlinear structured population model of tumor growth with quiescence[END_REF][START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Diekmann | The second half-with a quarter of a century delay[END_REF]).

Our goal is to study the existence of solutions to (1) and their long time behavior. Before we consider the steady states and the linear equation corresponding to X ≡ 0.

Steady states

Let J ≥ 0 and assume the simple situation [START_REF] Magal | Center manifold for semilinear equations with non-dense domain and applications on Hopf bifurcation in age structured models[END_REF]. A function A J (s) associated with the neural activity X * J , is a unit mass stationary solution to (1), (3) if and only if

       ∂A J ∂s + p(s, X * J )A J = 0, A J (0) = 1 + s * (X * J ) -1 , X * J = J 1 + s * (X * J ) -1 . (12)
Indeed we have

A J (s) = A J (0)e - R s 0 p(σ,X * J )dσ , A J (0) = ∞ 0 e - R s 0 p(σ,X * J )dσ ds -1 = 1 + s * (X * J ) -1 .
Consequently, the third equation of ( 12) defines steady states and, from our assumptions on s * (x), for all J there is at least one steady state. It is easy to see that there can be several steady states when s * (x) has large enough derivative.

Uniqueness is equivalent to find a unique X * J such that

X * J = J 1 + s * (X * J ) -1 .
Therefore uniqueness always holds true for J small enough. Uniqueness of the steady state also holds true, for all J ≥ 0, under the condition

J |s * ′ (x)| 1 + s * (x) 2 < 1, ∀x > 0,
but such a regime is very restricitive. We can relax it and notice that Lemma 3.1 With the assumptions (8), the steady state is unique for J large enough under the condition D < 1 in (9).

Proof of Lemma 3.1. Indeed, using (8), a steady state satisfies X * J ≥ J 1 + s * + -1 . If there are two steady states at least, then there is a point Y in between such that

1 = - J s * ′ (Y ) 1 + s * (Y ) 2 which means J ≥ J 0 , the largest value of J such that sup x≥J/(1+s * + ) J s * ′ (x) 1 + s * (x)
2 ≥ 1 which exists thanks to our assumption on D.

4 Asymptotic stability of the steady states for J = 0

For J = 0 the system is linear and reads, with the short hand notation p(s) = p(s, 0),

       ∂n(s,t) ∂t + ∂n(s,t) ∂s + p(s)n(s, t) = 0, N (t) := n(s = 0, t) = ∞ 0 p(s)n(s, t)ds, n(s, 0) = n 0 (s) ≥ 0, ∞ 0 n 0 (s)ds = 1. (13)
This linear equation admits a unique one parameter multiplicative family of steady states, we normalize to achieve unit mass

A(s) = A(0)e - R s 0 p, A(0) = ∞ 0 e - R s 0 pds -1
.

A first step is to show that this state is exponentially stable, i.e., that for long times all solutions to (13), whatever the initial data, converge exponentially to A(s) (thanks to the conservation property ( 4)). This property is known to follow from Laplace transform methods [START_REF] Feller | An introduction to probability theory and applications[END_REF] which are not easy to extend to nonlinear situations and which do not provide a norm to establish the convergence. Another approach is through delay equations. For the case of neural models, see [START_REF] Hien | Periodic solutions and exponential stability for shunting inhibitory cellular neural networks with continuously distributed delays[END_REF] and for a general approach see [START_REF] Diekmann | Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars[END_REF] . We rather use the entropy method introduced in [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]. The difficulty is that we need some kind of spectral gap property to arrive to an exponential convergence rate, which leaves some space for treating the nonlinear case later.

Theorem 4.1 (J = 0) Assume p(s) = 1 {s>s * } with s * < 1, then there is a µ > 0 and a function

ψ(s) ≥ B > 0 such that ∞ 0 ψ(s) n(s, t) -A(s) ds ≤ e -µt ∞ 0 ψ(s) n 0 (s) -A(s) ds.
Proof of Theorem 4.1. We set n(s, t) = n(s, t)-A(s). It still satisfies the equation ( 13) by linearity. Therefore, we also have

∂| n(s, t)| ∂t + ∂| n(s, t)| ∂s + p(s)| n(s, t)| = 0.
Let ν ∈ (0, 1), µ > 0 and ψ(s) > 0 satisfy (we still have to show this is possible and this is the lemma 4.2 below) ∂ψ ∂s

-(p(s) -µ)ψ = -ψ(0)|p(s) -ν|. (14)
Then we have (recall the conservation (4) which here tells us that

∞ 0 n(s, t)ds = 0) d dt ∞ 0 ψ(s)| n(s, t)|ds + ∞ 0 - ∂ψ(s) ∂s + p(s)ψ(s) | n(s, t)|ds = = ψ(0) ∞ 0 p(s) n(s, t) = ψ(0) ∞ 0 [p(s) -ν] n(s, t) ≤ ψ(0) ∞ 0 |p(s) -ν| | n(s, t)| .
Then, the supposed equation on ψ gives

d dt ∞ 0 ψ(s) | n(s, t)|ds = -µ ∞ 0 ψ(s) | n(s, t)|ds
and the result follows.

Lemma 4.2 Let p(s) given by p(s) = 1 {s>s * } with s * < 1 and 0 < ν ≤ 1 . Then, there exists µ > 0 such that equation ( 14) has a uniformly positive solution ψ normalized with ψ(0) = 1. Moreover, for the range

0 < s * -≤ s * ≤ s * + < 1, there is a uniform constant B > 0 such that ψ(s) ≥ B.
Proof of Lemma 4.2. Let 0 < µ < 1 and choose ψ(0) = 1. On (0, s * ), we have ψ ′ (s) = -µψ -ν, i.e.,

ψ(s) = e -µs (1 + ν µ ) - ν µ . On (s * , ∞), we have ψ ′ (s) = (1 -µ)ψ -(1 -ν)
, and this gives exponential growth as soon as

(1 -µ)ψ(s * ) = (1 -µ) e -µs * (1 + ν µ ) - ν µ > 1 -ν. ( 15
)
Thanks to Taylor expansion in µ at 0, we obtain that

(1 -µ)(1 -νs * + ε(µ)) > 1 -ν, with lim µ→0 ε(µ) = 0. Hence it is enough to check that (1 -µ)(1 -νs * ) > 1 -ν.
This holds true for µ small enough, because s * < 1. Lemma 4.2 is proved.

Existence and a priori estimates

Before we study its stability, we first establish a general existence theorem for the nonlinear problem under consideration.

Theorem 5.1 Assume (2). Assume furthermore that there exists q > 1 such that α ∈ L q (R + ). Then, there exists a unique solution to (1), [START_REF] Diekmann | Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars[END_REF],

n ∈ C R + ; L 1 (R + ) and 0 ≤ n(s, t) ≤ 1, ∀t, s ≥ 0, ( 16
) 0 ≤ N (t) ≤ 1, X(t) ∈ C(J), ∀t ≥ 0 ( 17
)
where C(J) is defined by [START_REF] Perthame | Transport equations in biology[END_REF].

The assumption α ∈ L q (R + ) with q > 1 is crucial in our proof. However the case with no delay and more generally fixed delay, α(s) = δ(s = τ ), can be handled by the same ideas when the firing rate p is Lipschitz continuous in s.

Proof of Theorem 5.1. To prove wellposedness of a solution to (1), we are going to use a fixed point argument.

More precisely, we prove that there is a T > 0 such that the following application

   F : L ∞ ([0, T ]) → L ∞ ([0, T ]) F (Y (• ))(t) := J t 0 α(u)N (t -u)du
admits a fixed point, where N is given by

N (t) := n(s = 0, t) = +∞ 0 p(s, Y (t)) n(s, t)ds
and where n is the solution of the equation ∂n(s,t) ∂t

+ ∂n(s,t) ∂s + p(s, Y (t)) n(s, t) = 0, N (t) := n(s = 0, t) = +∞ 0 p(s, Y (t)) n(s, t)ds,
completed with the initial data n 0 satisfying (2). A fixed point of F , based on the Banach-Picard fixed theorem gives a unique solution on [0, T ] to the equation (1). Iterating on T (which does not depend upon the initial data as we see it below), we deduce that in fact we have global existence and uniqueness of solutions of equation [START_REF] Brunel | Dynamics of sparsely connected networks of excitatory and inhibitory spiking networks[END_REF].

We are reduced to proving the following Lemma.

Lemma 5.2 Let J ≥ 0. Then, there exists T > 0 such that for all (X 1 , X 2 ) ∈ L ∞ ([0, T ]), the following estimate holds

F (X 1 ) -F (X 2 ) L ∞ ([0,T ]) ≤ 1 2 X 1 -X 2 L ∞ ([0,T ]) . ( 18
)
Proof of Lemma 5.2. We use the characteristics. For all s ≥ 0, the function n s (t) := n(s + t, t) is a solution of the equation

d dt n s (t) = -p(t + s, Y (t)) n s (t).
We deduce that n s (t) = n 0 (s)e -

R t 0 p(t ′ +s,Y (t ′ ))dt ′ and thus n(s, t) = n 0 (s -t)e - R t 0 p(t ′ +s-t,Y (t ′ ))dt ′ for s ≥ t. (19)
On the other hand, for all t ≥ 0 we call n t (s) := n(s, s + t). Then,

d ds n t (s) = -p(s, Y (t + s)) n t (s).
We deduce that n(s, s

+ t) = N (t)e - R s 0 p(s ′ ,Y (s ′ +t))ds ′ ,
and, as before, we deduce that

n(s, t) = N (t -s)e - R s 0 p(s ′ ,Y (s ′ +t-s))ds ′ for s ≤ t. (20)
We denote by n i , i = 1, 2 two solutions of the equation ∂n i (s,t) ∂t

+ ∂n i (s,t) ∂s + p(s, X i (t)) n i (s, t) = 0, N i (t) := n i (s = 0, t) = +∞ 0 p(s, X i (t)) n i (s, t)ds,
still with the initial data n 0 . From ( 19) and (20), we deduce that

N i (t) = t 0 p(s, X i (t))N i (t-s)e - R s 0 p(s ′ ,X i (s ′ +t-s))ds ′ ds+ +∞ t n 0 (s-t)p(s, X i (t))e - R t 0 p(t ′ +s-t,X i (t ′ ))dt ′ ds.
We deduce that

F (X 1 )(t) -F (X 2 )(t) = J t 0 α(u)(H + E)(t -u)du
where

H(t) = t 0 N 1 (t -s)p(s, X 1 (t))e - R s 0 p(s ′ ,X 1 (s ′ +t-s))ds ′ ds - t 0 N 2 (t -s)p(s, X 2 (t))e - R s 0 p(s ′ ,X 2 (s ′ +t-s))ds ′ ds,
and E(t) denotes the remaining term.

To prove Lemma 5.2, it is enough to prove that there exists a constant C such that for all t ∈ [0, T ],

H L ∞ ([0,T ]) + E L ∞ ([0,T ]) ≤ C X 1 -X 2 L ∞ ([0,T ]) . (21)
Indeed, we have with the Hölder inequality

F 1 -F 2 L ∞ ([0,T ]) ≤ JT 1 q ′ α L q H L ∞ ([0,T ]×[0,T ]) + E L ∞ ([0,T ]×[0,T ]) hence F 1 -F 2 L ∞ ([0,T ]) ≤ CT 1 q ′ X 1 -X 2 L ∞ ([0,T ]) .
By taking T small enough, we deduce estimate (18). This is where we use q > 1.

Study of H. We have

H = H 1 + H 2 + H 3
where

H 1 = t 0 (N 1 -N 2 )(t -s)p(s, X 1 (t))e - R s 0 p(s ′ ,X 1 (s ′ +t-s))ds ′ ds H 2 = t 0 N 2 (t -s) p(s, X 1 (t)) -p(s, X 2 (t)) e - R s 0 p(s ′ ,X 1 (s ′ +t-s))ds ′ ds H 3 = t 0 N 2 (t -s)p(s, X 2 (t)) e - R s 0 p(s ′ ,X 1 (s ′ +t-s))ds ′ -e - R s 0 p(s ′ ,X 2 (s ′ +t-s))ds ′ ds.
To deal with H 1 , we have to control more precisely N 1 -N 2 . We have

N 1 -N 2 (t) = F 1 + F 2 (t)
where

F 1 (t) = +∞ 0 p(s, X 1 )(t)(n 1 (s, t) -n 2 (s, t))ds F 2 (t) = +∞ 0 n 2 (s, t)(p(s, X 2 (t)) -p(s, X 1 (t)))ds.
To control the term F 2 , we use the Hölder inequalities, which tells us that

|F 2 (t)| ≤ n 2 (t) L ∞ L{s ∈ R + such that p(s, X 2 (t)) -p(s, X 1 (t)) = 0}
where L denotes the Lebesgue measure. But we have

L s ∈ R + such that p(s, X 2 (t)) -p(s, X 1 (t)) = 0 ≤ |s * (X 2 (t)) -s * (X 1 (t))| ≤ s * ′ L ∞ |X 1 (t) -X 2 (t)|. (22)
Therefore we have obtained that there exists a constant C such that for all

t ∈ R + |F 2 (t)| ≤ C|X 1 (t) -X 2 (t)|. ( 23
)
To control the term F 1 , we study n := n 1 -n 2 . Using computations made in Section 4, we deduce that there exists µ > 0 such that

f ′ (t) ≤ -µf (t) + C|X 1 (t) -X 2 (t)| ( 24
)
where f (t) := +∞ 0 ψ| n|ds. As n 1 (s, 0) = n 2 (s, 0), we deduce that f (0) = 0 and using equation (24), we deduce that for all t ≥ 0,

f (t) ≤ 2 C µ |X 1 (t) -X 2 (t)|.
By consequence, there exists a constant C such that

F 1 (t) ≤ C|X 1 (t) -X 2 (t)|,
which proves that there exists a constant C such that for all T > 0,

H 1 L ∞ ([0,T ]) ≤ C X 1 -X 2 L ∞ ([0,T ]) .
Concerning the term H 2 , we obtain with the Hölder inequality

H 2 (t) ≤ N 2 L ∞ (R + ) L{s ∈ R + such that p(s, X 2 (t)) -p(s, X 1 (t)) = 0}.
Hence there exists a constant C such that for all T > 0, for all t ∈ [0, T ]

H 2 (t) ≤ C X 1 -X 2 L ∞ ([0,T ]) .
To deal with the term H 3 , we use the fact that there exists a constant C such that

e - R s 0 p(s ′ ,X 1 (s ′ +t-s))ds ′ -e - R s 0 p(s ′ ,X 2 (s ′ +t-s))ds ′ ≤ C s 0 p(s ′ , X 1 (s ′ + t -s)) -p(s ′ , X 2 (s ′ + t -s)) ds ′ .
We deduce that there exists a constant C such that

|H 3 (t)| ≤ C t 0 t 0 |p(s ′ , X 1 (s ′ + t -s)) -p(s ′ , X 2 (s ′ + t -s))|ds ds ′ .
With the change of variable v = s ′ + t -s, we deduce that

|H 3 (t)| ≤ C t 0 t s ′ |p(s ′ , X 1 (v)) -p(s ′ , X 2 (v))|dv ds ′
and hence

|H 3 (t)| ≤ C t 0 t 0 |p(s ′ , X 1 (v)) -p(s ′ , X 2 (v))|dv ds ′ .
Using the fact that there exists a constant C such that for all v ∈ [0, t]

L s ∈ R + such that p(s, X 2 (v)) -p(s, X 1 (v)) = 0 ≤ s * ′ L ∞ |X 1 (v) -X 2 (v)|,
we deduce that there exists a constant C such that

|H 3 (t)| ≤ C X 1 (t) -X 2 (t) L ∞ ([0,T ]) .
Study of E. We have

E(t) = +∞ t n 0 (s -t) p(s, X 1 (t))e - R t 0 p(t ′ +s-t,X 1 (t ′ ))dt ′ -p(s, X 2 (t))e - R t 0 p(t ′ +s-t,X 2 (t ′ ))dt ′ ds.
As n 0 L ∞ ≤ 1, it is enough to prove that there exists a constant C such that for all t ∈ [0, T ]

G := +∞ t p(s, X 1 (t))e - R t 0 p(t ′ +s-t,X 1 (t ′ ))dt ′ -p(s, X 2 (t))e - R t 0 p(t ′ +s-t,X 2 (t ′ ))dt ′ ds ≤ C X 1 -X 2 L ∞ ([0,T ]) .
We have

G ≤ G 1 + G 2
where

G 1 = +∞ t p(s, X 1 (t)) -p(s, X 2 (t)) e - R t 0 p(t ′ +s-t,X 1 (t ′ ))dt ′ ds, G 2 = +∞ t p(s, X 2 (t)) e - R t 0 p(t ′ +s-t,X 2 (t ′ ))dt ′ -e - R t 0 p(t ′ +s-t,X 1 (t ′ ))dt ′ ds.
Using estimate ( 22), we deduce that there exists a constant C such that

G 1 ≤ C X 1 -X 2 L ∞ ([0,T ]) .
We have

G 2 ≤ +∞ t t 0 |p(t ′ + s -t, X 2 (t ′ )) -p(t ′ + s -t, X 1 (t ′ ))|dt ′ ds.
By the change of variables v = t ′ + s -t, we deduce that

G 2 ≤ C +∞ 0 t 0 |p(v, X 2 (t ′ )) -p(v, X 1 (t ′ ))|dvdt ′
which proves that there exists a constant C such that

G 2 ≤ C X 1 -X 2 L ∞ ([0,T ])
using estimate (22). All these estimates together complete the proof Lemma 5.2.

It remains to prove estimate (17) of Theorem 5.1. Firstly, we have

N (t) = +∞ 0 p(s, X(t))n(s, t)ds ≤ +∞ 0 n(s, t)ds = 1.
Secondly, for X(t), we write

X(t) = J +∞ 0 α(u)N (t -u)du ≤ J N L ∞ +∞ 0 α(u)du ≤ J.
On the other hand, we have

N (t) ≥ +∞ s * + n(s, t)ds because p(s, • ) ≡ 1 for all s ∈ [s * + , +∞). But, s * + 0 n(s, t)ds ≤ s * + n L ∞ ≤ s * + max( n 0 L ∞ , n 0 L 1 ) ≤ s * + .
As for all t ≥ 0, we know that +∞ 0 n(s, t)ds = 1, we deduce that

+∞ s * + n(s, t)ds ≥ 1 -s * + .
We deduce that X(t) ≥ J(1 -s * + ) which completes the proof of Theorem 5.1.

6 Nonlinear stability of the steady state for J ≈ 0 and J ≈ +∞.

In this section we prove the nonlinear asymptotic stability of the steady state on the one hand for J small enough, and on the other hand for J ≈ +∞. For J ≈ +∞ our analysis is based on the decay property (10) of s * (x) for D small enough.

Our approach uses the spectral gap property arising from Lemma 4.2. Given a steady state X * J , we can consider the value s * (X * J ) ∈ (s * -, s * + ), and the corresponding function ψ J ≥ B > 0 as in Lemma 4.2. We recall that the constants ν > 0, µ > 0 and B > 0 can be chosen uniformly positive, bounded, and independent of J. For this reason we use the notation ψ in place of ψ J in this section.

The main theorem of this section is the following Theorem 6.1 Assume ( 2), ( 8) with s * + < 1 and

∞ 0 ψ(s) n 0 (s) -A J (s) ds < +∞
where A J is given by equation ( 12) and consider the solution n to (

Then, for J small enough, the following estimate holds

lim t→+∞ ∞ 0 ψ(s) n(s, t) -A J (s) ds = 0.
For J large enough, and assuming that estimate (10) holds with D small enough, the following estimate holds lim t→+∞ ∞ 0 ψ(s) n(s, t) -A J (s) ds = 0.

We recall, from section 3 that these are regimes where the steady states are unique. We first treat the case with no delay which is simpler and allows us to give the idea of the proof. Then, we consider time delay.

The case with no delay

Theorem 6.2 Assume (2), ( 8) with s * + < 1 and that there is no delay, i.e., X(t) = JN (t). Then, there is a constant β > 0 such that for J small enough, the following estimate holds

∞ 0 ψ(s) n(s, t) -A J (s) ds ≤ e -βt ∞ 0 ψ(s) n 0 (s) -A J (s) ds
where A J is given by equation [START_REF] Pham | Activity in sparsely connected excitatory neural networks: effect of connectivity[END_REF].

Proof of Theorem 6.2. The function n = n -A J satisfies the equation

∂ t n + ∂ s n + p(s, X * J ) n + [p(s, X(t)) -p(s, X * J )]n = 0.
We deduce that

∂ t | n| + ∂ s | n| + p(s, X * J )| n| ≤ |p(s, X(t)) -p(s, X * J )|n.
In order to simplify notations, we define f by

f (t) := ∞ 0 ψ| n(s, t)|ds = ∞ 0 ψ(s) n(s, t) -A J (s) ds.
Then, according to the calculation in the proof of Theorem 4.1,

d dt f (t) ≤ -µf (t) + ∞ 0 I s∈[s * (X(t)),s * (X * J )] ψn(s, t)ds. (25)
As the function s * (•) is Lipschitz continuous, and 0

≤ n ≤ 1, if we note C J = s * ′ L ∞ (C(J))
(where C(J) is given by ( 11)); we deduce that there exists a constant C such that

∞ 0 I s∈[s * (X(t)),s * (X * J )] ψn(s, t)ds ≤ C J |X(t) -X * J | ψn L ∞ (s * -,s * + ) ≤ CC J |X(t) -X * J |.
But, we have also

X(t) -X * J = J +∞ 0 p(s, X(t)) nds + J +∞ 0 [p(s, X(t)) -p(s, X * J )]A J (s)ds.
We deduce that |X(t) -

X * J | ≤ J B f (t) + C J J |X(t) -X * J | A J L ∞ ,
and hence, since A J L ∞ = A J (0) < 1, we have as long as C J J < 1,

|X(t) -X * J | ≤ J B(1 -C J J) f (t).
Back to (25), we obtain

d dt f (t) ≤ -µf (t) + C C J J B(1 -C J J) f (t)
which proves Theorem 6.2 under the condition C C J J B(1-C J J) < µ; that is to say that C J J is small enough. This is the case as soon as J is small enough because C J ≤ s * ′ L ∞ (R + ) remains bounded. This is also the case when J is large enough and if condition (10) holds with D small enough.

Case with delay

Proof of Theorem 6.1. In this setting, it is a priori not true that the solution converge exponentially to the steady state given by equation [START_REF] Pham | Activity in sparsely connected excitatory neural networks: effect of connectivity[END_REF]. We are going to prove that the solution tends to the steady state when t → +∞ without informations of the rate of convergence. This is due to the fact that here, we add a delay pondered by the function α in the formula of X with respect to the preceding case, and the way that the solution tends to the steady state at infinity seems depends strongly of the function α. To prove Theorem 6.1, the idea is to use an iteration proceed.

Using formula (25) in the proof of Theorem 6.2, we already know that

d dt f (t) ≤ -µf (t) + CC J |X(t) -X * J |.
The problem here is to control the term |X(t) -X * J | which do not trivially tends to 0 when t tends to +∞ because the formula of X(t) takes in consideration all the value of f between 0 and t contrarily to the case with no delay. But we have the following control on X and X * J , see (17),

X(t) ∈ C(J) and X * J ∈ C(J), (26)
and therefore

d dt f (t) ≤ -µf (t) + C(C J J). (27)
Hence, if J is small enough or if J is big enough and if condition (10) holds with D small enough, then C J J is small. We can deduce that f is also not to big if t is big enough. More precisely

f (t) ≤ max(f (0), C(C J J) µ ), f (t) ≤ 2 C(C J J) µ for t ≥ T 0 large enough. ( 28
)
Now we can estimate more accurately the term

X(t) -X * J = J +∞ 0 α(u) +∞ 0 p(s, X(t -u)) n(s, t -u)dsdu + J +∞ 0 α(u) +∞ 0 [p(s, X(t -u)) -p(s, X * J )]A J (s)dsdu.
We claim that in fact if t is large enough then a constant C exists such that

|X(t) -X * J | ≤ C C J J 2 µ ( 29
)
To prove it, we set

I 1 (t) := +∞ 0 α(u) t u=0 p(s, X(t -u)) | n(s, t -u)| dsdu, I 2 (t) := t u=0 α(u) ∞ 0 |p(s, X(t -u)) -p(s, X * )|A J (s)dsdu.
Concerning, the second term I 2 (t), using the fact that at t and u fixed, the set of s ∈ R + such that |p(s, X(t -u)) -p(s, X * )| = 0 is less than |s * (X(t -u)) -s * (X * J )|; we deduce via (26) (which holds for all t) that for all (t, u)

∈ R + × [0, t] |s * (X(t -u)) -s * (X * J )| ≤ (C J J).
Hence, we deduce that 0

≤ I 2 ≤ 2(C J J) A J L ∞ ≤ 2(C J J).
Remark 6.3 Let us mention that here, (26) holds for all t ∈ R + and hence we do not see the problem of the delay. For the second iteration, we control estimate (29) only if t is big enough so we can not apply exactly the same strategy and we have to deal with the delay. The question of the delay is treated in the next estimate of I 2 where the problem of the delay appear in the first iteration.

Concerning, the first term I 1 , we have

0 ≤ I 1 (t) ≤ 1 B t u=0 α(u) ∞ 0 ψ(s) | n(s, t -u)| dsdu.
leq Here, we must be careful of the delay because here at t fixed, f is integrated over all the points t ′ = t-u with u ∈ [0, t] that is to say that all the values of f between 0 and t are taken in consideration.

But we only know that f (t ′ ) is small if t ′ is big enough. This is in fact not a problem because the function α ∈ L 1 . Indeed, when t is big, t -u is small (and hence bad) only when u is near to t. But α being integrable, we have lim

A→+∞ +∞ A α(u)du = 0.
More precisely, if t is large enough we have

α * f ≤ 2(C J J) µ because α ∈ L 1 and f L ∞ ≤ C. Indeed, we split the integral +∞ 0 α(u)f (t -u)du
in two parts. As α ∈ L 1 , we deduce that there exists a T 1 > 0 such that

+∞ T 1 α(u)du ≤ C J J max(1, f L ∞ ([0,T 0 ]) ) ,
Let us mention, using (27) that the denominator is controlled because

f L ∞ (R + ) ≤ max f (0), 2( CC J µ ) .
Now, we choose t big enough such that t -T 1 ≥ T 0 and we have

T 1 0 α(u)f (t -u)du ≤ α L 1 f L ∞ ([T 0 ,+∞]) ≤ C(C J J) µ .
Back to our estimate, we obtain

d dt f (t) ≤ -µf (t) + C (C J J) 2 µ .
and hence for t large enough

f (t) ≤ C (C J J) 2 µ 2 .
By taking J small enough or big enough assuming estimate (10) with D small enough, we obtain that

C(C J J) µ < 1
and hence by iteration, we deduce Theorem 6.1.

Remark 6.4 Starting the second iteration, when we deal with the term I 1 , we also have to split the integral of α in two subintegrals as done for the study of I 2 for the first iteration because estimate (29) holds only if t is large enough.

Periodic solutions and numerical results

We have seen that the network connectivity J in the neuron system (1) determines both the number of steady states and their stability in the range J small or large. We now study, by means of direct numerical simulations, if other solutions may exist in the middle range of J. In particular, despite the dissipative nature of each individual neuron leading to relaxation towards an asynchronous state (J = 0), is it possible that the network connectivity leads to natural periodic oscillations as observed experimentally and described by other network models ( [START_REF] Brunel | Dynamics of sparsely connected networks of excitatory and inhibitory spiking networks[END_REF] and the references therein). Existence of periodic solutions for age structured equations has been obtained through Hopf bifurcation, see [START_REF] Magal | Center manifold for semilinear equations with non-dense domain and applications on Hopf bifurcation in age structured models[END_REF], but the mechanism here is different because it seems to stem from the appearance of several steady states according to section 3.

We have chosen the framework of the particular firing rate given by formula (8) with s * (x) = 2 -x 4 1 + x 4 , and we study either the case with no delay or a Poisson law for the pulse propagation delay α α(u) = e -a u .

(30)

The numerical solutions are depicted in Figure 1 . They put in evidence that an intermediate regime exists for appropriate values of J where periodic solutions occur.

The effect of delay is generically to stabilize the dynamics and the periodic solutions do not seem to exist for J = 3 for instance. The numerical solutions for J = 2.5 are depicted in Figure 2 and show that periodic solutions persist for large enough values of a in (30).

Figure 1 :

 1 Figure 1: Total neural activity N (t) in the case with no delay. Three different values of the network connectivity have been used, left: J = 2, center J = 3, right J = 3.5.

Figure 2 :

 2 Figure2: Total neural activity N (t) in the case with delay. We used the network connectivity J = 2.5. Left: no delay, center a = 5, right a = 4.


Professor Emeritus, Alain Alcouffe

F M N Mourad

Paolo Buchleitner

F Tortese

A Squadri

T Castoldi

D Solano

Valenze

Economic Crisis and Unemployment in 19 th Century literature

A dramatic increase in unemployment followed the crash of 1929 and signaled the downward economic slide initiating the depression; thereafter unemployment (meant as involuntary unemployment) became the most significant indicator of the economic crisis. In the General Theory, Keynes distinguished three categories of unemployment: frictional, voluntary and a third category,

which he defines as "involuntary unemployment" and asserted that Classical economics "do not admit the possibility of the third category (Keynes, 1936, p.6). This focus on cyclical unemployment was an important feature of the Keynesian Revolution, it led to the reconstructed vision of "Classical" economics and it did not include any representation of cyclical unemployment. The many controversies about Say's law have clearly established that the existence of involuntary unemployment had been recognized very early on in the long lasting discussion about poverty. Classical economists, as well as their critics, generally considered an excess of labor supply to be very common, especially as a consequence of the introduction of machinery. Despite the fact that they did not deal at length with the short-term fluctuations in employment, JB Say himself was perfectly aware of the consequences of the crisis with workers, and he went so far as to advocate Keynesian policies of public works in order to cope with unemployment [START_REF] Baumol | Retrospectives[END_REF].

Nevertheless, it was only in the last decades of the 19 th century that cyclical unemployment became a major issue in economic literature, while the first insurance schemes were being implemented. Several authors confused the implementation of these remedies to unemployment with the "discovery" or the invention of unemployment (Salais et alii (1986), [START_REF] Topalov | Naissance du chômeur[END_REF], [START_REF] Walters | Unemployment and government: genealogies of the social[END_REF], Alberti, M. (2013), [START_REF] Lipowski | Die Frage der Arbeitslosigkeit in der klassischen Nationalökonomie in[END_REF], [START_REF] Promberger | Eine kurze Geschichte der Arbeitslosigkeit * Teil 1: Vom Mittelalter bis zur Industrialisierung[END_REF]: Zimmermann, B. ( 2001)). It is nevertheless true that there was no room for cyclical unemployment in the opinion of some neoclassical economists [START_REF] Hicks | The theory of wages[END_REF], 1963) but decades before the Great Depression and the General Theory, Gustav Cassel, commenting on (?) British unemployment statistics, concluded that "The real significance of these statistics is that they are a symptom of trade cycles, and represent a kind of barometer of crises" (Cassel, 1918). Preprint 2

I Unemployment and the crisis in economics

Early recognition of involuntary unemployment in economics

It would be vain to look for the origin of the word or the idea of « involuntary unemployment », which has its roots in the treatment of poverty. It is very plausible that it was always common knowledge that among poor people, some were able and willing to work. The Poor Law system classified people into categories: those considered deserving and those who were not considered deserving of poor relief.

Four categories were distinguished:

The impotent poor

The able-bodied poor -with an able body but were unwilling to work.

The idle poor

Vagrants or beggars, sometimes termed "sturdy rogues"

The second category included those who were unable to find work -either due to cyclical or long-term unemployment in the area, or due to a lack of skills. Attempts to assist these people, and move them out of this category, varied over the centuries, but usually consisted of relief either in the form of work or money. In a wide range of literature, we can pick up the position sustained by the novelist Henry Fielding (1707Fielding ( -1754) ) and the economist and statesman Turgot. The author of the famous novel Tom Jones (published in 1749) was also the author of an Enquiry into the Causes of the late Increase of Robbers (1751), written at the request of the City of London. Fielding who is often considered as a democrat and a radical social reformer held views on unemployment which were, to say the least, unsympathetic.

The poor emulated the vices of the rich when they sat idle: aspiring still to a degree beyond what belongs to them, and not being able by the Fruits of honest Labor to support the State which they affect, they disdain the Wages to which their Industry would entitle them; and abandoning themselves to Idleness, the more simple and poor-spirited betake themselves to a State of Starving and Beggary, while those of more Art and Courage become Thieves, Sharpers, and Robbers (ibidem, p. 4).

Like most works on the subject, Fielding's Enquiry attempted to classify the poor (defined as having no source of 'gaining a comfortable subsistence') into distinct groups. His three 'divisions' were drawn from Elizabethan statutory distinctions: 'Such Poor as are unable to work', 'Such as are able and willing to work', and 'Such as are able to work, but not willing'. Fielding judged that the first class was very small and it would not be difficult to provide them some relief. The second, on the contrary: "though not in fact, equally objects of the regard of the compassionate man, and much more worthy the care of the politician; and yet, without his care, they will be in a much more worse condition than the others a: for they have none of these incitements of pity that fill the pockets of the artful beggar, and procure relief for the blind, the lame, and other visible subjects of compassion. Such, therefore, without a law, and without an honest and sensible execution of that law, must languish under, and often perish with want. [..] To say the truth, this affair of finding a universal employment for the industrious poor is of great difficulty, [..] And yet difficult as it is, it is not, I hope, not impracticable" Fielding reached the conclusion that "it is a task surely not above the reach of the British parliament to find work for the hands" willing to work (ibidem, p.52).

For Turgot, the task seems altogether possible and necessary 2 . When Turgot was "Intendant" of the Limousin region, France suffered one of the last subsistence crises of the eighteenth century, and the experience of the crisis exercised a profound influence on Turgot's subsequent policies. During 1769-1770, following a series of bad harvests food prices increased sharply in the Limousin region, and mortality began to increase, especially in remote rural areas. The freedom of the corn trade could not prevent scarcity "in the first years when it is established", as Turgot concluded; "if commerce is to be able to prevent the scarcities entirely", he wrote to Dupont de Nemours, "the people would already have to be rich". The prospects of the landless poor were evidently insecure while a charity of individuals (or their "moral economy") provided insufficient security during the crisis of 1770. There was a tendency for prosperous farmers to send away their sharecroppers, Turgot wrote, and to "turn out their domestics and servants" so that unemployment grew. His principal strategy, in the Limousin, was to provide short-term employment for public works, and later as Minister of Finance, he attempted to generalize his policies in other regions, his reform of relief policies aimed to reduce the numbers of people who were unemployed for long periods. Therefore, he established a system of "Charity Offices and Workshops" on the grounds that the poor, who are able to work, "need wages, and the best and most useful alms consist of providing them with the means of earning". He laid special emphasis on "the employment of women", which he described as "an objective no less worthy of attention" than the employment of men; he proposed that the Charity Offices should providespinning wheels to "poor women", and pay for spinning lessons in each village.

Crises and unemployment in classical economics

The spread of Keynes' criticisms addressed to the Classics led to the reconstruction of Classical economics as if these authors had ruled out the crisis and the resulting unemployment. During the last decades, several authors (notably Kates, Baumol and Hollander,) have proved that this was not the case and Baumol, completing Kates' argument demonstrated that Say advocated a clear Keynesian remedy for unemployment: public works, in line with the mercantilists' recommendations.

The main concern of Say, Malthus, and Ricardo in their discussion of unemployment was related to technical progress. In Say's view, it is necessary to distinguish between the short and long run: 2 Turgot's welfare policies have been analyzed by [START_REF] Rothschild | United Nations Research Institute for Social Development[END_REF] and Clément (2004) Preprint 4

The ultimate effect is wholly in its favor. . .The multiplication of a product commonly reduces its price, that reduction extends its consumption; and so its employment to more hands than before. It is beyond question, that the manufacture of cotton now occupies more hands in England, France, and Germany, than it did before the introduction of the machinery that has abridged and perfected this branch of manufacture to such a remarkable degree. . .When printing was first brought into use, a multitude of copyists were of course immediately deprived of occupation; for it may be fairly reckoned, that one journeyman does the business of two hundred copyists. We may, therefore, conclude that 199 out of 200 were thrown out of work. What followed? Why, in a little time, the greater facility of reading printed than written books, the low price to which books fell, the stimulus this invention gave to authorship, whether devoted to amusement or instruction, the combination, in short of all these causes, operated so effectually as to set at work, in a very little time, more journeymen printers than there were formerly copyists. (Say, 1814, 88).

But Say was perfectly aware of the evils produced by the introduction of technical progress and he concluded that new machinery might not only depress earnings but was also likely to lead to unemployment. As he put it:

Whenever a new machine or a new more expeditious process is substituted in the place of human labor previously in activity, part of the industrious human agents, whose service is thus ingeniously dispensed with, needs are thrown out of employ. Whence many objections have been raised against the use of machinery that has been often obstructed by popular violence, and sometimes by the act of authority itself.

. . .This...however advantageous to the community at large, as we shall presently see, is always attended with some painful circumstances. For the distress of a capitalist, when his funds are unprofitably engaged or in a state of inactivity, is nothing to that of an industrious population deprived of the means of subsistence [Say, 1814, p. 86].

And consequently, in his view public policies to alleviate them seemed to him not only legitimate but also perfectly appropriate: "Government could, in the beginning, restrict the introduction of new machinery to some regions where labor is rare and already employed by other industries. It could prepare in advance how to offer employment for the idle hands, by creating at its own expenses, public utilities, as a canal, a road, a big building. It could also organize a colonization, a transfer of population from one place to another" (Say 1814, 1: 55n. & 1817, 1: 55n.)

From Say to the Ateliers Nationaux

If Say devoted significant development to the unemployment caused by commercial crises as well as technological progress, this concern was not elaborated by Say's French liberal economists followers.

For example, the Dictionnaire edited by Coquelin and Guillaumin includes a long article on "Crises commerciales" which mentions only in passing the consequences of crises on labor. First, in the introduction we have a general description of a crisis, the last consequence of which is that workshops stop or slow down their activities, releasing a part of its employees. Therefore, labor suffers and wages decrease [..] The whole society is impacted and a universal disarray develops during the duration of this crisis". Then the Dictionnaire reproduced Say's comments on the English crisis of 1825, where unsurprisingly Say mentions that a multitude of workers remain out of work.

Nevertheless, it is noticeable that the remedies to cure technological unemployment were advocated by Michel Chevalier (1806-1879), a former student of Polytechnique and a Saint-Simonian economist. In his Cours, he advocates the introduction of machinery taking on Sismondi's case for technological unemployment that was famously illustrated by Sismondi's vivid image of "the King, remaining alone on an island, by constantly turning a crank, [producing] through automata, all the output of England" [START_REF] Simonde | New Principles of Political Economy: of Wealth in its Relation to Population[END_REF](Sismondi, ( , [1827]]), p. 563, footnote).

Chevalier does not deny that under exceptional circumstances, a crisis could trigger unemployment and he discusses the example of the consequences of the 1837 crisis in the United States on Lyon's industry where "20,000 workers were put out of work", a very dangerous situation for the workers, their employers, the city, and the whole of France. Therefore, a committee was set up to deal with this evil. It launched a subscription, which could help some workers to move in order to find subsistence elsewhere but the subscription was unable to provide support for the remaining workers. Therefore, the committee auctioned on behalf of the city, the army, the administration of bridges and roads, for the construction of a warehouse, a slaughterhouse, a road, a cemetery, several fords and a dyke, works which whould be run in all circumstances. It was the hope that the unfortunate workers would find in these undertakings something to do and a fair return for their work". (Chevalier, p. 128) and Chevalier concludes "It is a duty for governments to have work plans prepared in advance where, if necessary, it is possible to employ the suddenly decommissioned laborers" (Ibidem, p.130). This idea that plans can be made to cope with unemployment was extended to its limit by Louis Blanc.

In the aftermath of the 1848 Revolution, he writes hastily on February 25 th , a decree proclaiming that "The provisional government of the French Republic undertakes to guarantee the existence of the workmen by work. It undertakes to guarantee work for every citizen". As a result, the ateliers nationaux (national workshops) were created in order to provide work and a source of income for the unemployed. They opened on February 27 th but were quickly overwhelmed by the flow of the unemployed whose numbers rose to 6000 by March 15 th , to 30,000 by March 30 th , 64,000 on April 15 th and reached a peak of 117,000 by April 30 th . They were closed on June 21 st by the Commission Executive, which replaced the provisional government after the conservatives won the elections, as they were horrified by the ateliers nationaux.

Mill and "

The more or less of the labor" Preprint 6 Despite its terrific ending (their closure pushed the proletariat into a rebellion that was squashed in a bloodshed), this public employment experiment was vindicated by J.S. Mill: "The declaration of the droit au travail was followed by the creation of ateliers nationaux; which, indeed, was its necessary consequence; since, in the great falling-off of the employment through the industrial stagnation consequent to the Revolution, it would neither have been honorable nor safe to make zero commencement of fulfilling the promise given, and circumstances did not allow improvising any better mode of temporary employment for the destitute. Some such measures would have been necessary after any revolution. In 1830, large sums were expended in setting the unemployed to work. It was the misfortune, not the fault, of the Provisional Government, that the numbers requiring employment were so much greater than any former period, and that the other circumstances of the case were to render the creation of these ateliers and eventually the greatest calamity of the time. Since it soon became impossible to provide funds for continuing them, while the first attempt to dissolve them was likely to produce, and did, in fact, produce, an outbreak in June. [START_REF] Mill | Vindication of the French Revolution of February 1848[END_REF] 1873, p.58).

In addition to this pamphlet, Mill considers in his article the Principles the droit au travail and its consequences in a rather benevolent way, despite the fact that "experience has shown the sort of work to be expected from recipients of public charity [..]. to extract real work from day-laborers without the power of dismissal is only practicable by the power of the lash". Therefore, Mill agreed with the supporters of the droit au travail in France; provided that "any unemployed laborer [is given] a right to demand support in a particular place or from a particular functionary" and that "the government undertakes to create additional employment [only] when there was a deficiency, and reserving, like other employers, the choice of its own workpeople". Mill, ([1848Mill, ([ ], 1965, pp. 357-8), pp. 357-8).

Actually, Mill's ideas about employment did not differ notably from Say. Mill quoting Smith argues that "a journeyman may be pretty sure of employment almost every day in the year that he is able to work" observing that "the interruptions of business arising from overstocked markets, or from a suspension of demand, or from a commercial crisis, must be excepted" (Mill, [1848(Mill, [ ] 1908, p. 387), p. 387). As [START_REF] Kates | Reassessing the Political Economy of John Stuart Mill[END_REF] has emphasized, Mill rejected the Malthusian argument that demand deficiency was the cause of the recession and a body of unproductive consumers was needed to raise the level of demand if everyone who wished to work was to find employment. Mill's answer was that tbuying goods and services would not increase employment, or, in Mill's own words: "Demand for commodities is not demand for labor. The demand for commodities determines in what particular branch of production the labor and capital shall be employed; it determines the direction of the labor, but not the more or less of the labor itself, nor of the maintenance or payment of the labor. (Mill, [1848(Mill, [ ], 1907, p.79), p.79).

With this radical stance, Mill precluded active employment policy through deficit spending, whilst unemployment was reduced to a severed formulation of "more or less of labor".

Unemployment during the Belle Époque

Despite these first "years of high theory" related to the marginal revolution, and the occurrence of several serious commercial crises, the academic economists paid very little attention to unemployment till the beginning of the 20 th century. Walras and Menger mention unemployment only in an exotic context (a war between Peru and Chile for the first, the interruption of cotton imports for the second) and the labor market did not raise any specific question that could not be solved by individual foresight. Therefore, Topalov concludes that « le processus d'émergence de nouvelles catégories ne coïncide en effet nullement avec le rythme court des dépressions industrielles » the process of the emergence of new categories does not coincide at all with the short pace of industrial depressions (Topalov, 1994 p.28). There are two exceptions to this academic indifference, the Historical School on the one hand, and Marshall on the other.

Almost nothing is to be found in his Principles, but his Correspondence includes two letters largely devoted to unemployment. The first letter addressed to the Economist exposes Marshall's views on real wages during business cycles: Under the present system, their real value 'falls when prices are rising and the purchasing power of money is falling, so that the employer pays smaller salaries and wages at the very time when his profits are largest in other ways, and vice versa, when trade is bad and prices low, wages and salaries of the old nominal value rise in real value above their normal level. This causes extra pressure on the employer just when he has the least power of bearing it, and thus increases the number of the unemployed in times of depression. Marshall (Letter 201, 16 March 1887, 233) Even if the first sentence sounds like a critic of the employers' behavior, one cannot help conclude from the second sentence that the stickiness of wages during the recession is the culprit of unemployment.

Marshall came back to the topic in 1903 when the debates about unemployment insurances were in full swing and there were a large number of unemployed. Marshall observes: "unemployment is a symptom of several distinct social maladies, which require different treatment". Firstly, Marshall distinguishes "the occasional unemployment of capable energetic workers of all grades" which he attributes to "the inability of beings of finite intelligence to forecast coming economic needs and opportunities with perfect precision". Curiously, Marshall thinks that this unemployment could be remedied by "the diffusion of the notion, that to spend the whole of one's income in prosperous times and to be without resource when the tide turns, is inconsistent with the respect that everyone owes to himself." Eventually, Marshall considers "systematic unemployment [..] caused by the existence of large numbers of people, who will not or can not work steadily or strongly enough to make it possible that they should be employed regularly". Thus, Marshall argues that the lack of employability is the culprit for this systematic unemployment. He suggests as a cure "de-urbanizing life, in the sense in which urbanized life is enfeebled life". He concluded his letter by a last eugenic argument: This [cure] should be supplemented by kindly but the severe discipline of those who are bringing up children under physical and moral conditions which will make them recruits to the great army of the habitually unemployed. Ibidem pp 3-4.

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The approach in Germany was very different as could be expected from Kathedersozialists (Socialist of the Chair). With the economic downturn after 1873, the Historical school had concluded that something needed to be done to provide workers with more effective insurance against the hazards of life. Lujo Brentano published several studies about unemployment and a thick book in 1879 in which he studies unemployment since the time of Charlemagne. He found that trade unions alone in the face of large-scale cyclical unemployment could not provide the needed cover. In light of these problems, he criticized the existing benefit funds, arguing that these were in effect outdated and not functioning, especially since they provided insurance only to those employed and were a hindrance to trade union development. He called instead for the creation of voluntary, nationally amalgamated insurance benefits funds supported by wage contributions that would provide old age, disability, health, life, and unemployment insurance for specific occupations. This concern with unemployment insurance lead to heated discussions not only in Germany but also in many other countries and culminated in the early decades of the 20th century especially the United Kingdom, which suffered severe depressions in 1900-4 and 1907-8. Then a new generation of economists devoted a great deal of attention to the unemployed, proof of which is the exuberance of publications (Pigou, Beveridge, Gide to quote only a few of them), whereas Wicksell and Cassel designed the first monetary policies intended to cure recessions and related unemployment.

II Zola and Unemployment -a literary perspective

During the first half of the 19 th century, two major novelists, Stendhal (1783-1842) and Balzac (1799-1850) considered their novels as competing with historians' works in order to depict and represent French society and its evolution. They also read economists, Stendhal read Smith and Say and Balzac -Sismondi. Simultaneously, Smith was very concerned by Rhetoric and Belles Lettres and JB Say wrote novels, resulting in the blurring ofthe frontiers between fiction and the social sciences. Balzac wrote that he was "competing with registry office" but as Christophe Reffait has emphasized, he certainly provided us with a remarkable analysis of speculations, financiers and stock markets and his novels are useful tools to understand the economy and sociology in the post-Napoleonic wars.

Émile Zola (1840Zola ( -1902) ) was the self-proclaimed champion of naturalism, a literary movement or tendency that used detailed realism to suggest that social conditions, heredity, and environment had an inescapable force in shaping human character. Zola was born in Paris in 1840. His father, François Zola (originally, Francesco Zolla), was an Italian engineer. With his French wife, Émilie Aubert, the family moved to Aix-en-Provence in the southeast when Émile was three years old. Four years later in 1847, his father died leaving his mother with a meager pension. In 1858, the Zolas moved to Paris, where Émile's childhood friend Paul Cézanne soon joined him. Zola started to write in the romantic style. His widowed mother had planned a law career for Émile but he failed his Baccalauréat examination. During his adolescence,/he dreamt of a literary career from an early age he dreamt very early of a literary career and he read many authors such as Molière, Montaigne and Shakespeare, but he also read a large number of texts by Jules Michelet, the famous French historian. He fell in love with a prostitute whom he dreamt to exfiltrate from her condition. He failed, but his efforts gave him a good knowledge of life in the slums for poor people. Meanwhile, he worked as a clerk in a shipping firm but unsatisfied by the job, he quit after a couple of months and remained unemployed for two years before he was hired on March 1 st , 1862as a clerk by Louis Hachette who owned a bookshop and a publishing house. There, he was quickly in charge of the advertising department and this was to influence his writing as well as his concern with modern distribution (Au bonheur des dames). He began to write papers or short stories for newspapers at the same time as his first novels and plays. Among his early books was Contes à Ninon, published in 1864. With the publication of his sordid autobiographical novel La Confession de Claude (1865) attracting police attention, Hachettedisquieted by the content of his novels -fired him. After his first major novel, Thérèse Raquin (1867), Zola started the series called Les Rougon Macquart, about a family under the Second Empire; but he also continued to make his living by writing for newspapers.

In 1870, the French-Prussian war caused hard times for Zola who wrote to E. de Goncourt on August 20 th : "This terrible war made me drop the pen from my hand. I'm like a lost soul. I beat the streets" (Zola, 1870, p.360). Even after the Republic was proclaimed on September 4 th at the Hotel de Ville in Paris, there was a blockade and newspapers could not be published and hence, Zola remained unemployed. Despite his links with several figures among Communards, Zola kept his distance with the revolutionary forces during this episode. Even if his personal situation improved dramatically under the 3 rd Republic, his concern with unemployment surged continuously in his works 3 . He devoted a short story tothis theme that was to appear in December 1872 under the title "Lendemains de Crise" (The Aftermath of the Crisis) in a radical newspaper "Le corsaire" and was inserted later in his new book Nouveaux contes à Ninon under the title "Le chômage" ("Unemployment").

The story began with a layoff of workers employed by a petit patron close to his employees, but the enterprise is condemned to close as he exposes them in his paternalistic way: "My good fellows, there is no work to-day -I receive no orders; countermands reach me from all quarters, I shall remain with goods on my hands. This month of December, on which I relied, this month that, in previous years, has been a very busy one, threatens to ruin the most stable firms -we must suspend everything.

Zola translated by Vizetely 1898

It is interesting to emphasize some points in the short story which illustrate the representation of unemployment related to the crises in the 19 th century and can help us to understand why this marginal point in Dictionnaire de Coquelin or Mill went to the fore as the barometer of the crisis in Cassel.

Firstly, it is important to understand that "chômage" (the French word for "unemployment") has a religious origin4 . "Chômage" represented the suspension of activities for religious motives (Sundays and other religious celebrations). It is by this extension that "chômage" went to mean any suspension of activities (even for machinery) but the religious connotation remains predominant to this day (among the French "jours chômés", all of them have a religious origin with the exception of May 1 st , November 11 th , May 8 th ). This suspension of activity was clearly out of the control of the individual (a worker or an employer) so that the word went on to be used to represent unemployment (and to distinguish among idle persons "chômeurs" represent involuntary unemployed). In this very short story, Zola succeeded in stressing several features of "unemployment".

Secondly, he insisted forcefully that the level of wages is not a factor, as even if the workers were ready to work with no compensation, they could not find a job. Nevertheless, discrimination on the labor market makes "unemployment" "more cruel for the women than the men". Thirdly, technical progress is not at stake either and "chômage" as well as hands strike the machinery 5 . Eventually Zola could see that cyclical unemployment was not a consequence of misbehavior of the employers and he exonerated the employer of any moral fault. It does not mean that they are not revolting against the inequality in the society and that the rich cannot understand how terrible unemployment could be for the poor. It is possible that this understanding was related to Zola's fourierist sympathy, which he was to develop later in his life (cf. Le Travail). In any case, he doesn't hide his support to the solidarity between producers ("patrons" and "compagnons") and his reluctance vis à vis communism. This explains why Friedrich Engels or Franz Mehring were very critical vis à vis Zola, in spite of the sympathy he expressed for the working class in many of his novels (just think of Germinal) and his criticism of the world of finance ("cowardly money").

We can now consider the contemporary translations of this novel? as indices of the evolution of public opinion in Europe vis-à-vis cyclical unemployment. The English translator and biographer of Zola, Ernest Vezetely, who was a journalist in Paris as well as in Berlin, completely failed to understand the true meaning when he translated "chômage" as "slack season". Similarly, the first German translation picked up a word (Geschäftstockung) related to depression, but it does not emphasize the dramatic consequences for workers. It took half a century before improved translations appeared in the GDR using the term "arbeitlos" and "arbeitlosigkeit" (unemployed and unemployment):

Conclusion

During the 19 th century, representation of crisis evolved in economic literature as well as in novels.

Say and Ricardo were well aware of the dramatic situation of workers when discussing gluts and their adherence to Say's Law did not prevent them from seeking remedies to this evil. Nevertheless, the unemployed were hardly distinguished from the ordinary poor. At the end of the period, various different positions could be found among economists: many denied or paid very little attention to involuntary unemployment while they focused on unionization that prevented the clearing of the labor markets, others seeking to introduce some dynamics, considered unemployment as the barometer of crisis, and another group tried to implement remedies to cure unemployment through insurance and to lessen the recessions.

On the other hand, Zola's short novel depicted the suffering of the unemployed and insisted on several features which are now taken into account in the economic analysis of crises.

Bibliography

  […]My position, also, is dreadful, more so perhaps than yours. I have lost fifty thousand francs in a week. I am stopping work to-day so as not to deepen the abyss, and I have not the first sou of the money I shall require meeting my acceptances on the 15 th ." Then Zola focused on the individual fate of the worker who embodies the painful fate of the unemployed. As the choice of the words is crucial, we present side-by-side, the original presentation by Zola and the first translation into English under the title "The Slack Season". Clearly the link established by Zola between crisis and unemployment has not been completely understood . II L'ouvrier est dehors, dans la rue, sur le pavé. Il a II The workman is outside, on the road, on the 3 See annex III appearances of 'chômage' in Zola's words Preprint 10 battu les trottoirs pendant huit jours, sans pouvoir trouver du travail. Il est allé de porte en porte, offrant ses bras, offrant ses mains, s'offrant tout entier à n'importe quelle besogne, à la plus rebutante, à la plus dure, à la plus mortelle. Toutes les portes se sont refermées. Alors, l'ouvrier a offert de travailler à moitié prix. Les portes ne se sont pas rouvertes. Il travaillerait pour rien qu'on ne pourrait le garder. C'est le chômage, le terrible chômage qui sonne le glas des mansardes. La panique a arrêté toutes les industries, et l'argent, l'argent lâche s'est caché. Zola, Le chômage pavement. He has been running about the streets for a week without being able to find employment. He has been from door to door, offering his arms, his hands, his whole self for any kind of labour, the most revolting, the hardest, the most fatal. All doors have been closed to him. Then, he was willing to work at half price. The doors did not open. Were he to work for nothing, he could not be employed. It is the slack season, the terrible slack season that sounds the deathknell of the garrets. The panic has stopped all trades, and money, cowardly money, has hidden away.

  IIEs ist die Geschäftstockung, die schreckliche Geschäftstockung, welche in den Mansarden das Todtenglöckchen läutet. Die Panik hat alle Industrien gehemmt, und das Geld, das niederträchtige Geld hat sich versteckt Die Geschäftstockung, in Neue Erzählungen an Ninon, Deutsch von Mathilde Günther, Budapest, Grimm, 1892 II Das macht die Arbeitslosigkeit. die schreckliche Arbeitslosigkeit. die in den Mansarden die Totenglocke läutet. Panik hat alle Industrien zum Stillstand gebracht. und das Geld. das feige Geld. hat sich versteckt. Arbeitslos, Erzählungen ; (Dt. v. Curt ̱ Noch. Mit Einf. v. Herbert Kühn) ([19]58)

See http://www.cnrtl.fr/lexicographie/chomage

In other places, Zola established some links between technological progress and unemployment (cf. Germinal for example) but he did not wholly adhere to Sismondi's vision.