Laura Pascual

Jiaxin Xu

Nelly Desplat

Benoit Valot

Mireille Faubert

Jean-Paul Bouchet

Dominique Brunel

Mathilde Causse
email: mathilde.causse@avignon.inra.fr

Exploring genetic diversity in tomato fruit with a systems biology approach

Introduction

Dissection of the genetic variation and inheritance of phenotypic trait is the first step for plant improvement. The cascade of effects from DNA variation to trait phenotype is organized in complicated biological networks (Kliebenstein 2010), where intermediate molecular phenotypes such as transcript and protein abundance also vary in populations (Rockman, 2006). However, most of the omics studies developed in crop species focused on one or two genotypes and failed to provide an insight into this natural variation. Tomato (Solanum lycopersicum) is a model organism for the fleshy-fruited plants. Its genome has been fully sequenced and annotated (Tomato Genome Consortium, 2012), opening new prospects for analysing biological systems and their complex functions at different levels including genomics, transcriptomics, proteomics, and metabolomics. Here, we carried out an extensive multi-level omic experiment in order to dissect fruit quality at several scales.

The study was carried out using four Solanum lycopersicum lines (Levovil, Stupicke Polni Rane, LA0147, Ferum) with large fruits and four cherry type tomato (S. l. var cerasiforme) lines (Cervil, Criollo, Plovdiv 24A, LA1420), as well as four F1 hybrids between lines of the two groups. Plants were phenotyped for fruit development traits. Besides, fruits were harvested and pericarp samples analysed at 2 stages (cell expansion and orange) and different scales. Proteome profiles were revealed by 2D-PAGE, variable spots were identified and sequenced by LC-MS. Gene expression was analysed by Digital Gene Expression, that is, sequencing short tags (16-17 bp) located at the end of the transcripts.

divergent tomato lines

Criollo Stupicke Polni Rane Ferum LA1420 Plovdiv 24A LA0147 Cervil Levovil

Conclusion

All these data revealed the high natural variation present in the tomato species, in spite of its low molecular diversity. They also showed that there is no clear relation between gene expression and protein amounts, except for a few genes. However, they permitted the discovery of several correlations within and between levels of analysis revealing the regulatory networks operating during fruit development. This information will be especially useful to analyse a multiallelic genetic intercross population derived from the 8 lines, and the genome of the 8 lines, that has been already sequenced. The detailed description provided here will help us to identify the genes under the QTL and to relate the genetic polymorphisms identified at the sequence level with their expression.

Bibliography

*

  Kliebenstein DJ. 2010. Systems biology uncovers the foundation of natural genetic diversity. Plant Physiol. 152: 480-486. Rockman MV; Kruglvqk L. 2006 Genetics of global gene expression. Nat Rev Genet. 7(11):862-72. Tomato Genome Consortium. 2012. The tomato genome sequence provides insights into fleshy fruit evolution. Nature. 485(7400):635-41. Cavanagh C., Morell M., Mackay I. And Powell W. 2008. From mutations to MAGIC: resources for gene discovery; validation and delivery in crop plants. Current Opinion in Plant Biology 11: 215-221The project is funded by the ANR Genomics, project MAGICTom-SNP, in collaboration with Vilmorin & Cie After filtering tags sequenced just one time


Mouhamad Jradeh
email: mouhamad.jradeh@univ-orleans.fr

Maïtine Bergounioux
email: maitine.bergounioux@univ-orleans.fr

On Solutions of a Non linear Wave Equation Derived From Brain Activity Modeling

Keywords: Neural fields, wave equation, Galerkin method, boundary value problem 1991 MSC: : 35L05, 35Q80

The paper deals with a boundary value problem governed by a damped non-linear wave equation. It generalizes the equation modelling cerebral activity that has been proposed by Jirsa and Haken [9]. Using a Galerkin approximation scheme, we prove existence of global solutions. In the 1-D case, we show the continuous dependence of solutions on the initial data and derive uniqueness.

Introduction

Non-invasive techniques such as functional Magnetic Resonance Imaging, Elec-troEncephaloGraphy (EEG) and MagnetoEncephaloGraphy (MEG) provide entry points to human brain dynamics for clinical purposes, as well as the study of human behavior and cognition. Each of these observation technologies provides spatiotemporal information about the ongoing neural activity in the cortex. Unfortunately measures are generally noisy, and it is difficult to identify the equation which governs the dynamics of neural activity. Several physicists [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF][START_REF] Nunez | The brain wave equation : A model for the eeg[END_REF][START_REF] Jirsa | Field theory of electromagnetic brain activity[END_REF][START_REF] Freeman | Tutorial on neurobiology: From single neurons to brain chaos[END_REF] have formulated continuous models called neural fields to predict neural activity using brain anatomy. Jirsa and Haken's model [START_REF] Jirsa | Field theory of electromagnetic brain activity[END_REF], which we will analyze in section 4 of this paper, generalizes the models [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF][START_REF] Nunez | The brain wave equation : A model for the eeg[END_REF] and leads to the following evolution problem that we investigate in this work: u tt -α∆u = a(u, p)u t + b(u, p, p t ), in [0, +∞) × Ω (1.1) u(0, •) = u 0 u t (0, •) = u 1 .

(1.2) u = 0, on [0, +∞) × ∂Ω (1.3) where Ω is an open bounded domain in R n (n ≤ 4) with sufficiently smooth boundary ∂Ω. The notation u t (respectively u tt ) stands for the first (respectively second) partial derivative of u with respect to time variable t, ∆ is the Laplace operator and p is a smooth given function modelling the external stimulus (input). For simplicity, we assume that α = 1.

Although equation (1.1) has a simple form, the existence and uniqueness of the solution is (to our knowledge) still an open problem, if we do not assume additional conditions on a, b or u 0 , u 1 . In addition, the existence of a global solution is not always assured (see for example [START_REF] Esquivel-Avila | The dynamics of a nonlinear wave equation[END_REF][START_REF] Zhou | A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in R N[END_REF][START_REF] Qi | A blow-up result for a nonlinear wave equation with damping: the critical case[END_REF]). Equation (1.1) belongs to a class of non-linear damped wave equations that have been widely studied by many mathematicians and engineers. Existence and uniqueness under conditions (1.2) and (1.3) has been considered using various methods such as semi-group theory [START_REF] Orive | Long-time behavior of solutions to a nonlinear hyperbolic relaxation system[END_REF][START_REF] López-Gómez | On the linear damped wave equation[END_REF][START_REF] Zhou | Kernel sections for non-autonomous strongly damped wave equations[END_REF][START_REF] Cholewa | Strongly damped wave equation in uniform spaces[END_REF] and fixed point methods [START_REF] Kiguradze | On bounded and time-periodic solutions of nonlinear wave equations[END_REF][START_REF] Bournaveas | Low regularity solutions for a class of nonlinear wave equations[END_REF]. In [START_REF] Feireisl | Global attractors for semilinear damped wave equations with supercritical exponent[END_REF][START_REF] Ebihara | On some nonlinear evolution equations with the strong dissipation[END_REF][START_REF] Yeoul | On solutions of quasilinear wave equations with nonlinear damping terms[END_REF][START_REF] Pitts | Global existence and nonexistence theorems for nonlinear wave equations[END_REF][START_REF] Zhijian | Initial boundary value problem for a class of non-linear strongly damped wave equations[END_REF], the authors prove existence and uniqueness of global solutions under conditions (1.2) and (1.3) using Galerkin methods. Zhou [START_REF] Zhou | A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in R N[END_REF] has studied a particular case of equation (1.1), without external input p. The function a is assumed to be constant and b is defined by

b(u) = |u| m-1 u
with condition (1.2) and Ω = R n ; he proved that, if 1 < m < n+2 n , the solution blows up in finite time. Similarly, authors in [START_REF] Orive | Long-time behavior of solutions to a nonlinear hyperbolic relaxation system[END_REF][START_REF] Yagdjian | Global existence in the Cauchy problem for nonlinear wave equations with variable speed of propagation[END_REF][START_REF] Zhou | A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in R N[END_REF][START_REF] Zhou | Global existence and nonexistence for a nonlinear wave equation with damping and source terms[END_REF][START_REF] Levine | Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation[END_REF][START_REF] Levine | Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy[END_REF][START_REF] Todorova | Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms[END_REF][START_REF] Todorova | Critical exponent for a nonlinear wave equation with damping[END_REF][START_REF] Esquivel-Avila | The dynamics of a nonlinear wave equation[END_REF][START_REF] Nishihara | Existence and nonexistence of time-global solutions to damped wave equation on half-line[END_REF][START_REF] Qi | A blow-up result for a nonlinear wave equation with damping: the critical case[END_REF] discussed the case when the solution blows up in finite time. The closest equation to the one we investigate here has been studied by Zhijian [START_REF] Zhijian | Initial boundary value problem for a class of non-linear strongly damped wave equations[END_REF], who considered:

u tt -∆u -∆u t = n i=1 ∂ ∂x i (σ i (u x i ) + β i (u x it )) +F (u, u t , ∇u, ∇u t ) on [0, ∞) × Ω (1.4)
with initial and boundary conditions (1.2) and (1.3). He proved that this problem has a unique classical solution, assuming the following about the initial data:

n i=1 ∂ ∂x i (σ i (u x i (0)) + β i (v x i (0))) + F (u(0), v(0), ∇u(0), ∇v(0)) H k < δ (1.5) v(0) 2 H k+1 + v(0) H k +
be changed in the sequel, where we shall use H 1 and H 2 . In addition, it was supposed that

|D β F | ≤ B(|λ 1 | γ 1 + |λ 2 | γ 2 + | λ1 | γ 3 + | λ2 | γ 4 + 1) and F (0) = 0. (1.7)
The RHS of (1.1) is a particular case of the RHS in (1.4): it involves an additional (parameter) function p that we call external input. This function physically describes a signal generated at the brain surface and transmitted to the neocortex. We have to mention here that the term ∆u t on the LHS of equation (1.4)

Existence of weak global solutions

We let

• denote the L 2 (Ω) norm, • ∞ the L ∞ -norm, • 1 the H 1 (Ω) norm, and (•, •) the L 2 -inner product. Definition 2.1 Let (u 0 , u 1 ) ∈ H 1 0 (Ω) × H 1 0 (Ω)
, and T > 0 be given. We shall say that u is a weak solution to problem (1.1)-

(1.3) if u ∈ L 2 ([0, T ), H 1 0 (Ω)), u t ∈ L 2 ([0, T ), H 1 (Ω)), u tt ∈ L 2 ([0, T ), L 2 (Ω))
and for almost every t ≥ 0, ∀ω ∈ H

and u(0, •) = u 0 , u t (0, •) = u 1 (2.2)
From now on, we do not indicate the dependence on t in order to make the paper more readable. First we give an existence result of weak solutions to problem (1.1)-(1.3).

Theorem 2.1 Assume that for some T > 0,

(A 1 ) a ∈ C 1 (R 2 , R) and b ∈ C 1 (R 3 , R)
, where a, ∇a and ∇b are (uniformly) bounded functions.

(A 2 ) |b(u, p, q)(t, x)| ≤ C 1 |u(t, x)| + C 2 a.e. on [0, T ) × Ω, where C 1 and C 2 are nonnegative constants. (A 3 ) p ∈ H 2 ([0, T ), L ∞ (Ω)). (A 4 ) (u 0 , u 1 ) ∈ H 1 0 (Ω) × H 1 0 (Ω) Then the problem (1.1)-(1.3) admits (at least) a weak solution u on [0, T ) and u ∈ H 2 ([0, T ), L 2 (Ω)) ∩ H 1 ([0, T ), H 1 0 (Ω)).
Proof -We need several steps to prove this result. We use the Galerkin method to construct a solution. Let

{λ k } ∞ k=1 be a sequence of eigenvalues of -∆ in Ω. Let {ω k } ∞ k=1 ∈ H 1 0 (Ω) ∩ H 2 (Ω) be the associated eigenfunctions such that {ω k } ∞
k=1 is a complete orthonormal system of L 2 (Ω). We construct approximate solutions u n as

u n (t) = n k=1 d n,k (t)ω k
where the functions d n,k are determined by the system of ordinary differential equations

(u n tt , ω k ) + (∇u n , ∇ω k ) = (a(u n , p)u n t + b(u n , p, p t ), ω k ) k = 1, . . . n. (2.3) d n,k (0) = u k 0 , d n,k t (0) = u k 1 (2.4) with u k 0 , u k 1 ∈ D(Ω), such that (u k 0 , u k 1 ) → (u 0 , u 1 ) in H 1 0 (Ω) × H 1 0 (Ω).
Here D(Ω) denotes the space of C ∞ functions with compact support in Ω. Now, (2.3) and (2.4) are equivalent to:

d n,k tt + λ k d n,k = f (d n,k , d n,k t , t) k = 1, . . . n, (2.5)
d n,k (0) = u k 0 , d n,k t (0) = u n 1 (2.6)
where

f (d n,k , d n,k t , t) := (a(u n , p)u n t + b(u n , p, p t ), ω k ) a.e. t ≥ 0 . (2.7)
For every n ≥ 1, (2.5), (2.6) is a second-order n × n Cauchy system of differential equations with continuous nonlinearities. So, it follows from the Cauchy-Peano Theorem [START_REF] Queffélec | Éléments d'analyse[END_REF] that the system has at least one solution d n,k defined on [0, T n ] for some T n > 0. Moreover, for every k such that 1 ≤ k ≤ n, we get

d n,k ∈ C 2 ([0, T n ]).
Lemma 2.1 The functions (u n ) satisfy

u n t (t) + u n (t) 1 ≤ M (T )
where M (T ) only depends on T . In particular, u n (t) can be extended to [0, T ).

Proof -We multiply equation (2.3) by d n,k t and sum over k=1,. . ., n. By assumptions A 1 and A 2 we obtain, for every t ∈ [0, T ),

(u n tt , u n t ) + (∇u n , ∇u n t ) = (a(u n , p)u n t + b(u n , p, p t ), u n t ) ≤ C(|u n t | + C 1 |u n | + C 2 , |u n t |).
Using the Cauchy-Schwartz inequality, we have

d dt ( u n t (t) 2 + ∇u n (t) 2 ) ≤ C( u n t (t) 2 + u n (t) u n t (t) + u n t (t) ) ≤ C( u n t (t) 2 + u n (t) 2 + u n t (t) 2 + u n t (t) 2 + 1) ≤ C( u n t (t) 2 + u n (t) 2 + 1) ≤ C( u n t (t) 2 + ∇u n (t) 2 + 1)
by the Poincaré inequality, where C denotes (here and in the sequel) a generic constant. Applying the Gronwall inequality yields

u n t (t) + ∇u n (t) ≤ M (T ); (2.8)
then, by the Poincaré inequality,

u n t (t) + u n (t) 1 ≤ M (T ) (2.9)
and

|f (d n,k , d n,k t , t)| = |(a(u n , p)u n t + b(u n , p, p t ), ω k )| a.e. t ≥ 0 ≤ C( u n t (t) 2 + u n (t) 2 ) ≤ C. So f (d n,k , d n,k t ,
•) is a bounded function, and the solution can be extended to [0, T ) [START_REF] Queffélec | Éléments d'analyse[END_REF] .

2

Lemma 2.2 The sequence of approximated solutions u n satisfies the following:

(1)

u n is bounded in L ∞ ([0, T ), H 1 0 (Ω)) (and in L 2 ([0, T ), H 1 0 (Ω)). (2) u n t is bounded in L ∞ ([0, T ), H 1 0 (Ω)) (and in L 2 ([0, T ), H 1 0 (Ω)). (3) u n tt is bounded in L ∞ ([0, T ), L 2 (Ω)) (and in L 2 ([0, T ), L 2 (Ω)).
Proof -The first item is an immediate consequence of Lemma 2.1. Differentiating (2.3) with respect to t gives, for k = 1, . . . n,

(u n ttt , ω k ) + (∇u n t , ∇ω k ) = (a u (u n , p)(u n t ) 2 + a p (u n , p)p t u n t + a(u n , p)u n tt + b u (u n , p, p t )u n t + b p (u n , p, p t )p t + b q (u n , p, p t )p tt , ω k ) k = 1, . . . n ≤ C((u n t ) 2 + |u n t | + |u n tt | + |u n t | + M, |ω k |),
with assumptions A 1 , A 2 and A 3 . Here, a u and a p denote the partial derivatives of a : (u, p) → a(u, p). Similarly, b u , b p and b q are the partial derivatives of b : (u, p, q) → b(u, p, q).

The method is similar to the one used in the previous Lemma. Replace now ω k by u n tt (t):

d dt ( u n tt (t) 2 + ∇u n t (t) 2 ) ≤ C (u n t (t)) 2 + u n t (t) + u n tt (t) + M u n tt (t) ≤ C( (u n t (t)) 2 u n tt (t) + u n t (t) u n tt (t) + u n tt (t) 2 + u n tt (t) ) ≤ C( (u n t (t)) 2 2 + u n tt (t) 2 + u n t (t) 2 + u n tt (t) 2 + u n tt (t) 2 + u n tt (t) 2 + 1) ≤ C( (u n t (t)) 2 2 + u n tt (t) 2 + u n t (t) 2 + 1)
Using the Sobolev-Poincaré inequality [START_REF] Adams | Sobolev spaces[END_REF],

u n t (t) L 4 ≤ C u n t (t) 1 ,
and the Poincaré inequality, we get

d dt ( u n tt (t) 2 + ∇u n t (t) 2 ) ≤ C( u n tt (t) 2 + u n t (t) 2 1 + 1) ≤ C( u n tt (t) 2 + ∇u n t (t) 2 + 1) .
Then by the Gronwall inequality we obtain

u n t (t) 1 + u n tt (t) ≤ M (T ) a.e. t ∈ [0, T[ , (2.10)
and the lemma is proved. 2

Thanks to estimations (2.9) and (2.10), we may extract subsequences of {u n } (denoted similarly in the sequel) such that

u n u star-weakly in L ∞ ([0, T ), H 1 0 (Ω)) and weakly in L 2 ([0, T ), H 1 0 (Ω)) , u n t u t star-weakly in L ∞ ([0, T ), H 1 0 (Ω)) and weakly in L 2 ([0, T ), H 1 0 (Ω)) , u n tt u tt star-weakly in L ∞ ([0, T ), L 2 (Ω)) and weakly in L 2 ([0, T ), L 2 (Ω)) .
By classical compactness results [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF],

u n → u strongly in L 2 ([0, T ), L 2 (Ω)).
(2.11)

With the above convergence results, we can extract subsequences {u n } such that On the other hand, for all v ∈ L 2 , we have

u n (t, x) → u(t,
|(a(u n , p)u n t -a(u, p)u t , v)| = |(a(u n , p)u n t -a(u, p)u n t +a(u, p)u n t -a(u, p)u t , v)| ≤ |(a(u n , p) -a(u, p))u n t , v)| + |(a(u, p)(u n t -u t ), v)| ≤ a(u n , p) -a(u, p) L 4 u n t L 4 v + |(u n t -u t , a(u, p)v)| ≤ C a(u n , p) -a(u, p) L 4 u n t 1 v + |(u n t -u t ,
([0, T ), L 2 (Ω)) ∩ H 1 ([0, T ), H 1 0 (Ω)) and for almost every t ∈ [0, T ), (u tt , ω) + (∇u, ∇ω) = (a(u, p)u t + b(u, p, p t ), ω) ∀ ω in H 1 0 (Ω) (2.20) u(0) = u 0 in H 1 0 (Ω), u t (0) = u 1 in L 2 (Ω) (2.21) 2
Corollary 2.1 Assume that the conditions of theorem 2.1 are fulfilled for every T > 0. Then u is a global weak solution of problem (1.1)-(1.3).

The 1-D case

In this part we suppose n = 1 and give additional properties of solutions that we are not able to prove if n ≥ 2. Indeed, we use the Sobolev inclusion H 1 0 (Ω) ⊂ C 0 (Ω) (space of bounded continuous functions) which is valid only if the space dimension is 1. The main result is a uniqueness result which is a direct consequence of a "local stability" result, namely the continuous dependence of weak solutions on the initial data. Theorem 3.1 Assume that n = 1 and that the assumptions of Theorem 2.1 are satisfied for some T > 0. Let u, v be two solutions of problem (1.1)-(1.3) corresponding (respectively) to initial data (u 0 , u 1 ) and (v 0 , v 1 ) ∈ H 1 0 (Ω) × H 1 0 (Ω). Then for almost every t ∈ [0, T ),

u t (t) -v t (t) 2 + u(t) -v(t) 2 1 ≤ M (T )( u 1 -v 1 2 + u 0 -v 0 2 1 ),
where M (T ) is a positive constant depending on T .

Proof -Let u and v be two solutions of problem (1.1)-(1.3) corresponding to initial data (u 0 , u 1 ) and (v 0 , v 1 ) respectively. We set ω = u -v and

ω i = u i -v i , i = 0, 1. Then, for every ϕ ∈ H 1 0 (Ω) we get (ω tt , ϕ) + (∇ω, ∇ϕ) = (a(u, p)u t -a(v, p)v t + b(u, p, p t ) -b(v, p, p t ), ϕ) ω = 0 on [0, T) × ∂Ω ω(0, x) = ω 0 , ω t (0, x) = ω 1 in Ω.
Replacing ϕ by 2ω t (u t and v t ∈ H 1 0 (Ω)) gives

2(ω tt , ω t ) + 2(∇ω, ∇ω t ) = (a(u, p)u t -a(v, p)v t + b(u, p, p t ) -b(v, p, p t ), 2ω t )
divided into three basic components [START_REF] Freeman | Tutorial on neurobiology: From single neurons to brain chaos[END_REF]: dendrites, cell body and axons. A neuron communicates with others at synapses and there are mainly two kinds of synapses [START_REF] Freeman | Tutorial on neurobiology: From single neurons to brain chaos[END_REF]: excitatory or inhibitory types. The information transfer between two neurons A and B can be described as follows:

• A pulse (electrical signal) arrives at a dendrite of neuron A (under potential action) and emits neurotransmitters (chemical substances) so that it is transformed into a wave. According to [START_REF] Freeman | Tutorial on neurobiology: From single neurons to brain chaos[END_REF], this conversion is a linear operation between neural sheets. • Then the wave reaches the so-called "trigger zone" (Fig. 1) and is converted to a pulse again. Now the conversion law is sigmoidal . The analysis of the conversion operations at the synapses and the trigger zones of neural ensembles allows us to derive a nonlinear partial differential field equation describing the spatio-temporal behavior of brain activity. The pulse and wave variables are classified as two sub-variables according to their excitatory or inhibitory characters: We have the excitatory pulse E(x, t) and inhibitory pulse I(x, t), the excitatory wave ψ e (x, t) and inhibitory wave ψ i (x, t). We can view the wave as

ψ k (x, t) = Ω f k (x, X)H k (x, X, t) dX. (4.1)
The functions H k (x, X, t), k = i, e are conversion operation outputs and the functions f k (x, X) are the corresponding spatial connectivity distributions. Ω denotes the neural sheet set (at the brain surface, which is supposed to be continuous). In what follows Ω ⊂ R: we have chosen a 1-D model for the brain geometry as in [START_REF] Jirsa | Connectivity and dynamics of neural information processing[END_REF]. We assume that these connectivity functions have the following form:

f k e(x, X) = 1 2σ e e -|x-X| σe (4.2)
where k = e, i and σ e and σ i are excitatory and inhibitory connectivity parameters (respectively). Moreover, σ i 1, since the corticocortical connections (between two distant neurons via corticocortical fibers) are only excitatory. On the other hand, the information propagation along corticocortical fibers implies a delay t = |x -X| v j , where v e and v i are the excitatory and inhibitory propagation speed, respectively, so that we obtain

H k (x, X, t) = a k E X, t - |x -X| v k , (4.3)
where k = e, i and a e and a i are real constants. Substituting equations (4.3) into (4.1), we obtain:

ψ e (x, t) = a e Ω f e (x, X)E X, t - |x -X| v dX (4.4) ψ i (x, t) = Ω f i (x, X)H i (x, X, t)dX ≈ f i (x, x)H i (x, x, t) = a i I(x, t) (because σ i 1) (4.5)
The pulse value is calculated from the wave value and the external input in the neural tissue (Fig 1 ); it has a sigmoidal form: E(x, t) = S e [ψ e (x, t)-ψ i (x, t)+p e (x, t)] , I(x, t) = S i [ψ e (x, t)-ψ i (x, t)+p i (x, t)] (4.6) where p k (x, t) is the external input. The sigmoid function S j is the following:

S j (n j ) = 1 1 + exp(-v j n j ) - 1 2 11
When n j is small enough, we have a third order approximation of S j :

S j (n j ) ≈ n j - v 3 j n 3 j 48 (4.7)
Because v e v i ≈ 100, we take only the linear part of (4.7) into consideration to obtain:

ψ i (x, t) ≈ a i v i 4 + a i v i (ψ e (x, t) + p i (x, t)) (4.8)
We finally obtain the following equation for the dynamics of the excitatory synaptic activity:

ψ e (x, t) = a e Ω f e (x, X) • S e ρψ e X, t - |x -X| v + p X, t - |x -X| v dX
where p(x, t) = p e (x, t) -

a i v i 4 + a i v i • p i (x, t), and ρ = 1 - a i v i 4 + a i v i .
We suppose hereafter that Ω := (0, L) is one-dimensional; that is, we connect all neuron sheets by a line (see [START_REF] Jirsa | Connectivity and dynamics of neural information processing[END_REF]), ψ e (0, t) = ψ e (L, t) = 0 as in [START_REF] Jirsa | Connectivity and dynamics of neural information processing[END_REF], and f e (x, X) is an exponential function depending only on the distance between two neural sheets. Therefore

ψ e (x, t) = a e Ω f e (x, X) • S e ρψ e X, t - |x -X| v + p X, t - |x -X| v dX .
Let us mathematicaly justify the computation of [START_REF] Jirsa | Field theory of electromagnetic brain activity[END_REF].

We set ρ(X, T ) = a e S e ρψ(X, T ) + p(X, T ) and

µ : L 1 (R × R) → R ϕ → < µ, ϕ >= R h(x)ϕ(x, |x| v )dx
with h(x) = 1 (0,L) e -|x| σ and 1 (0,L) the indicator function of (0, L). Then

µ * ρ(x, t) = R h(X)g(X, |X| v )dX with g(X, T ) = ρ(x -X, t -T ) = R h(X)ρ(x -X, t - |x| v )dX = Ω f e (x, X)ρ(X, t - |x -X| v )dX and µ * ρ(x, t) = ψ e (x, t).
The extended Fourier transform gives: ψ e = µ ρ.

Now, we prove that

µ = T f , f ∈ L 1 loc (R × R). Let ϕ ∈ D(R × R). < µ, ϕ >=< µ, ϕ > = R h(X) ϕ(X, |X| v )dX = R h(X) R R ϕ(x, t)e -2iπ(xX+t |X| v ) dxdt dX = R R R h(X)e -2iπ(xX+t |X| v ) dX ϕ(x, t)dxdt = R R f (x, t)ϕ(x, t)dxdt with f (x, t) = 1 2σ e Ω e -|X| σ e -2iπ(xX+t |X| v ) dX. Then ψ e (x, t) = f (x, t) ρ(x, t) ∀(x, t) ∈ R × R. (4.9) As +∞ 0 e -(a+iα)ζ dζ = 1 a + iα , (4.10)
supposing

ω 0 = v σ , we obtain f (x, t) = 1 2 1 1 σe + i t v -x + 1 1 σe + i t v + x = 1 2
ω 0 ω 0 + i(t -xv) + ω 0 ω 0 + i(t + xv) = ω 2 0 + iω 0 t (ω 0 + it) 2 + x 2 v 2 ∈ L 1 loc (R × R).
Equation (4.9) becomes

ψ e (x, t) = ω 2 0 + iω 0 t (ω 0 + it) 2 + x 2 v 2 ρ(x, t). (4.11)
Developing equation (4.11), we obtain

ω 2 0 ψ e (x, t) + 2ω 0 ∂ψ e ∂t (x, t) + ∂ 2 ψ e ∂t 2 (x, t) -v 2 ∂ 2 ψ e ∂x 2 (x, t) = ω 2 0 + ω 0 ∂ ∂t • ρ(x, t)
Applying the inverse Fourier Transform, we formally obtain the following partial differential equation: We complete this equation with ψ e (t, 0) = ψ e (t, L) = 0 and Cauchy initial condition. Thanks to the previous sections' results, we have the following: In this paper, we prove the existence of solutions to a new class of wave equations which is motivated by brain activity modeling. In the 1-D spatial dimension case, the solution depends Lipschitz-continuously on the initial data, and so it is unique. This means that the responses to small perturbations are small as well, which is coherent from a physical point of view. We have simplified the model by assuming that:

∂ 2 ψ e ∂t 2 + (ω 2 0 -v 2 ∂ 2 ∂x 2 )ψ e + 2ω 0 ∂ψ e ∂t = (ω 2 0 + ω 0 ∂ ∂t ) • ρ (4.
• the geometry of the brain surface area is 1-D. The 2-D geometry is more consistent but introduces additional difficulties, • the connectivity functions (4.2) between the neural sheets are homogeneous.

In fact more than 50 % of neural sheets have heterogeneous connections, so we should add another term representing the heterogeneity, • the external input p belongs to H 2 ([0, T ), L ∞ (Ω)). In reality, p may be less smooth or non-local.

In our future work, we will try to generalize our model in order to satisfy the general constraints which are introduced above.

Fig 1 .

 1 Fig 1. Dendritic trees form thousands of synapses. Excitatory current flows inward at excitatory synapses and outward at the trigger zone. Inhibitory loop current flows in the opposite direction.

Theorem 4 . 1

 41 Assume p ∈ H 2 ([0, T ), L ∞ (0, L)). Then equation(4.13) with ψ e (t, 0) = ψ e (t, L) = 0 and Cauchy initial condition, with (ψ e,0 , ψ e,1 ) ∈ H 1 0 (0, L)× L 2 (0, L), has a unique weak solution inH 2 ([0, T ), L 2 (0, L))∩H 1 ([0, T ), H 1 0 (0, L)).Proof -Equation (4.13) is a particular case of equation (1.1), with a(u, p) = a e ω 0 ρS e ( ρu + p) -2ω 0 and b(u, p, p t ) = -ω 2 0 u + a e ω 2 0 S e ( ρu + p) + a e ω 0 S e ( ρu + p)p t . If we suppose p ∈ H 2 ([0, T ), L ∞ (0, L)), it is easy to see that functions a and b satisfy the conditions of Theorem 2.1 and consequently, by (2.20) and (3.3), the above equation has a unique weak solution. 2 5 Conclusion

  is crucial in the proof estimates: the solution is bounded with respect to the H 2 norm.

	Now, the question is: if the initial data does not satisfy (1.5), (1.6) or (1.7)
	and if no "regularizing" term ∆u t appears, does problem (1.1)-(1.3) admit
	a global solution? If yes, is the problem mathematically well posed: is the
	solution unique and continuous with respect to initial data?
	In this paper, we prove that under "realistic" assumptions, problem (1.1)-
	(1.3) has global weak solutions. Moreover, if the problem is a unidimensional
	one, the solutions depend continuously on the intial data; therefore we get
	uniqueness. Required assumptions are motivated by the underlying physical
	cerebral activity model that leads to a particular case of equation (1.1).
	The outline of the paper is the following: existence results are stated in Section
	2. Uniqueness and stability are discussed in Section 3. Finally, in Section 4,
	we present and analyze the cerebral activity model and we apply the general
	result.

  Developing (4.12) and summing the similar terms gives∂ 2 ψ e ∂t 2 -v 2 ∂ 2 ψ e ∂x 2 =a e ω 0 ρS e ( ρψ e + p) -2ω 0 ∂ψ e ∂t -ω 2 0 ψ e + a e ω 2 0 S e ( ρψ e + p) + a e ω 0 S e ( ρψ e + p)p t . (4.13)

[START_REF] Levine | Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy[END_REF]

with ρ(x, t) = a e S e ρψ e (x, t) + p(x, t) .

u(0) H k + < u(0), v(0) > H k < γ (1.6)where δ and γ are two real numbers. In the present section, H k stands for the usual Sobolev space and • H k denotes the H k -norm. This notation will

On the other hand, as in the proof of (2.10) we have

Therefore, as

Using the Sobolev embedding theorem [START_REF] Adams | Sobolev spaces[END_REF], we get Applying the Gronwall inequality to (3.2), we get

Application to cerebral activity model

We now quickly present a cerebral activity model stated by Jirsa and Haken [START_REF] Jirsa | Field theory of electromagnetic brain activity[END_REF], which we justify mathematically. This model equation is a particular case of equation (1.1). The elementary unit of the nervous system is the neuron (Fig. 1), which is


domain: sdv shs

text:
Ophélie Robineau
email: robineau@supagro.inra.fr

Main industries linked to agriculture (processing of cotton, industrial breweries) Today, -in the heart of the coton basin of Burkina Faso Industries -largest commercial hub for exports of livestock -in the heart of the main area of vegetable production of the country Consumption and export, wholesale market Economic development of the city

Regional agricultural dynamics

Market gardens from within the city to >100km from the city Many interactions between regional agricultural dynamics, urban dynamics, and urban agriculture dynamics Complexity of interactions within the urban food system (and we saw part of them…)

Conclusion and discussion

Conclusion and discussion

Combinaison of different « sub-food systems », at different scales Urban agriculture contributes to the urban food system. Its existence is linked to a multitude of multilevel interactions These multiple interaction have to be known to draw up appropriate policies for designing urban food systems

•

  Regional vegetable production: export • vegetable production: local urban consumption Beef cattle, large-scale pig breeding: export • Small scale pig breeding: local urban consumption • Cereals and cotton: processed and linked to urban livestock production

Links city -agriculture of the hinterland -urban agriculture

Cotton production

Cereals

Bobo-Dioulasso Livestock (cattle)

Vegetables

Urban consumption

Boosts the production

Beef cattle producers

Ivorian Coast, Ghana etc. Urban vegetable production (introduced in colonial times)

Hinterland

Links city -agriculture of the hinterland -urban agriculture

Hinterland

Urban livestock production (pig, cattle, small ruminants)

Manure

Urban vegetable production

Subsistence crops in the urban fringe

Urban wastes

Within the city


domain: sdv info

text:
M Mcevoy

M O'donovan

L Delaby

Effect of cultivar on sward structural characteristics in a rotational grazing system during the spring and summer period

Introduction

For several years the focus on perennial ryegrass breeding has been to increase annual DM yield. It is widely accepted that both management and other factors can influence animal performance from grazed swards. These factors include ploidy, heading date, leaf to stem ratio and stage of growth, among others, which have been shown to influence intake and the milk yield of grazing ruminants. The objective of this study was to investigate the phenotypic variation among four perennial ryegrass cultivars for sward structural characteristics considered to be important for intake and production of lactating dairy cows. The milk production data is reported by [START_REF] Mcevoy | Proc. Ag. Res. Forum[END_REF].

Material and Methods

Four cultivars of perennial ryegrass were sown in a split plot design with 4 reps. The cultivars included 2 tetraploids: Bealey (heading date 24 th May) and Astonenergy (31 st May) and 2 diploids: Spelga (22 nd May) and Abermagic (28 th May). Cultivars were sown in 2009. From mid March to May 2011 (P1), paddocks were grazed with lactating dairy cows for an eight week period. Cows were offered 16 kg DM cow -1 d -1 . From May to August (P2), paddocks were grazed for a 12 week period. Cows were offered 17 kg DM cow -1 d -1 . A similar rotation length was maintained for each cultivar. During weeks 2, 4, 6 and 8 in P1, and weeks 3, 6, 9 and 12 in P2, sward measurements were completed. Within each measurement period the following was completed: pre and post grazed sward height; herbage mass; extended tiller height and sheath heights was measured on 100 tillers in the pre and posts grazed sward; herbage was sampled to ground level for each cultivar and a subsample of 50g fresh weight was separated into leaf stem and dead content, before being dried at 80°C overnight. Height was measured daily, with all other measurements occurring twice during each measurement week. Data were analysed using Proc Mixed in SAS. The model included period, season, cultivar and the interaction between season and cultivar.

Results and Discussion

There was no difference in pre or post grazing sward heights between cultivars, which were 9 and 4.3 cm, respectively. Sward results are presented in Table 1. Sward density was lower for Bealey and Astonenergy, than for Spelga and Abermagic. There was no difference in the pre-grazing extended tiller height between cultivars, however the higher pseudostem height of Spelga and Abermagic resulted in these cultivars having proportionately less leaf available in the grazing horizon, compared to Astonenergy. Although not statistically significant, the leaf proportion of Astonenergy was 7% higher than the other three cultivars. Higher leaf proportions have been shown to result in higher intakes from these swards. Milk yield and milk solids production was higher on the Bealey and Astonenergy swards compared to the diploids [START_REF] Mcevoy | Proc. Ag. Res. Forum[END_REF].

Conclusions

The cultivars used in this study showed significant differences in sward characteristics which have previously been shown to affect intake including sward density, tiller height, pseudostem height and sward composition. In conclusion, differences do occur between cultivars when maintained under similar rotation lengths and managements and this has the potential to influence animal performance.

Table 1 .

 1 Differences sward structural characteristics between four perennial ryegrass cultivars

	P-value

Acknowledgements

The authors are grateful for the assistance of Moorepark farm staff and in particular Mr. M. Feeney for assistance with measurements.


domain: math info

text:
Gauthier Quesnel
email: gauthier.quesnel@toulouse.inra.fr

Ronan Trépos
email: ronan.trepos@toulouse.inra.fr

Éric Ramat
email: eric.ramat@lisic.univ-littoral.fr

Observations of Discrete Event Models

Keywords: Simulation, Discrete Event Systems, DEVS formalism, Observation, Methodology

The observation of a simulation is an important task of the modeling and simulation activity. However, this task is rarely explained in the underlying formalism or simulator. Observation consists to capture the state of the model during the simulation. Observation helps understand the behavior of the studied model and allows improving, analyzing or debugging it. In this paper, we focus on appending an observation mechanism in the Parallel Discrete Event System Specification (PDEVS) formalism with guarantee of the reproducible simulation with or without observation mechanism. This extension to PDEVS allows us to observe models at the end of the simulation or according to a time step. Thus, we define a formal specification of this extension and its abstract simulators algorithms. Finally, we present an implementation in the DEVS framework VLE.

INTRODUCTION

In the Modeling and Simulation (M&S) activity, observation of the behavior of models is an important aspect. An observation captures the state or the evolution of the state during or at the end of the simulation. It allows the modeler to test, prove, validate, or generate data from simulations by connecting simulation software application to output streams like files, databases, unit test or visualization software. Observations contribute to the modelers representation of its studied system.

In environmental and agronomic modeling domain, we need to couple heterogeneous models. These models can be continuous or discrete. In addition, these models co-exist in different temporal and spatial scales. Thus, to solve this problem, we use the Discrete Event Specification System (DEVS) formalism [START_REF] Zeigler | Theory of modeling and simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems[END_REF]. DEVS can be seen as a common denominator for multi-formalism hybrid systems modeling [START_REF] Vangheluwe | DEVS as a common denominator for hybrid systems modelling[END_REF]. DEVS takes place in the M&S theory defined by B. P. Zeigler. M&S theory tends to be as general as possible. It addresses major issues of computer sciences from artificial intelligence to model design and distributed simulations. M&S theory aims to develop a common framework, formal and operational, for the specification of dynamical systems.

Modeling the observation process is dependent on the underlying formalism. However, for many for-malisms, the observation does not exist and the outputs of models are used directly. For example, the numerical integration method Euler used for solving ordinary differential equations does not allow catching variable between two time-steps. The observation process is generally not explicit nor specified in the formalism. Specify the observation process is generally not useful. However, it becomes necessary when we want to observe multi-models with the same observation mechanism.

However, the DEVS formalism does not propose observation of simulation models. Several solutions exist without changing the behavior of DEVS. For example, in the first picture of the figure 1, outputs of the models A and B are connected to an observation model O. O has a reactive behavior. Pow-erDEVS [START_REF] Bergero | PowerDEVS: a tool for hybrid system modeling and real-time simulation[END_REF] chooses this solution. It is simple and has great flexibility. However, it can be very expensive. Indeed, in the case of experimental frames (to calibrate or estimate parameters or to optimize parameters on criteria), only the final values of the observations are necessary. All the values received are useless. The interpretation of these values can be costly. In the second picture of the figure 1, an observation model O generates events to observed models A and B. These models send their outputs to the observation model. This solution is computationally less expensive, but requires mixing observation and normal behavior of the model. This mechanism can induce a difficulty in reproduc- We chose in this work to modify the DEVS formalism by adding an extension. This addition does not change the behavior of DEVS. It adds a layer of observation to DEVS. The DEVS trajectories of the simulations are the same with or without this extension. However, some alternative exist to add an observation mechanism. Thus, rather than developing the observation process in the formalism, researchers developed this process in the software part of their simulators. It is the choice made by JAMES 2 (Himmelspach and Uhrmacher, 2009) and OSA [START_REF] Ribault | OSIF: a framework to instrument, validate, and analyze simulations[END_REF]. JAMES 2 uses the design pattern observation; OSA uses AOP (Aspect-oriented programming. The design pattern observation links for each model a list of observation component. This list has a function called notify. The model calls this function when it wants to send observation value. The observation component has a reactive behavior. This type of behavior can increase the calculation to perform the treatment of observation values from multimodels (e.g. Models that use formalism with different time advance function). Of course, through a combination of the visitor and observer design patterns, we can make active the behavior of the observation component.

Comment

In this paper, we propose to extend the PDEVS formalism with an observation mechanism. This extension to DEVS provides a generic way to observe heterogeneous models and it guarantees the same simulations when this mechanism is activated and when it is not.

This paper is divided into three parts. In the first part, we explain the PDEVS formalism. This formalism is the context of our work. Then, we develop our contribution in two parts. (1) We develop the formal specification of the observation extension. This specification guarantees to conserve the behavior of PDEVS. (2) We present the abstract simulators of the observation extension. Developers of PDEVS simulators software can use these abstract simulators to introduce the extension observation in their simulators. The last sections show an implementation of this extension in the software VLE and an example of use. Finally, we conclude this paper with conclusion and perspectives.

Method

DEVS [START_REF] Zeigler | Theory of modeling and simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems[END_REF] is a well-known and accepted formalism for the specification of complex discrete or continuous systems abstracted as a network of concurrent, timed and interacting models. DEVS provides a hierarchical and modular approach to modeling and simulation. The formalism supports two types of models: atomic and coupled models. Atomic models interact thought a well-defined set of input and output values enabling a software component based orientation to model representation. Atomic models can be combined to form coupled models, providing the constructs to the representation of complex models.

PDEVS formalism

PDEVS [START_REF] Chow | Parallel DEVS: a parallel, hierarchical, modular, modeling formalism[END_REF] extends the Classic DEVS essentially by allowing bags of inputs to the external transition function. Bags can collect inputs that are built at the same date, and process their effects in future bags. This formalism offers a solution to manage simultaneous events that could not be easily managed with Classic DEVS. For a detailed description, we encourage the reader to read the section 3.4.2 in chapter 3 and the section 11.4 in chapter 11 of Zeigler's book [START_REF] Zeigler | Theory of modeling and simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems[END_REF].

PDEVS defines an atomic model as a set of input and output ports and a set of state transition functions:

M = X,Y, S, δ int , δ ext , δ con , λ,
ext : Q × X b → S is the external transition function Q = {(s, e)|s ∈ S, 0 ≤ e ≤ ta(s)}
Q is the set of total states, e is the time elapsed since last transition X b is a set of bags over elements in X δ con : S × X b → S is the confluent transition function, subject to δ con (s, / 0) = δ int (s) λ : S → Y is the output function If no external event occurs, the system will stay in state s for ta(s) time. When e = ta(s), the system changes to the state δ int . If an external event, of value x, occurs when the system is in the state (s, e), the system changes its state by calling δ ext (s, e, x). If it occurs when e = ta(s), the system changes its state by calling δ con (s, x). The default confluent function δ con definition is:

δ con (s, x) = δ ext (δ int (s), 0, x)
The modeler can prefer the opposite order:

δ con (s, x) = δ int (δ ext (s,ta(s), x))
Indeed, the modeler can define its own function. Every atomic model can be coupled with one or several other atomic models to build a coupled model. This operation can be repeated to form a hierarchy of coupled models. A coupled model is defined by:

N = X,Y, D, {M d }, {I d }, {Z i,d }
Where X and Y are input and output ports, D the set of models and:

∀d ∈ D, M d is a PDEVS model ∀d ∈ D ∪ {N}, I d is the influencer set of d : I d ⊆ D ∪ {N}, d / ∈ I d ∀d ∈ D ∪ {N}, ∀i ∈ I d , Z i,d is a function, the i-to-d output translation: Z i,d : X → X d , if i = N Z i,d : Y i → Y, if d = N Z i,d : Y i → X d , if i = N and d = N
The influencer set of d is the set of models that interact with d and Z i,d specifies the types of relations between models i and d.

Extension Observation: Formal Specification

The development of simulation software based on DEVS formalism implies to observe models and their evolution during the simulation. Observation of DEVS models involves watching or capturing their states. These captures can be done when a change of state occurs (in DEVS terminology, in δ int , δ ext or δ con transition functions) or at specific dates.

In DEVS, the solution generally used is to connect models, both inputs and outputs, to an observation model (see figure 2). This model has in charge to send observation messages and to process the responses of the models. For example, to build a discrete observation, the observation model relies on a constant ta function in order to send, at each time step, an event to observed models. Observed models compute observations values and send them to the observer. This solution forces the modelers to merge the state graphs between observation and behavior of their models (See figure 3 for an explanation).

In addition, by merging the state graphs, the modeler might make the results of its models dependent to his observation. If model is observed asynchronously with its behavior, the computation of the elapsed time e when an external event disturbs an atomic model may introduce a floating-point error. Indeed, coordinator needs to call the external transition of the simulator with s, e, x parameters where e = ttl with t the current time of the simulation (time of the perturbation) and tl the time of last event. If an internal state variable in S is based on e to compute its value in the δ ext transition, its internal state variables can give different results with or without observations. Precisely, If all models use integer as representation of the time (i.e. all time advance functions return duration in N), the floating-point problem would not exist. However, in the domain of Environmental Modeling, the models use continuous and discrete event formalisms such as the QSS2 formalism [START_REF] Kofman | A second order approximation for devs simulation of continuous systems[END_REF] which solves in R a second order polynomial function in order to compute the duration in a state. They can hardly rely on a representation of the time in N.

Comment
The solution proposed does not change the value of e in the case of observation.

Finally, in DEVS, a model can have several states at the same date (when at least one call to the ta(s) function returns 0). These states are called transitory states. It seems obvious that the observation of the transitory states makes no sense. However, this solution does not guarantee to provide only observations of real model states. For example, if the observation is required at a specific time of simulation, the state observed can be one of these transitory states.

In this section, we propose a formal and operational representation of observations in the PDEVS formalism. This extension is based on a function of observation in atomic models and a second graph of connections in coupled models. This extension to the PDEVS formalism integrates necessary characteristics:

• First is to ensure that this extension does not disturb the simulation i.e. we need to conserve the same result with or without the observation.

• Second is to provide several types of observation.

We propose, in this paper, two modes: time step and finish that define respectively: observations when a model reaches a time and observations at the end of the simulation (see figure 4).

• Third, the extension should not take its values directly from the state of the current model but from a function (see figure 4).

• Finally, in DEVS, state variables may change several times at the same date (when its ta returns 0 or when an external event disturb it). In this case, we observe the last state of the model.

Thus, to develop this extension we need to add function to the atomic models, to add new set in coupled model and add specific observation models. We develop these changes in the next section.

PDEVS Atomic model

To develop this extension in the PDEVS formalism, we extend the PDEVS atomic model with new sets of input and output ports (X obs? and Y obs ). The first one is used for observation request; the second one is used for routing values returned by modeler. We attach to this port a new output function called λ obs . The new PDEVS atomic model is defined such as:

M = X,Y, S, X obs? ,Y obs , δ int , δ ext , δ con , λ, λ obs , ta
Where:

X obs? are the ports used to catch observation requests. Y obs are the ports used to send observation values. This definition allows us to represent both observed models and classical PDEVS models as defined in section 2.1.

Indeed, the model X,Y, S, X obs? ,Y obs , δ int , δ ext , δ con , λ, λ obs , ta with no observations 1 is equivalent to the classical PDEVS atomic model X,Y, S, δ int , δ ext , δ con , λ, ta . In fact, we should distinguish observed models from observer models.

Observer models

In this paper, we suggest two types of observers. O timed , and O finish to respectively, send observation following a given time-step or at the end of the simulation.

• The discrete time observer model, O timed , needs to send output events on its output port Y obs every 1 X obs? = / 0, Y obs = / 0 and λ obs is a null function.

∆ t unit time. O timed reads observation events from its port X obs and stores data transported by event. This model is : λ obs is a null function

O timed = X
The function observation is a user defined function that identifies in Y obs? the observation request that should be sent to the observed models.

Coupled model

As seen earlier in the introduction and in the section 2.1, the I d variable and the function Z i,d define the graph of connections of the models by calculating the influenced models. Our observation extension of the PDEVS formalism uses the same principle. It proposes two subsets I o and I r that, respectively identify the influences of atomic models (responses of the observations) and the influences of observer models (models sending the request of observation).

We extend the PDEVS model coupled in order to introduce a second connection network (see the figure 5). This second network is dedicated to the observer models and to the classical PDEVS atomic models to observe through their ports X obs? and Y obs . The new PDEVS coupled model is defined as:

N = X,Y, D, O, R, X obs? , X obs ,Y obs? ,Y obs , {M d }, {I d }, {Z i,d }, {M o }, {O d }, {Z o,d }, {M r }, {R d }, {Z r,d }
Where X obs , Y obs and X obs? , Y obs? are input and output ports to route observation and request events. O is the set of names of observed models

O ⊆ D, {M o } is the set of observed models {M o } ⊆ {M d } , R is the set of names of observer models R ⊆ D and {M r } is the set of observer models {M r } ⊆ {M d }. ∀r ∈ R ∪ {N}, I r is the influencer set of observed models of r I r ⊆ O ∪ {N}, r / ∈ I r ∀o ∈ O ∪ {N}, I o is the influencer set of observed models of o I o ⊆ R ∪ {N}, o / ∈ I o
The observation network is however very constrained. Thus, the functions Z o,d and Z r,d are very different and provide additional constraints from the classical Z i,d function. These additional constraints ensure that a DEVS atomic model cannot send an observation event to another DEVS atomic model, and an observation model cannot send request observation to another observation model. The figure 5 shows an example of connections between observed and observer models. the coupled model), the PDEVS functions have the same content (See chapter 7 in [START_REF] Zeigler | Theory of modeling and simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems[END_REF] to the complete formalization). The closure under coupling property of the extension observation for parallel DEVS is still verified.

∀r ∈ R ∪ {N}, ∀d ∈ I r , ∃ an output translation function Z r,d : Z r,d : X obs? → X obs? d , if r = N Z r,d : Y obs? r → Y obs? , if d = N Z r,d : Y obs? r → X obs? d , if r = N and d = N ∀o ∈ O ∪ {N}, ∀d ∈ I o , ∃ an output translation function Z o,d : Z o,d : X obs → X obs d , if o = N Z o,d : Y obs o → Y obs , if d = N Z o,d : Y obs o → X obs d , if o = N

Algorithms

Root coordinator

The root-coordinator implements the overall simulation loop. It sends messages to its direct subordinate (simulator or coordinator). The root-coordinator first sends an initialize message (i-message), and loop on internal transition (*-message) from its child to perform the simulation cycles until some termination conditions.

As describe in the listing 1, the root coordinator is unchanged in this extension.

Coordinator

In the coordinator algorithm, we add an additional variable tb, which indicates the last date of the simulation. This variable tb detects the last PDEVS bags to ensure to observe the latest state of a model. We add a new bag, called mailobs to store all observation events to atomic models at a specific date. When tb detect a change, the mailobs is flushed into the observation network using the functions Z r,d and Z o,d .

The coordinator's algorithms are described in the listings 2. From l. 1 to l. 38: routes the message or stores in mailobs. From l. 40 to the end: to update the tb variable and to send xobs?-message to the atomic models.

Simulator

Lastly, this last section on the abstract simulators develops algorithm for the simulator of the atomic model. As presented previously, the management of the observations events is very simple for the modeler since only one function is called λ obs . This function is called only for classical atomic models when they receive an X obs? event. When observation models receives X obs event, they use the classical way when receiving input message (See x-message in PDEVS abstract simulators).

The listing 3 describes the simulators algorithms. In line 12: when receives an xobs?-message from the parent, simulator computes the observation in the λ obs function and sends the data (wrapped into an yobsmessage) to the parent coordinator.

To validate this work, we made an implementation of this extension in the DEVS open source simulator VLE.

RESULT

These works have started with multi-disciplinary issues emerging from the interaction between computer scientists and biologists. Considering these works, we think that the integration of heterogeneous models and the respect of the M&S cycle are the key issues to provide a complete and reliable software environment for natural complex systems modeling. Therefore, VLE has evolved on a complete multi-modeling and simulation environment [START_REF] Quesnel | The Virtual Laboratory Environment -An operational framework for multi-modelling, simulation and analysis of complex dynamical systems[END_REF] and is now a generic environment for M&S, in Environmen-tal Sciences, in Industry or Medicine. Many projects from two major French research institutes INRA and Cirad use VLE. For example, the RECORD project [START_REF] Bergez | Observation of discret event models[END_REF] is a large platform designed for developing and analyzing models of cropping systems.

VLE: Virtual Laboratory Environment

VLE is a set of programs and libraries software. Figure 6: DEVS BUS and multi-modeling. The events (x,t), (y,t), ( * ,t), (done,tN) represent the bus.

Implementation

In our implementation of the PDEVS abstract simulator, we define an atomic model as a classical C++ class. Modeler must this class inherits this class to override the observation function and to benefit to the observation extension developed in the previous section The listing 4 shows a simplified representation (without constructor, destructor, and other members) of the class which permits to develop atomic model behavior:

• l. 3-14: the classic PDEVS functions: ta, δ int , δ ext , λ and δ con .

Comment citer ce document : Quesnel, G., Trépos, R., Ramat, É. ( 2012 • l. 15: the observation method is called when an observation event arrived on a input observation port X obs . λ obs is also a constant function to prevent user to modify the state of its model. This function returns a Value (simple type as integer, real, Boolean, string, and complex type as set, dictionary, matrix.).

Thus, the observation function of the simulator that implements the Euler method (a numerical procedure for solving ordinary differential equations) in VLE, the C++ code is: The function getEstimatedValue calculates an approximation of the value of the variable. It uses the current value of the state variable and the time elapsed since the last transition. This observation function does not change the current state of the variable. Thus, we respect the reproducibility of the simulation. However, we create additional cost calculations for each observation. This extra cost can be important if the observation function is complex and if called often. However, we believe that observing a model often is only needed when debugging the model.

Using models with VLE

The study of the models is very important in the modeling and simulation cycle. Generally, this analysis is performed with an experimental setting. Experimental frames as sensitivity analysis, replicas generation are used to study or to validate models. One important motivation in formalizing the observation process in PDEVS is to develop generic experimental frames, which ones rely on intensive observation of models.

VLE is an environment for M&S. It relies on a set of libraries for modeling, simulation and analysis. These libraries developed in C++ are available for several programming languages. Thus, in order to collaborate with users of statistical tools, we provide an interface to the program R (R Development Core Team, 2012). R is a tool and language for statistical computing. This package named RVLE allows the user to run simulations, changes its parameters and retrieves simulation results (from the extension observation). This package combined with another packages of R as the classical sensitity package permit to build experimental frames to validate, explore and optimize models. On this principle, we also offer program and PYVLE JVLE to develop web services based on simulation.

CONCLUSION

In this paper, in section 2, we extend the PDEVS formalism to introduce the observation of models in the formalism. This work was motivated by the separation between the dynamics of the system and its observation as a fundamental issue. This addition to the formalism is closed under coupling and allows building observations in a hierarchical and modular manner. The abstract simulators necessary for this extension were provided. In addition, we provided an implementation of this observation extension in the VLE simulators. This DEVS extension is crucial for us to develop experimental design and to abstract observation from models dynamic. However, a type of observation is missing in these works. It concerns the event observation of atomic models. In PDEVS terminology, after each change in δ int , δ ext or δ con transition functions. We work on the formalization of this new type of observation.

  citer ce document : Quesnel, G., Trépos, R., Ramat, É. (2012). Observation of discret event models. In: Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (p. 32-41). Presented at 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2012), Rome, ITA (2012-07-28 -ing simulations and can increase the complexity of model.

Figure 1 :

 1 Figure 1: Classical reactive and proactive observation mechanism in DEVS.

Figure 2 :

 2 Figure 2: This picture shows the classic observation mechanism to observe models in the DEVS formalism. Three atomic models A, B and C are observed by an observation model O which sends event for request observation on output Y obs and waits the results on input port X obs in order to store it in output files or databases.

Figure 3 :

 3 Figure 3: This picture show state graphs of an atomic model. In left picture, we show the model without observation and in the right part, the same model with the capacity to be observed at any time (dotted lines represent external transitions). In the computation of the observation states Obs init , Obs A or Obs B (in δ ext , δ int , δ conf or ta functions), simulators can introduce floating-point errors when computing the time elapsed since last transition (e in DEVS terminology) for example.

λFigure 4 :

 4 Figure 4: The first figure shows the evolution of the state S of a system over time t. The second figure shows the model of this system in DEVS. This figure shows the trajectory of the state of the DEVS model. The last two figures show the extension observation described in this paper in time-step (∆ t = 1) or in finish mode. This extension allows the modeler to provide a function (observation) that computes an observation value. Dots are observation values. These values are computed by interpolation of the current DEVS state of s and e, the elapsed time since the last transition.

  ∀d ∈ Data : δ int ((WAIT, d)) = (END, d) ∀d ∈ Data : δ int ((END, d)) = (END, d) ∀s ∈ {WAIT, END} : δ ext ((s, d), e, X obs ) = (s, d ) where d is the state of the output stream once X obs has been stored, Comment citer ce document : Quesnel, G., Trépos, R., Ramat, É. (2012). Observation of discret event models. In: Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (p. 32-41). Presented at 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2012), Rome, ITA (2012-07-28 -X obs? = / 0 Y obs = / 0 ∀d ∈ Data : ta((WAIT, d)) = end where end is the duration of simulation ∀d ∈ Data : ta((END, d)) = +∞ ∀s ∈ S : λ(s) = observation(s,t, x)

  X, Y , D, {M d }, {I d }, {Z i,d }) list of element (d, tn d ) sorted by tn d 8 mail // output mail bag 9 mailobs // observation mail bag 10 O event // event-list of event observation model 11 y parent // output message bag to parent 12 y d // set of output message bags for each child d 13 14 when receive xobs-message (x obs ,t) 15 if not (tl ≤ t ≤ tn) then 16 error: bad synchronisation 17 receivers = {o|o ∈ children, o ∈ I r } 18 for each o ∈ receivers 19 send x-message (Z r,d (x) with input value Z r,d to o 20 21 when receive xobs?-message (x obs? ,t) 22 if not (tl ≤ t ≤ tn) then 23 error: bad synchronisation 24 receivers = {r|r ∈ children, r ∈ I o } 25 for each r ∈ receivers 26 add r, x obs? to mailobs 27 28 when receive yobs-message (y obs ,t) from o 29 for each child d ∈ Z o,d 30 send xobs-message to d 31 for each d ∈ Z o,d and d = N 32 send yobs-message to parent 33 34 when receive yobs?-message (y obs? ,t) from r 35 for each child d ∈ Z r,d 36 send yobs-message to d 37 for each d ∈ Z r,d and d = N 38 send yobs?-message to parent 39 40 // finally, we append these algorithms 41 // in the following message 42 43 when receive i-message (i,t) at time t 44 [...] // The same algorithms than PDEVS Comment citer ce document : Quesnel, G., Trépos, R., Ramat, É. (2012). Observation of discret event models. In: Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (p. 32-41). Presented at 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2012), Rome, ITA (2012-07-28 -?-message (x obs? ,t) at t 14 with input x obs? 15 y obs = λ obs (x obs? ,t) 16 send yobs-message (y obs ,t) to parent coordinator 17 18 end Parallel-Devs-Simulator

  obs ,Y obs? , S, X obs? ,Y obs , δ int , δ ext , δ con , λ, λ obs , ta Where: X obs is the set of observation values, Y obs? is the set of observation requests,

	X obs is the set of observation values,
	Y obs? is the set of observation requests,
	S : {IDLE, SENT} × Data, with:
	IDLE, SENT: automata finite states,
	Data: output stream states.
	X obs? = / 0
	Y obs = / 0

∀d ∈ Data : δ int ((IDLE, d)) = (SENT, d) ∀d ∈ Data : δ int ((SENT, d)) = (IDLE, d) ∀s ∈ {IDLE, SENT } :

δ ext ((s, d), e, X obs ) = (s, d )

where d is the state of the output stream once X obs has been stored,

∀d ∈ Data : ta((IDLE, d)) = ∆ t (the time step) ∀d ∈ Data : ta((SENT, d)) = 0 ∀s ∈ S : λ(s) = observation(s,t, x)

λ obs is a null function • The finish observer model, O finish , produces observation events at the end of the simulation on it Y obs port and wait observation events from its port X obs . O finish is the same model as O timed but, its ta function returns the end of the simulation to build observation event, or +∞ after.

O timed = X obs ,Y obs? , S, X obs? ,Y obs , δ int , δ ext , δ con , λ, λ obs , ta Where: S : {WAIT, END} × Data, with:

WAIT, END: automata finite states, Data: output stream states.

  Figure 5: In this picture, we illustrate the second network in the coupled model to deal with observations graph. Plain connections are observation requests from output port Y obs? of observation models, to atomic model input port X obs? or to coupled model's output port Y obs? , or from coupled model input port X obs? to atomic model X obs? . Dashed connections constitute response of observation request from extended PDEVS atomic model (on output port Y obs ) to observation model X obs or coupled model output port Y obs . The atomic model B is observed by an external observation model on its port X obs2 . It sends observations to its output port Y obs2 . For this figure and for a better understanding, we distinguish observation ports for the model B.

	Listing 1: Root coordinator algorithms
	Parallel-Devs-Root-Coordinator variables: t // current simulation time child // direct subordinate devs-simulator // or devs-coordinator
	t = t 0 send initialization message (i,t) to child t = tn of its child loop send ( * ,t) message to child t = tn of its child until end of simulation end Parallel-Devs-Root-Coordinator
	Closure under coupling
	Since we keep unchanged the internal S of coupled or atomic models nor all sets of the PDEVS defini-

and d = N tion (IMM, INF(s), CONF(s), EXT(s) and UN(s) of Comment citer ce document : Quesnel, G., Trépos, R., Ramat, É. (2012). Observation of discret event models. In: Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (p. 32-41). Presented at 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2012), Rome, ITA (2012-07-28 -

  These components are used to model, to simulate and to analyze multimodel. VLE offers a set of formalism developed with a DEVS BUS[START_REF] Kim | A heterogeneous distributed simulation framework based on DEVS formalism[END_REF].

	The sub-formalisms are DESS, QSS 1 and QSS 2 numerical procedure for solving ordinary differential equations, Petri nets, finite state automaton (moore, mealy, FDDEVS and Harel statechart (Harel, 1987)), cellular automaton (Cell-DEVS and Cell-QSS) and agent decision making. These extensions are devel-oped using a DEVS BUS i.e. these extensions are de-veloped such as atomic model. Finally, VLE uses an open-source license and all the source code are avail-able on its website including this observation exten-sion.
	DEVS BUS
	• Scheduller, • Network.	Petrinet	QSS 2
		Simulator	Simulator
			(done, tN)
			( * , t)
			(y, t)
			(x, t)

  Listing 4: The C++ interface of the simulators in VLE

	class Dynamics {	1
	public:	2
	virtual Time timeAdvance() const;	3
	virtual void internalTransition(	4
	const Time& time);	5
	virtual void externalTransition(	6
	const Time& time,	7
	const ExternalEventList& lst);	8
	virtual void output(	9
	const Time& time,	10
	ExternalEventList& out) const;	11
	virtual void confluentTransition(	12
	const Time& time,	13
	const ExternalEventList& lst);	14
	virtual Value* observation(	15
	const ObservationEvent& event) const;	16
	};	17

). Observation of discret event models. In: Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (p. 32-41). Presented at 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2012), Rome, ITA (2012-07-28 -