David Martins

Fanny Grumbach

Charles Manière

Pierre Sallot

Katia Mocellin

Michel Bellet

Claude Estournès
email: estournes@chimie.ups-tlse.fr

In-situ creep law determination for modeling Spark Plasma Sintering of TiAl 48-2-2 powder

Keywords: Spark plasma sintering, Ti-48Al-2Cr-2Nb, Creep, Densification parameters, Norton-Green model

Spark Plasma Sintering (SPS) is a process which allows powder densification, applying simultaneously a uniaxial external load and pulsed direct current of very high intensity through tools. This process is attracting significant attention, with a tremendous increase of studies in the metal powder densification field. Its growing popularity lies in the very fast heating rate and short cycle time driven by the Joule effect, which limits grain growth.

However, this process implements different coupled electrical, thermal and mechanical phenomena. All this makes the process difficult to develop and to apply for routine industrial production, which has motivated the development of numerical simulation tools in order to understand and optimize the process. Up to now, very few models integrating the coupling between heat generation, electric transfer and mechanics have been proposed. In particular, a numerical predictive model for powder densification requires a good understanding of the mechanical behavior, in our case a viscoplastic compressive law (Abouaf mechanical model). In this article, we will discuss the characterization of the material during densification, focusing on the creep behaviors of dense and porous state materials used to simulate sintering in the Abouaf framework. Validations of the creep law parameters and also of the densification parameters will be presented and subsequently discussed.

Introduction

Titanium Aluminide intermetallics are now a reality for structural applications such as aeroengines, replacing nickel based superalloys [START_REF] Ranjit | Numerical Analysis of Powder Compaction to Obtain High Relative Density in '601AB'Aluminum Powder[END_REF][START_REF] Besson | Simulation numérique de la mise en forme des céramiques -Application à la compaction isostatique à chaud[END_REF][START_REF] Manière | Identification of the Norton-Green compaction model for the prediction of the Ti-6Al-4V densification during the Spark plasma sintering process[END_REF]. The combination of a relatively high yield strength at high temperatures with good creep resistance, excellent resistance to oxidation / corrosion and their low density (3.9-4.1 g/cc) have contributed to their introduction on aeroengines' low pressure turbine blades or turbochargers in the automotive industry [START_REF] Geindreau | Constitutive behaviour of metal powder during hot forming. Part I: Experimental investigation with lead powder as a simulation material[END_REF][START_REF] Green | A plasticity theory for porous solids[END_REF].

Several processes are available to produce TiAl alloys, such as classical fusion processes (Vacuum Arc Remelting (VAR)), hot working techniques [START_REF] Ranjit | Numerical Analysis of Powder Compaction to Obtain High Relative Density in '601AB'Aluminum Powder[END_REF][START_REF] Besson | Simulation numérique de la mise en forme des céramiques -Application à la compaction isostatique à chaud[END_REF][START_REF] Manière | Identification of the Norton-Green compaction model for the prediction of the Ti-6Al-4V densification during the Spark plasma sintering process[END_REF][START_REF] Wolff | Modeling of conventional hot compaction and Spark Plasma Sintering based on modified micromechanical models of porous materials[END_REF] and casting [START_REF] Jarvis | IMPRESS Integrated Project-An overview paper[END_REF]. The first step involves mixing starting raw materials and melting them in VAR. Then, shaping is achieved by the casting technique using centrifugation or gravity depending on the complexity of the parts to produce. However, chemical homogeneity as well as microstructure control could be an issue when using casting, and can imply dispersions of mechanical properties. Powder Metallurgy is a good alternative to deal with part of these drawbacks and is worth studying; SPS can be compared to Hot Pressing (HP) but it is considered as a nonconventional densification process. Like in HP, a uniaxial external load is applied to the powder, which leads to more rapid densification compared to free sintering [START_REF] Guyon | Evolution des microstructures et mécanismes de densification d'un alliage TiAl lors du frittage par Spark Plasma Sintering[END_REF]. The major difference between the two techniques lies in the fact that in SPS the heating source is provided by the Joule effect, produced by a pulsed current applied through the tools.

Depending on their relative electrical conductivity, heat is generated in the sample itself (i.e. if the sample is more conductive than the tool) or in the tools [START_REF] Munir | The effect of electric field and pressure on the synthesis and consolidation of materials: a review of the spark plasma sintering method[END_REF][START_REF] Molenat | Temperature control in Spark Plasma sintering: an FEM approach[END_REF]. Heating rates and external applied pressures can be as high as 1000 K/min and 150 MPa respectively, depending on the tools. This method can densify a large range of materials such as polymers, metals, ceramics and their composites. The major advantage of this process is the control of the microstructure generated and its homogeneity in the bulk after sintering [START_REF] Orrù | Consolidation/synthesis of materials by electric current activated/assisted sintering[END_REF]. Densification rates as well as microstructure evolutions are closely linked to process parameters such as temperature and pressure distribution within the whole volume, and notably when considering complex-shaped parts. Thus, finite element modeling (FEM) of the whole process is a very efficient way to further optimize tooling design and SPS parameters.

Two main phenomena are usually modeled in SPS: heat generation by the Joule effect and densification. The electro-thermal component of the process can be modeled by a simple resistive approach with very good accuracy [START_REF] Molenat | Temperature control in Spark Plasma sintering: an FEM approach[END_REF][START_REF] Matsugi | Temperature distribution at steady state under constant current discharge in spark sintering process of Ti and Al 2 O 3 powders[END_REF][START_REF] Anselmi-Tamburini | Fundamental investigations on the spark plasma sintering/synthesis process: II. Modeling of current and temperature distributions[END_REF]. At the different interfaces between the tools and the powder, electrical and thermal contacts have to be considered as they greatly affect the temperature field [START_REF] Munir | The effect of electric field and pressure on the synthesis and consolidation of materials: a review of the spark plasma sintering method[END_REF][START_REF] Zavaliangos | Temperature evolution during field activated sintering[END_REF][START_REF] Maizza | Moving finite-element mesh model for aiding spark plasma sintering in current control mode of pure ultrafine WC powder[END_REF][START_REF] Vanmeensel | Field assisted sintering of electro-conductive ZrO 2-based composites[END_REF][START_REF] Maniere | Pulse analysis and electric contact measurements in spark plasma sintering[END_REF][START_REF] Maniere | Finiteelement modeling of the electro-thermal contacts in the spark plasma sintering process[END_REF].

Considering densification under an external applied pressure field, viscoplastic laws are often used to describe the sintering process. Famous expressions developed by Abouaf [START_REF] Abouaf | Modélisation de la compaction de poudres métalliques frittées -Approche par la mécanique des milieux continus[END_REF],

Olevsky [START_REF] Olevsky | Fundamental aspects of spark plasma sintering: II. Finite element analysis of scalability[END_REF][START_REF] Manière | Spark Plasma Sintering : couplage entre les approches Modélisation[END_REF] or Camclay [START_REF] Ranjit | Numerical Analysis of Powder Compaction to Obtain High Relative Density in '601AB'Aluminum Powder[END_REF] are gateways between dense and porous mechanic fields. To simulate densification using such approaches, identification of creep law parameters for both dense and porous material is crucial. These data are usually very time-consuming to acquire and require many interrupted experiments at different temperatures and pressures on dense and porous samples. These parameters are often determined using HIP (Hot Isostatic Pressure) cycles [START_REF] Besson | Simulation numérique de la mise en forme des céramiques -Application à la compaction isostatique à chaud[END_REF][START_REF] Abouaf | Modélisation de la compaction de poudres métalliques frittées -Approche par la mécanique des milieux continus[END_REF][START_REF] Brisson | Etude experimentale du frittage assemblage d'un composite conducteur en Ag SnO2 sur un support en cuivre par courants pulsés[END_REF] which require several hours of handling [START_REF] Besson | Simulation numérique de la mise en forme des céramiques -Application à la compaction isostatique à chaud[END_REF] and thus restrict the identification of such parameters to coarse microstructures.

Being able to run such experiments using SPS is very attractive for several reasons. The short processing time associated with SPS allows an identification of the parameters on controlled microstructures. It opens a way to take into account, in simulations, the impact of microstructure evolution on sintering. A growing number of authors use reverse analysis to determine the powder compaction parameters from SPS tests [START_REF] Abouaf | Modélisation de la compaction de poudres métalliques frittées -Approche par la mécanique des milieux continus[END_REF][START_REF] Olevsky | Fundamental aspects of spark plasma sintering: II. Finite element analysis of scalability[END_REF][START_REF] Wei | Spark Plasma Sintering of Commercial Zirconium Carbide Powders: Densification Behavior and Mechanical Properties[END_REF]. Manière et al [START_REF] Manière | Identification of the Norton-Green compaction model for the prediction of the Ti-6Al-4V densification during the Spark plasma sintering process[END_REF] have developed an original methodology, based on the work of Geindreau et al [START_REF] Geindreau | Constitutive behaviour of metal powder during hot forming. Part I: Experimental investigation with lead powder as a simulation material[END_REF], to determine by a set of experiences, the parameters of the creep law for TA6V processed using SPS in-situ. This methodology will be applied in the present study to TiAl 48-2-2.

The constitutive model: Norton-Green compressible viscoplasticity

The densification of a metal powder during electrically activated sintering results from different physical phenomena: the plastic deformation of powder particles ( pl ε  ), the mass transport between particles by diffusion under load ( dl ε  ), the mass transport between particles by electro-migration ( ̇ ) and the mass transport due to surface tension ( st ε  ) [START_REF] Olevsky | Constitutive modeling of spark-plasma sintering of conductive materials[END_REF]. As a consequence, the global deformation rate of the powder material, considered as a compressible continuum, can be expressed as the sum of the five contributions:

th st em dl pl ε ε ε ε ε ε            (1)
Where ( th ε  ) is the thermal dilatation.

At high temperature, which is the domain of interest for effective densification in SPS conditions, it is assumed that elastic effects can be neglected. Moreover, and as a first approximation, the respective contributions of electro-migration and transport by diffusion are neglected. Equation (1) reduces then to:

th vp ε ε ε      (2)
Where pl ε  is replaced by vp ε  assuming that particles deformation is purely viscoplastic at high temperature.

The constitutive model of the powder material should provide a relationship between the Cauchy stress tensor σ and the viscoplastic part of the strain-rate tensor ( vp ε  ). The selected constitutive model is the one proposed by Shima and Oyane [START_REF] Shima | Plasticity theory for porous metals[END_REF], which consists of an extension of the viscoplastic power law usually used for high temperature creep of metallic alloys in the dense state. The two next paragraphs contain a brief reminder of this power law and a description of its extension to porous (not dense) materials.

Viscoplastic power law for creep in the dense state

Metallic alloys generally obey a viscoplastic power law when deformed at high temperature:

n T A   ) (   (3)
Where s s : 2 The viscoplastic behavior is characterized by the temperature dependent function A and by n, a constant coefficient. Function A is assumed to follow an Arrhenius law:

( ) ( ) (4)
Where is a pre-exponential term, the activation energy ( ), R the universal gas constant and the absolute temperature ( ).

Equation [START_REF] Manière | Identification of the Norton-Green compaction model for the prediction of the Ti-6Al-4V densification during the Spark plasma sintering process[END_REF] shows that the generalized strain-rate derives from a stress potential:

1 1 ) ( ) , (    n n T A T           (5)
The tensor constitutive equation can then be obtained by the normality rule:

σ σ ε         σ σ vp    (6)
Leading to:

s ε 1 ) ( 2 3   n vp σ T A  (7)

Extension to the case of a porous metallic alloy

This extension relies first on the definition of the equivalent stress, as proposed by Green [START_REF] Green | A plasticity theory for porous solids[END_REF][START_REF] Mondalek | Numerical modeling of Spark Plasma Sintering process[END_REF]:

2 / 1 2 ) (tr : 2 3         σ s s f c  (8)
With c and f two decreasing functions of the relative density r  .

When densification tends to be completed ( 1  r 

), the limit values of c and f should be 1 and 0, respectively, in order to retrieve the standard von Mises expression for . Different expressions for c and f functions can be found in the literature ( [START_REF] Geindreau | Constitutive behaviour of metal powder during hot forming. Part I: Experimental investigation with lead powder as a simulation material[END_REF][START_REF] Wolff | Modeling of conventional hot compaction and Spark Plasma Sintering based on modified micromechanical models of porous materials[END_REF]27]). On the interval  

1 ;

,crit r  for the relative density, there are the general expressions:

                  crit r r r f K f f c , 1 1 (9)
With material parameters.

In the equation [START_REF] Molenat | Temperature control in Spark Plasma sintering: an FEM approach[END_REF], are dimensionless with no physical meaning, whereas can be considered as the green relative density .

As proposed by to Shima and Oyane [START_REF] Shima | Plasticity theory for porous metals[END_REF] and by Abouaf [START_REF] Abouaf | Modélisation de la compaction de poudres métalliques frittées -Approche par la mécanique des milieux continus[END_REF], for a porous media, the tensor form of the constitutive equation is written in the same form as for the dense material, Eq.( 6) and using the same potential function  as in the dense state , but with the modified expression of the generalized stress (Eq. (8)). We obtain:

          I σ s ε ) (tr 2 3 1 f c σ σ vp   (11)
Taking the deviatoric part and the trace of this expression, we have:

) dev( 3 2 1 vp σ c σ ε s             ) tr( 3 ) tr( 1 vp σ f σ ε σ             (12)
Where dev stands for the deviatoric part. Injecting those two expressions in the equation for the equivalent stress, we obtain:

2 1 2 ) (tr 9 1 ) dev( : ) dev( 3 2             vp vp vp f c σ ε ε ε     (13)
This suggests that the expression on the right hand side should be used for the equivalent or generalized strain rate in the context of a porous material. Thus, the power of deformation per unit of volume may be expressed as

     σ f c σ σ w vp             2 ) (tr : 2 3 1 : σ s s ε σ (14)
Finally, the viscoplastic compressible equation takes the following form:

                  I σ s I σ s ε ) (tr 2 3 ) (tr 2 3 1 0 f c e A f c n RT Q vp      (15)
Different material parameters must be identified for this constitutive equation. Parameters A 0 , Q and n are associated with the intrinsic viscoplastic behavior of the metal that constitutes the powder particles. Those three parameters can be identified with tests performed on a fully densified specimen, provided that its microstructure is representative of the powder particles, especially in terms of metallurgical grain size. Conversely, the functions c and f describe the impact of the relative density  r on the mechanical deformation of the powder material. They are assumed to depend on  r exclusively. For this reason, they can be identified by performing different compaction tests conducted on porous samples with different relative densities.

Before defining precisely an identification strategy, we will examine what the relationships are between stress and strain rate for two specific cases which will be used afterwards: free compression and confined compression.

Model equations in different configurations and definition of a strategy for parameters identification

Equations for free compression in the dense state

Free compression along the vertical direction z is supposed to be induced by a constant force F, applied to a plain cylindrical specimen, which is maintained at a constant temperature. Assuming a perfect lubrication, the stress state is found to be uniaxial and uniform through the specimen. In a coordinate system ( ), the stress tensor writes:

           zz  0 0 0 0 0 0 0 0 (16)
As the temperature is constant, Eq. ( 2) reduces to

vp ε ε   
, and we have, from Eq. (7):

n zz RT Q n zz zz σ e A σ A ε      0  (17)

Equations for free compression in the porous state

A similar situation, but for a compressible porous material, will now be considered. The stress tensor is equally uniaxial. From Eq. (8) we have:

zz σ f c    (18)
Applying the flow rule, Eq. ( 15), we obtain:

n zz n zz σ f c T A ε 2 1 ) )( (      (19)

Equations for confined compression in the porous state

Suppose now that the compression is operated with a tooling which restrains the specimen from any diameter variation. Still assuming a uniform strain rate, we have:

           zz ε   0 0 0 0 0 0 0 0 ε (20)
From Eq. ( 13), the equivalent strain rate writes:

f c ε ε zz 1 4 3     (21)
Applying Eq. (12), we obtain:

zz zz ε σ c s    9 4  and zz ε σ f    3 1 ) tr(  σ (22)
This leads to:

n zz n n n zz n n zz zz zz ε f c T A ε f c A ε f c σ s 1 1 1 2 1 1 1 1 1 1 1 1 4 3 1 ) ( 1 4 1 9 1 1 4 9 tr 3 1                                                                σ (23)
Or, equivalently, to:

n zz n n zz f c T A ε  ) 1 ( 2 1 1 1 4 ) ( 3                (24)

Definition of the identification strategy

The usual strategy for the identification of the different parameters combines different tests Like in the usual strategy developed in [START_REF] Manière | Identification of the Norton-Green compaction model for the prediction of the Ti-6Al-4V densification during the Spark plasma sintering process[END_REF], the dense state tests allow the determination of parameters A 0 , Q and n. The difference lies in the fact that it is now a combined analysis of the results of the other tests (free and confined uniaxial compressions) that will lead to the determination of c and f functions. The experimental tests, together with their rough results, are presented in the next section.

Experimental

All the SPS experiments (i.e. : densifications and in-situ creep tests) were carried out on a Dr.

Sinter 2080 device of the Plateforme Nationale CNRS de Frittage Flash located at the Université Toulouse III Paul Sabatier at Toulouse.

The TiAl composition involved in this study is the Ti-48Al-2Nb-2Cr (at%) alloy, currently used on GE90 or LEAP engines. The powder was supplied by Safran Tech and developed by General Electrics. The main chemical species are titanium and aluminum, with chromium and niobium as alloying elements.

The shape and the grain size distribution of the powder were controlled by Scanning Electron Microscope (SEM) characterizations (Figure 1). Concerning the morphology of the powder, grains are mostly spherical. Some heterogeneities, such as elongated grains or satellites are noticeable on Figure 1-a. We also note the presence of several grain size particules (Figure 1-b) between 30 and 144 µm. Laser granulometry analysis of the powder revealed that 10% of the volume of the powder had a size less than 46,6 μm. The 50% of the volume of the powder has a size smaller than 67,6 μm. For large particles, 10% of the volume studied is larger than 97,9 μm (Table 1)

The characteristics of the powder are given on

Uniaxial free compression tests on full dense and porous material

In both cases, samples were obtained by SPS compaction in graphite molds. They are all 8mm in diameter. Dense specimens are 10 mm in height whereas the porous ones are taller (Table 2). Porosity-controlled samples were produced using seven interrupted SPS sintering cycles performed under vacuum (Table 2). The heating rate was 100 K/min and the pressure applied was 50 MPa without any dwell. With this SPS cycle and for each of the seven specimens, the only difference was the maximum temperature reached ( The average relative density was estimated from the geometrical method. For samples exhibiting higher relative density than 0,95 verification by Archimedes' method was also performed.

Uniaxial free compression tests were performed in a graphite mold, 20 mm in diameter, used to avoid any contact between the creeping sample and the mold. During the tests the temperature of the sample and the load applied to the punches, as well as the lower punch displacement were continuously recorded.

All tests were performed in the SPS using indirect heating, i.e. the Joule effect was induced in the graphite tools surrounding the samples which were subsequently heated by radiation. To achieve indirect heating, graphite sheets (papyex) of 10 mm diameter with a thickness of 0.2 mm were coated with boron nitride on each side. They were placed between the sample and the punches for electrical insulation, in order to avoid direct Joule heating of the sample. The circular surfaces of the pistons in contact with these graphite sheets were also coated with boron nitride. Another graphite sheet was placed between the punches and the mold (Figure 2a). Boron nitride is also a very good lubricant which avoids the barrel effect of the sample. All this allows a homogeneous distribution of the temperature during the experiment [START_REF] Manière | Identification of the Norton-Green compaction model for the prediction of the Ti-6Al-4V densification during the Spark plasma sintering process[END_REF]. First, according to the methodology developed by Maniere et al., a partially open mold was used (Figure 2-b) to run the tests, but it was found that the temperature was not homogeneous in the sample (temperature given by thermocouple n°2 was higher than that given by n°1). A closed mold was finally used (Figure 2-c) and tests indicated that the temperature on thermocouple n°1 and 2 was the same. All the following free compression tests were performed with this closed mold.

A thermocouple was positioned at mid height on the samples surface (thermocouple n°2

Confined compression on porous material

Isothermal confined compression tests were carried out for various levels of applied pressure until complete densification was reached. The temperature was controlled with thermocouple n°2 (Figure 3). To assess the real powder temperature during densification, the SPS cycle was reproduced a second time with a sacrificial thermocouple (n°1 Figure 3) introduced in the middle of the powder. Due to the thermal gradient between the powder and the tools, the real compression temperature was then determined. The experimental setup is schematized (Figure 3). Between graphite (pistons and mold) and powder, a sheet of papyex graphite was used (blue lines in the Figure 3). This prevents powder from reacting with the mold and avoids any friction between the powder and the mold [START_REF] Manière | Powder/die friction in the spark plasma sintering process: Modelling and experimental identification[END_REF].

. Hypothesis and method

The uniaxial free compression test for dense samples was conducted under uniform and stable thermal conditions. The value of the parameter is then constant during the whole experiment.

We get from the Equation ( 17):

zz zz n T A   ln ) ( ln ln    (25)
The strain rate was computed using the time evolution of the lower punch displacement ( ) as follows:

                               dt h h d zz 0 ln  (26)
Where and are the height at the initial stage and at a given time respectively.

The load recorded by the SPS being the load which must be converted to stress using the section of the specimen. Considering a fully dense sample, its volume is constant and the section is deduced from:

0 2 0 0 0 h h R F S F h S Sh zz        ( 27
)
Where and are the value of the section at a given time and initial. is the initial radius of the sample.

Identification of creep parameters

First, 3 specimens were fully densified in the SPS at 1200 °C (50MPa, 3') and subsequently cooled to room temperature after completion of the experiments. Creep tests were performed on the temperature range achievable for TiAl compression by SPS, i.e. (925-975C°). Below this temperature range, the load to be applied in order to have some significant creep displacement is too high and reaches graphite tool limits 7,53 kN (ie 150 MPa).

When creep tests were performed at higher temperatures, the displacement of the pistons was too rapid. The section evolves too much for each step so the constraint is not constant. The data recorded during a creep test of a full dense sample at 975 °C is shown in the Figure 4 when the isothermal state is reached and stabilized, and then in green the evolution of the displacement of the lower punch. To ensure a proper contact between the samples and the tools, a very low load was imposed during heating. It should be noted that the evolution of the load before reaching the isothermal step results from the thermal expansion of the different part constituting the SPS column. Indeed, at the very beginning of the process, a contact force is applied and the punch height is recorded. During heating the tools and the specimen expand and so in order to keep this punch position the value of the force increases.

After each creep test, all the samples were examined to verify that their cylindrical shape was retained and thus, if uniaxial conditions were respected. In Figure 5 explained by the rapid and strong increase of the specimen sections due to large creep rates. In these cases, the stress cannot be assigned with an average value and thus this part of the curve is not being used. Finally, 2 other steps remain noisy, which could generate some errors for the creep parameters identification.

As a consequence, at 975 °C, only 5 steps will be considered from the 10 observed in Figure 5-a. Using a linear regression method on these data, creep parameter values were determined as: = 3,14 10 8 , = 414 163 J. mol -1 and = 2,81.

Validation of creep parameters

To validate the creep law established, the free compression test, performed at 975 °C, was simulated with the finite element software FORGE® using the parameters A 0 , and determined previously. The imposed boundaries conditions and the initial geometry of the sample at the different force applied were extracted from experimental creep data. Figure 6 compares the height evolution of the sample during the creep test as recorded experimentally and as calculated with FORGE®. The stresses simulated is plotted in the figure 6 too.

Considering the time and load range explored, the simulated final displacement is in firm agreement with the experimental one. The same comparison was made for tests performed at 925 and 950 °C, also with strong agreement. Thus, it validates the determination of creep parameters and the identification strategy described in section 4.2.

Figure 6 : Simulated and experimental creep test performed on a full dense sample at 975 °C

Identification of functions and

To further model material densification, the 2 functions ( ) and ( ) must be determined using two kind of tests performed on non-dense specimens: uniaxial free compression and confined compression.

For a given relative density ρ r , the values of and functions can be determined by the resolution of the equation for the uniaxial compression and for the confined compression, combined in the following system [START_REF] Manière | Powder/die friction in the spark plasma sintering process: Modelling and experimental identification[END_REF]. The sum is obtained from porous creep tests (with configuration described in Figure : 2c) and the sum ⁄ ⁄ from non-dense confined compression compaction tests (with configuration described in Figure 3).

{ ( | ̇ | | | ) ( | || ̇ | ( ) ) (28)
Provided that , and are known, the resolution of the system (28) gives a unique solution. For , the creep test is performed at constant temperature, thus is directly computed using the earlier coefficients. During the experiment, the stress is computed using eq. ( 27). After each free compression test, the diameter and the relative density of the sample were measured and no significant changes were observed. Therefore, the necks keep the same geometry all the time meaning that no mass transfer takes place.

̇ was calculated using the displacement of the lower punch and finally, the value was calculated and summarized in Figure 8-a (blue dots). Contrary to the creep test on dense material, a single load dwell was applied to porous samples. The explanation lies in the fact that for porous material, volume conservation hypothesis is no longer valid and the actuation of specimen section through eq. ( 27) is no longer possible.

The free compression test conditions on porous material are summarized in The second part of the system (28) involves several isobar stages. Thus, confined compression tests were carried out at 950 °C (ie outer mold temperature) for various levels of applied pressure until complete densification was reached. The second set of experiments (see section 3.2) allowed determination of the real temperature of the powder. The relative density is computed using the displacement of the lower punch. So for each relative density, the value of

⁄

⁄ can be calculated. The initial value of relative density is 0,65.

Figure 7 : Evolution of ⁄ ⁄ as a function of the relative density computed with the compaction test with a zoom for = 0,9 in the box In Figure 7,⁄ ⁄ values are represented during the confined compression. A lower punch displacement during application of an isobar stage implies that there has been densification.

On each of the load bearings sufficiently large for the densification to take place, three different steps are discernible. In the first step, just after the application of the load, the displacement speed of the lower piston is not constant. This therefore leads to a variation in the rate of deformation. This movement reaches a steady state, as a result of which this deformation rate is constant. Finally, the applied force is no longer large enough to generate the densification phenomenon. The value of the displacement of the lower piston decreases and tends towards 0. Surrounded in red are located the values of ⁄ ⁄ computed according to the degree of porosity for the second step of each stage. Here the more the relative density values increase, the more the ⁄ ⁄ values increase. Under 0,85, the value is very low due to and being very high. Only one value was found at 0,75 equal to 0,06.

Then, empirical expressions for and have to be used to deconvoluate both functions. The retained expressions for and functions are [START_REF] Wolff | Modeling of conventional hot compaction and Spark Plasma Sintering based on modified micromechanical models of porous materials[END_REF][START_REF] Abouaf | Modélisation de la compaction de poudres métalliques frittées -Approche par la mécanique des milieux continus[END_REF][START_REF] Mondalek | Numerical modeling of Spark Plasma Sintering process[END_REF] given in [START_REF] Munir | The effect of electric field and pressure on the synthesis and consolidation of materials: a review of the spark plasma sintering method[END_REF]. An optimization script was developed with Matlab ® to determine the material parameters present in the equations [START_REF] Munir | The effect of electric field and pressure on the synthesis and consolidation of materials: a review of the spark plasma sintering method[END_REF]. Minimizing the residual error from the least squares, the quadruple ( , , , ) was determined. Figure 8 presents the fit obtained between experimental values and simulated ones as well as the final shape of functions and versus relative density. This will be done in another study. A final comment that can be done on Figure 9 is that, despite the fact that only the mass transfer arising from the plastic deformation was considered here (as outlined in section 1), the agreement with the experimental data is excellent even at the end of the densification. This seems to indicate that for such metal densification, plastic deformation is definitely the first order phenomenon, the contribution of diffusion under load and surface tension remaining of minor importance. This will need confirmation through future extensive comparison between experimental observation and numerical solution.

Conclusion

This study focused on the protocol for the in-situ determination of the creep parameters of TiAl 48-2-2 case, and this approach could be generalized.

[ 3 ]

 3 . First, uniaxial compression tests are conducted on the material in the dense state, at different temperatures, from which A 0 , Q and n can be determined. Second, isostatic pressing tests are performed in the non-dense state on specimens with different relative densities. Such tests can provide a direct determination of the function ( ). Finally, complementary tests, still performed in the non-dense state and for different relative densities, are used to determine a combination of c and f, from which the function c( ) can be determined. In the framework of the present study, a similar strategy for the identification of the full set of material parameters has been used. The difference lies on the fact that it is based on the sole use of tests which could be carried out in the SPS pressing device. The different testing conditions are the following ones:  uniaxial free compression at different temperatures of cylindrical specimens made of densified powder. The tests are conducted in a creep mode, by prescribing the axial force;  uniaxial free compression of non-dense cylindrical specimen, at uniform temperature, and for different values of imposed axial force;  confined compression of non-dense cylindrical specimen in a die, at uniform temperature, and for different values of imposed axial force.

Figure 1 :

 1 Figure 1 : Ti-48Al-2Nb-2Cr powder under the Scanning Electron Microscope (SEM) (a),Grain size curve distribution obtained by laser granulometry analysis (b)

Figure 2 -

 2 Figure 2-c) to allow the control of temperature.

Figure 2 :

 2 Figure 2 : Experimental configurations in situ in the SPS for (a), uniaxial free compression for a dense or porous material, (b) partially open mold and (c) closed mold with thermocouple positions.

Figure 3 :

 3 Figure 3 : Powder compaction test in the SPS

Figure 4 :Figure 4

 44 Figure 4 : Load, displacement and temperature evolution during a free compression test in-situ in the SPS at 975°C

Figure 5 :

 5 Figure 5 : Time evolution of the height of the sample and the applied stress at 975°C (a) and evolution of ln( ̇) against ln( ) (b) for the 3 different creep tests performed on the full dense material in-situ SPS

Figure 5 -

 5 Figure 5-b represents the evolution of ln( ̇) as a function of ln( ) for valid tests at three temperatures : 925, 950 and 975 °C with the load ranges respectively of [2,78-5,10 kN], [2,60-5,25 kN] and [1,55-4,94 kN].

  Height of the sample computed with experimental data Height of the sample computed wit Forge(r) Computed stress with Forge(r)

Figure 8 :

 8 Figure 8 : Experimental (dots) and fits (lines) for (O; --) and ⁄ ⁄ (O; --) (a) and evolution of the functions (b) for the relative density between 0,65 and 1.

Figure 8 -

 8 Figure 8-a) shows that decreases with the relative density and tends towards 1 in the dense state. Fitted curves are in good agreement with the experimental data. Thus, it is now possible to calculate the value of the functions c and for any relative density between 0,65 and 1. In the Figure 8-b, the evolutions of these two functions are presented and they both decrease when the relative density increases.

Figure 9 :

 9 Figure 9 : Experimental and simulated densification

  SPS-sinteredTiAl 48-2-2 powders. This experimental protocol, associated with a proper model description of the sintering phenomenon provides acceptable and encouraging results towards the simulation of TiAl densification. This experimental methodology was developed in the framework of the Norton-Green powder compaction model, involving a relatively limited number of parameters, such as , and for the creep law of dense material and ( , ) functions describing porous material behavior. The unique combination of in-situ SPS creep tests on dense and porous samples as well as confined compaction experiments of powders opens a wide range of opportunities to model sintering of different kind of alloys. This potential has been demonstrated and validated on

Table 1

 1

Table 1 :

 1 Grain size distribution and interstitial contamination

	Grain size (µm)	Interstitial contamination (wppm)
	D10	46,6	C	76
	D50	67,6	O	760
	D90	97,9	N	< 5

Table 2 )

 2

	Temperature max (°C)	935	975	995	1005	1035	1100	1200
	Obtained relative density	0,77	0,88	0,91	0,93	0,96	0,99	1 (dense samples)
	Obtained height (mm)	12,98	11,49	10,98	10,87	10,41	10,1	10

Table 2 :

 2 Sintering conditions to obtain samples for uniaxial free compression tests

Table 3 :

 3

	Relative density	0,77	0,88	0,91	0,93	0,96	0,99	1,00
	Temperature (°C)	775	825	850	850	875	950	950
	Load (kN)	4,2	4,6	4,95	4,85	4,2	4,37	4,43

Table 3 :

 3 Conditions for free compression test on porous material conditions

Table 4 .

 4

	Parameters	(	)	(J. mol -1 )
	Values	3,14 10 8		414 163	2,81
		(	)

Table 4 :

 4 Values of the creep parameters and and functions

	6. Numerical validation of the proposed model
	Since the creep law of TiAl 48-2-2 was determined in previous sections, the following section
	focuses on the validation of this model by comparing experimental and simulated sintering
	cycles.

A sample 8 mm in diameter and 10 mm in height was sintered by SPS at 1050 °C (50MPa and 3'). The sintering cycle resumes to a temperature ramp of 100 K/min, no dwell in temperature and a constant applied load of 50 MPa. The evolution of the relative density as function of time is given in Figure

9

. Complete densification is obtained in 740 s. No densification is observed for the first 400 s as temperature is too low.

Acknowledgments

The support of the Plateforme Nationale CNRS de Frittage Flash (PNF2/CNRS) is gratefully appreciated. D.M., F.G., P.S., K.M, M.B. and C.E. thank the French National Research Technology Association (ANRT) and Safran for financial support of this study.


Keywords: Neurone integre et tire, integre et tire à conductances synaptiques, mouvement brownien, statistiques de spikes

Statistiques de trains de spikes pour quelques types de modeles de neurones simples

Résumé : Ce rapport de recherche decrit les résultats preliminaires d'un travail visant a caracteriser les distributions statistiques des trains de spikes, pour certains modèles de neurones couramment utilisés, en présence de bruit stochastique et avec des entrées déterministes. Le principal angle d'attaque du problème est l'utilisation du calcul stochastique et la représentations des martingales (locales) comme un mouvement brownien via un changement de l'échelle de temps ¶ .

Introduction

The dynamics of the discharge of neurons in vivo is greatly influenced by noise. It is generally agreed that a large part of the noise experienced by a cortical neuron is due to the intensive and random excitation of synaptic sites. The impact of noise on neuronal dynamics can be studied in detail in a simple spiking neuron model, the integrate-and-fire (IF) neuron [START_REF] Tuckwell | Introduction to theoretical neurobiology[END_REF]. For more complicated models the authors usually make use of the framework of the Fokker-Planck equation associated to a set of stochastic differential equations describing the dynamics of the neuron membrane potential in the presence of synaptic noise [START_REF] Risken | The Fokker-Planck equation: Methods of solution and applications[END_REF]. Since this equation cannot in general be solved analytically, the authors resort to some plausible approximations to obtain analytical results in various extreme case [START_REF] Brunel | Firing frequency of leaky integrate and fire neurons with synaptic current dynamics[END_REF][START_REF] Fourcaud | Dynamics of the firing probability of noisy integrate-and-fire neurons[END_REF]. In this paper we first give a general overview of methods arising from the stochastic calculus which we think can be very useful for studying this kind of problems. We then choose two of these methods and apply them to the description of the statistics of the inter-spikes time intervals for any input current and for a variety of synaptic noise types.

Stochastic calculus methods

Densities of hitting times

Stochastic diffusion models for neuron membrane potentials are widely admitted in the literature. For such a model the time of the first spike is the first time at which the diffusion process hits a determined threshold θ > 0. Numerous authors consider that the probability density function (pdf) of this first hitting time conveys an important biological information.

In this section we briefly present a few mathematical tools which allow one, in some cases, to explicit this density function. As, in the sequel of the paper, we will deal with the Integrate and Fire model and variations thereof, we limit ourselves to consider diffusion processes of the type

X t = X 0 + t 0 (α(s) -X s )ds + σW t ,
where W is a a standard one dimensional Brownian motion W , σ is a positive real number, and α(s) is a real valued function of time. We search the pdf of the stopping time T X,θ = inf {s ≥ 0; X s = θ}. As shown in section 3 below, this question is equivalent to finding the pdf of the stopping time

T W,φ = inf {s ≥ 0; W s = φ(s)}
for some function φ which can easily be explicited in terms of α(s) and θ. Notice that Strassen [START_REF] Volker Strassen | Almost sure behavior of sums of independent random variables and martingales[END_REF] has actually shown that, when φ is of class C 1 , then T W,φ has a continuous pdf. Unfortunately there are only a few cases where one can explicit this pdf as we now explain.

We first consider the case where the function α(s) is such that φ is constant: φ(s) = a for all s. We write T a for T W,φ . It is well known that the Laplace transform of a random variable fully characterizes its distribution. It is thus natural to search an explicit formula for the function u(x) := E x exp(-λT a ) for all positive real number λ, where E x is the expectation of the law of the Brownian motion W issued from x at time 0. To this end we consider the elliptic partial differential equation on (-∞, a):

   1 2 d 2 u dx 2 (x) -λu(x) = 0 for all x < a, u(a) = 1, (1)
with the limit condition:

lim x→-∞ u(x) = 0.
The smooth solution to this PDE is u

(x) = exp[ √ 2λ(x-a)].
Itô's formula applied to exp(-λt)u(W t ) shows that this process is a martingale. Thus the Optional Stopping Theorem (see, e.g., [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]) implies

E x [exp(-λT a )] = u(x) = exp[ √ 2λ(x -a)].
It now remains to invert the Laplace transform in order to get the density function of T a :

P x (T a ∈ dt) = a -x √ 2πt 3/2 exp - (a -x) 2 2t .
From this formula and Girsanov's theorem it is easy to deduce the pdf when φ is of the type µt + c. For more general functions φ we have to solve the two dimensional elliptic PDE

     ∂ t u(t, x) + 1 2 d 2 u dx 2 u(t, x) -λu(t, x) = 0 for all x < φ(t), u(t, φ(t)) = 1, (2)
with the limit condition lim

x→-∞ u(t, x) = 0.
If a smooth solution exists, we may apply Itô's formula to the process exp(-λt)u(t, X t ) and prove that it is a martingale. We then get

E x exp(-λT W,φ ) = u(0, x).
The difficulty is to exhibit explicit solutions of (2) and to compute their inverse Laplace transform. However, Groeneboom [START_REF] Groeneboom | Brownian motion with a parabolic drift and Airy functions[END_REF] succeeded to treat the case φ(t) = -γ 2 t 2 and xγ > 0. Using Girsanov's Theorem, one easily sees that it is then enough to solve the one dimensional PDE

   1 2 d 2 u dx 2 (x) -(γx + λ)u(x) = 0 for all x > 0, u(0) = 1, (3)
INRIA with the limit condition: lim x→∞ u(x) = 0. The solution can be explicited in terms of the decreasing sequence (v k ) k≥0 of the negative zeros of the Airy function Ai (see, e.g., Borodin and Salminen for a definition). The Laplace transform inversion formula then leads to

P x T W,φ ∈ dt = 2γ 2 1/3 e -γ 2 t 3 6 × ∞ k=0 Ai v k + (2γ) 1/3 x Ai ′ (v k ) e 2 -1/3 α 2/3 v k t dt.
A completely different methodoly is the Method of Images (see Lerche [START_REF] Rudolf | Boundary crossing of Brownian motion[END_REF]). It consists in considering the function h defined by

h(t, x)dx = P(T W,φ > t, W t ∈ dx).
Notice that h solves the heat equation

   ∂ ∂t h(t, x) = 1 2 ∂ 2 ∂x 2 h(t, x) for all t > 0, x < φ(t), h(t, φ(t)) = 0, h(0, x) = δ 0 (x) for all x < φ(0). (4)
Here we suppose that φ is concave and φ(0) > 0.

Suppose now that the following representation holds true: for some a > 0,

h(t, x) = 1 √ t η x √ t - 1 θ ∞ 0 1 √ t η x -s √ t F (ds), (5)
where η(x)

:= 1 √ 2π exp(-x 2 /2) and F (ds) is a positive σ-finite measure such that ∞ 0 η( √ ǫs)F (ds) <
∞ for all ǫ > 0. In Lerche [START_REF] Rudolf | Boundary crossing of Brownian motion[END_REF], it is shown that, if z = φ(t) is the unique solution of h(t, z) = 0 for all fixed t, then

P 0 T W,φ ≤ t = 1 -η φ(t) √ t + 1 θ ∞ 0 η φ(t) -s √ t F (ds). (6)
The difficulty is to explicit the measure F which seems to have been possible for very particular functions φ only: see Patie [START_REF] Patie | On some First Passage Time Problems Motivated by Financial Applications[END_REF] for a survey.

In conclusion, PDEs and Method of Images allow one to explicitly get the pdf of T W,φ for restrcited families of functions φ. These families can be slightly enlarged by using the technical Doob's transform: see Alili and Patie [START_REF] Alili | On the first crossing times of a Brownian motion and a family of continuous curves[END_REF].

An alternative approach consists in searching approximations of the pdf. This approach has been successfully developed by Durbin for concave or convex functions φ: see Theorem 1 below. An important challenge is to extend Durbin's results to general functions φ.

Monte Carlo simulations

As seen in section 2.1, explicit exact formulae or approximations of the pdf of T W,φ are generally difficult to obtain. Monte Carlo simulations apply without restriction to the model and the function φ. Indeed, simulations of the stochastic model (or of accurate discretizations of the model) allow one to obtain empirical density functions of T W,φ .

For example, consider:

dX t = (-X t + 1 + sin(2πt))dt + 2dB t X 0 = 0.
And consider the stopping time T = inf {t ≥ 0; X t ≥ 2} We obtain numerically an approximation of the pdf of T (see Fig. 1). This method permits us to control the error (see Cramer bound). This method is robust, it does need no assumption of convexity.

Integrate and fire with instantaneous synaptic conductances

The simplest model we consider is the integrate and fire where the membrane potential u follows the stochastic differential equation

τ du = (µ -u(t))dt + I e (t)dt + σdW,
with initial condition u(0) = 0, where τ is the time constant of the membrane, µ a reversal potential, I e (t) the injected current and W (t) a Brownian process representing synaptic input. The neuron emits a spike each time its membrane potential reaches a threshold θ. The membrane potential is then reinitialized to the initial value, i.e. 0. We are interested in characterizing the sequence {t i }, i = 1, • • • , t i > 0, t i+1 > t i when the neuron emits spikes.

INRIA

The time of the first spike

The problem of characterizing the first time t 1 when the membrane potential reaches the threshold θ is defined as

t 1 = inf {t : t > 0, u(t) = θ},
where u(t) is given by the following expression

u(t) = µ(1 -e -t τ ) + 1 τ t 0 e s-t τ I e (s) ds + σ τ t 0 e s-t τ dW (s)
The condition u(t) = θ can be rewritten as

t 0 e s τ dW = τ σ (θ -µ)e t τ + µ -1 τ t 0 e s τ I e (s) ds ≡ b(t) (7)
In order to characterize t 1 we need the following

Lemma 1 Let X(t) = t 0 e s τ dW (s)
The stochastic process X(t) is a Brownian motion if we change the time scale:

X(t) = W τ 2 e 2 t τ -1 .
Proof This lemma is in fact a direct consequence of the Dubins-Schwarz theorem [START_REF] Karatzas | Brownian motion and stochatic calculus[END_REF]. We provide an elementary proof for completeness. Let r = τ 2 " e 2 t τ -1 « , it is a monotonously increasing function of t equal to 0 for t = 0. For all times 0 < r1 < r2 < • • • < rn, the random variables X(r1), X(r2) -X(r1), . . . X(rn) -X(rn-1) are independent because W is a Brownian motion. Finally, it is easy to see that X(t2) -X(t1) is distributed as N (0, R t 2 t 1 e 2 s τ ds) which implies that X(r2) -X(r1) is distributed as N (0, r2 -r1).

We can now rewrite the threshold crossing condition above as

W (r) = τ σ (θ -µ) 2 τ r + 1 + µ -1 τ r 0 Ĩe (s) ds , where Ĩe (s) = I e ( τ 2 log 2 τ s + 1 )
2 τ s + 1 The time t 1 at which the membrane potential reaches the threshold θ is obtained from the time r 1 at which the Brownian motion W reaches for the first time the curve a(r) defined by the equation

y = a(r) = τ σ (θ -µ) 2 τ r + 1 + µ -1 τ r 0 Ĩe (s) ds ,
by the formula

t 1 := τ 2 log( 2 τ r 1 + 1) RR n°5950
The corresponding problem has been studied in particular by Durbin [START_REF] Durbin | The first-passage-density of a continuous Gaussian process to a general boundary[END_REF][START_REF] Durbin | The first-passage density of the Brownian motion process to a curved boundary[END_REF] who provides an integral equation for the probability density function (pdf) of r 1 . From this integral equation he deduces a series approximation of the pdf and proves convergence when the curve is concave or convex. This result is summarized in the Theorem 1 (Durbin) Let W (τ ) be a standard Brownian motion for τ ≥ 0 and y = a(τ ) be a boundary such that a(0) > 0 and a(τ ) is continuously differentiable for τ ≥ 0. The first-passage density p(t) of W (τ ) to a(t) is solution of the following integral equation

q 0 (t) = p(t) + t 0 p(r)( a(t) -a(s) t -s -a ′ (t))f (t|s) ds,
which can be written as

p(t) = k j=1 (-1) j-1 q j (t) + r k (t),
where

q j (t) = t 0 q j-1 (s)( a(t) -a(s) t -s -a ′ (t))f (t|s) ds j ≥ 1.
a ′ (t) is the derivative of a(t) and q 0 is given by

q 0 (t) = ( a(t) t -a ′ (t))f 0 (t),
where f 0 (t) is the density of W (t) on the boundary, i.e.

f 0 (t) = (2πt) -1/2 exp(-a(t) 2 /2t), and f (t|s) is the joint density of W (s) and W (t) -W (s) on the boundary, i.e.

f (t|s) = f 0 (s)(2π(t -s)) -1/2 exp(-(a(t) -a(s)) 2 /(2(t -s))).
The remainder r k (t) goes to 0 if a(τ ) is convex or concave.

As an application of the above, we consider two examples.

Constant intensity

In this case the membrane potential is the realization of an Ornstein-Uhlenbeck (OU) process. The function a(r) is convex, hence the hypotheses of Durbin's theorem are satisfied. The moments of the pdf of the hitting time can be computed analytically [START_REF] Ricciardi | First-passage time density and moments of the ornstein-uhlenbeck process[END_REF], see the next theorem.

INRIA

Theorem 2 Let us define α := µ σ and β := σ θ √ τ and the three following functions:

Φ 1 (z) := 1 2 ∞ n=1 2 β n 1 n! Γ( n 2 )(z -α) n Φ 2 (z) := 1 2 ∞ n=1 2 β n 1 n! Γ( n 2 ) Ψ( n 2 ) -Ψ(1) (z -α) n Φ 1 (z) := 3 8 ∞ n=1 2 β n 1 n! Γ( n 2 )(z -α) n ρ (3)
n where Γ is the gamma function, Ψ(z) = Γ ′ (z) Γ(z) is the digamma function, and

ρ (3) n = Ψ( n 2 ) -Ψ(1) 2 + Ψ ′ ( n 2 ) -Ψ ′ (1)
If T is the hitting time of an OU process starting at 0 to the barrier θ, we have: 0.88 0.97 1.00 0.99 Table 2: Values of the integral of the estimated pdf for I e = sin(2πt).

E[T ] = τ (Φ 1 (1) -Φ 1 (0)) E[T 2 ] = τ 2 (2Φ 1 (1) 2 -Φ 2 (1) -2Φ 1 (1)Φ 1 (0) + Φ 2 (0)) E[T 3 ] = τ 3 6Φ 1 (1) 3 -6Φ 1 (1)Φ 2 (1) + Φ 3 (1) -6(Φ 1 (1) 2 -3Φ 2 (1))Φ 1 (0) + 3Φ 1 (1)Φ 2 (0) -Φ 3 (0) A comparison

Periodic intensity

We choose I e (t) = sin(2πf t). Table 2 shows the values of the integral of the estimated pdf. The parameters are the same as in the previous example, f = 1. It is seen that Durbin's series converges very quickly. Figure 2 shows the shape of the pdf of the first passage time and the first four terms in the series approximation. Figure 2: Four terms of the series approximation of the pdf when I e (t) = sin(2πt) and the resulting pdf (the horizontal scale is in r units).

The times of the next spikes

The previous analysis and results can be extended to the times t 2 , . . . t n of the next spikes. We discuss how to determine t n given t n-1 , i.e. how to compute p(t n |t n-1 ). The scenario is similar to INRIA the one used to compute t 1 . For t > t n-1 , u(t) is given by the following expression

u(t) = µ(1 -e -t-tn-1 τ ) + 1 τ t tn-1 e s-t
τ I e (s) ds

+ σ τ t tn-1 e s-t τ dW (s)
After some algebra, the condition for crossing the threshold can be written

X(t) = t tn-1 e s τ dW (s) = b(t) + h(t n-1 ),
where

h(t n-1 ) = τ σ 1 τ tn-1 0 e s τ I e (s) ds + µ(e tn-1 τ -1) ,
and b(t) is defined in (refeq:bt). Consider the stochastic process

Y (t ′ ) = X(t ′ + t n-1 ), us- ing once again the Dubins-Schwarz theorem, Y (t ′ ) = W (r ′ ) where r ′ = t ′ 0 e 2 s ′ +t n-1 τ ds ′ = e 2tn-1 τ τ 2 (e 2t ′
τ -1). The next threshold crossing time t n is the first crossing time of W (r ′ ) with the boundary b(t

′ + t n-1 ) + f (t n-1 ) = b( τ 2 log( 2 τ e -2tn-1 τ r ′ + 1) + t n-1 ) + h(t n-1 ).

Integrate and fire with exponentially decaying synaptic conductances

We modify the model of section 3 to include exponentially decaying synaptic conductances.

τ du = (µ -u(t))dt + I e (t)dt + I s (t)dt τ s dI s = -I s (t)dt + σdW
We can integrate this system of stochastic differential equations as follows. The first equation yields Proof By exchanging the order of integration in the definition of X(t) (Fubini's theorem) we obtain

u(t) = µ(1 -e -t τ ) +
X(t) = Z t 0 e s ′ τs "Z t s ′ e s α ds « dW (s ′ ) = α Z t 0 e s ′ τs (e t α -e s ′ α ) dW (s ′ ),
This expression is of the form X(t) = R t 0 f (t, s) dW (s) and therefore X(t1) -X(t2) is not independent of X(t2) for 0 ≤ t2 < t1. Because of the properties of the Brownian motion it is a continuous zero-mean Gaussian process whose covariance function ρ(r, t) for 0 ≤ r ≤ t is

E[X(r)X(t)] = E »Z r 0 f (r, s) dW (s) Z t 0 f (t, s) dW (s) - = E »Z r 0 f (r, s)f (t, s) d < W, W >s - = Z r 0 f (r, s)f (t, s) ds
and the result follows from the computation of the last integral.

In the same line of idea as in section 3, we can express the problem of characterizing the time t 1 at which the membrane potential reaches the threshold θ as that at which the Gaussian process X reaches for the first time the curve a(τ ) defined by the equation

y = a(τ ) = - ατ s σ I s (0)(e t α -1)+ τ τ s σ (θ -µ)e t τ + µ - 1 τ t 0 e s τ I e (s) ds ,
This problem has been studied by Durbin [START_REF] Durbin | The first-passage-density of a continuous Gaussian process to a general boundary[END_REF] and is summarized in the following theorem

(t) = f 0 (t)b 1 (t), where b 1 (t) = a(t) ρ(t, t) ∂ρ(s, t) ∂s | s=t -a ′ (t) f 0 (t)
is the density of X(t) on the boundary and f (t|s) is the conditional density of X(t) at a(t) given that X(s) = a(s).

The times of the next spikes

As in the case of instantaneous synaptic conductances, we can extend our analysis and compute the conditional probabilities p(t n |t n-1 ), in effect p(t n |t n-1 , I s (0)), as follows. For t > t n-1 . The expression for the membrane potential is The analysis proceeds as follows using theorem 3 instead of 1.

Conclusion

We have sketched out a number of techniques borrowed from the field of stochastic calculus that we believe to be useful in the analysis of spikes trains generated by neural models. We have applied two variations of one of these methods to the computation of the pdfs of the spikes times of two instances of the integrate and fire neuron model with synaptic conductances. The first variation is based upon representing the membrane potential as the sum of a deterministic function and a local martingale. Due to a theorem by Dubins and Schwarz, by changing the time scale we can turn the local martingale into a Brownian motion and the problem of computing the pdfs of the spikes times into that of computing the first-passage density of the Brownian motion to a curved boundary. This particular problem can be solved through a method due to Durbin [START_REF] Durbin | The first-passage density of the Brownian motion process to a curved boundary[END_REF] which provides a series approximation of the pdf. Numerical experiments show that the series converges rapidly. A comparison with the results obtained by Monte-Carlo simulation is provided. The second variation is based upon representing the membrane potential as the sum of a deterministic function and a zeromean continuous Gaussian process. Due to another theorem by Durbin the problem is shown to be equivalent to solving a Volterra integral equation of the second kind. These ideas can probably be extended to more complex neuron models [START_REF] Brette | Adaptive exponential integrate-and-fire model as an effective description of neuronal activity[END_REF][START_REF] Izhikevich | Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting[END_REF].

2 Figure 1 :

 21 Figure 1: Approximation of the pdf of T with 10E6 Monte Carlo simulations

1 τ

 1 u(t) = µ(1 -e -t-tn-dW (s ′ ) ds

Table 1 :

 1 of the values of the first three moments computed from the pdf of the hitting time obtained from Durbin's theorem or by simulation with the analytical values (obtained by truncating the series Φ i ) is shown in table 1. This table shows that the theoretical values can be closely Values of the first 3 moments of the OU process and the empirical values, for the parameters: θ = σ = 2, µ = τ = 1 approximated if sufficiently many terms are taken into account in Durbin's series expansion.

	method	E[T ]	E[T 2 ]	E[T 3 ]
	theoretical values 1.9319289 7.1356162 40.0830265
	Durbin, 30 terms, 1.9292822 7.1269290 39.8541918
	T max = 10 36 ,
	step = 10 -2
	Monte-Carlo,	1.932180	7.139402	40.079556
	10 6 realisations,
	step = 10 -4
	RR n°5950

  Table 2 indicate that a very good approximation of the pdf can be obtained with only 5 terms in the series. The total computation time is 8 seconds on a 2GHz computer for 800 sample points.

	0.5
	0.4						q3 q2
							pdf
	0.3						q1 q4
	0.2
	0.1
	0	2	4	6	8	10	12	14	16	18	20

RR n°5950