domain: chim spi sde

text:
J A Mbey
email: mbey25@yahoo.fr

COMPONENTS INTERACTIONS CONTROLLING STARCH-KAOLINITE COMPOSITE FILMS PROPERTIES

A c c e p t e d M a n u s c r i p t 2

Abstract

In order to relate the primary filler-matrix interactions to the macroscopic properties of starchkaolinite composite material, these interactions are monitored through homo-or heterocoagulation experiments involving both components. Turbidity measurement and Infrared spectra confirm the extreme weakness of the interactions. The addition of calcium cations shows that these weak interactions between starch and kaolinite are due to the combination of the electrostatic repulsion and hydrogen bonds formation between this two negatively charge components. Some possible relationships between the starch-kaolinite interactions and starchkaolinite composite films properties are proposed. Keys words: Kaolinite; Starch; Composite; Coagulation; Interface.

: Introduction

Starch and kaolinite are readily available, low cost materials that have already found uses in several domain such as composite materials (Kaewtatip and Tanrattanakul, 2012 ; Mbey et al., 2012a; Chen and Evan, 2005; de Carvalho et al., 2001; Whilem et al., 2003), paper coating (Husband, 1998), flotation of iron ore [START_REF] Ma | Starch-kaolinite Interactions[END_REF][START_REF] Ma | Role of hydrolyzable metal cations in starch-kaolinite interactions[END_REF][START_REF] Ma | The effect of pH and ionic strength on starch-kaolinite interactions[END_REF]Liu et al., 2000;[START_REF] Weissenborn | Selective flocculation of ultrafine iron ore. Part 1. Mechanism of adsorption of starch onto hematite[END_REF], water treatment [START_REF] Shogren | Flocculation of kaolin by waxy maize starch phosphates[END_REF][START_REF] Bolto | Organic polyelectrolytes in water treatment[END_REF]Krentz zt al._2006 ;Järnström et al. 1995a ;Järnström et al. 1995b). In some cases, starch undergoes some modifications prior to its use in a particular domain. Cationic starch derivatives for instance are usually of interest in flocculation of solid matter from water [START_REF] Bratskaya | Starch derivatives of high degree of functionalization 10. Flocculation of kaolin dispersions[END_REF][START_REF] Chen | A kinetic investigation of cationic starch adsorption and flocculation in kaolin suspension[END_REF][START_REF] Wei | Synthesis and flocculating properties of cationic starch derivatives[END_REF].

In the domain of polymer-clay composite, the filler-polymer matrix interactions are one major key for properties change understanding. Those interactions are influenced by the filler-matrix interface and/or interphase, the filler particles orientation, the filler dosages, the filler particles anisotropy, the surface properties of the filler or the filler dispersion [START_REF] Fu | Effects of particle size, particle/matrix interface adhesion and particle loading on mechanical properties of particulate-polymer composites[END_REF][START_REF] Tran | Dispersion of silicate nano-plates within poly(acrylic acid) and their interfacial interactions[END_REF][START_REF] Wu | Tensile performance improvement of low nanoparticles filled-polypropylene composites[END_REF]Pukansky, 1990). In those materials, the interactions between the organic matrix and the inorganic filler determine structural and aging properties. Strong interactions guarantee rigidity and resistance to abrasion, weak interactions are required for flexibility. (Lopez et al., 2014;[START_REF] Müller | Thermoplastic starch/wood composites: Interfacial interactions and functional properties[END_REF][START_REF] Jandas | Surface treated banana fiber reinforced poly (lactic acid) nanocomposites for disposable applications[END_REF][START_REF] Reddy | Biobased plastics and bionanocomposites: Current status and future opportunities[END_REF].

Hence, the explanation of properties changes in composites are discussed on the basis of the filler-polymer matrix interactions. Lopez at al. (2014) associated the ductility conservation in starch-talc composite to low interactions between starch and talc particles. [START_REF] Müller | Thermoplastic starch/wood composites: Interfacial interactions and functional properties[END_REF], used wood particles to improve thermoplastic starch properties. They show that the wood particles having large aspect ratio improve the stiffness and the strength of the A c c e p t e d M a n u s c r i p t 3 composite. Also, strong interfacial adhesion between the wood particles and the starch matrix decreases the starch chains mobility and reduces the water uptake of the composite. In our previous works [START_REF] Mbey | Cassava-starch kaolinite composite film. Effect of clay content and clay modification on film properties[END_REF]2014), unmodified or DMSO-intercalated kaolinite are used as filler in thermoplastic starch based films. The filler particle orientation and the low kaolinite dosage, together with weak kaolinite-starch interactions better account for the plasticity increase within the film. Better dispersion of the DMSO-intercalated kaolinite shows larger properties changes compared to unmodified kaolinite filled films. It is then believed that analysing the primary interactions between the components of the composite may be useful to understand the composite properties. Hence, in the case of kaolinitethermoplastic starch composite, the study of starch-kaolinite interactions is of interest.

To date, general studies on the kaolinite-starch interactions are rare [START_REF] Ma | Starch-kaolinite Interactions[END_REF][START_REF] Ma | Role of hydrolyzable metal cations in starch-kaolinite interactions[END_REF][START_REF] Ma | The effect of pH and ionic strength on starch-kaolinite interactions[END_REF], especially in the field of starch-clay composites. In fact, those interactions are described as very weak, since the essentially non ionic hydroxylic functions available on both starch and clay interface give rise to physical sorption through hydrogen bonds.

For this study, we will attempt to correlate the starch-kaolinite interactions to the properties of starch-kaolinite composite films.

This study aims at monitoring the adsorption of cassava starch onto kaolinite as a way to understand the starch-kaolinite films properties. The dispersion or the coagulation of the kaolinite phase during composite processing will largely depend on the kaolinite-starch interactions. Absence of interactions could favour phase separation between the two components, whereas strong interactions may induce the coagulation of the clay within the polymer matrix or increase the polymers chain rigidity within the composite. On this basis, it is then obvious that weak to medium interactions are preferred for viscoelastic composite films making. To access the starch-kaolinite interactions, the sedimentation/flocculation test of dilute kaolinite suspension in presence of natural cassava starch is used.

: Experimental section

2-1: The Kaolinite material

The raw kaolinite was collected from a deposit situated in Mayouom (western Cameroon) [START_REF] Njoya | Genesis of Mayouom kaolin deposit (western Cameroon)[END_REF]. The sample used in the present study was taken at 3 m depth. The %. The fine clay fraction was collected through sedimentation after Stokes' law. The particle size distribution of the collected fraction was analysed using a Sympatec laser diffraction granulometer equipped with the HELOS optical system and the WINDOX software for data acquisition. The average particle size (D50) was found to be 4.5 µm and more than 80 % of the material has size less than 7.5 µm. The BET specific surface area, determined by nitrogen adsorption using an automatic homemade apparatus, was (25.9 ± 0.1) m 2 /g. The cation exchange capacity (CEC) was measured using hexaminecobalt(III) chloride [Co(NH 3 ) 6 Cl 3 ].

The amount of hexaminecobalt(III) sorbed by the solid phase was determined by colorimetric measurement at 472 nm using UV-vis spectroscopy and the CEC was found to be 6.0 meq/100g.

2-2: The starch material

The cassava starch was obtained by aqueous extraction from cassava tubers produced in Mambando (Centre Cameroon). Starch material was manually ground and sieved at 100 µm.

The sieved starch was stored in high density polyethylene container at ambient temperature.

The moisture content, determined by drying to constant weight at 105°C, is 14 %. Content in mineral ashes, determined by ignition at 550 °C in a muffle furnace, is 0.3 %. Elemental analysis using the CHNS analyser Carlo Erba 1108, indicated respective contents of carbon (39.1 %), hydrogen (7.5 %) and nitrogen (0.05 %). The H/C ratio, 5.2 is common for polysaccharides. The very low nitrogen content, and the absence of the characteristic amine band in the FTIR spectrum (not presented) indicate negligible content of proteins.

2-3: Experimental procedures

The kaolinite suspension was prepared by dilution of the aqueous extract of fine clay fraction to a solid concentration of 0.5 mg/mL of turbidity 235 NTU (HACH 2100 N turbidimeter).

The starch solution was prepared by dissolving 100 mg of starch in 5 mL of analytical grade DMSO (Sigma Aldrich 99.99 %) under heating at 60 °C during 5 min. The mixture was then diluted with deionised water (18.2 Mcm 1 ) to obtain 100 mL of solution (starch concentration 1 mg/mL). The dissolution with DMSO prevents alteration of the chemical functions and degradation of the carbohydrate chains [START_REF] Han | Structural changes of corn starches by heating and stirring in DMSO measured by SEC-MALLS-RI system[END_REF]. The working pH was selected after analysing its effect on the sedimentation of the kaolinite suspension.

Typically 30 mL of the initial suspension having 235 NTU turbidity was taken and used at the during 40 min after which 15 mL of the suspension supernatant was taken using a micropipette for turbidity measurements. On figure 1, are reported the results obtained. It can be noted that the turbidity is stabilized around 160 NTU. This reduction of turbidity is caused by the settling of large particles that arise from collision between the clay particles subject to Brownian motion. The acid or alkaline value of pH induced either protonation or deprotonation of edge functions (=AlOH and =SiOH). Given that the permanent charge of the clay is negative, hence deprotonation of the edge functions, in alkaline pH, increasing the global negative charge of the suspension which result in an increase electrostatic repulsion that prevent particles collision and hence, stabilized the suspension. Conversely, in acid pH, the edge functions are positively charge inducing then a reduction of the global electrostatic repulsion. This charge reduction favours collision that result in increase particle size that settled under their weight. This explains why, acid pH favours the sedimentation and alkaline pH slightly increases suspension stability. Intermediate pH, reached spontaneously by the kaolinite suspension (pH = 6.2), was considered as an optimal starting condition for further coagulation experiments.

The proton surface excess of both kaolinite and starch was measured by acid-base titration using a Metrohm Titrando 809 station with two dosing units Dosino 800. The system is automatically driven by the software Tiamo (version 1.2.1 light). The titration of the suspension was carried out under magnetic stirring and argon stream to avoid carbon dioxide dissolution in the suspension. Electrophoretic mobility of the kaolinite and the starch materials was measured in NaNO 3 background electrolyte (10 -3 M, 10 -2 M and 10 -1 M) and in the presence of varying amounts of CaCl 2 using a Zetaphoremeter V by CAD Instrumentation (France).

For the sedimentation tests, 30 mL of the kaolinite suspensions was magnetically stirred for 5 min after addition of known dosages of starch and CaCl 2 . The samples were then placed in vertical test tubes for sedimentation. After 40 min, 15 mL of the supernatant suspensions was taken for turbidity measurement using HACH 2100 N turbidimeter. The remaining suspensions were freeze-dried and the sediment conditioned in KBr pellets were analysed by FTIR. The FTIR spectra were recorded using a Bruker IFS 55 interferometer in transmission mode. The sedimentation tests were done in absence or in presence of calcium. A preliminary test of coagulation by calcium ions was done to determine the dosage to be used.

Uv-vis spectrometer, SHIMADZU UV 2101 PC, was used for kinetic monitoring of the

: Result and discussion

3.1: Starch-Kaolinite interactions

The titration curves presented in figure 2, show the pH-dependence of the net proton surface excess. For the starch, the specific amount of acidic groups is around 43 µmol/g (figure 2b).

In order to evaluate the spatial distribution of charges on the starch macromolecule, we can assume that the latter is composed of glucose residues of molar mass 180 g/mol. The obtained charge is then 7.7 mmol of protons per mole of glucose residue, which represents 0.77 % of carboxylated residues (less than 1 % of glucose residues are bind to carboxylate group).

The titration curves at different ionic strength are virtually superimposed, which indicates a very low charge.

The titration curve of kaolinite quantifies the pH-dependent charges on the edges of the particles (=AO and =SiO), resulting in a proton surface excess of about 50 µmol/g (figure 2a). This surface charge is in accordance with the commonly observed cationic exchange capacity for kaolinites, generally below 10 meq/100g (Eslinger andPevear, 1988, after Martin and[START_REF] Martin | Effective Qv by NMR core tests[END_REF]. The fact that no common intersection point is observed on the titration curves at different ionic strength indicates the presence of small amount of permanent charge due to isomorphic substitutions in the crystalline structure of the kaolinite. Similar behaviour was observed and modelled for montmorillinites and illites by [START_REF] Delhorme | Modelling acid-base properties of 2:1 clays. The role of electrostatics[END_REF] and could be attributed to the 10% of illite contained in the sample.

From electrophoretic mobility measurements (figure 3), it is shown that the surface charge of both starch and kaolinite is negative in the pH range of 3 to 10. The electrophoretic mobility of the starch (figure 3b) shows positive values for pH below 2.1, which indicates the presence of cationic functions such as ammonium functions in proteins, which were detected in very low amount by elemental analysis ( §2.2). The negative values and pH dependence of the electrophoretic mobility of starch in the pH domain 2.5 -7 is probably due to the dissociation of carboxylic groups on the starch macromolecules.

The electrophoretic mobility of kaolinite (Figure 3a) shows very little dependence on ionic strength and also on pH in the domain above pH 6, which confirms the presence of permanent charges that were detected from the titration curves. In the domain below pH 6, strong pH dependence of the electrophoretic mobility results from the protonation of the hydroxylic functions on the edges of the clay particles.

A c c e p t e d M a n u s c r i p t 7 Adsorption isotherms are generally used to study the mechanism and strength of interaction between a mineral interface and an organic (macro)molecule. In the present case, such approach is ineffective for at least two reasons. Firstly, the weakness of the interactions does not allow strong amounts of organics to be adsorbed on the solid, as shown in published studies [START_REF] Liu | Polymer modified clay minerals: A review[END_REF][START_REF] Mpofu | Investigation of the effect of polymer structure type on flocculation, rheology and dewatering behaviour of kaolinite dispersions[END_REF]Järnström et al. 1995a). Secondly, in the present study, cassava starch was used in the native form, therefore heterocoagulation experiments are more adapted than adsorption isotherms to study organo-mineral interactions. In figure 4, the turbidity of a kaolinite suspension is independent of the amount of added starch up to 2.5% (higher amounts would increase the turbidity due to the granular character of starch). Addition of 10 -5 M CaCl 2 results in significant and rapid (40 minutes) sedimentation of the suspension, which confirms that electrostatic repulsions must be overcome to achieve attractive interactions, as also reported by [START_REF] Ma | Role of hydrolyzable metal cations in starch-kaolinite interactions[END_REF]. It is obvious that for dosage between 1 % and 1.5 %, the effect of starch on the sedimentation is maximal. In this study, the calcium ion acts as neutralizer of the kaolinite surface charge, and the screening of the kaolinite charge favours the starch adsorption on the kaolinite surface.

The increase in adsorption of starch is evidenced by the decrease in the turbidity (increase sedimentation of the clay) for the same starch concentration in presence of calcium ion (figure 4). The starch adsorption is almost maximal early at 0.33 % dosage of starch. It seems that adsorption of starch at low dosage is possible and this helps the kaolinite sedimentation.

However, the adhesion forces are not strong enough and this explains the turbidity fluctuation observed around 1 %. Weak interactions take place in the system due to combine effect of hydrogen bonds between the two components and electrostatic repulsion between starch and kaolinite due to their negative surface charge. The hydrogen bonds could be beneficially increase by the reduction of the interaction sphere (that is, a reduction of the volume in which starch and kaolinite interact). Hence, the screening of the kaolinite surface charge by the calcium cations induce then an increase in starch adsorption. The effect of calcium ions on the electrostatic interactions between the clay and the starch is illustrated in figure 3 (c andd).

While kaolinite showed to be relatively insensitive to monovalent cations (Na + ) (Figure 3a), its negative electrophoretic mobility dramatically decreases in the presence of 10 5 M Ca 2+ .

Comparatively, the mobility of starch is less sensitive to the presence of Ca 2+ , the complexation of which with carboxyl and eventually dissociated hydroxyls being much weaker than on dissociated surface sites on clay. The charge interaction between kaolinite and starch is hence, reduced and this results in a decrease of the interaction sphere between kaolinite and starch that favours starch adsorption. However, one can easily note that the clay A c c e p t e d M a n u s c r i p t 8 surface charge remains negative even in the presence of calcium (figure 3c) as well as the starch surface (figure 3d). The hypothesis to explain the remaining negative charge of the kaolinite in the presence of calcium is that, edge-edge aggregation that leads to loose aggregates takes place within the kaolinite suspension in presence of calcium. These loose aggregates enclose part of the negative charge of kaolinite particles preventing then further screening because of the repulsion between the calcium cation at the external shell of these aggregates. Hence, even if the added calcium cations screened both starch and kaolinite surface, the overall surface of both component remain negative and the electrostatic repulsion between the two components is dominant and favours the stability of the system. FTIR spectra of the raw kaolinite and aggregated sediments are presented in figure 5. The presence of the band at 1383 cm 1 characterising the C-H bending in the pyranose ring [START_REF] Velraj | Fourier Transform Infrared Spectroscopic Study on Glycoalkaloid Concentration in Varieties of Solanum tuberosum[END_REF]) is a proof of some starch adsorption onto kaolinite particles. However, as concluded from turbidity measurements, the amount of starch adsorbed is low. The sediments are mostly made of kaolinite as shown by the similarity between the raw kaolinite spectra and that of the freezed-dried sediments. The absence of the band at 1383 cm 1 on the sediment obtained with 2 % starch, is probably due to the loose nature of the aggregate which agree with the previously mentioned weak adhesion force between the two components. It is proposed that natural gravimetric sedimentation is dominant and the contribution of potential coagulation due to both starch and calcium is low.

The kinetic evidence of the influence of starch on the kaolinite sedimentation is presented on figure 6. The difference between the blank system and systems containing starch, calcium or starch + calcium is obvious. The starting of sedimentation is effective between 10 min and 15 min. The kaolinite sedimentation is increase in the presence of starch. The addition of calcium ameliorate the starch adsorption on the clay which better promotes the kaolinite sedimentation. The increase sedimentation due to starch adsorption is rather low indicating weak adhesion interactions that are clearly related to the surface state of both components which are negatively charged.

3.2: Relating the starch-kaolinite interactions to the starch-kaolinite films properties

In the aim of understanding the properties of composite films, a global view of weak starchkaolinite interfacial interactions can explain the good dispersion of the kaolinite within the starch matrix. This view of weak interactions is regarded as a result of the predominance of the electrostatic repulsion between these two negatively charged components and the H-bonds formation due to the existence of hydroxyl groups on the surface of both components. In particular, the existence of electrostatic repulsion is favourable to the clay dispersion. The A c c e p t e d M a n u s c r i p t 9 presence of kaolinite between the starch chains with weak kaolinite-starch interactions facilitates the mobility of the starch chains, giving rise to a plasticizing effect of the clay within the starch matrix. These hypotheses correlate well with the decrease of the glass transition temperature reported in a previous work [START_REF] Mbey | Cassava-starch kaolinite composite film. Effect of clay content and clay modification on film properties[END_REF]. The glycerol molecules, present at the starch-kaolinite interface, may also participate in the interactions within the films network.

Due to the electrostatic repulsion between the two components, the clay particles are better oriented in the starch matrix in the direction that minimizes the repulsion between the two components. Because kaolinite charge is supposed to be more dense on edges surfaces, a preferential basal orientation of the kaolinite particles is expected within the films (figure 7a).

Such an orientation may also contribute to an increase in starch chain mobility through an increase of the plasticizer diffusion and prevention of plasticizer escape. The optical micrograph presented in figure 7b, actually corroborates such an orientation. This orientation also agrees well with barrier properties to water uptake, heat diffusion or UV-light transmission [START_REF] Mbey | Cassava-starch kaolinite composite film. Effect of clay content and clay modification on film properties[END_REF]. More recently [START_REF] Mbey | Cassava-starch kaolinite composite film. Thermal and mechanical properties related to filler-matrix interactions[END_REF], tensile test measurements indicate a decrease in ultimate tensile strength and elastic modulus, in accordance with the decrease of the glass transition temperature previously reported [START_REF] Mbey | Cassava-starch kaolinite composite film. Effect of clay content and clay modification on film properties[END_REF]. These changes also corroborate the weak starch-kaolinite interactions that allows better gliding of the starch chains and a basal orientation of the clay particles which are consistent with the increase plasticizing effect due to the kaolinite. On figure 8, the experimental evaluation of some critical films properties, namely glass transition temperature and ultimate tensile strength evolution with clay content in the composite films, are given [START_REF] Mbey | Cassava-starch kaolinite composite film. Effect of clay content and clay modification on film properties[END_REF]2014) to reinforce the propose qualitative correlation of the interfacial interactions with the macroscopic properties.

: Conclusion

This study shows possible links between filler-polymer matrix interactions and starchkaolinite composite films properties. Weak starch-kaolinite interactions, that result from electrostatic repulsion combined to hydrogen bonds formation, evidenced from coagulation tests corroborate the weak interfacial interactions in the composites films. These weak interactions are favourable to preferential basal orientation of clay particles within the film and improve dispersion of the filler. All of which agree with the reported improve plasticization and water uptake, UV light and heat diffusion barrier [START_REF] Mbey | Cassava-starch kaolinite composite film. Effect of clay content and clay modification on film properties[END_REF]2014). Hence, the dependency of the material properties of the starch-kaolinite film to components interactions is better evidenced. As reported in the literature, this work also

  fraction < 40 µm, labelled K3, was collected by means of wet sieving. Using inductive coupled plasma by atomic emission spectrometry (ICP-AES) to analyze the major elemental composition of the sample, an approximation of the structural formula of the kaolinite phase

  be (Al 1.94 Fe 0.06 )(Si 1.98 Fe 0.02 )O 5 (OH) 4 (Mg 0.02 Ca 0.002 ). The mineralogical composition of the sample is as follows: Kaolinite 83.3 %; Illite 10.4 %; Titanium oxide 3.4

  by the suspension (6.02) or after adjusting the pH value by adding aliquot of 0.1 M solution of either NaOH or HNO 3 . The samples are left for sedimentation

  The measurements were done at 300 nm.

Figure 1 :

 1 Figure 1: Analysis of the pH effect on the sedimentation of the kaolinite suspension

Figure 2 :Figure 3 :

 23 Figure 1

Figure 4 :

 4 Figure 3

Figure 6 :

 6 Figure 4

  Figure 6

Figure 7 :

 7 Figure 7: (a) Illustration of clay-starch interface in the composite film (b) optical micrograph of 10 % loaded kaolinite cassava starch composite film analyzed for form enhancement using image J.

Acknowledgements:

Celine Caillet is greatly thanked for her help in titrations experiments.


José Antonio Carrillo
email: carrillo@imperial.ac.uk

Benoît Perthame
email: benoit.perthame@upmc.fr

Delphine Salort
email: del-phine.salort@upmc.fr

Didier Smets
email: smets@ann.jussieu.fr

Qualitative Properties of Solutions for the Noisy Integrate & Fire model in Computational Neuroscience

Keywords: Integrate and fire, Neural networks, Fokker-Planck equations, Global existence Mathematics Subject Classification 35K60, 35Q84, 82C32, 92B20

The Noisy Integrate-and-Fire equation is a standard non-linear Fokker-Planck Equation used to describe the activity of a homogeneous neural network characterized by its connectivity b (each neuron connected to all others through synaptic weights); b > 0 describes excitatory networks and b < 0 inhibitory networks. In the excitatory case, it was proved that, once the proportion of neurons that are close to their action potential V F is too high, solutions cannot exist for all times. In this paper, we show a priori uniform bounds in time on the firing rate to discard the scenario of blow-up, and, for small connectivity, we prove qualitative properties on the long time behavior of solutions. The methods are based on the one hand on relative entropy and Poincaré inequalities leading to L 2 estimates and on the other hand, on the notion of 'universal super-solution' and parabolic regularizing effects to obtain L ∞ bounds.

Introduction

Large networks composed by individual neuron activity models have been proposed in the computational neuroscience literature [START_REF] Bressloff | Waves in neural media[END_REF][START_REF] Tuckwell | Introduction to Theoretical Neurobiology[END_REF] to capture coherent structures such as synchronisation and stationary distributions with applications in decision making in behavioral neurosciences for instance [START_REF] Ratcliff | A comparison of sequential sampling sampling models for two-choice reaction time[END_REF][START_REF] Carrillo | Complexity reduction of rate-equations models for two-choice decision-making[END_REF]. Most of these models start from stochastic differential equations for each of the neurons coupled through their spiking times. These ideas go back to the early works of [START_REF] Lapicque | Recherches quantitatives sur l'excitation électrique des nerfs traitée comme une polarisation[END_REF] modeling neurons as electrical circuits for the evolution of the action potential. These spikes produced, when the action potential surpasses a certain threshold value V F , induce electrical discharges in the neuronal network inhibiting or exciting other neurons depending on their connectivity rate and strength. Several assumptions have been proposed in the random distribution of the spiking times of neuronal network, but the diffusion approximation has been used in many different applications [START_REF] Brunel | Fast global oscillations in networks of integrate-and-fire neurons with long firing rates[END_REF][START_REF] Brette | Adaptive exponential integrate-and-fire model as an effective description of neural activity[END_REF][START_REF] Brunel | Dynamics of sparsely connected networks of excitatory and inhibitory spiking networks[END_REF][START_REF] Mattia | Population dynamics of interacting spiking neurons[END_REF][START_REF] Renart | Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks[END_REF]. This approximation consists in assuming that all the neurons in the network follow an average stochastic differential equation of the form

τ dV = -(V -V L )∆t + µ C dt + σ C dB t (1.1)
where V L < V F is the leak potential and τ is the typical relaxation time for a neuron. The final parameters µ C and σ C take into account the interaction with the rest of the network and are of the form µ C = bN (t) and σ 2 C = 2a in the absence of external inputs, with a > 0 and b constants in its simplest form. The parameter b represents the connectivity parameter being positive for excitatory networks and negative for inhibitory networks. Here, N (t) takes into account the average number of spikes per unit of time within the network usually referred as the firing rate of the network. This basic model is called an integrate and fire model since the action potential is integrated in time following the SDE (1.1) upto the time in which the action potential surpasses V F . At that time, the neuron spikes and it is assumed that it relaxes instantaneously to the reset potential V R . Typically the values of the action potentials involved are such that V L < V R < V F . As usual in SDEs, we can at least formally write the evolution equation for the probability density of neurons having an action potential v at time t > 0 leading to a Fokker-Planck like equation that we called the noisy Integrate and Fire (NIF) equation:

∂p ∂t (v, t) + ∂ ∂v -v + bN (t) p(v, t) -a ∂ 2 p ∂v 2 (v, t) = N (t)δ(v -V R ), v ≤ V F . (1.2)
Here, we have assumed that without loss of generality V L = 0 < V R < V F . With this choice of units, physiological values for the action potentials are V R = 10mV and V F = 20mV with typical time units in miliseconds, where τ was absorbed in the time units. The initial and boundary conditions associated to such model are p(V F , t) = 0, p(-∞, t) = 0, p(v, 0) = p 0 (v) ≥ 0 .

(1.3) since neurons are not allowed at the threshold potential V F . Finally, the right-hand side of (1.2) incorporates the percentage of neurons that surpasses the threshold value at time t and thus are reset to potential V R . In fact, in order to have an evolution of a probability density, we need to verify that solutions to (1.2) satisfy

V F -∞ p(v, t) dv = V F -∞ p 0 (v) dv = 1 . (1.4)
It is straightforward to check that relation (1.4) together with the boundary conditions (1.3) defines the firing rate of the network as the flux of probability at V F . This gives

N (t) := -a ∂p ∂v (V F , t) ≥ 0 . (1.5)
More complicated models incorporating the evolution for the voltage conductance have also been proposed , see for instance [START_REF] Cai | Kinetic theory for neuronal network dynamics[END_REF][START_REF] Cáceres | A numerical solver for a nonlinear fokker-planck equation representation of neuronal network dynamics[END_REF][START_REF] Perthame | On a voltage-conductance kinetic system for integrate & fire neural networks[END_REF].

In this work, we plan to advance the analysis of properties of the solutions to the problem (1.2)-(1.3)- (1.5). The main difficulty of analyzing the behavior of solutions to (1.2)-(1.3)-(1.5) lies in the nonlinearity due to the poinwise value of the derivative appearing in the drift of the equation in voltage variable. These difficulties in analyzing the behaviour of solutions of (1.2) are already present in the stationary states given by

       ∂ ∂v [(-v + bN ∞ )p ∞ (v)] -a ∂ 2 p ∞ ∂v 2 (v) = N ∞ δ(v -V R ), v ≤ V F , p ∞ (V F ) = 0, N ∞ = -a ∂p ∞ ∂v (V F ), V F -∞ p ∞ (v)dv = 1.
(1.6)

The existence of stationary states for this problem was analysed in [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF]Theorem 3.1] where it is proved that for inhibitory networks, b < 0, and b ≥ 0 small enough, there is a unique stationary state while the non-uniqueness of stationary states and the nonexistence can happen for larger values of the connectivity parameter b > 0. Moreover, the authors in [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF] were able to show that no matter how small b > 0 is, there are initial data very concentrated on the firing voltage V F such that the firing rate diverges in finite time. This blow-up phenomenon makes the dynamics for excitatory networks interesting since it is not clear how to understand the coexistence of steady states and the blow-up.

Well-posedness of classical solutions have been obtained locally in time for b > 0 and global in time for b ≤ 0 in [START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF]. The extension criteria for classical solutions is the pointwise control of the firing rate N (t). The solutions exist as long as N (t) remains bounded. The existence of solution and its derivation from stochastic processes have also been studied in [START_REF] Delarue | Global solvability of a networked integrate-and-fire model of McKean-Vlasov type[END_REF]. However, no information about the long-time asymptotics of the firing rate is obtained and the local stability near stationary states was not discussed neither in the excitatory or inhibitory case. Let us mention that, even if global in time solutions was proved for the inhibitory case, the inhibitory dynamic is also complex to deal, in particular for strong inhibitory connections, because, the bigger |b| is, the stronger the nonlinearity is. In this work, we give some answers of these above problems.

The rest of the paper is organized as follows. In section 2, we prove local asymptotic stability for stationary states via entropy inequality, for small connectivity (|b| small). In the third section, we first prove, for the inhibitory case, uniform L 2 bounds for the firing rate in time which are independent of b ≤ 0 (and so of the strength of the nonlinearity) and the initial data. In the excitatory case, given an initial data, we show uniform bounds on N and the solution in L 2 as soon b ≥ 0 is small enough. These uniform in time bounds lead also to the local asymptotic stability result in Section 2 with weaker conditions on the initial data. The techniques here make extensive use of the general entropy method (GRE) as in [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF][START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF]. Section 4 is devoted, given b ≤ 0 and a bounded initial data, to find uniform L ∞ bounds both for the density and for the firing rate of classical solutions of the problem (1.2)-(1.3)-(1.5). We make use of maximum principle tools adapted to the case of parabolic equations with drift together with a change of variables already used in [START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF] to reduce the problem locally to standard parabolic regularity theory.

Local Asymptotic Stability of stationary states

As mentioned earlier, solutions of the Equation (1.2) can blow-up as proved in [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF]. In our first result, we prove that if the strength of interconnections |b| is small enough, then by choosing an initial data close enough to the unique stationary state defined by the Equation (1.6), exponential convergence of the solution to the stationary state holds. Let us recall that the existence and uniqueness of solution for the stationary problem for b ≤ 0 or small enough b > 0 was shown in [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF]Theorem 3.1]. These apriori bounds can also be used to get global existence results of weak solutions, direction that we will not pursue here.

Theorem 2.1. Assume that |b| is small enough, then there is a constant µ > 0 such that if the initial data satisfies

V F -∞ p ∞ p 0 -p ∞ p ∞ 2 (v)dv ≤ 1 2|b| , then, for all t ≥ 0 V F -∞ p ∞ p -p ∞ p ∞ 2 (v, t)dv ≤ e -µt V F -∞ p ∞ p 0 -p ∞ p ∞ 2 (v)dv.
The method is a direct application of relative entropy methods and Poincaré inequalities combined with a control of boundary terms discovered in the same context in [START_REF] Cáceres | Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activity[END_REF]. We first recall some facts. The relative entropy differential inequality, generalized from [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF] for the case b = 0, relates the quantities

h(v, t) := p(v, t) p ∞ (v) , and
ν(t) := N (t) N ∞ . (2.1)
The computation for classical solutions gives, for any smooth convex function G : R + -→ R,

d dt V F -∞ p ∞ (v)G h(v, t) dv = -a V F -∞ p ∞ (v) G ′′ h(v, t) ∂h ∂v 2 (v, t) dv -N ∞ G ν(t) -G h(V R , t) -ν(t) -h(V R , t) G ′ h(V R , t) (2.2) + b(N -N ∞ ) V F -∞ ∂p ∞ ∂v (v) G h(v, t) -h(v, t)G ′ h(v, t) dv.
Another tool that we wil use is the following Poincaré inequality which can be found in [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF]Appendix]; there is a constant γ > 0 depending on b such that for any function h(v) satisfying

V F -∞ p ∞ (v)h(v)dv = 1, we have γ V F -∞ p ∞ (v) h(v) -1 2 dv ≤ V F -∞ p ∞ (v) ∂h ∂v 2 (v) dv. (2.3)
Notice that even if the Poincaré inequality stated in [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF]Proposition 4.3] was only for b = 0, its proof in [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF]Appendix] is valid for b small enough since it only uses the behavior or

p ∞ at v = V F and v = -∞.
In fact, p ∞ is implicitly given by the formula

p ∞ (v) = N ∞ a e -(v-bN∞) 2 2a V F max(v,V R ) e (w-bN∞) 2 2a dw , (2.4)
and thus, the behavior of

p ∞ (v) ≃ V F -v at v = V F and p ∞ (v) ≃ e -v 2 2a at v = -∞
does not depend on b. Therefore, the proof in [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF]Appendix] applies equally well for b small enough, leading to (2.3). Also we will use both notations ∂ v and ∂ ∂v to denote the partial derivative in v. After this preliminary material, we can give the proof of the asymptotic behavior.

Proof of Theorem 2.1. We choose G(h) = (h -1) 2 and manipulate the three terms in the right hand side of (2.2). The first term is negative and gives the main control when combined with the Poincaré inequality.

We begin our arguments with the second term which is also negative by the convexity property. But we can also use it to provide us with a control of ν(t). We claim that for some constant c 0 > 0 we have

-N ∞ [G (ν(t)) -G (h(V R , t)) -ν(t) -h(V R , t) G ′ (h(V R , t)) ≤ -c 0 G (ν(t)) + a 2 V F -∞ p ∞ (v) ∂h ∂v 2 (v, t) dv. (2.5)
Indeed, for the function G(•) at hand, we have, for all 0 < ε < 1/2,

G (ν(t)) -G (h(V R , t)) -ν(t) -h(V R , t) G ′ (h(V R , t)) = ν(t) -h(V R , t) 2 ≥ ε(ν(t) -1) 2 -2ε(h(V R , t) -1) 2 .
Now, using the Sobolev injection of L ∞ (I) in H 1 (I) for a sufficiently small neighborhood I of V R where p ∞ is bounded below due to its expression in (2.4), and the Poincaré inequality (2.3), we obtain that there exists a constant C such that

h(V R , t) -1 2 ≤ C V F -∞ p ∞ (v) ∂h ∂v 2 (v, t) dv which proves Estimate (2.5), choosing ε < 1/2 such that 2CN ∞ ε ≤ a/2.
We now control the last term in the right hand side of (2.2) which is of cubic nature because it comes from the nonlinerity of the NIF equation. We first observe that G(h) -hG ′ (h) = 1 -h 2 , and thus

V F -∞ ∂ v p ∞ (v) G h(v, t) -h(v, t)G ′ h(v, t) dv = 2 V F -∞ p ∞ [∂ v h(v, t) (h(v, t) -1) + ∂ v h(v, t)] dv ,
where we take into account the Dirichlet boundary condition on p in (1.3). The interest of forcing the term h(v, t) -1 in the above equality can be seen when we handle the full cubic term. We can estimate it with two quantities under control as follows. Using the Cauchy-Schwarz inequality, we obtain that

|b||N (t) -N ∞ | V F -∞ p ∞ ∂ v h(v, t) (h(v, t) -1) dv ≤ |b| a N 2 ∞ (ν(t) -1) 2 + a|b| V F -∞ p ∞ (∂ v h(v, t)) 2 dv V F -∞ p ∞ (h(v, t) -1) 2 dv.
For the other term, we write that

|b||N -N ∞ | V F -∞ p ∞ |∂ v h(v, t)| dv ≤ 2b 2 a N 2 ∞ (ν(t) -1) 2 + a 2 V F -∞ p ∞ (v) (∂ v h(v, t)) 2 dv.
We can now conclude the proof of the Theorem 2.1. We include the previous estimates in the entropy equality (2.2) and obtain

d dt V F -∞ p ∞ (v) h(v, t) -1 2 dv ≤ -c 0 (ν(t) -1) 2 + C(b 2 + |b|)(ν(t) -1) 2 -a V F -∞ p ∞ (v) ∂ ∂v h(v, t) 2 dv 1 -|b| V F -∞ p ∞ (v) (h(v, t) -1) 2 dv . (2.6)
We now take b such that C(b 2 + |b|) ≤ c 0 and the initial data such that 3 Improved uniform estimates in L 2 and of long term convergence.

|b| V F -∞ p ∞ (v) (h(v, t) -1) 2 dv < 1/2, then V F -∞ p ∞ (v) h(v, t
3.1 Uniform estimates of N on L 2 .

The aim of this part is to prove L 2 bounds on the global activity N (t) for general initial data. In the inhibitory case, we show that for large enough time intervals, these uniforms in time bounds can be chosen independently of the choice of the initial data and of the interconnection strength b ≤ 0, see Theorem 3.1 below. This is one more indication toward the understanding of the long time asymptotics in the inhibitory case. We will see in next section that uniform in time pointwise bounds can also be obtained under more stringent conditions on the initial data. We remind the reader that in the inhibitory case, there is only a unique stationary solution for each value of b < 0, see [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF]Theorem 3.1]. However, for b < 0 with large |b|, we are unable yet to show that this unique stationary state attracts the global dynamics of the problem (see Theorem 3.5 for the proof of convergence of stationary state for small |b|, b ≤ 0 with very low condition on the initial data).

In the case of excitatory networks, for any b > 0, it was proved in [5, Theorem 2.2], that if the initial data is enough concentrated near from V F , blow-up occurs. Then the results obtained in Theorem 3.1 in the inhibitory case can not be extended to the excitatory case. However, we prove uniform in time bounds of N in L 2 under the constraint that the initial data, closed to V F , is small enough with respect to b > 0. Other approaches to control the firing rate in the excitatory case can be seen in [START_REF] Delarue | Global solvability of a networked integrate-and-fire model of McKean-Vlasov type[END_REF]. Let us remind the reader that for b > 0 small enough, it was shown in [5, Theorem 3.1] the existence of stationary states, for b within a range determined by V F , V R , and a.

Our main results are based on the idea of estimating the localized relative entropy of the solution p(v, t) toward a given stationary state p 1 ∞ for an excitatory case b 1 > 0. They can be summarized as follows.

Theorem 3.1. Let b 1 > 0 such that there exists a stationary state of Equation (1.2) and let p 1 ∞ be the corresponding stationary state. Let

V M with V F > V M > V R and define S(b 1 , V M ) := V F V M p 0 2 p 1 ∞ dv
with an initial data chosen such that S(b 1 , V M ) < +∞. Then: i) There exists a constant C independent of S(b 1 , V M ) and there exists a time T > 0 depending only on V M and S(b 1 , V M ), such that for all intervals I ⊂ (T, +∞) and for all b ≤ 0,

I N (t) 2 dt ≤ C(1 + |I|) (3.1)
holds.

ii) Assume b > 0 is small enough depending on S(b 1 , V M ) and V M , then there exists a constant C such that for all intervals

I ⊂ R + , I N (t) 2 dt ≤ C(1 + |I|) (3.2)
holds.

Remark 3.2. The proof of Theorem 3.1 only involves the dynamics of the probability density solution of Equation

(1.2) closed to V F , that V F -∞ p(t, v)dv L ∞ (R + ) < +∞
and the existence of stationary states of Equation (1.2) for some b 1 > 0 strictly decreasing in the neighborhood of V F . Hence, the proof makes use of only a small part of the structure of Equation (1.2), and therefore may be applied in more general settings.

Remark 3.3. The proof is based on an entropy inequality in the spirit of (2.2) with however a main difference: the solution of Equation (1.2) is not compared to a stationary state associated to Equation (1.2) with the same b, but with a stationary state p 1 ∞ of Equation (1.2) associated to a given b 1 > 0 fixed independently of b.

As mentioned earlier these results are based on the localization of a relative entropy toward a fixed steady state for a given b 1 > 0. In order to estimate these localized estimates for solutions to (1.2), we will compute the evolution of weighted quantities localized around V F of the form

V F -∞ W (v, t)γ(v)dv with W (v, t) = (p(v, t)) 2 p 1 ∞ (v) = p 1 ∞ (v)(H(v, t)) 2 , (3.3)
where the notation H := p/p 1 ∞ is used and with a localization weight γ(v) to be chosen. Actually, in both cases we need to localize near V F , and thus the weight γ ∈ C ∞ (-∞, V F ) is such that the support of γ ⊂ (α, V F ] with α > V R > 0. Remember we can assume V R positive without loss of generality. Let us define the function γ explicitly. For α ∈ (-∞, V F ), let us fix the notation β = (V F -α) 2 . Then, we define our cutoff function γ as follows

γ(v) = e -1 β-(V F -v) 2 if v ≥ α, γ(v) = 0 else. (3.4)
The following technical Lemma on γ holds.

Lemma 3.4. Let V F > 0 and let α ∈ (-∞, V F ). Then γ given by (3.4) is a positive increasing function on (α, V F ) and the following properties holds: i)

lim v→V F γ ′ γ (v) = 0.
ii) There exists a constant C such that on (α, V F ), we have

γ ′2 + γ ′′2 + γ ′′′2 ≤ Cγ. (3.

5)

iii) There exists a constant α < δ < V F such that

γ ′′ (δ) = 0 and γ ′′ (v) ≤ 0 on (δ, V F ). (3.6)
Proof of Lemma 3.4. A direct computation leads to

γ ′ (v) = 2(V F -v)) β -(V F -v) 2 γ(v) ,
and so γ is increasing and

lim v→V F γ ′ γ (v) = 0.
Moreover, we have

γ ′ (v) 2 = γ 2 (v) 2(V F -v)) β -(V F -v) 2 2 with lim v→α γ 2(V F -v)) β -(V F -v) 2 2 = 0.
We deduce that there exists a constant C such that γ ′2 ≤ Cγ. Computing γ ′′ (v), we obtain

γ ′′ (v) = 4(V F -v) 2 (β -(V F -v) 2 ) 4 - 2 (β -(V F -v) 2 ) 2 - 8(V F -v) 2 (β -(V F -v) 2 ) 3 γ.
We deduce the last point of Lemma 3.4 since the signs of the second derivative are different at α and V F , and that there exists a constant C such that γ ′′2 ≤ Cγ. A similar tedious but trivial argument computing γ ′′′ (v) leads to find that there exists a constant C > 0 such that (γ ′′′ ) 2 ≤ Cγ which ends the proof of Lemma 3.4.

3.1.1 Proof of Theorem 3.1 in the inhibitory case.

The proof of Theorem 3.1 in the inhibitory case can be reduced to prove that for all α < V F close enough to V F , there exist three positive constants C 1 , C 2 , and

C 3 independent of b ≤ 0 such that d dt V F -∞ W (v, t)γ(v)dv ≤ -C 1 N (t) 2 N 1 ∞ + C 2 + (bN (t) -C 3 ) V F -∞ W (v, t)γ(v)dv. (3.7)
Indeed, let us assume that estimate (3.7) holds and let us prove Theorem 3.1. Let us define

I(t) := V F -∞ W (v, t)γ(v)dv.
By assumption we have I(0) < +∞. Since b ≤ 0, (3.7) implies that I ′ (t) ≤ C 2 -C 3 I(t), and thus, assuming t large enough depending on the initial value I(0), we deduce

V F -∞ W (v, t)γ(v)dv ≤ 2 C 2 C 3 ,
which implies Theorem 3.1 by integrating (3.7) on any interval I.

Let us now prove estimate (3.7). By using (1.2) and (1.6) for b 1 > 0, it is straightforward to check that the function W satisfies

∂ t W + ∂ ∂v (-v + bN (t))W -a ∂ 2 ∂ 2 v W + 2ap 1 ∞ ∂H ∂v 2 + bN (t) -b 1 N 1 ∞ H 2 ∂p 1 ∞ ∂v = N 1 ∞ δ(v -V R ) 2N N 1 ∞ -H H , (3.8)
where we have used the notation (3.3). Multiplying equation (3.8) by the weight γ ∈ C ∞ (-∞, V F ) and taking into account that the support of γ ⊂ (α, V F ] with α > V R > 0, we deduce after integration in v that

d dt V F -∞ W (v, t)γ(v)dv = V F -∞ (-v + bN (t))W (v, t)γ ′ (v)dv + a ∂W ∂v (V F , t)γ(V F ) -aW (V F , t)γ ′ (V F ) -2a V F -∞ p 1 ∞ (v) ∂H ∂v (v, t) 2 γ(v)dv + a V F -∞ W (v, t)γ"(v)dv -bN (t) -b 1 N 1 ∞ V F -∞ ∂p 1 ∞ ∂v (v)H(v, t) 2 γ(v)dv . (3.9)
At this stage, we use the structure of p 1 ∞ closed to V F due to its almost explicit expression given in [5, Section 3]. We observe that if α is close enough to V F , then

∂ v p 1 ∞ ≤ -p 1 ∞ on the support of γ. Indeed, p 1 ∞ is C ∞ closed to V F with p 1 ∞ (V F ) = 0 and ∂ v p 1 ∞ (V F ) < 0.
This implies that, as soon b ≤ 0 and α close enough to V F , the non linear term can be controlled by

-bN (t) -b 1 N 1 ∞ V F -∞ ∂p 1 ∞ ∂v (v)H 2 (v, t)γ(v)dv ≤ bN (t) -b 1 N 1 ∞ V F -∞ W (v, t)γ(v)dv. (3.10)
Moreover, since γ is increasing, b ≤ 0, and α > V R > 0, we get

V F -∞ ((-v + bN (t))W ) γ ′ (v)dv ≤ 0.
Finally, the boundary conditions in (1.3)-(1.5) and (1.6) together with L'Hopital rule, implies that

-a ∂W ∂v (V F , t) = N (t) 2 N 1 ∞ and W (V F , t) = 0.
Collecting all the last three estimates, using that γ is nonnegative and increasing and that γ ′′ (v) ≤ 0 for v ∈ (δ, V F ) as proven in Lemma 3.4, we deduce from (3.9) that

d dt V F -∞ W (v, t)γ(v)dv ≤ - N (t) 2 N 1 ∞ γ(V F ) -2a V F -∞ p 1 ∞ (v) ∂H ∂v (v, t) 2 γ(v)dv + a δ -∞ W (v, t)γ"(v)dv + bN (t) -b 1 N 1 ∞ V F -∞ W (v, t)γ(v)dv. (3.11)
Hence, to prove (3.7), and so Theorem 3.1 part i), it remains to prove the following claim: there exists a positive constant C such that for all ε > 0 small enough,

δ -∞ W (v, t)γ"(v)dv ≤ εC V F -∞ ∂H ∂v (v, t) 2 p 1 ∞ (v)γ(v)dv + V F -∞ W (v, t)γ(v)dv + C ε (3.12)
holds. Indeed, assuming (3.12), taking α close enough to V F such that estimate (3.10) holds, and taking ε small enough, we find estimate (3.7) from (3.11), which proves Theorem 3.1 part i).

We are now reduce to show (3.12). The main difficulty of the proof of this claim is that

lim v→α γ" γ = +∞
and so, the choice of the structure of γ in Lemma 3.4 here is crucial. To control the right-hand side of (3.12), we write

δ -∞ W (v, t)γ ′′ (v)dv = δ -∞ H(v, t) ∂ ∂v v -∞ p(t, w)dw γ ′′ (v)dv.
After integration by parts, recalling that γ ′′ (δ) = 0 due to Lemma 3.4, we find

δ -∞ W (v, t)γ ′′ (v)dv = - δ -∞ ∂H ∂v (v, t) v -∞ p(t, w)dw γ ′′ (v)dv - δ -∞ H(v, t) v -∞ p(t, w)dw γ ′′′ (v)dv ,
and so, using that

V F -∞ p(v, t)dt = 1, ∀t ≥ 0 , we obtain δ -∞ W (v, t)γ ′′ (v)dv ≤ δ -∞ ∂H ∂v (v, t) |γ ′′ (v)|dv + δ -∞ H(v, t)|γ ′′′ (v)|dv .
Using that on (α, δ), there exists a constant C > 0 such that p 1 ∞ (v) ≥ C > 0, estimate (3.5) of Lemma 3.4, and the inequality cd

≤ εc 2 + 1 ε d 2 , we finally conclude δ -∞ W (v, t)γ"(v)dv ≤ εC δ -∞ ∂H ∂v (v, t) 2 p 1 ∞ (v)γ(v)dv + δ -∞ W (v, t)γ(v)dv + C ε
leading to (3.12) by the positivity of the integrands.

3.1.2

Proof of Theorem 3.1 in the excitatory case.

The proof of Theorem 3.1 in the excitatory case can be reduced to prove that there exist four positive constants C 1 , C 2 , C 3 , and C 4 depending only on γ defined in (3.4), and p 1 ∞ such that for all t ≥ 0,

dI dt ≤ N (t) 2 C 1 b 2 - γ(V F ) N 1 ∞ + b 2 C 2 I(t) + C 3 -C 4 I(t) . (3.13)
Here, as in the previous subsection, we use the notation

I(t) := V F -∞ W (v, t)γ(v)dv.
Indeed, assuming estimate (3.13) and giving I 0 < +∞, define

M := max I 0 , C 3 C 4 .
Let us chose b > 0 small enough such that

C 1 b 2 - γ(V F ) N 1 ∞ + b 2 C 2 M < 0.
Then, in this case, for all t ≥ 0, we have

I(t) = V F -∞ W (v, t)γ(v)dv ≤ M ,
which implies Theorem 3.1 in the excitatory case.

Let us now prove estimate (3.13). Coming back to Equation (3.9), since b > 0 we can proceed as in the previous case to have sign control on the terms

V F -∞ -vW (v, t)γ ′ (v)dv ≤ 0 (3.14) and b 1 N 1 ∞ V F -∞ γ(v) ∂p 1 ∞ ∂v (v)H(v, t) 2 dv ≤ -b 1 N 1 ∞ V F -∞ γ(v)W (v, t)dv , (3.15)
for α close enough to V F . The term with γ ′′ (v) in (3.9) is treated analogously to the previous subsection. Hence, when b > 0, we have to deal differently with the terms bN (t)

V F -∞ W (v, t)γ ′ (v)dv ≥ 0 and -bN (t) V F -∞ γ(v) ∂p 1 ∞ ∂v (v)H(v, t) 2 dv ≥ 0.
We now make the following claim: let γ defined as in (3.4). Choosing α close enough to V F , there exists a constant C such that bN (t)

V F -∞ W (v, t)γ ′ (v)dv ≤ ε C V F -∞ γ(v)p 1 ∞ (v) ∂H ∂v (v, t) 2 dv + V F -∞ W (v, t)γ(v)dv + C(bN (t)) 2 ε V F -∞ W (v, t)γ(v)dv + C(1 + (bN (t)) 2 ) ε (3.16)
and

-bN (t) V F -∞ γ(v) ∂p 1 ∞ ∂v (v)H(v, t) 2 dv ≤ ε C V F -∞ γ(v)p 1 ∞ (v) ∂H ∂v (v, t) 2 dv + V F -∞ W (v, t)γ(v)dv + C(1 + (bN (t)) 2 ) ε + C(bN (t)) 2 ε V F -∞ W (v, t)γ(v)dv , (3.17)
for all ε > 0. The proof of (3.13), and thus the proof of Theorem 3.1 in the excitatory case, is obtained from (3.9) taking into account (3.12), (3.14), (3.15), (3.16), and (3.17).

We are then reduced to show the claims (3.16) and (3.17). Let us first prove estimate (3.16) which follows the idea of the proof of (3.12). The difference is that in our present case γ ′ ≥ 0 on (α, V F ), so that we have to split γ ′ in two pieces, close and far from V F . More precisely, using estimate (3.6) of Lemma 3.4, we obtain that there exists α < ω < V F such that γ ′ ≤ γ on (ω, V F ). We introduce two nonnegative functions γ 1 , γ 2 ∈ C ∞ (-∞, V F ) with γ 1 + γ 2 = 1 and such that, for ε 1 > 0 small enough:

• γ 1 is decreasing on (-∞, V F ) with γ 1 = 0 on (ω + ε 1 , V F ).
• γ 2 is an increasing function with γ 2 = 0 on (-∞, ω).

With this partition of unity, we obtain that

V F -∞ W (v, t)γ ′ dv = V F -∞ W (v, t)γ ′ (v)γ 1 dv + V F -∞ W (v, t)γ ′ (v)γ 2 dv.
As γ 2 = 0 on (-∞, ω), we have

V F -∞ W (v, t)γ ′ (v)γ 2 (v)dv ≤ V F -∞ W (v, t)γ(v)dv
using that γ ′ ≤ γ on (ω, V F ), and so for all ε > 0, bN (t)

V F -∞ W (v, t)γ ′ (v)γ 2 (v)dv ≤ ε + b 2 N 2 (t) ε V F -∞ W (v, t)γ(v)dv.
Let us now control the term

V F -∞ W (v, t)γ ′ (v)γ 1 (v)dv.
Following the proof of (3.12), we find that

V F -∞ W (v, t)γ ′ (v)γ 1 (v)dv = V F -∞ H(v, t) ∂ ∂v v -∞ p(w, t)dw γ ′ (v)γ 1 (v)dv.
After integration by parts, we obtain that

V F -∞ W (v, t)γ ′ (v)γ 1 (v)dv ≤ V F -∞ ∂H ∂v (v, t) γ ′ (v)γ 1 (v)dv + V F -∞ H(v, t)|(γ ′ γ 1 ) ′ (v)|dv . To control bN (t) V F -∞ ∂H ∂v (v, t) γ ′ (v)γ 1 (v)dv, we apply inequality cd ≤ εc 2 + 1 ε d 2 with c = ∂ v Hγ ′ γ 1 and d = bN (t) to obtain that for all ε > 0, bN (t) V F -∞ ∂H ∂v (v, t) γ ′ (v)γ 1 (v)dv ≤ ε V F -∞ ∂H ∂v (v, t) 2 (γ ′ (v)γ 1 (v)) 2 dv + 1 ε (bN (t)) 2 (V F -α).
Using Lemma 3.4, we obtain that there exists a constant C such that (γ ′ γ 1 ) 2 ≤ Cγ. Moreover, as

p 1 ∞ (v) is a decreasing function on (α, V F ) if α close enough to V F , we have p 1 ∞ (v) ≥ p ∞ (ω + ε 1 ) > 0 on the support of γ 1 . We deduce that for all ε > 0, bN (t) V F -∞ ∂H ∂v (v, t) γ ′ (v)γ 1 (v)dv ≤ Cε p 1 ∞ (ω + ε 1 ) V F -∞ p 1 ∞ (v) ∂H ∂v (v, t) 2 γ(v)dv + (bN (t)) 2 (V F -α) ε .
We proceed similarly with the term bN (t)

V F -∞ H(v, t)|(γ ′ γ 1 ) ′ (v)|dv, to show that bN (t) V F -∞ H(v, t)|(γ ′ γ 1 ) ′ (v)|dv ≤ Cε p 1 ∞ (ω + ε 1 ) V F -∞ W (v, t)γ(v)dv + (bN (t)) 2 (V F -α) ε .
which finally proves estimate (3.16).

Let us now prove estimate (3.17). Integration by parts implies that

V F -∞ γ(v) ∂p 1 ∞ ∂v (v)H(v, t) 2 dv = - V F -∞ γ ′ (v)W (v, t)dv -2 V F -∞ ∂H ∂v (v, t)H(v, t)γ(v)p 1 ∞ (v)dv.
The term bN (t)

V F -∞ γ ′ (v)W (v, t)dv
is estimated using (3.16). To control the term

J(t) := bN (t) V F -∞ ∂H ∂v (v, t)H(v, t)γ(v)p 1 ∞ (v)dv,
we use the inequality cd ≤ εc 2 + 1 ε d 2 with c = |∂ v H| and d = 2bN (t)H. We deduce that there exists a constant C such that for all ε > 0

J(t) ≤ Cε V F -∞ ∂H ∂v (v, t) 2 p 1 ∞ (v)γ(v)dv + (CbN (t)) 2 ε V F -∞ γ(v)W (v, t)dv ,
which ends the proof of estimate (3.17). As a consequence, we conclude the proof of the claims, the estimate (3.13), and Theorem 3.1 part ii).

Application of Theorem 3.1 to convergence toward stationary states.

In this part, we prove, using Theorem 3.1, that the conclusion of Theorem 2.1 still holds when we relax the conditions on the initial data.

Theorem 3.5. Given p ∞ a stationary state solution to (1.6) with associated coupling constant b. Let V M < V F and assume that

S b,V M := V F V M p ∞ p 0 -p ∞ p ∞ 2 (v)dv < +∞.
i) There exist a positive constant C, depending only on S b,V M and V M , and µ > 0 such that

V F -∞ p ∞ p -p ∞ p ∞ 2 (v, t)dv ≤ e -νt V F -∞ p ∞ p 0 -p ∞ p ∞ 2 (v)dv , for all 0 < b ≤ C and t ≥ 1.
ii) There exist a positive constant C, independent of S b,V M , and T > 0 such that

V F -∞ p ∞ p -p ∞ p ∞ 2 (v, t)dv ≤ e -ν(t-T ) V F -∞ p ∞ p -p ∞ p ∞ 2 (v, T )dv ,
for all t ≥ T + 1 and for all 0 < -b ≤ C.

Proof of Theorem 3.5. The proof of this Theorem is close to the proof of Theorem 2.1. Let us follow the notations (2.1) of Theorem 2.1 and recall that G(h) = (h -1) 2 . Our starting point is the identity (2.2). By integrating by parts in the last term of (2.2), we obtain

-b(N (t) -N ∞ ) V F -∞ ∂p ∞ ∂v (v)h 2 (v, t)dv = 2b(N (t) -N ∞ ) V F -∞ p ∞ (v)h(v, t) ∂h ∂v (v, t)dv .
Using the inequality cd

≤ εc 2 + 1 ε d 2 with c = √ p ∞ ∂ v h, d = 2b(N (t) -N ∞ ) √ p ∞ h, and ε = a in Equation (2.
2), we find that

d dt V F -∞ p ∞ (v)G (h(v, t)) dv ≤ -N ∞ G (ν(t)) -G (h(V R , t)) -(ν(t) -h(V R , t)) G ′ (h(V R , t)) -a V F -∞ p ∞ (v) ∂h ∂v (v, t) 2 dv + 4|b(N (t) -N ∞ )| 2 a V F -∞ p ∞ (v) (h(v, t) -1) 2 dv + 4|b(N -N ∞ )(t)| 2 a .
Applying Poincaré inequality (2.3), we deduce

d dt V F -∞ p ∞ (v)G (h(v, t)) dv ≤ -N ∞ G (ν(t)) -G (h(V R , t)) -(ν(t) -h(V R , t)) G ′ (h(V R , t)) + -νa + 4|b(N -N ∞ )| 2 a V F -∞ p ∞ (v)G (h(v, t)) dv + 4|b(N -N ∞ )| 2 a .
Using the proof of (2.5), it is easy to check that there exists C > 0 such that for all 0 < ε < 1 2 , we have

G (ν(t)) -G (h(V R , t)) -ν(t) -h(V R , t) G ′ h(V R , t) ≥ ε(ν(t) -1) 2 -2εC V F -∞ p ∞ (v) ∂h ∂v (v, t) 2 dv .
Poincaré inequality (2.3) again implies that

d dt V F -∞ p ∞ (v)G (h(v, t)) dv ≤ -νa + εC + 4|b(N -N ∞ )| 2 a V F -∞ p ∞ (v)G (h(v, t)) dv + (N -N ∞ ) 2 N ∞ 4b 2 N ∞ a -ε .
Then, taking ε small enough and |b| small enough such that

4b 2 N ∞ a -ε ≤ 0 we finally conclude that d dt V F -∞ p ∞ (v)G (h(v, t)) dv ≤ -νa + εC + 4|b(N -N ∞ )| 2 a V F -∞ p ∞ (v)G (h(v, t)) dv .
Applying estimate (3.1) of Theorem 3.1 when b > 0 and estimate (3.2) of Theorem 3.1 when b < 0, we obtain that there exists a constant C > 0 as in Theorem 3.5 and there exists T ≥ 0 and µ > 0 such that for all t ∈ (T + 1, +∞),

t T -νa + εC + 2|b(N -N ∞ )| 2 a (s)ds ≤ -µ(t -T ) .
We remind that T could be arbitrary in the b > 0 case. This concludes the proof of Theorem 3.5.

Uniform L ∞ bounds for the inhibitory case

So far, we have obtained controls for the solutions in L 2 spaces. Another method, based on the comparison principle adapted to the nonlinear drift, gives time-uniform controls in L ∞ . We present this method in several subsections leading to the following main result:

Theorem 4.1. For b < 0 and p 0 ∈ L 1 + ∩ L ∞ (-∞, V F ) ∩ C 1 ((-∞, V F ]) with p 0 (V F ) = 0, given a classical solution p to (1.2)-(1.5), it satisfies sup t≥0 p(t) ∞ < ∞ and sup t≥0 N (t) < ∞.
By classical solutions of (1.2), we mean a function p(v, t) which is for all t > 0 C 1 in time and

C 2 in the voltage variable v ∈ (-∞, V R ) ∪ (V R , V F ],
it satisfies the Dirichlet boundary conditions in v at -∞ and at V F , and such that the first derivatives from left and right exist at v = V R and from the left at v = V F defining the firing rate in (1.5). Moreover, p(v, t) satisfies (1.2) in the classical sense for v ∈ (-∞, V R ) ∪ (V R , V F ) and t > 0, and p(v, t) satisfies (1.2)-(1.3) in the distributional sense.

Notice that the last condition is equivalent to say that p(v, t) satisfies (1.2) in the classical sense for v ∈ (-∞, V R ) ∪ (V R , V F ) and t > 0, and that ∂p ∂v (v, t) has a jump discontinuity at v = V R of value -N (t)/a. We point out that classical solutions were shown to exist under the conditions [START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF]. They used a suitable change of variables translating (1.2)-(1.5) to a free boundary problem, see subsection 4.2, together with an integral equation for the firing rate solved by fixed point arguments.

p 0 ∈ L 1 + ∩ C 1 (-∞, V F ) in
Moreover, the authors show that this solution can be extended in a unique way as soon as the firing rate N (t) is bounded. They also prove that solutions exist globally in time for b < 0 by a contradiction argument on this extension criteria using the firing rate. Here, we will show the global in time existence of a unique classical solution by showing apriori L ∞ bounds on the solution and its firing rate N (t).

Let us finally comment that, as it will cleared later on, the regularity assumptions on the initial data can be improved to

p 0 ∈ L 1 + ∩ L ∞ (-∞, V F ) such that lim sup v→V F p 0 (v) V F -v < ∞ .
In the rest of this section, we always assume that all the functions of the voltage variable under consideration are non-negative, vanish at v = V F and at v = -∞, and are smooth separately on (-∞, V R ] and [V R , V F ]. By smooth, we precisely mean that solutions are at least

C 2 ((-∞, V R )) ∩ C 1 ((-∞, V R ]) and C 2 ((V R , V F )) ∩ C 1 ([V R , V F ]
). The notation C 1 (J) with J a closed interval means that the one-sided derivatives exist and are continuous up to the boundary of the interval. If functions are time dependent, then they are assumed to be at least C 1 for t > 0. In other words, that the functions are in the same functional framework as the classical solutions of the problem (1.2)-(1.5) constructed in [START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF].

Comparison principle and super-solutions

4.2. Let b < 0, V 0 ∈ [-∞, V F ) and T > 0. We say that p is a universal (classical) super-solution to (1.2) on [V 0 , V F ] × [0, T ] if ∂ p ∂t (v, t) - ∂ ∂v v p(v, t) -a ∂ 2 p ∂v 2 (v, t) ≥ N (t)δ(v -V R ) (4.1)
on (V 0 , V F ) × (0, T ) in the distributional sense and on ((V 0 , V F )\{V R }) × (0, T ) in the classical sense, where N (t) := -a ∂ p ∂v (V F , t) ≥ 0 and p(•, t) non-increasing on

[V 0 , V F ] ∀t ∈ [0, T ].
Notice that, as for classical solutions, the condition that the super-solution p satisfies (4.1) is equivalent to say that p satisfies (4.1) on ((V 0 , V F )\{V R }) × (0, T ) in the classical sense, and that ∂ p ∂v (v, t) has a decreasing jump discontinuity at v = V R of at least N (t)/a length.

p(v, 0) ≥ p(v, 0) ∀v ∈ [V 0 , V F ] and that p(V 0 , t) ≥ p(V 0 , t) ∀t ∈ [0, T ]. Then p ≥ p on [V 0 , V F ]×[0, T ] and if p(•, 0)-p(•, 0) is not identically zero then p > p in (V 0 , V F )×(0, T ].
Proof. We treat the case V 0 < V R , the other one being simpler. Since N (t) ≤ N (t) whenever p(•, t) ≤ p(•, t), as long as the latter holds we may write

∂ p ∂t (v, t) - ∂ ∂v v -bN (t) p(v, t) -a ∂ 2 p ∂v 2 (v, t) ≥ N (t)δ(v -V R ) + bN (t) ∂ p ∂v (v, t) ≥ N (t)δ(v -V R ),
where we have used the fact that p is a universal super-solution, that b ≤ 0 and that ∂ p ∂v (v, t) by assumption. Therefore if we set w := p -p then as long as w(•, t) ≥ 0 we obtain

∂w ∂t (v, t) - ∂ ∂v v -bN (t) w(v, t) -a ∂ 2 w ∂v 2 (v, t) ≥ 0
and it follows from the weak and strong forms of the maximum principle for linear parabolic equations that w(•, t) ≥ 0 holds for all t ∈ [0, T ] and then that for t ∈ (0, T ] that w(•, t) > 0 on (V 0 , V F ) unless w is identically zero.

We shall consider two classes of universal super-solutions. For the first one, we choose V 0 = -∞ and we explicitly define

P (v, t) =    exp(t) for v ≤ V R , exp(t) V F -v V F -V R for V R ≤ v ≤ V F . ( 4
(1.2) on [-∞, V F ] × R + .
Proof. This function P is clearly non-increasing. By linearity, it is enough to check the inequality for α = 1. Indeed, for v ≤ V R since P is independent of v, expanding the equation it is enough to check that ∂P ∂t -P ≥ 0, which actually holds with an equality. At v = V R the jump in the derivative exactly matches the derivative at V F . Finally, for v ≥ V R we have

d dt exp(t) V F -v V F -V R + exp(t) v V F -V R -exp(t) V F -v V F -V R = exp(t) v V F -V R ≥ 0.
0 ∈ L 1 + ∩ L ∞ (-∞, V F ) ∩ C 1 ((-∞, V F ]
) with p 0 (V F ) = 0, classical solutions to (1.2)-(1.5) are globally defined in time with p ∈ L ∞ (-∞, V F ) × (0, T ) and N ∈ L ∞ (0, T ) for all T > 0.

Proof. Unique classical solutions to (1.2)-(1.3) are constructed in [START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF] in a maximal time interval T * by fixed point arguments, where T * is characterized as

T * = sup{t > 0 : N (t) < ∞} .
Now, it suffices to consider a sufficiently large constant α so that p 0 (v) ≤ αP (v, 0) for all v ∈ (-∞, V F ) that is possible under the assumptions on p 0 . To check this, just realize that due to the continuity and derivability from the left assumptions at v = V F , there is M > 0 such that lim sup

v→V F p 0 (v) V F -v ≤ M
and thus, we can choose α large enough so that M < α V F -V R and p 0 (v) ≤ αP (v, 0) for all v ∈ (V F -ǫ 0 , V F ) for some ǫ 0 > 0. Taking α even larger, it is easy to show that p 0 (v) ≤ αP (v, 0) for all v ∈ (-∞, V F ) using that p 0 is bounded in (-∞, V F ). Now, we can use Lemmas 4.3 and 4.4 to deduce that p(v, t) ≤ αP (v, t) for all v ∈ (-∞, V F ) and t ≥ 0, and thus

N (t) ≤ exp(t) a V F -V R
for all t > 0. This shows that T * = ∞ and the stated uniform time dependent bounds.

The second class of universal super-solutions is time independent but is only defined for v ≥ V 0 := 0. We consider the solutions Q 1 and Q 2 to the problems

-aQ ′ 1 -vQ 1 = a on (V R , V F ), Q 1 (V F ) = 0, -aQ ′ 2 -vQ 2 = 0 on (0, V R ), Q 2 (V R ) = Q 1 (V R ), (4.3)
and we define

Q equal to Q 1 on [V R , V F ] and to Q 2 on [0, V R ].
Notice that as mentioned in the introduction, we can assume without loss of generality that 0 < V R < V L Proof. The positivity and monotonicity properties are straightforward. Since

vQ 1 = vQ 2 at v = V R , and since Q 1 (V F ) = 0, we have -a ∂(βQ 1 ) ∂v (V R ) + a ∂(βQ 2 ) ∂v (V R ) = aβ = -a ∂(αQ 1 ) ∂v (V F ),
and in particular N (t) ≡ N = aβ. This together with the definition of Q in (4.3), implies that

∂ ∂v -a ∂(βQ) ∂v -vβQ = N δ(v = V R ).
The conclusion follows.

Change of variable

The next step is to reduce equation (1.2), away from the singularity at v = V R and with N (t) considered as a data, to the linear heat equation on a time varying domain. This allows us to later use standard regularizing effects. This kind of change of variables was at the basis of the existence result in [START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF] since the whole problem (1.2)-(1.5) is equivalent to a free boundary problem for the heat equation with a Dirac Delta source in a moving point. This connection to a Stefan-like free boundary problem clarifies the fact that for b < 0 we have global existence of solutions while for b > 0 there is blow-up for any value of b for suitably chosen initial data. We refer to [START_REF] Friedman | Free boundary problems for parabolic equations. I. Melting of solids[END_REF][START_REF] Fasano | A critical case for the solvability of Stefan-like problems[END_REF][START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF] for further discussions on this connection to free boundary problems. The construction of the change of variables is well-known in kinetic theory, since it is the one making equivalent the heat equation and the linear Fokker-Planck equation, see [START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF] for more references and nonlinear versions of it, together with the method of characteristics to eliminate a linear drift term, we only include the details here for the sake of completeness and to do it adapted to our purposes. Since we want to use it in the regions where the classical solutions are smooth, we will avoid writing the right-hand side in the new variables, we refer to [START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF] for this. On a formal level we first write q(y, τ ) := f (τ )p(h(y, τ ), g(τ )) for some arbitrary smooth functions f, g and h. Direct computations lead to the equality 1 f (τ )g ′ (τ )

[∂ τ q -a∂ yy q] (y, τ )

= ∂ t p -a (∂ y h(y, τ )) 2 g ′ (τ ) ∂ vv p + f ′ (τ ) f (τ )g ′ (τ )
p + ∂ τ h(y, τ ) -a∂ yy h(y, τ ) g ′ (τ ) ∂ v p (h(y, τ ), g(τ )).

(4.4)

To annihilate the right-hand side of (4.4), in view of (1.2), we therefore impose the conditions (∂ y h(y, τ )) 2 = g ′ (τ ), ∂ τ h(y, τ ) g ′ (τ ) = -h(y, τ ) + bN (g(τ )), and f ′ (τ ) = -f (τ )g ′ (τ ). (4.5)

Notice that the first equality actually implies that ∂ yy h vanishes. In terms of t = g(τ ), the second equation in (4.5) translates into d dt h(y, g -1 (t)) = -h(y, g -1 (t)) + bN (t), whose solution is given by the variation of constants formula (choosing a reference time t 0 ≥ 0) h(y, τ ) = h(y, g -1 (t 0 )) exp(-(t -t 0 )) + t t 0 bN (s) exp(-(t -s)) ds.

In particular, (∂ y h(y, τ )) 2 = (∂ y h(y, g -1 (t 0 ))) 2 exp(-2(t -t 0 )).

In view of the invariance of the heat equation by translation and by the scaling (y, τ ) → (λy, λ 2 τ ), and since ∂ y h(y, g -1 (t 0 )) does not depend on y, there is no loss of generality in requiring that h(y, g -1 (t 0 )) = y and that g -1 (t 0 ) =

Definition

Lemma 4 . 3 .

 43 Let V 0 ∈ (-∞, V F ), b ≤ 0,and T > 0. Let p be a universal classical super-solution and p be a classical solution to (1.2)-(1.5) on [V 0 , V F ] × [0, T ], and assume that

. 2 ) 4 . 4 .

 244 Lemma Given arbitrary α > 0 and b ≤ 0, the function αP in (4.2) is a universal super-solution to

Corollary 4 . 5 .

 45 For b ≤ 0 and p

Lemma 4 . 6 .

 46 The function Q is non negative and strictly monotone decreasing on [0, V F ]. Moreover, given any β > 0 the function βQ is a universal classical super-solution to equation (1.2) on [0, V F ]×R + with N (t) = aβ.

Lemma 4 . 7 . 1 2 1 2

 4711 Let p be a classical solution to (1.2)-(1.5) and 0 ≤ t 0 ≤ T . Then the function q defined by (4.6) is a classical solution to the heat equation∂ τ q -a∂ yy q = 0 on the set Ω t 0 of (τ, y) in R 2 such that 1 2 exp(-2t 0 ) ≤ τ ≤ 1 2 exp(2(T -t 0 )), y < √ 2τ V F -log(2τ ) 0 bN (s + t 0 ) exp(s) ds, and y = √ 2τ V R -log(2τ ) 0 bN (s + t 0 ) exp(s) ds.

  1 2 , so that the first equation in (4.5) is met if and only if

	g(τ ) = t 0 +	1 2	log(2τ ),
	and then the third one if and only if	f (τ ) = Cτ -1
					t
					t 0	bN (s) exp(-(t -s)) ds, t ,	(4.6)
	where t = g(τ ) = t 0 + 1 2 log(2τ ), or equivalently
	τ =	1 2	exp(2(t -t 0 )) .	(4.7)
	Summarizing, we have

2

for some arbitrary constant C which only reflects the linearity of the equation. Choosing it conveniently we finally write q(y, τ ) = exp(-(t -t 0 ))p exp(-(t -t 0 ))y +

Acknowledgment: JAC acknowledges support from projects MTM2011-27739-C04-02, 2009-SGR-345 from Agència de Gestió d'Ajuts Universitaris i de Recerca-Generalitat de Catalunya, the Royal Society through a Wolfson Research Merit Award, and the Engineering and Physical Sciences Research Council (UK) grant number EP/K008404/1. BP and DS are supported by the french "ANR blanche" project Kibord: ANR-13-BS01-0004 funded by the French Ministry of Research.

Remark 4.8. Let us finally point out, althought not needed for our purposes, that q also satisfies ∂ τ q -a∂ yy q = M (τ )δ v -√ 2τ V R + 1 2 log(2τ ) 0 bN (s + t 0 ) exp(s) ds in the distributional sense for (τ, y) in R 2 such that 1 2 exp(-2t 0 ) < τ < 1 2 exp(2(T -t 0 )) and

where

This is the Stefan-like free boundary problem that was used for the well-posedness theory in [START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF].

A uniform bound

) with p 0 (V F ) = 0 be an initial datum for (1.2)-(1.5). Then, there exists α 0 sufficiently large (depending on p 0 ) so that

as it was done in the proof of Corollary 4.5. Let then β 0 be sufficiently large so that

We extend the function Q to the interval (-∞, 0) being constant equal to its value at zero. Since P (•, 1) is also constant on that interval, we have

We are going to show the following lemma which obviously shows Theorem 4.1.

Lemma 4.9. If β 0 is sufficiently large (depending now only on a, b, V R , p 0 ∞ , and the derivative from the left of

and for all t ≥ 0.

Assume by contradiction that this is not the case and let t 0 be the first time for which there exists v 0 ∈ (-∞, V F ) such that p(t 0 , v 0 ) = β 0 Q(v 0 ). Since P is a universal super-solution to (1.2) which is decreasing in v and increasing in t, it follows from Lemma 4.3 (with the choice V 0 = -∞) that p(•, t) ≤ α 0 P (•, t) for all t ≥ 0. Therefore from (4.8), we deduce that t 0 ≥ 1. Also, since β 0 Q is a universal supersolution on (0, V F ), we infer once more from Lemma 4.3 (with the choice V 0 = 0) that we can choose v 0 ≤ 0, and thus

For t ≤ t 0 , we have p(•, t) ≤ β 0 Q and in particular

Let q be defined from p by the change of variable (4.6). In view of the previous estimate on N (t) and since b ≤ 0, we have, for

In particular, we obtain

, where

and as as consequence, we have the inclusion of the parabolic cylinders

where Ω t 0 is defined in Lemma 4.7, provided

We are now in position to make use of the regularizing effect of parabolic equations.

Lemma 4.10. If q is a solution of the heat equation ∂ τ q -a∂ yy q = 0 on the cylinder Λ v 0 ,r , then we have the estimate

where K > 0 is a universal constant.

Proof. Using the scaling (y, τ ) → (y -v 0 )/r, (τ -1 2 )a/r 2 one reduces the estimate to the case a = 1, r = 1, on the unit cylinder [-1, 1] × [-1, 0], which yields the constant K by standard parabolic regularization. Lemma 4.10 together with the property that p(t, •) is a probability density give

) ≤ Kar -3