Véronique Gernigon

Patrick Lévêque

Cyril Brochon

Jean-Nicolas Audinot

Nicolas Leclerc

Rony Bechara

Fanny Richard

Thomas Heiser
email: thomas.heiser@unistra.fr

Georges Hadziioannou

Fullerene-grafted block copolymers used as compatibilizer in P3HT/PCBM bulk heterojunctions: morphology and photovoltaic performances

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The active layer of polymer/fullerene bulk heterojunction (BHJ) solar cells consists of an interpenetrating network formed by the electron-donor (D) π-conjugated polymer and the electron-acceptor (A) fullerene derivative. The currently most investigated devices use blends of regioregular poly(3-hexylthiophene) (rr-P3HT) and [6,6]-phenyl-C61-butyric acid methyl ester (PCBM) and have reached power conversion efficiencies (PCE) up to 5% [1]. In these devices, the active layer nanostructure determines to a great extend the overall device performances [2][3][4][5][6]. Ideally, the donor and acceptor materials should form nanoscale interpenetrating networks to ensure an efficient exciton dissociation and charge transport. It is thus of considerable importance to control the blend morphology and in particular to avoid macro-phase separation during film deposition as well as during long term device operation. Previous attempts to control the blend morphology include exposure to solvent vapour, thermal annealing, as well as the inclusion of additives (such as selective solvents) during the film formation procedure [7][8][9][10]. The use of D/A block-copolymers [11][12][13][14][15] as compatibilizer in polymer/fullerene blends offers another particularly attractive strategy to control the morphology. Sivula et al. first succeeded to avoid thermal phase segregation by introducing small amounts of a diblock copolymer into the P3HT/PCBM blend [13]. Their polymer blocks incorporated a P3HT macromonomer and a grafted fullerene monomer, respectively, giving rise to a selective affinity of both blocks with the corresponding blend components. More recently, Lee et al. used a fullerene end-capped P3HT as compatibilizer in P3HT/PCBM blends and were able to suppress phase segregation after prolonged annealing [14]. Furthermore, Kim et al. explored the influence of the thiophene block length of a thiophene-C 60 dyad that was used as compatibilizer in P3HT/PCBM blends. Their results highlight the strong dependence of the compatibilizer molecular structure on its effectiveness to stabilize the blend morphology [15]. In the present study we use as compatibilizer a rod-coil block copolymer based on a regio-regular P3HT rod-block and a poly(butylacrylate-stat-C 60methylstyrene) [P(BA-stat-C 60 MS)] coil-block. Both, the common chemical nature of the rod block with the electron-donor polymer (P3HT) and the expected strong affinity between the grafted fullerene and PCBM, make this copolymer a good candidate for this application. We have investigated the impact of the compatibilizer on the photovoltaic (PV) device performances as a function of annealing time and of the amount of compatibilizer. Also, the thin film morphology was studied by using atomic force microscopy (AFM) and spatially resolved secondary ion mass spectrometry (SIMS) in order to correlate the behavior of the PV devices to modifications in the thin film morphology.

Experimental section

Materials

The structure of the P3HT-b-P(BA-stat-C 60 MS) copolymer used as compatibilizer is shown in Scheme 1. The rod-coil block copolymer has been obtained by a "living"/control radical polymerization of a PBA-co-CMS statistical block from a P3HT macro-initiator. The latter has been obtained by the Grignard metathesis (GRIM) polymerization method, as described by McCullough et al. [16]. The final block copolymer was obtained by grafting fullerenes onto the CMS moieties via an azidation step. More details on the synthesis procedure can be found in reference [17]. The rod and coil block characteristics are summarized in Table 1.

The P3HT and PCBM were commercial products and were used as received. The P3HT was purchased at two different suppliers (Aldrich and Merck). Regio-regularity and molecular weight, measured respectively by 1H NMR and by GPC, were 90 % of regioregularity and 24.3 kg.mol -1 for the Aldrich batch and 97 % of regioregularity and 20 kg.mol -1 for the Merck batch.

The Aldrich material was used for the investigation of the device thermal stability, while the Merck polymer was used for all other experiments.

Thin film deposition and characterization

For AFM investigations, thin films of P3HT /PCBM/Compatibilizer ternary blends were spincoated from a diluted ortho-dichlorobenzene solution onto freshly cleaved mica substrates. The PCBM weight fraction was fixed to 0.5, while the P3HT and compatibilizer fractions were set to 0.5-x and x, respectively, where x was varied between 0 and 0.1. Thermal annealing of the samples has been done under nitrogen atmosphere.

The film surface morphology was investigated by atomic force microscopy using a VEECO Multimode microscope in tapping mode. Additionally, ternary and binary organic layers have been imaged with the NanoSIMS50 set-up from CAMECA [18]. A Cs (1 pA) primary beam (with impact energy of 16 keV) was rastered over a 20µm*20µm area on the sample surface with a spatial resolution of 50 nm (spot size). The two-dimensional images were recorded with 256*256 pixels, using an acquisition time of 5 ms per pixel and a parallel detection of four different negative secondary ions ( 12 C, 12 C 1 H, 16 O and 32 S).

Photovoltaic device elaboration and characterization

Bulk heterojunction devices were elaborated using P3HT as electron donor and PCBM as electron acceptor as well as polyethylenedioxythiophene:polystyrenesulfonate (PEDOT:PSS) as highly conductive interfacial layer. The device structure was the following: ITO/PEDOT:PSS(~40nm)/active layer (~100 nm)/Al(~120 nm). Indium tin oxide-coated glass with a surface resistance lower than 20 Ω/sq was used after sequential ultrasonic cleaning in acetone, isopropyl alcohol, and deionized water. After an additional UV-ozone cleaning step of the ITO, the PEDOT:PSS was spin-coated (60 second at 1200 rpm and 120 seconds at 2000 rpm at a constant 200 rpm sec -2 acceleration) from an aqueous solution and dried for 30 min at 120 °C under vacuum before being transferred to the nitrogen-filled glovebox. The dichlorobenzene solutions were stirred at 70 °C for at least 24 h before spincoating. The solution had a total concentration of 40 mg/mL with a polymer/fullerene/compatibilizer weight fraction ratio similar to those used for AFM investigations. The active layer thickness (around 100 nm) was measured using a Dektak Veeco profilometer. Finally, a 120 nm thick aluminum layer was thermally evaporated and used as cathode. A post-elaboration thermal annealing step of 15 minutes at 140°C has been performed systematically. The device active area was 9 mm 2 , while each sample included four independent diodes. Current versus voltage (J-V) characteristics were measured under darkness and under AM1.5 (100 mW/cm 2 ) illumination using an Oriel 150W solar simulator.

Results and discussion

Photovoltaic devices

The device performances as a function of the active layer compositions are summarized in Table 2. On block copolymer free reference devices a short-circuit current (J sc ) of 8.1 mA/cm², an open-circuit voltage (V oc ) of 0.59 V, a fill factor (FF) of 50 % and a power conversion efficiency (PCE) of 2.5 % have been measured. Adding 2 wt% of compatibilizer into the blend increases significantly all the photovoltaic parameters and leads to a 3.2 % power conversion efficiency. This surprising result shows that the presence of a compatibilizer, besides stabilizing the morphology, may also enhance the device performances for a given experimental procedure. The origin of this behavior may be due to changes in the morphology, that favor charge generation (leading to higher J sc ) and charge transport (higher FF). Although similar or even higher conversion efficiencies have been obtained with pure P3HT/PCBM blends in optimized devices [1,19], adding the compatibilizer may enlarge the choice of processing parameters that lead to optimal operation, making the device elaboration procedure more robust. For higher copolymer amounts however, the device performances decrease significantly. The origin of this drop in performances will be discussed further below.

To investigate the influence of the copolymer on the device stability, the PCE has been measured on devices as a function of annealing time at 140°C. The copolymer weight fraction was either 0, 2.5 or 5 wt%. The P3HT purchased at Aldrich was used for this study. As shown in Figure 1, the reference sample PCE starts to increase and decreases after 420 minutes annealing. This initial increase points out that the as-deposited morphology of the reference blends was still far from optimum. A prolonged annealing time was necessary to reach similar performances than the devices including a small fraction of compatibilizer. This behavior corroborates the positive impact of the compatibilizer on the blend morphology at short annealing times, as suggested above.

The device containing a 2.5 wt% copolymer fraction was stable up to 300 minutes, but its performances decreased rapidly for longer annealing times. This drop in efficiency occurs faster with higher copolymer contents. A similar evolution holds for both homopolymers. Indeed, the device performances summarized in table 2 (using Merck P3HT), show a loss in performances that increases with copolymer fractions above 4%. To understand the origin of both, the initial device improvement and the premature unexpected device failure induced by the copolymer, we have further investigated the thin film morphology as a function of composition and annealing time.

Thin film morphology

The AFM phase images obtained on ternary blend thin films with copolymer fractions of 0 (reference blend), 2 wt% and 5 wt% after a short heat treatment (15 minutes at 140°C) are shown in Figure 2. The morphology of the copolymer free P3HT/PCBM sample appears to be composed of a dense network of lamellae structures. A similar morphology has been observed for P3HT/PCBM blends before and is believed to be due to P3HT lamellae separated by amorphous regions where most PCBM molecules are localized [20,21]. Adding the block copolymer to the blend leads to significant changes in the morphology. For the 2 wt% copolymer sample, lamellae-like structures are still observed but appear to be less densely packed. The 2D rather than 1D nature of these features is corroborated by their dominant parallel alignment (see Figure 2b, black circle). In the 5 wt% copolymer film, a more disordered network of elongated structures is seen. The recurrent crossing of these structures (indicated by arrows in Figure 2c) point out that the film is composed, at least in the near surface region, of 1D fibrils rather than of 2D lamellae.

The observed morphologies are consistent with the idea that the strong affinity between the copolymer rod block and the homopolymer drives the P3HT rod preferentially into the P3HT lamellae, while the fullerene-grafted coil-block tries to minimize its interaction with P3HT by being localized mostly in the PCBM rich regions. For the lowest copolymer fractions, we may expect that the P3HT lamellar structure can be preserved, as it is seen for the 2 wt% copolymer sample (Figure 2b). For higher copolymer fractions however, a coreshell 1D fibril composed of a P3HT core and surrounded by the fullerene-grafted coil block as well as free PCBM molecules, may become energetically favorable. We therefore tentatively attribute the 1D features seen in the 5 wt% copolymer sample (Figure 2c) to such core-shelltype fibrils.

Surprisingly, for the highest copolymer ratio, micrometer-sized objects can be observed all over the film. The latter are absent in the as-deposited as well as shortly annealed films of blends with lower copolymer content. Their irregular angular shapes suggest these objects to be crystalline. To investigate further the nature of these domains we performed spatially resolved secondary ion mass spectrometry [18]. Both, a reference and a blend with 10 wt% compatibilizer were analyzed by NanoSIMS50. The measurements were done on as-deposited as well as on annealed films. The distribution of P3HT, corresponding either to the homopolymer or the copolymer rod-block, could be visualized by following the sulfur (32S) isotope distribution, which is absent in PCBM. Additionally, as PCBM contains one oxygen atom and as the molar fraction of PCBM is seventy seven times higher than the copolymer fraction, the distribution of the 16O isotope can also be used to map PCBM. Note however, that the presence of oxygen impurities in the P3HT domains cannot be avoided and reduces the contrast.

The results of the NanoSIMS50 acquisition are shown in Figure 3. The variation of the grey scale color (black to white) is proportional to the intensity of the element detected. In the reference sample, a homogenous 32S distribution was found on the as-deposited film (Figure 3(a1) as well as on the shortly annealed (15 minutes at 140°C) film Figure 3 (a2)). We can therefore conclude that the P3HT (and consequently PCBM) are homogenously distributed on a length scale larger than 50 nm (the spatial resolution of our equipment). In contrast, on the film that has been annealed for 3 hours, a 32S depletion is observed within a micrometersized region (Figure 3(a3)). This depletion necessarily goes along with an increase in PCBM content since all other major blend components include sulfur. The 16O map, shown in Figure 3(a4), confirms the high PCBM content within the sulfur poor region. We therefore may identify this region as a PCBM aggregate. Moreover, its irregular shape suggests the PCBM aggregate to be crystalline. Note that the PCBM aggregate appears to be linked to a second specific region visible in the 32S image by a slightly brighter contour. The same region can also be seen in the 16O map (again due to a slightly brighter contour). This feature could correspond to border effect due to the morphologic of the crystalline. In this case, the secondary ions detected come from the border and from the surface don't have the same angular distribution, and so the same secondary intensity.

The behavior of the compatibilized blend is considerably different. A relatively large number of small PCBM aggregates can be seen already in the as-deposited film (Figure 3(b1)). After annealing, the size of the aggregates increases significantly while their density decreases simultaneously. These results indicate a local PCBM depletion and suggest that diffusion of PCBM molecules at high temperature allows the aggregates to grow by an Ostwald ripening process.

The SIMS results allow us to conclude that the microdomains seen by AFM in the 10 wt% copolymer sample (Figure 2d) are composed of PCBM, in a presumably crystalline phase. Similar features have been seen by optical microscopy (not shown) in samples with lower content but after longer annealing times. In other words, the number of PCBM crystallites increases with the compatibilizer content for a given heat treatment. However, only for the highest copolymer fraction (10 wt%) could the crystallites be seen in the asdeposited films.

Interestingly, for the compatibilized blend, no impurity or defect similar to those seen in the reference blend (Figure 3(a3)) could be detected in the proximity of the aggregates. We may thus conclude that nucleation of PCBM crystallites in the compatibilized films is not induced by the presence of impurities. Rather, since the number of aggregates scales roughly with the copolymer content, we believe that the copolymer itself, and more specifically the fullerene-grafted coil block, acts as nucleation center for PCBM crystallization.

From these morphology investigations we may conclude that the rapid drop in the photovoltaic performances, observed upon long term annealing, for the devices including the compatibilizer (Figure 1) is most likely due to the enhanced nucleation of PCBM micrometer crystals. Indeed, both, the depletion of PCBM in the neighboring regions and the macrodomains induced by the crystal growth, reduce considerably the D/A interface effective area and thereby lower the charge photogeneration rate as well as the overall conversion efficiency.

Conclusion

We have described the influence of P3HT-b-P(BA-stat-C 60 MS) block copolymer used as compatibilizers in P3HT/PCBM blends. A strong correlation has been demonstrated between the morphology and the photovoltaic performances. In particular, a significant improvement in power conversion efficiency could be obtained by adding a small fraction of compatibilizer. Although a similar efficiency has been reported for pure P3HT/PCBM blends after optimization of the film deposition, our results point out that the introduction of compatibilizers may lead to an appropriate morphology even in, otherwise, non-optimized processing conditions.

Extended isothermal annealing of compatibilized blends was found to lead to rapid formation of PCBM aggregates and to cause preliminary device failure. This phenomenon increases with the block copolymer concentration and reveals that grafted fullerene behave as nucleation centre for the PCBM crystal growth. The use of a coil block that does have a strong affinity towards PCBM without being fullerene grafted (for instance poly(4vinylpyridine)), may be an interesting alternative [22,23].

Figure caption:

Figure 1. Normalized PCE as a function of annealing time at 140°C for devices with copolymer weight ratios of 0 wt% (triangles), 2.5 wt% (circles) and 5 wt% (squares), respectively. P3HT/PCBM/10 wt% block copolymer bulk heterojunctions. The grey scale of the different images goes from black to white with increasing intensity. a1 and b1 correspond to as deposited films; a2 and b2 where measured after 15 minutes at 140°C, and a3 and b3 after an additional 180 minutes at 140°C, a4 and b4 are the 16 O distributions of the same samples than a3 and b3, respectively.

Figure 2 .

 2 Figure 2. AFM phase images on thin films after 15 minutes at 140°C as a function copolymer content: a) P3HT/PCBM, b) P3HT/PCBM/2 wt% copolymer, c) P3HT/PCBM/5 wt% copolymer, d) AFM height image for P3HT/PCBM/10 wt% copolymer. The scan area is 1 µm x 1 µm for a), b) and c) while it is 20 µm x 20 µm for d).

Figure 3 .

 3 Figure 3. NanoSIMS50 images showing the 32 S distribution in a) P3HT/PCBM and b)P3HT/PCBM/10 wt% block copolymer bulk heterojunctions. The grey scale of the different images goes from black to white with increasing intensity. a1 and b1 correspond to as deposited films; a2 and b2 where measured after 15 minutes at 140°C, and a3 and b3 after an additional 180 minutes at 140°C, a4 and b4 are the16 O distributions of the same samples than a3 and b3, respectively.

  Figure 1.

Table caption :Table 1 .

 caption1 Properties of the copolymer

Table 2 .

 2 Evolution of the photovoltaic device performances, after 15 minutes at 140°C, for increasing copolymer contents.

	Schema 1.
	C 6 H 13
		S
	Br	S
		C 6 H 13

Table 1 .

 1

	Copolymer Mn	Rod Mn	Rod rr	Coil Mn	C 60 /chain C 60 weight in copolymer
	[Kg/mol]	[Kg/mol]	[%]	[Kg/mol]		[%]
	14.0	2.0	93	9.4	6.3	28%

Table 2 .

 2

	Copolymer fraction	Voc	Jsc	FF	PCE
	[weight %]	[V]	[mA/cm²]	[%]	[%]
	0	0.59	8.1	50	2.5
	1	0.6	8.2	50	2.5
	2	0.62	9.4	55	3.2
	4	0.57	9.3	42	2.2
	5	0.57	9.2	39	2.0
	8	0.56	4.7	24	0.6

Acknowledgements: The authors thank the Region Alsace and the French Research Agency (ANR) for financial support and are grateful to Damien Vadillo for his contribution to the manuscript writing.


Loic Coquelin
email: loic.coquelin@lne.fr

Nicolas Fischer

Tatiana Macé

C Motzkus

François Gensdarmes

Gilles Fleury

Laurent Le Brusquet

T Mace

L Lebrusquet

Scanning Mobility Particle Sizer: Fast data Inversion and uncertainty analysis

Keywords: SMPS, uncertainty, data inversion, sensitivity analysis

come

Introduction

SMPS (Scanning Mobility Particle Sizer) is the most widely used device to characterize particle sizes below 0.1 µm and it takes part in a large range of applications such as toxicology testing, indoor air quality measurement, etc.

Yet, a dimensional uncertainty on the size distribution isn't provided to SMPS users when the instrument operates under scanning mode. Since it has been proved that ultra-fine particles can pass through human barriers (lung, blood-brain barrier, etc.), assessing uncertainties on the estimated size distribution has become of primary interest. Present study aims at simulating SMPS measurements and at proposing a fast and accurate inversion technique to estimate the particle size distribution, as well as providing a sensitivity analysis to determine the influent parameters of the system. A 95% confidence region on the output size distribution is finally obtained by performing Monte-Carlo simulations.

Model and Inversion

Aerosol measurement using a SMPS is based on particle electrical mobility, that is defined as the ratio of the particles drift velocity to the magnitude of the electric field; particle electrical mobility is then linked to its mobility diameter d p and electrical charge p. Once selected by DMA (Differential Mobility Analyser), particles are gathered into canals, enlarged and detected by a CPC (Condensation Particle Counter). Wang and Flagan (1990) developed the theory of particle classification when the instrument operates under scanning mode, and precision on the voltage ramp is given by Collins et al. (2004).

Raw data extracted from the SMPS represent a concentration over a certain class of mobility diameters. If we denote r i as the concentration over the class i, n the size distribution, p the number of charges carried by a particle, d 1,i and d 2,i respectively the left and right extremity of the class i, t i the counting time for the class i, t 0,i being the time at which the count begins in the class i, f describes particle electrical mobility selected according to scanning time and k i the kernel function (product of the charge distribution, the detection efficiency of the CPC, the DMA transfer function), then:

  2, 0, , , 0, 1 ( ), , ( )
1,i d i i i i i i p p p d t t t i r k f t p d n d dd dt t                .
Starting from the deterministic model, next step is to focus on the variability of the measurements. Indeed, a model of noise based on experimental results that takes into account the CPC counting modes (single particle counting, live-time counting and photometric mode) is created.

Knowing the concentrations for each class, data inversion consists in retrieving the size distribution. It is a minimization problem with the constraint of positivity of the solution. Regularization technique of [START_REF] Tikhonov | Solution of Ill-posed Problems[END_REF] that penalizes non-smooth solutions is performed. In our case, the inversion has been proved to be fast and stable to perturbations in the input space and can then be used for the sensitivity analysis.

Primary Results

Since the number of parameters involved in the model is high, sensitivity analysis would require too much computing time, that's why analysis based on the decomposition of the variance is carried out by using indices defined by Sobol (1993). Then, according to experts in aerosol physics, the range of variation for each influent parameter of the system is chosen and a 95 % confidence interval is computed using Monte-Carlo simulations as shown in figure 1 for a typical atmospheric aerosol size distribution [START_REF] Whitby | [END_REF].

Figure 1 .

 1 Figure 1. 95% confidence region when simulating SMPS measurement of an atmospheric aerosol


Hassan Nasser

Olivier Marre

Selim Kraria

Thierry Viéville

Bruno Cessac

Analyzing large-scale spike trains data

published or not. The documents may come

Analyzing large-scale spike trains data with spatio-temporal constraints Recent experimental advances have made it possible to record several hundred neurons simultaneously in the retina as well as in the cortex. Analyzing such a huge amount of data requires to elaborate statistical, mathematical and numerical methods, to describe both the spatio-temporal structure of the population activity and its relevance to sensory coding. Among these methods, the maximum entropy principle has been used to describe the statistics of spike trains. Recall that the maximum entropy principle consists of fixing a set of constraints, determined with the empirical average of quantities ("observables") measured on the raster: for example average firing rate of neurons, or pairwise correlations. Maximising the statistical entropy given those constraints provides a probability distribution, called a Gibbs distribution, that provides a statistical model to fit the data and extrapolate phenomenological laws. Most approaches were restricted to instantaneous observables i.e. quantities corresponding to spikes occurring at the same time (singlets, pairs, triplets and so on. See [START_REF] Tkacik | Spin glass models for a network of real neurons[END_REF], [START_REF] Shlens | The structure of multi-neuron firing patterns in primate retina[END_REF] and [START_REF] Schneidman | Weak pairwise correlations imply strongly correlated network states in a neural population[END_REF]). The corresponding Gibbs distributions are only successful in predicting the probability of patterns lying within one time bin [START_REF] Marre | Prediction of spatiotemporal patterns of neural activity from pairwise correlations[END_REF] but fail to predict the temporal statistics of the neural activity.

Extension of the maximum entropy to spatiotemporal statistics

We first define the Gibbs potential as:

H β = K k=1 β k O k , (1)
where the β k s are real numbers and free parameters. O k s are the observables that defines the constraints. Fitting the data to this Gibbs potential model aims to reproduce the probability distribution of the data, i.e, finding a theoretical distribution whose observables probabilities are close to the empirical probabilities measured on the raster, using the maximum entropy principle. We have extended this principle considering spatio temporal constraints (i.e. neurons interactions with time lags) and allowing the characterization of spatio-temporal patterns statistics. This approach, based on a transfer matrix technique, avoids the computation of the normalization factor (partition function) and directly provides the moment generating function (free energy density). We first developed a numerical method to learn the parameters of a maximum entropy model with spatio-temporal constraints on real spike train data [START_REF] Vasquez | Entropy-based parametric estimation of spike train statistics[END_REF]. However, this exact method could only be applied to small subsets of neurons, since it required construction of huge matrices whose size increases exponentially with the number of neurons.

To circumvent this combinatorial explosion, we then adopted a Monte-Carlo approach to compute the observable averages for a given set of model parameters (see [START_REF] Nasser | Spatio-temporal spike trains analysis for large scale networks using maximum entropy principle and monte-carlo method[END_REF] for more details). We first tested the efficiency of this new algorithm in the case where a raster plot is generated from a known Gibbs distribution [Fig. 1]. We then make the algorithm find back this Gibbs distribution from that raster. This allowed us to quantify both convergence rate and model accuracy as a function of the number of neurons. The main advantage of this new technique is its computational speed. Fig. 2 shows the difference between the computation time taken using transfer matrix technique and the Monte-Carlo technique: The computation time with Monte-Carlo increases linearly with the number of neurons while it increases exponentially with transfer matrix based technique.

Learning the parameters β k

We have also developed an efficient algorithm to compute the parameters β k defining the Gibbs distribution, even for large data sets. We adopted the method used in [START_REF] Dudík | Performance guarantees for regularized maximum entropy density estimation[END_REF] and we extended it to the spatio-temporal case. Their method is based on the minimization a criterion, the negative likelihood, given by (Eq. 2)

L π (T ) ω (β) = -π (T ) ω [log µ β ], (2)
where β k s are the parameters to be inferred, µ β is the probability distribution estimated using Monte-Carlo technique, π (T ) ω is the empirical distribution and finally. L π (T ) ω (β) represent the likelihood between the data and the model, to be minimized. We have tested the method considering large spatio temporal data sets where the number N of neurons is large (N ∼ 100) as well as the model range R, where the exact transfer matrix computation becomes intractable. In Fig. 3 we present the maximal distance (Eq. 3) between the exact value of the coefficient and the estimated value, averaged on a set of 10 random Gibbs potentials.

The error is given by the following formula:

Error = 1 L k=1...L β k -βk (3)
where β k is the exact value of coefficient k, βk is the estimated value and L is the number of terms in the potential.

Conclusion

We have extended the maximum entropy technique to handle spatio-temporal temporal constraint and developped a Monte-Carlo based method which allows to infer statistical properties for large data set, in less time than the transfer matrix based technique. This work will help to analyse spike train sorted from large MEA (Multi-Electrode array) recordings and discover the structure of the hidden statistics behind the data. The choice of the Gibbs potential form (what are the terms that we put in the potential) remains a challenge and will be subject for future studies. Since for each (N, R) value the potential is not the same, then the maximum number of monomials (L) will not be the same and by consequence we observe that the curves do not have all the same number of points.

Figure 1 :

 1 Figure1: Comparison between the real (x -axis) and estimated (y -axis) values of observable averages (Points represent the mean value of the observable averages and segments represents the standard deviation, over 10 trials).Given a random Gibbs potential, we computed the probability distribution of observables with the transfer-matrix based technique and with the Monte-Carlo based technique. Since the transfer matrix cannot handle for large sets, we were limited to a small number of neurons in order compare the empirical and theoretical averages. The values obtained with Monte-Carlo are closed to those obtained with the transfer matrix technique since they are all aligned to the y = x axis.

Figure 2 :

 2 Figure 2:The CPU time necessary to obtain the observable averages presented in Fig.1, for the Monte-Carlo Average (MC Av.) and for the theoretical (Th. Av.), as a function of N (Here, the theoretical average is exact since the potential is generated synthetically with known probability distribution). The full lines correspond to fit. The computation time with Monte-Carlo increases linearly with the number of neurons while it increases exponentially with transfer matrix based technique.

Figure 3 :

 3 Figure3: Max distance between the exact values of coefficients and the estimated values, averaged on a set of 10 random potentials for several values of N, R. Since for each (N, R) value the potential is not the same, then the maximum number of monomials (L) will not be the same and by consequence we observe that the curves do not have all the same number of points.

Acknowledgements: This work was supported by the INRIA, ERC-NERVI number 227747, KEOPS ANR-CONICYT and BrainScales projects.


L Coquelin
email: loic.coquelin@lne.fr

N Fischer

C Motzkus

T Mace

F Gensdarmes

L Le Brusquet

G Fleury

Aerosol size distribution estimation and associated uncertainty for measurement with a Scanning Mobility Particle Sizer (SMPS)

Scanning Mobility Particle Sizer (SMPS) is a high resolution nanoparticle sizing system that has long been hailed as the researcher's choice for airborne nanoparticle size characterization for nano applications including nanotechnology research and development. SMPS is widely used as the standard method to measure airborne particle size distributions below 1 µm. It is composed of two devices: a Differential Mobility Analyzer (DMA) selects particle sizes thanks to their electrical mobility and a Condensation Particle Counter (CPC) enlarges particles to make them detectable by common optical counters. System raw data represent the number of particles counted over several classes of mobility diameters. Then, common inversion procedures lead to the estimation of the aerosol size distribution. In this paper, we develop a methodology to compute the uncertainties associated with the estimation of the size distribution when several experiences have been carried out. The requirement to repeat the measure ensures a realistic variability on the simulated data to be generated. The work we present consists in considering both the uncertainties coming from the experimental dispersion and the uncertainties induced by the lack of knowledge on physical phenomena. Experimental dispersion is quantified with the experimental data while the lack of knowledge is modelled via the existing physical theories and the judgements of experts in the field of aerosol science. Thus, running Monte-Carlo simulations give an estimation of the size distribution and its corresponding confidence region.

1. Introduction Among devices using electrical mobility methods to measure aerosol particle size distribution, the Scanning Mobility Particle Sizer (SMPS) is the most widely used to characterize particles smaller than 1µm. Since it can be applied in fields like: indoors air quality measurement, vehicle exhaust pipes, atmospheric studies, toxicology testing, etc., it has become of interest to assess the uncertainty associated with its outputs. So far, although some works have been carried on the adequacy of particle size measurements made by this device, such a dimensional feature uncertainty cannot be provided. SMPS is composed of a Differential Mobility Analyser (DMA) and a Condensation Particle Counter (CPC). First one selects particles mobility diameters while the other is a counter used for the detection. First, aerosol passes through an inertial impactor to avoid largest particles to enter the DMA column, then the aerosol enters the charge neutraliser to be conditioned. Indeed, aerosol particles are usually charged. In this case, a bipolar charge distribution is created by exposing the aerosol to a radioactive source, so, particles that carry several charges lose their charge excess. Once aerosol is well conditioned particles are selected using electrical classification inside DMA column: an electric field is created, and the airborne particles drift in the DMA according to their electrical mobility. Once selected, particles reach the CPC to be counted. In this paper, we focus on providing the users with a methodology to compute the uncertainties associated to the size distribution estimated by an inversion procedure. The article is organized as follows: section 2 defines the technical protocol to produce the data, section 3 describes the physical model to relate the experimental data to the size distribution and the inversion procedure to estimate the size distribution. Improvements are mainly brought in section 4 where the sources of uncertainty arising from both the measure and the estimation of the size distribution are decomposed and propagated. Thus, the experimental dispersion is represented by errors on the raw data. Section 4.1 explains how the model of these errors is deduced from some observed signals. Then, section 4.2 describes how the estimation of the size distribution is realized via an inversion model whose variability will refer to the lack of knowledge on the parameters and other physical laws of the system. Preliminary results obtained when this methodology is carried out on SiO 2 airborne nanoparticles are presented in section 5 and it is followed by a discussion about the limits of the methodology and the possible improvements.

Technical protocol

Measurements using a SMPS can be performed via several kind of devices. The measurement process can be split into phases: the generation of the aerosol, the neutralization of the charges, the selection of the particles and their detection. In this case, the measurement is assured by a pneumatic atomizer 3076, a charge neutraliser using Krypton source ( 85 K r ), a Long-DMA 3081, and a butanol based CPC 3022. Each equipment comes from TSI company. As an example, an other choice could have been an electrospray, a corona charger, a nano-DMA and a water based CPC. The results that will be shown later in this paper refer to the chosen devices. The protocol that is used is defined as follows:

• Instruments are warmed up for at least 30 minutes • TSI 3076 operates in the following conditions: air flow rate: 3 L.min -1 with inlet pressure at 2.5 kg.cm -2 , liquid flow rate: 20 cm 3 .min -1 at 2.5 kg.cm -2 which are nominal settings for the device • Nanoparticles generated by the atomizer pass through the charge neutralizer 3077, and then, are introduced into the DMA • DMA settings only consist in defining the following equalities for the flow rates: Q sh = Q exc and Q a = Q cpc so that the flows are symmetric inside the classifier • CPC low flow mode is activated such that the inlet flow is 5 cm 3 .s -1

Model definition and Inversion procedure

The recovery of size distributions, which is referred as data inversion, requires the solution of the set of Fredholm integral equations. Let y i be the number of particles counted by the CPC in the i th channel, k i the nonnegative kernel function for the time range corresponding to the channel i, d p the mobility diameter of the particle, u the logarithm function, i the measurement error in the i th channel and finally f the aerosol size distribution, then:

y i = +∞ 0 k i (d p )f (u(d p ))d(u(d p )) + i , i = 1, ..., m (1)
where each kernel function is expressed as:

k i (d p ) = Q a t i c +∞ p=1 Φ(p, d p )W (d p ) Ωi (d p , p). (2)
Components Φ, W and Ωi of the kernel function respectively represent the bipolar charge distribution function, the CPC detection efficiency and the mean transfer function for the time range corresponding to the channel i, p being the number of charges carried by the particle, Q a the aerosol flow rate entering the DMA and t i c the counting time for the i th channel. The principle of selection is based on the relation between the voltage applied at the center rod of the column of the classifier and the mobility diameter. This relationship is expressed via the electrical mobility. The experimental electrical mobility, denoted as Z v , can be computed according to the voltage V , the flow rates (Q sh the sheath flow rate entering the DMA and Q ex the excess flow rate exiting the DMA) and the geometry of the DMA (L the length of the column, r 1 and r 2 respectively the inner and outer radii of the electrodes). The theoretical electrical mobility, noted Z d , relates Z v to a specific mobility diameter d p . Finally considering the equality of both quantities, one can get the mobility diameter associated to the applied voltage.

Z v = (Q sh + Q ex )log(r 2 /r 1 ) 4πLV , Z d = peC c (d p ) 3πµ g d p , ( 3
)
where e is the charge of the electron, µ g the dynamic gas viscosity and C c the slip correction factor. The expression of the slip correction factor is based on the triplet (α, β, γ) and its expression is as follows:

C c (d p ) = 1 + K n (d p ) [α + β exp(-γ/K n (d p ))] , (4)
K n (d p ) = (2λ m )/d p , (5)
where K n is called the Knudsen number and λ m is the mean free path of a particle.

Since the DMA voltage is being continuously scanned, the times of the system become crucial. Indeed, particles are selected by the DMA and after transport through the tubing (transport time is denoted as t d ), the counter detects them. An error in the time specification corresponding to the time for particles to pass from the DMA exit slit to the CPC counting chamber lead to an error in the estimation of f on the diameter space. The same kind of statement can be made in what concerns the time for particles to pass through the DMA column (denoted as t f ). This time is computed as the time for the aerosol to pass through the DMA.

The discrete form of equation 1 after applying a numerical quadrature to approach the integral numerically can be written as:

y i = j=n j=1 w ij k ij f j + i , i = 1, ..., m, (6)
where

k ij = k i (d p,j ), f j = f (d p,j
), (d p,j ) j=1,...,n being the vector of points used for the discretization of the diameter space. Once the relationship between the inputs and the outputs of the measurement system is well-defined, next step is to estimate each component of the vector f . Several algorithms lead to comparable estimations of the size distribution. Most commonly used algorithms in the field of aerosol are those developed by Seinfeld [1], Twomey [2], [3] and more recently Collins et al. [4] or Talukdar [5]. Following Seinfeld's method, inversion with regularization techniques has been chosen here. It leads to a fast and accurate estimation of f and it also takes into account the judgements of the experts through the prior. Let H the matrix such that H = (w ij k ij ) j=1,...,n i=1,...,m ∈ R m×n , y = (y i ) i=1,...,m ∈ R m and f = (f j ) j=1,...,n ∈ R n , then the inversion procedure becomes the minimization problem: find f * ∈ R n such that

f * = arg min f ≥0 Hf -y 2 2 + λ D 2 f 2 2 , (7)
= arg min

f ≥0 {G (H, f, y, λ)} , (8)
where D 2 is the second-order difference matrix and f ≥ 0 means f j ≥ 0, ∀j .

Here the prior penalizes solutions that are not smooth. This type of penalty seems justified in this application because aerosol processes often tend to smooth rough distributions. The parameter λ will be a trade off between the prior we associate to the solution to be estimated and the solution that minimizes the quadratic error term. The L-curve method is chosen here to determine the regularization parameter and it is computed via the pruning algorithm developed in [6]. To ensure robustness, λ is calibrated through the all set of data. The solution of the minimization is computed via the fast non negative least-square algorithm (fnnls [7]) algorithm that is an optimized version of Lawson and Hanson non negative least-square algorithm (nnls [8]).

Several references [1], [2], [3], [4], [5] define how to estimate the size distribution, this article does not intend to bring improvements in the inversion procedure but mainly in the following section dedicated to the uncertainty analysis.

Uncertainty analysis

In this part, the aim is to quantify and to propagate the uncertainties coming from both the measurement process and the inversion. The experimental dispersion is difficult to model since each component of uncertainty that it involves is not accurately modelled yet. Thus, choice has been made to quantify it with experimental data. It can be seen as a behavioural model and it has the advantage that it warranties a realistic quantification of this source. On the other hand, the uncertainty associated with the inversion procedure is modelled. Unlike for the experimental dispersion, the uncertainty it induces cannot be track down into the data because the error it produces is systematic. The major source is the lack of knowledge on the physical models to be used to define the matrix H. Indeed, several theories compete to define the factors of Cunningham, the bipolar charging law or the diffusive transfer function. Moreover, elements of this group are different in terms of nature: scalars and functions. Some scalars can be modelled with corresponding uncertainty because they can be measured (L, r 1 , r 2 ). Some cannot be measured but the existing theories can be used to bound them (α, β, γ) or concerning the functions φ and Ω, comparisons between data and models have been performed under controlled conditions. In this paper, we will only consider the ignorance on parameters, so Fuch's theory [START_REF] Fuchs | Geofis[END_REF] is chosen to define φ and Stolzenburg's work [START_REF] Stolzenburg | An Ultrafine Aerosol Size Distribution Measuring System Ph[END_REF] is used to model the diffusional transfer function. To sum up, the experimental dispersion, denoted as the first group of uncertainty represents 3 major components:

• the instability of the aerosol generation,

• the partial variability of the physical parameters (flow rates, environmental conditions, etc.),

• the noise coming from the CPC counting process, while the inversion uncertainty that is considered as the second group is made of:

• the lack of knowledge associated with the choice of the physical parameters,

• the variability induced by the choice of the regularization parameter,

• the bias due to the method of regularization.

The evaluation of the bias requires an aerosol of reference from a metrological point of view. Since it is not available due to the instabilities of the generation process, it is not yet possible to account for it in the uncertainty budget.

The experimental dispersion

In practice, a set of measurements have been made and the user's got observations (y il ) l=1,...,n l i=1,...,m , i being the index for the class of diameter and l the index that corresponds to the number of experiences. We consider that the experiences have been repeated n l times, so that enough information can be extracted from the data. The idea we propose is to infer a model based on this information by considering simulated data modelled as a correlated Gaussian process noted ỹ. Experiences tend to justify the Gaussian assumption and authors usually abound in that sense. Let C be the empirical correlation matrix, the autocorrelated Gaussian kernel is computed by performing a SVD (Singular Value Decomposition) on C such that it can be decomposed as C = U ΣV * . ỹ is then created as follows:

ỹ = U √ Σz σ + µ, (9)
where z = (z 1 , z 2 , ..., z m ) with z i ∼ N (0, 1), µ and σ are respectively the mean and standard deviation of the real observations:

µ = (µ 1 , µ 2 , ..., µ m ) ∈ R m such that µ i = 1 n l l=n l l=1 y il , σ = (σ 1 , σ 2 , ..., σ m ) ∈ R m such that σ i = 1 n l -1 l=n l l=1 (y il -µ i ) 2 .
To propagate the uncertainty arising from the experimental dispersion, the methodology is to generate a set of simulated data via Monte-Carlo simulations and to estimate the vector f via the minimization described in (eq 7). Let ỹ[q] ∈ R m be the simulated data generated for the q th simulation, n q the total number of simulations and λLcurve the regularization parameter calibrated thanks to the set of real data y via the L-curve algorithm, then the estimation of the solution vector f for the q th simulation denoted as f [q] ∈ R n is computed as:

f [q] = arg min f ≥0 G H, f, ỹ[q]
, λLcurve , q = 1, ..., n q .

(10)

The inversion uncertainty

A parametric model is developed in order to take into account the uncertainty associated with the lack of knowledge on parameters to be used for the definition of the matrix H. These parameters are selected such that they are not well-known or because several theories compete to define them and especially because they have a significant effect on the final estimation. In a previous paper [START_REF] Coquelin | Instrumentation and Measurement Technology Conference (I2MTC[END_REF], sensitivity analysis has been carried out to determine the most significant parameters, significant means here that a small error in their definition produces a large variability on the reconstruction of the solution vector. The chosen parameters are: L, r 1 , r 2 , α, β and γ. Nominal values for α, β and γ are taken from experiments made by Kim et al.. [START_REF] Kim | [END_REF]. Experiences realized by Hutchins et al.. [13] and Allen and Raabe [14] are chosen as bounds around the nominal values. Subscripts k , h and ar refer to each author. Nominal values for the geometrical parameters of the DMA are given by the constructor and their associated uncertainty is taken from the ISO standard (reference ISO 15900:2009(E)). To account for the uncertainty the selected parameters produce, they are modelled as random variables and propagate through the inversion via Monte-Carlo simulations as for the experimental dispersion. Table 1 summarizes the choices made for the definition of the random variables. Let H[q] be the matrix drawn by the random variables for the q th simulation, for a chosen experimental measurement y = (y il ) l=l i=1,...,m , f [q] is now computed as:

f [q] = arg min f ≥0 G H[q] , f, y, λLcurve , q = 1, ..., n q . ( 11
)
Table 1. Definition of the parameters that model the lack of knowledge. U and N respectively refer to Uniform and Normal distributions. Subscript 0 refers to the nominal value given by the constructor for the corresponding parameter.

Name Law Mean Standard deviation Min Max

r 1 U r 1,0 - r 1,0 -0.001r 1,0 r 1,0 + 0.001r 1,0 r 2 U r 2,0 - r 2,0 -0.0006r 2,0 r 2,0 + 0.0006r 2,0 L U L 0 - L 0 -0.005L 0 L 0 + 0.005L 0 α N α k max( α h -α k , αar-α k ) α k - - β N β k max( β h -β k , βar-β k ) β k - - γ N γ k max( γ h -γ k , γar-γ k ) γ k - -
The decomposition of the sources of uncertainty as presented above is required when the purpose is to characterize their respective effect on the reconstruction of the solution vector f . Nevertheless, the final result is the estimation of f with the associated uncertainty and it has to be computed when both sources are simultaneously considered. Combining the sources is straightforward since the procedure is the same as for each source taken individually. Thus, when both sources are combined, the estimation of the solution vector f for the q th simulation is obtained as:

f [q] = arg min f ≥0 G H[q] , f, ỹ[q] , λ [q],Lcurve , q = 1, ..., n q , (12)
where λ [q],Lcurve is the regularization parameter computed for the q th simulation.

Results

Starting from real observations, the aim of this part is to show the uncertainty associated with the mean estimate, denoted f , that is computed with the methodology presented above for each source taken individually and combined. As an example, SiO 2 airborne nanoparticles are presented. The aerosol presents 2 populations in terms of size: the first one is around 30 nm while the second population stands around 40 nm. Dispersions around the peaks are similar. These observations will prove the robustness of the inversion procedure, indeed, both peaks are really close and a bad estimation of the regularization parameter shall lead to over-smooth the solution. Figure 2 shows the real observations y (l = 40 in this case) and the corresponding simulated data generated via Monte-Carlo simulations. Following the developments proposed in section 4, experimental dispersion and its effects on the estimation of the vector f as well as the lack of knowledge and its impact on the reconstruction are illustrated in figure 3. Combined uncertainties of both sources are also presented in this figure. The uncertainties are shown through a confidence region with a confidence level of 95 %. The results suggest that the experimental dispersion affects the estimation of f on the y axis (concentration of particles) with the same amplitude through the all diameters while the lack of knowledge on the physical phenomena does not significantly affect the concentration of particles but it induces a translation of the reconstruction. Other experiences are actually being realized on different aerosols in order to confirm these results.

Conclusion

A methodology has been presented to compute the confidence region on the estimated size distribution when several experiences have been carried out to characterize an aerosol. The statistical model that accounts for the experimental dispersion has been inferred from real observations because spatial correlations are difficult to model precisely. The statistical inversion procedure takes into account the uncertainties coming from the lack of knowledge on parameters that are used to relate the data to the size distribution through the matrix of kernel functions. An example of aerosol composed of SiO 2 airborne nanoparticles is treated using this methodology. Results reveal that the uncertainty arising from each source affects respectively the reconstruction on the concentration of particles and on the diameters. Some sources still need to be added especially the impact of the prior in the minimization problem and the bias that it introduces on the estimation of f . Next step will be to consider analytical size distributions to estimate the bias induced by the prior since an aerosol of reference is not yet available. An other improvement would be to precisely model and propagate the uncertainties when only one experience is available.

Figure 1

 1 Figure 1 presents a schematic diagram of the experimental setup.

Figure 1 .

 1 Figure 1. Schematic diagram of the experimental setup

Figure 2 .Figure 3 .

 23 Figure 2. Real data (top) versus simulated data (bottom) for SiO 2 airborne nanoparticles


Carlos Carvajal
email: carlos.carvajal@inria.fr

Thierry Viéville

Frédéric Alexandre

Konio Pathway: An Instinctive Visual Mechanism for Survival and Decision Making?

Keywords: Konio, visual mechanism, survival, decision making, pathway interplay

Survival is one of the evolutionary priorities for every species on Earth. It is strongly linked to making good decisions, even if these are unconsciously made. In the visual system, interactions between conscious and unconscious processing have been discussed, depending on the main two information flows originating from the retina: the Parvo and Magno pathways. We consider here a possible role of the Konio pathway; less studied and with different characteristics, such as large receptive fields for global complex event detection, and diffuse projections which may allow to quickly inform several areas in the brain of something important present in the visual world. To test these hypotheses, we developed a computational systemic model of the early visual system proposing one type of Konio cells, which respond to certain spectral patterns and change the dynamics of the rest of the system accordingly. When examining the model's responses to static inputs, we observe that the system exhibits desirable reactions, together with time cost and efficiency characteristics. We also discuss the implications of the hypotheses for more complete survival and decision making processes, as the Konio pathway may only be a foundation stone for them in terms of evolution.

Introduction & Objectives

Physiologically, in the early visual system, the Parvo and Magno pathways are better understood [START_REF] Callaway | Structure and function of parallel pathways in the primate early visual system[END_REF] than the Konio pathway [START_REF] Hendry | The koniocellular pathway in primate vision[END_REF]. Furthermore, the interactions between these pathways lack understanding. The standard Magno and Parvo pathways arrive at the primary visual cortex (V1), and the temporal and parietal pathways, via the core lateral geniculate nucleus (LGN); whilst the non-standard konio pathway projects to several cortical areas, but at the same time to the amygdala via the superior colliculus. The Parvo pathway is associated to details such as color and shapes, whereas the Magno pathway is related to flashing or moving images. The Konio pathway has been found in each of the Old World primates [START_REF] Hendry | The koniocellular pathway in primate vision[END_REF] and some studies [START_REF] Yoonesi | Functional assessment of magno, parvo and koniocellular pathways; current state and future clinical applications[END_REF] have proposed that it may be involved with seasonal mood changes [START_REF] Glickman | Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (leds)[END_REF] and color consistency mechanism [START_REF] Yoonesi | Comparison of sensitivity to color changes in natural and phase-scrambled scenes[END_REF] [START_REF] Paul | The role of retinal photoreceptors in the 6 regulation of circadian rhythms[END_REF]. Whereas these studies propose certain roles, there are 3 Konio layers to consider, and these studies seem to focus on the middle Konio layer, that carries information from central blue cones to superficial color blobs in layer 3 of V1 [START_REF] Hendry | The koniocellular pathway in primate vision[END_REF]. Figure 1 sketches their main features, including the matrix representation in the thalamus for the Konio flow and the thalamic reticular nucleus (TRN) described as a modulator for thalamo-cortical connections. Depending on the characteristics that we implement in our computational systemic model for the Konio pathway, our goal is to investigate how the system changes its dynamics in case the information signaled the presence of an important natural element in the visual world, which might be a predator, a possible sexual partner or food.

Figure 1: Overview of the model. An implicit hypothesis here is that the matrix part of the pulvinar uses an analogous mechanism to that of the Konio cells in the lateral geniculate nucleus (LGN). Feedback here is considered from cortical layer 6 of V1 and V2 with a relay at TRN for the core pathway, and from V2 to V1 through superficial layers.

Methods

We built our model from scratch using Python. Its architecture is inspired from the model proposed by [START_REF] Rodriguez | Derivation and analysis of basic computational operations of thalamocortical circuits[END_REF] [START_REF] Granger | Engines of the brain: the computational instruction set of human cognition[END_REF], and con-siders facts from other studies, including [START_REF] Callaway | Local circuits in primary visual cortex of the macaque monkey[END_REF] [START_REF] Callaway | Feedforward, feedback and inhibitory connections in primate visual cortex[END_REF] [START_REF] Callaway | Structure and function of parallel pathways in the primate early visual system[END_REF] [START_REF] Sherman | The thalamus is more than just a relay[END_REF] [START_REF] Masland | The unsolved mystery of vision[END_REF] [START_REF] Nassi | Parallel processing strategies of the primate visual system[END_REF].

Its implementation is based on three-dimensional Gabor filters for cortical levels, on Difference of Gaussians filters for core thalamic structures, and on a statistical analysis of oriented features for the matrix thalamus. We only worked with horizontal and vertical orientations for simplicity.

The input to the system corresponds to a series of black & white images of 324x324 px, which are treated (in detail) 3 times, as a saccadic movement is done about 3 times per second [START_REF] Ballard | Deictic codes for the embodiment of cognition (with commentary)[END_REF] and humans may process complex natural images in about 120 [ms] [START_REF] Thorpe | Speed of processing in the human visual system[END_REF] [START_REF] Vanrullen | The time course of visual processing: from early perception to decision-making[END_REF].

Figure 2 shows how artificial inputs are analyzed by the system topographically by extracting patterns. The global analysis made by the Konio system allows to choose the order of the hierarchical decomposition, whereas the TRN inhibits information already analyzed.

The output of the system is a series of messages that represent a certain motor output (Figure 3). They are based on the result of a pattern recognition from the Konio, V1 and V2 information, and in how critical this pattern is for the system.

Results

We have postulated that our system analyzes 3 times a frame, and that we have different criticality parameters for different stimuli. One of the first tests that we did then was to check whether the model could extract the patterns following a certain order given the responses from the proposed Konio cells and how critical this pattern was for the system. For this, we created a fake pattern that contained 3 others, which were the ones interesting for our model. To do this we consider only the cell activation maps1 at every level rather than to also consider the motor output. It is validated that the system prioritizes critical patterns if found and just then passes to less critical options if found. Already analyzed patterns are inhibited through cortical feedback relaying at the thalamic reticular nucleus to allow for a less costly treatment. Figure 2 shows this process.

Another needed study was noise resistance to test the capability of the system to demonstrate that a more developed system is needed as the processing undertaken by the Konio cells is global and simple so it can arrive at the desired areas without spending too much time at the thalamus. Figure 3 shows this process, where a noncritical pattern is used to reflect the idea that the system allows itself to wait for a more detailed representation as the error at lower visual levels is too high to recognize the pattern as present. This figure also shows that the simulation time for the Konio mechanism proposed here is about 4 times faster than the Parvo mechanism implemented, which was expected as its goals are to communicate a simple information to several areas of the brain, either to trigger a defensive action, or to bias the conscious visual processing as something naturally interesting seems to be present.

Discussion

The Konio pathway has been found in each of the Old World primates [START_REF] Hendry | The koniocellular pathway in primate vision[END_REF]. It projects to higher cortical stages and to the amygdala via the superior colliculus. This connection is relevant as the superior colliculus has a role in responses to events and emergencies [START_REF] Dean | Event or emergency? two response systems in the mammalian superior colliculus[END_REF], and the amygdala [START_REF] Ledoux | Amygdala[END_REF] is involved in several cognitive and behavioral functions, such as attention, explicit and emotional memory, and decision making [START_REF] Gupta | The amygdala and decision making[END_REF]. We have not yet addressed these structures nor their links, but we're keen to study them as the Konio pathway is just one starting point for them.

Conclusions and Perspectives

Our model shows that architectural and functional of characteristics of the Konio system can bring flexibility to a local and hierarchical analysis of the perceptive world. These properties could be replicated at higher levels of the matrix system, like in the pulvinar. It might be also important to consider other crucial extra-cortical inputs like the ones stemming from the superior colliculus or the amygdala.

Perspectives

• Work with natural images.

• Extend the model to higher cortical areas and the Magno pathway to account for motion.

• Include the path to the amygdala to exploit, e.g., higher level decision making processes.

• Enhance the model to further study the interactions between pathways and the role of the other layers of Konio cells.

• Examine other stages of the food chain: how a predator would be helped by such a prewired network, and the duality (being and having predators).

• Explore different disorders [START_REF] Yoonesi | Functional assessment of magno, parvo and koniocellular pathways; current state and future clinical applications[END_REF] to which the visual pathways (Parvo, Magno and Konio) are related, such as multiple sclerosis (Murav'eva, Deshkovich, & Shelepin, 2009) [START_REF] Thurtell | Evaluation of optic neuropathy in multiple sclerosis using low-contrast visual evoked potentials[END_REF], schizophrenia [START_REF] Delord | Psychophysical assessment of magno-and parvocellular function in schizophrenia[END_REF] [START_REF] Butler | Early-stage visual processing deficits in schizophrenia[END_REF]), Parkinson's [START_REF] Silva | Independent patterns of damage within magno-, parvo-and koniocellular pathways in parkinsons disease[END_REF]) and Alzheimer's [START_REF] Guo | Alzheimers disease and retinal neurodegeneration[END_REF]) [START_REF] Sartucci | Dysfunction of the magnocellular stream in alzheimer's disease evaluated by pattern electroretinograms and visual evoked potentials[END_REF] disease, and many others [START_REF] Gutowski | Early impairment of foveal magno-and parvocellular pathways in juxta chiasmal tumours[END_REF]) [START_REF] Papia | Reduced soma size of the m-neurons in the lateral geniculate nucleus following foetal alcohol exposure in non-human primates[END_REF].

Figure 2 :

 2 Figure 2: Pattern extraction and hierarchical decomposition. Using 3 konio cells of the proposed type and a fake pattern as input, that contains the 3 interesting subpatterns, we observe how the model analyzes the frame one pattern at a time (T-P-U here), inhibiting what has been already analyzed at each iteration. A The 3 interesting patterns. B Rows: Each iteration for the single frame in A. Columns: Input, cell mapping at V1 and V2, and the remainder after extracting the analyzed pattern.

Figure 3 :

 3 Figure 3: Detail & Motor output. The system recognizes the pattern and triggers an action depending on how critical the pattern is. Top: Input and cell activation mapping at V1 and V2. Bottom: Reduced set of messages produced by the motor output block.

A map that shows cell activation rather than to show the visual representation at that stage of the system.

Acknowledgements

This work is supported by the ANR/CONICYT KEOpS project, the Lorraine Region and the CORTINA associated team.

* This work is supported by the ANR/CONICYT KEOpS project, the Lorraine Region and the CORTINA associated team.


Sylvain Kubler
email: sylvain.kubler@cran.uhp-nancy.fr

William Derigent

Eric Rondeau

André Thomas

Éric Rondeau

Embedding information on communicating materials from context-sensitive information analysis based on Fuzzy AHP theory

Keywords: Product Lifecycle Management, Data Dissemination, Fuzzy AHP, Pervasive Computing

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

New challenges and opportunities arise with concepts such as internet of things (IoT), artificial intelligence, or still ubiquitous computing [START_REF] Weiser | The computer for the 21st century[END_REF]. The idea of using intelligent products to ensure an information continuum all along the product life cycle (PLC) is more and more shared today [START_REF] Wong | The Intelligent Product Driven Supply Chain[END_REF]. Främling et al. [START_REF] Främling | Agent-based model for managing composite product information[END_REF] argue that it is a formidable challenge to link the product related information to the products themselves, making the information of all the product components easily achievable. To do so, several solutions and platforms have emerged such as EPCglobal, ID@URI or WWAI [START_REF] Meyer | Intelligent products: A survey[END_REF]. However, products are only given an identifier (e.g. via a RFID tag) which provides a network pointer to a linked database; these solutions do not answer the questions of (i) what information is relevant to users? and (ii) where information should be stored? (on the product or on the database) [START_REF] Mcfarlane | Intelligent products in the supply chain -10 years on[END_REF].

As a result, we propose a data dissemination process in two steps to disseminate information between databases and communicating products:

• Process step 1 aims at selecting, at a given moment of the PLC, relevant information from the database that should be stored on the product, • Once data that must be stored on the product is identified, it is necessary to develop tools to write/read this data on/from the product. This is achieved via process step 2 which was the subject of previous work [START_REF] Kubler | Storage of information on manufactured products using "communicating materials[END_REF], This paper takes focus on the process step 1 and an approach is proposed in section II and III to select contextsensitive information (from the database), that should be stored/replicated on the product. Our approach uses the fuzzy-AHP theory for aggregating multiple points of view.

II. TOWARDS THE USE OF THE FUZZY-AHP THEORY

This paper develops an approach to select contextsensitive information. This approach takes as input the Logical Data Model (LDM). Figure 1 gives insight into a part of such a LDM. One entity from the LDM corresponds to a relational table as depicted in Figure 1 (entity/table: Material), where the attributes listed in each entity correspond to the table columns and, each table row is referred to as a tuple/instance of the relation. Our approach consists in assessing the relevance of storing a given "data item" (i.e. one table cell) on the product. Figure 1 gives the relevance value of all data items from the table Material (e.g. the relevance of the data item located at row 3, column 1, noted T MD{3,1} , is equal to 0.2). The higher the relevance value, the higher the probability that the data item will be stored on the product. Finally, data items can therefore be classified in order of relevancy. In the PLC, several experts are concerned by selecting data that must be stored on the product, therefore generating different points of view. All points of view are legitimate and must be taken into account. Accordingly, this paper proposes an approach which combines the use of the analytic hierarchy process -AHP (to structure the multi-criteria decision making problem -MCDM) [START_REF] Saaty | The Analytic Hierarchy Process[END_REF] with the fuzzy logic (to express and aggregate multiple points of view). [START_REF] Chang | Applications of the extent analysis method on fuzzy ahp[END_REF] introduced a new approach for handling fuzzy-AHP, with the use of triangular fuzzy numbers for pairwise comparison scale and the use of the extent analysis method (FEAHP).

III. DATA RELEVANCE COMPUTATION USING FEAHP

In a classical set A, an element belongs entirely or not to A. In a fuzzy set Ã, an element can have different degrees of membership to Ã. For each element of the referential x, a degree of membership to the fuzzy set à is assigned, noted µ Ã(x). Thus, when µ Ã(x) = 1, x belongs entirely to Ã; when µ Ã(x) ∈]0; 1], x belongs more or less to Ã. FEAHP uses triangular fuzzy numbers, noted [L, M, U ] (see Figure 5) and consists of five stages as depicted in Figure 2. Each stage is described and detailed in the five next sections.

A. AHP structure

Our MCDM problem is broken down into the hierarchical structure depicted in Figure 3. The alternatives are the data items (cf. level 3) which must be assessed and ranked in term of relevancy (cf. level 1). Three criteria are defined at level 2: Enumeration, Contextual and Data Size which are respectively abbreviated C e , C c , C s and are detailed in the next stage/section.

B. Information collection + Aggregation

In this section, we provide information about the way to evaluate each alternative with respect to each criterion (C e , C c , C m ) and the way to obtain the criteria importance.

1) C e : Through this criterion, the user may enumerate information (s)he judges important to store on the product. To do so, each decision maker p enumerates attributes from tables. Let t a table of T and v an attribute of t. If the attribute v is enumerated by p, the enumeration score s p (v, t) = 1, otherwise 0, as noted in equation 1. Then, all decision makers' opinion s p (v, t) are aggregated in equation 2 through a triangular fuzzy number s(v, t). Let us remind ourselves that an attribute corresponds to a column of a table, which in turn, consists of one or many data items. As a result, if a data item l belongs to v, its score with respect to the criterion C e , noted φe (l), will therefore be equal to s(v, t).

s p (v, t) = 1 enumerated 0 not enumerated , ∀ p = {1..pmax} (1)
s(v, t) = L M U min(s p (v)) pmax p=1 s p (v) pmax max(s p (v)) (2)
2) C c : The previous criterion allows users to specify information that must be stored on the product. However, they could omit important information. Indeed, they might not be aware of all the data needed by the downstream actors (along the PLC). As a result, a new criterion referred to as contextual (C c ) is integrated in order to moderate and to balance C e . First, let us note that a multitude of information systems exist over the PLC (e.g. ERP, PDM, MES) which are not concerned by the same data (i.e. the same entities from the LDM). The idea is to identify specific "entity groups" through the LDM according to, for instance, the information systems and, therefore, to evaluate their importance over the PLC. In our study, each decision maker p performs pairwise comparisons between entity groups as in equation 3 with z the number of groups defined trough the LDM. The importance of entity group i over entity group j evaluated by the decision maker p is noted s p ij . This evaluation is based on the 1 to 9-point scale from [START_REF] Saaty | The Analytic Hierarchy Process[END_REF]: {1, 3, 5, 7, 9}. s p ij =1 means that groups i and j are equal in importance and s pij =9 means that group i is strongly favored over j. Then, all score s p ij are aggregated through a triangular fuzzy number, noted sij , in equation 4. Equation 5 is finally applied in order to acquire a unique score/triangular fuzzy number φc (G i ) for each entity group i. φc (G i ) actually indicates the relative importance of information involves in G i . If a data item l is contained in a table included in G i , its score with respect to C c , noted φc (l), is therefore equal to φG (G i ).

   G 1 • • • Gz G 1 1 • • • s p 1z . . . . . . . . . . . . Gz s p z1 • • • 1   , s p ji = s p ij i = j (s p ij ) -1 i = j (3) sij = L M U min(s p ij ) pmax p=1 s p ij pmax max(s p ij ) (4)
φc(Gi) = z j=1 s(i, j) z k=1 z j=1 s(k, j) (5)
3) C s : This criterion favors the storage of information on the product according to the data size. Since products are often memory-constrained, data relevance should decrease when data size increases. Such a behavior can be obtained via equation 6, with d the size of a data item l and k p a constant adjusted by the expert p. Figure 4 shows two functions according to the data size d and for two different k p . It can be observed that the smaller k p is, the bigger the data authorized to be stored on the product is (e.g., data with sizes > 60 bytes are neglected when k = 1.08).

(k p ) -d k p ∈ R + ; d ∈ N + (6)
The coefficient k p must be adjusted by each decision maker p as in equation 7. Then, all coefficients s p k are aggregated through a triangular fuzzy number, noted sk , in equation 8. The score of an alternative l with respect to C c , noted φc (l), is therefore computed in equation 9 with d the size of the data item l in bytes.

= L M U min(s p k ) pmax p=1 s p k pmax max(s p k ) (7) sk
φs(l) = L M U sk (U ) -d sk (M ) -d sk (L) -d (9) (8)
4) Criteria importance: Experts specify the criteria importance via pairwise comparisons as in C c . The relative importance of each criterion, noted φ(C x ) (see equation 10), is computed based on the same equations than previously (i.e. equations 3 to 5).

φx =

j={Ce,Cm,Cs} s(x, j) k={Ce,Cm ,Cs} j={Ce,Cm,Cs} s(k, j)

C. Fuzzy judgment matrix Ã

After getting all alternative scores with respect to all criteria, we form the fuzzy judgment matrix à as in equation 11, where φx (l) denotes the judgement score of alternative l with respect to criterion x.

An φe(n) φc(n) φm(n) φs(n)    (11)

D. Fuzzy Performance Matrix H

At this stage, only scores of alternatives with respect to criteria are taken into account, without considering the relative importance of criteria. As a result, we synthesize the fuzzy matrix à with the criteria importance in a fuzzy performance matrix H. The performance score hx (l) consists in multiplying the fuzzy set φx (l) by the criterion importance itself φx as in equation 12. Finally, the fuzzy performance matrix H is obtained as depicted in Figure 13.

E. Defuzzification and alternative ranking

This stage aims at ranking alternatives according to their fuzzy sets. To do so, defuzzification is first applied to transform each fuzzy set into a crisp value. Then, alternatives will be ranked by considering the crisp values based on the TOPSIS method. Defuzzification and TOP-SIS methods are respectively presented in what follows.

1) Defuzzification step: Defuzzification is executed (i) by using the α-cut method on the fuzzy performance matrix H and (ii) by taking into account the risk index β to compute the crisp performance matrix H α β . i. The α-cut level of an alternative l, noted h α lx , is computed via equations 14 to 16. h α lxa and h α lxa denotes respectively the left and right point of the level of the triangle as shown in Figure 5 (0 ≤ α ≤ 1). The α-cut value reflects the degree of confidence of the decision makers when they subjectively evaluate alternative and criteria scores for the MCDM problem. If the decision makers set up a high degree of confidence, it means that they have gathered enough knowledge to support their decisions.

α-cut L lx h α lxa h α lxb U lx M lx x 0 0.5 1 u Ã(x) h α lxa = L lx + α (M lx -L lx ) (14)
h α lxb = U lx -α (U lx -M lx ) (15)
h α lx = h α lxa h α lxb (16)
Figure 5. α-cut performed on each fuzzy performance score hlx

ii. In our approach, the risk index β is applied as a defuzzifier. The crisp performance matrix H α β is calculated in equation 18 based on equation 17, where h α lxβ denotes the crisp performance score of alternative l with respect to the criterion C x under a degree of confidence α and a risk level β. In equation 17, when β continuously increases, the crisp performance score progressively approaches the left point of the interval. A risk index β = {0, 0.5, 1} respectively indicates an optimistic, moderate and pessimistic viewpoint of the decision maker choice. 2) Alternative ranking: In TOPSIS, the ideal solution h α+ xβ and the negative ideal solution h α- xβ are respectively defined as the best and the worst crisp performance scores among all alternatives on a criterion, as specified in equation 19. Subsequently, the distance between the ideal and the negative ideal solution for each alternative is respectively calculated in equations 20 and 21, where S α+ iβ and S α- lβ represent the distance between the crisp performance scores h α lxβ of an alternative l with respect to all criteria C x and all the ideal and negative ideal solutions h α+ xβ , h α- xβ , respectively.

Figure 1 .

 1 Figure 1. View of a Logical Data Model (LDM) and a relational table

1 axFigure 2 .

 12 Figure 2. Fuzzy AHP consisting of 5 stages

Figure 3 .

 3 Figure 3. General architecture of the hierarchy

Figure 4 .

 4 Figure 4. Adjustment of k p performed by a decision maker p

∀ 2 l 2 l

 22 l = 1..n (19) S α+ lβ = x h α lxβh α+ xβ = 1, 2, .., n(20)S α- lβ = x h α lxβh α- xβ = 1, 2, .., n

A prior alternative has a longer distance to the negative ideal solution and a shorter distance to the ideal solution.

Consequently, the relative closeness to the ideal solution for each alternative can be formulated as in equation 22, where R α lβ denotes the final performance score. The larger the score R α lβ , the more relevant the alternative l. All alternatives/data items can therefore be ranked from the more relevant to the less relevant.

IV. CASE STUDY In this section, a specific scenario is considered in which 225 data items are assessed (data items coming from a LDM of 19 entities). Results about their relevance and ranking are commented hereafter.

Figure 6 provides the resulting list of data items ordered from the highest relevant (see R α lβ ) to the smallest. The two more relevant data items come from MaterialLot (from attributes Quantity and IDMaterialLot) and the 3 rd is T MD{3,3} (cf. Figure 1).

In our scenario, the 19 entities are clustered in 4 entity groups. The Equipment and Personal data groups, noted respectively G 1 and G 2 , report information about equipments and persons which/who are somehow related to the product (e.g. equipments used for manufacturing it). The Product and Production data groups, noted respectively G 3 and G 4 , relate respectively information about the product composition (e.g. raw material) and operations (e.g. production rule). In our scenario, adjustments made by experts concerned by the contextual criterion indicates that information included in G 3 is of the highest importance comparatively to the other groups. Moreover, the experts indicate that this criterion (C c ) is one of the most important at this stage of the PLC, with C e . No more than the first 155 data items from the list can be embedded on the product because its memory is limited (highlighted in Figure 6). The pie chart in Figure 7 shows the percentage of data items among the first 155 which belong to each entity group. We can observe that 49% (i.e. ≈ 75 data items) are included in tables clustered in G 4 and 38% in tables clustered in G 3 . This is largely due to the choices made by the experts. Finally, the first 155 data items from the list will be stored on the product thanks to process step 2 (cf. section I).

V. CONCLUSION

The idea of using intelligent products to ensure an information continuum all along the product life cycle 49% 38% 9% 4%

Personal data group (G2) Product data group (G3) Equipment data group (G1)

Production data group (G4)

Figure 7. Result of data item relevance in form of a pie chart (PLC) is more and more shared today. However, it is not that easy to identify the information that must be linked to the product. As a result, this paper proposes an information dissemination process for selecting information sensitive to the context of use of the product. This information is then stored on the products themselves using a new type of augmented material, referred to as "communicating material". The approach developed in this paper uses the fuzzy-AHP theory for aggregating multiple expert's points of view on the product, which are all legitimate.