Teddy Caderby
email: teddy.caderby@univ-reunion.fr

Eric Yiou

Nicolas Peyrot

Bruno Bonazzi

Georges Dalleau

Detection of swing heel-off event in gait initiation using force-plate data

published or not. The documents may come

Detection of swing heel-off event in gait initiation using force-plate data

Introduction

Gait initiation, which is the transition between the upright stance and the steady-state gait, is a classical paradigm to investigate the coordination between posture and movement. In this process, the swing heel-off (HO) time delineates the postural phase (the so-called ''anticipatory postural adjustments'' (APAs)) with the execution phase [START_REF] Brenie `re | Are dynamic phenomena prior to stepping essential to walking[END_REF]. The exact detection of the HO event is crucial for quantifying APA features, i.e. duration and amplitude.

In most studies, force-plates are used for biomechanical analysis of gait initiation, e.g. to determine the dynamics of the centre of pressure (CP) [START_REF] Brenie `re | Are dynamic phenomena prior to stepping essential to walking[END_REF]2] and the centre of gravity (CG) [START_REF] Brenie `re | Are dynamic phenomena prior to stepping essential to walking[END_REF]. Furthermore, additional hardwares such as pressure sensors or footswitches are usually used to detect the HO event [START_REF] Brenie `re | Are dynamic phenomena prior to stepping essential to walking[END_REF][START_REF] Corbeil | Combined effects of speed and directional change on postural adjustments during gait initiation[END_REF]. Although these devices are effective, they may hinder natural gait development [START_REF] Ghoussayni | Assessment and validation of a simple automated method for the detection of gait events and intervals[END_REF]. They also reduce the number of channels available for the simultaneous collection of analogue data (e.g., electromyography) [START_REF] Mickelborough | Validity and reliability of a kinematic protocol for determining foot contact events[END_REF]. An alternative method using kinematic data has been proposed [START_REF] Mickelborough | Validity and reliability of a kinematic protocol for determining foot contact events[END_REF], but the cost and the complexity of collecting measurements of motion capture systems limit its use [START_REF] Martinez-Mendez | Detection of anticipatory postural adjustments prior to gait initiation using inertial wearable sensors[END_REF].

To avoid these limitations, many studies use only force-plate data to detect HO. Several methods based on CP displacement [2,[START_REF] Yiou | Effects of medio-lateral postural perturbation induced by voluntary arm raising on the biomechanical organization of rapid step initiation[END_REF][START_REF] Rogers | Triggering of protective stepping for the control of human balance: age and contextual dependence[END_REF] or CG vertical dynamics [START_REF] Ledebt | The build-up of anticipatory behaviour. An analysis of the development of gait initiation in children[END_REF] have been described in the literature. Nevertheless, it appears from these studies that there is no clear consensus on the method to be used to detect this instant. The questions of whether the accuracy and reliability of these methods are equivalent and how they compare with the footswitch, the hardware regularly used, remain open.

The aim of this study was to assess the accuracy and reliability of four force-plate methods (three methods previously reported in the literature and a novel method) to detect HO and to compare their performance with a reference method using a footswitch.

Methods

Ten healthy subjects (eight males and two females, age 23 AE 2 years (mean AE standard deviation [SD]), height 173 AE 9 cm, weight 72 AE 11 kg) were recruited for this study after being fully informed of the experimental procedure approved by the local ethics committee.

Each subject initially stood upright, barefoot and motionless on a force-plate (40 cm  60 cm, AMTI, USA). Beforehand, a footswitch (25 mm, Biometrics, France) was secured to the force-plate under the heel of the swing foot. To ensure that subjects returned to the same initial position after each trial, the foot positions were outlined on the force-plate. A second force-plate was positioned in front of the subject at a distance appropriate for the swing foot to step onto it. At the experimenter's signal, subjects initiated gait and continued walking straight ahead along a 5-m walkway. The swing leg was self-selected and remained the same throughout the experiment. Each subject performed series of 18 trials in two gait speed conditions: a self-selected speed and a maximal speed. Three practice trials were performed in each condition before the recordings. The order of the speed conditions was randomized across subjects.

The first force-plate was used to detect the HO time; the second force-plate was simply used to obtain all reaction forces during the entire gait initiation time-window (Fig. 1). The force-plate and footswitch signals, sampled to 1000 Hz, were transmitted to an acquisition system and synchronized. Before analysis, the force-plate data were low-pass filtered at 10 Hz [START_REF] Corbeil | Combined effects of speed and directional change on postural adjustments during gait initiation[END_REF] with a fourthorder zero-lag Butterworth filter. The mediolateral CP coordinate (yP) was

A B S T R A C T

This study investigated the accuracy and reliability of four methods using force-plate data for detecting the swing heel-off (HO) time in gait initiation. Results of these methods were compared to those obtained by means of a reference method using a footswitch. Ten young healthy adults performed 18 forward gait initiation trials at self-selected speed and at maximal speed. Results showed that the method based on vertical impulse was the most accurate and reliable in determining HO in both speed conditions. The mean error obtained with this method was À8 AE 10 ms in the self-selected speed condition (À7 AE 10 ms in the maximal speed condition), with no significant effect of gait speed (P > 0.05). These findings suggest that this method based on force-plate data is valid and reliable for detecting HO in forward gait initiation in the absence of additional hardware. calculated from force-plate data and was corrected by the 250 ms baseline preceding the start signal [START_REF] Corbeil | Combined effects of speed and directional change on postural adjustments during gait initiation[END_REF]. Vertical impulse (Iz) was calculated by integrating (trapezoidal rule) the vertical ground-reaction force (Fz), after weight correction (Fig. 1). Calculation was performed with integration constant equal to zero, i.e. initial impulse null [START_REF] Brenie `re | Are dynamic phenomena prior to stepping essential to walking[END_REF].

The reference HO time (HO ref ) was determined from the footswitch signal, as shown in Fig. 1(a). Four methods based on force-plate data were used to estimate the HO time. For each method, the HO event was detected manually by two experimented raters as illustrated in Fig. 1: -Method 1 detected HO when yP first peaked towards the swing limb [2,8] (b).

-Method 2 identified HO when yP changed sign [7] (c).

-Method 3 predicted HO when Fz peaked downwards (based on [9]) (d).

-Method 4 was custom-designed and detected HO when Iz peaked downwards (e).

The mean and standard deviation of the differences (errors) between the HO instant estimated by each of the four force-plate methods and that obtained by the footswitch method were calculated to assess the accuracy and reliability of these methods [START_REF] Leitch | Identifying gait events without a force plate during running: a comparison of methods[END_REF]. A positive value indicated that the HO time was delayed compared with the reference method while a negative value indicated it was early. The differences were expressed in milliseconds and as a percentage of APA duration (dAPA). APA duration was defined as the delay between the first dynamic phenomena (t 0 ) and HO time [START_REF] Brenie `re | Are dynamic phenomena prior to stepping essential to walking[END_REF]. After checking for data normality with Kolmogorov-Smirnov test, Student's t-tests for paired samples were used to assess the effect of gait speed on the error made with each method. Threshold for significant difference was P < 0.05.

Results

Table 1 shows the differences (mean AE SD) between the instants of HO predicted by each force-plate method and the reference timing identified from the footswitch sensor. The errors distribution for each method is illustrated as histogram in Fig. 2. On average, the methods detected HO earlier than the reference method (Table 1). In both gait speed conditions, Method 4 provided the smallest mean error and SD among the tested methods (Table 1). The mean error on HO detection obtained with this method was À8 AE 10 ms in the self-selected speed condition and À7 AE 10 ms in the maximal speed condition, with no significant difference across conditions (P > 0.05). Note that this error corresponded to solely À1 AE 2% of APA duration. In absolute terms, the errors of this method were less than 30 ms in 96% of trials in the self-selected speed condition (97% in the maximal speed condition; Fig. 2g andh).

In contrast to Method 4, statistical analysis revealed that the mean error (expressed in percentage of APA duration) significantly decreased with gait speed for Method 1 (t = À4.562, P < 0.01), Method 2 (t = À2.762, P < 0.05) and Method 3 (t = À3.250, P < 0.05). When expressed in milliseconds, it significantly decreased with gait speed only for Method 1 (t = À2.974, P < 0.05).

Discussion

Globally, the methods detected HO earlier than the reference method. Some authors [START_REF] Martinez-Mendez | Detection of anticipatory postural adjustments prior to gait initiation using inertial wearable sensors[END_REF] had already reported this trend for Method 2. In the current study, Method 4 presented the smallest error and SD whatever the gait speed, indicating that this method was the most accurate and reliable method to detect HO in those tested. The errors of this method were smaller than 30 ms in 96% of trials in the self-selected speed condition (97% in the maximal speed condition). It is noteworthy that these results are slightly better than those found using a kinematic method where errors are less than 30 ms in over 92.5% of the cases [START_REF] Mickelborough | Validity and reliability of a kinematic protocol for determining foot contact events[END_REF].

The high reliability and accuracy of Method 4 might be ascribed to a causal link between the HO time and the marker used to detect Table 1 The means and standard deviations (SD) of the differences between the instants of swing heel-off (HO) predicted by the various methods and the reference timing identified from the footswitch sensor, expressed in milliseconds (ms) and as a percentage of APA duration (%dAPA). Method 1: algorithm identifying HO when the mediolateral displacement of the CP first peaked towards the swing limb. Method 2: algorithm identifying HO when the mediolateral displacement of the CP changed sign. Method 3: algorithm using the vertical ground reaction force. Method 4: algorithm using the vertical impulse. a Significant difference self-selected versus maximal speed, P < 0.05. b Significant difference self-selected versus maximal speed, P < 0.01. this time (peak of downward Iz). HO is indeed associated with swing leg rise, which could act to attenuate the fall of the CG (classically described using the CG vertical velocity [START_REF] Welter | Control of vertical components of gait during initiation of walking in normal adults and patients with progressive supranuclear palsy[END_REF]) during the subsequent execution phase. In other words, it could be that the reversal of the negative vertical impulse might be ascribed to an effect of the swing leg rise associated with HO. The fact that the accuracy and reliability of Method 4 are speed-independent (in contrast to the other methods) is in agreement with this hypothesis.

Results further showed that SD values ranged from 10 ms (i.e. 2% of APA duration) with Method 4 to 86 ms (13% of APA duration) with Method 1 (Table 1). This range of SD values suggests that the reliability of the HO time measure (and therefore the reliability of APA spatio-temporal features) may strongly depend on the method used. Future studies on APA in gait initiation should therefore be cautious with regard to the method used to detect HO. Further investigation is also required to determine whether the present results apply to populations with gait disorders.

Conclusion

The results showed that the method based on vertical impulse was the most accurate and reliable for detecting HO time from force-plate data. In the absence of additional hardware, this method constitutes an attractive alternative to detect the swing heel-off time in forward gait initiation.

Fig. 1 .

 1 Fig. 1. Illustration of the various methods to detect the swing heel-off (HO) event.Traces are shown for one representative subject initiating gait at self-selected speed with the right leg. FS signal, yP, Fz, Iz: footswitch sensor signal, mediolateral displacement of centre of pressure, vertical ground reaction force, vertical impulse, respectively. These traces were corrected by the 250 ms baseline preceding the start signal. t 0 , HO ref , FC were the onset variation of the yP trace from the baseline, reference swing heel-off instant, and swing foot-contact, respectively. F, B, ST, SW, UP and DOWN indicate forward, backward, stance limb, swing limb, upward and downward respectively. The footswitch sensor signal was used to identify the reference HO time (HO ref ) (a). Method 1 predicted HO when yP reached its maximal value towards the swing limb (b). Method 2 predicted HO when yP changed sign (the first frame was considered) (c). Method 3 identified HO when Fz peaked downwards (peak preceding the last downward peak) (d). Method 4 identified HO when Iz peaked downwards before the last (usually maximal) downward peak (e).

Acknowledgements

This work was supported by a grant from the University of La Reunion. The authors wish to thank Mrs. Kytte Tameire for his technical assistance.

Conflict of interest statement

None of the authors have financial or other conflicts of interest in regards to this research.


Stefen Chan

Wai Tim

Michele Rombaut

Denis Pellerin

Multi-layer Dictionary Learning for Image Classification

This paper presents a multi-layer dictionary learning method for classification tasks. The goal of the proposed multi-layer framework is to use the supervised dictionary learning approach locally on raw images in order to learn local features. This method starts by building a sparse representation at the patch-level and relies on a hierarchy of learned dictionaries to output a global sparse representation for the whole image. It relies on a succession of sparse coding and pooling steps in order to find an efficient representation of the data for classification. This method has been tested on a classification task with good results.

Introduction

Sparse coding is the approximation of an input signal by a linear combination of a few number of dictionary elements. Dictionary learning and sparse representations have received a lot of focus in the recent years because they have led to state-of-the-art results in many applications, in particular in image processing. One reason for its success is that it can efficiently learn the underlying patterns in the data, leading to good performances for example in denoising [START_REF] Elad | Image denoising via sparse and redundant representations over learned dictionaries[END_REF], [START_REF] Mairal | Sparse representation for color image restauration[END_REF] or inpainting [START_REF] Aharon | K-svd: an algorithm for designing overcomplete dictionaries for sparse representation[END_REF]. The sparse codes obtained can also be seen as a new representation of the input or as features in classification tasks [START_REF] Mairal | Task-driven dictionary learning[END_REF], [START_REF] Raina | Self-taught learning: Transfer learning from unlabeled data[END_REF], [START_REF] Bradley | Differential sparse coding[END_REF], [START_REF] Wright | Robust face recognition via sparse representation[END_REF], [START_REF] Wai | Rejection-based classification for action recognition using a spatio-temporal dictionary[END_REF]. In such cases, the dictionary is often learned in an unsupervised way. Then, the sparse codes obtained can either be used directly for classification [START_REF] Qiu | Sparse dictionary-based representation and recognition of action attributes[END_REF], or as features fed to a classifier [START_REF] Wai | Rejection-based classification for action recognition using a spatio-temporal dictionary[END_REF] (i.e SVM).

Recent researches have emphasized the advantages of learning discriminative sparse models [START_REF] Mairal | Task-driven dictionary learning[END_REF], [START_REF] Bradley | Differential sparse coding[END_REF], [START_REF] Yang | Supervised translation-invariant sparse coding[END_REF] instead of purely reconstructive ones. It is usually done by learning conjointly the sparse representation and the classifier. In practice, each input image is matched with a label and the dictionary is learned in a supervised setup.

Generally, in image processing applications, dictionary learning and sparse coding are computed on a small portion of an image (i.e image patches) because learning a dictionary directly on high resolution images is computationally intensive. There is no particular problem doing so in denoising, however, in the case of classification, a mean to fuse efficiently the representation of the patches into an image-level descriptor is needed (i.e pooling [START_REF] Yang | Supervised translation-invariant sparse coding[END_REF] or Bag of words [START_REF] Wang | Evaluation of local spatio-temporal features for action recognition[END_REF]).

In this paper, we propose to learn discriminative dictionaries for classification (similarly as in [START_REF] Yang | Supervised translation-invariant sparse coding[END_REF]) while working at a patch-level in a supervised framework by using an architecture which combines many layers of sparse coding and pooling in order to reduce the dimension of the problem.

Method framework

In the proposed multi-layer architecture, the sparse codes obtained by encoding signals on a dictionnary are used as inputs to a subsequent coding layer. Each additional layer of dictionary encoding changes the representation by projecting the features into a new space. The prospective objective is to increase the discriminability of the features by building a hierarchy of dictionaries.

In Section 2, we recall the dictionary learning framework going from the unsupervised setup to the supervised dictionary learning setup. In Section 3, we introduce our multi-layer dictionary learning setup. In Section 4, we present the experiments and their results.

Dictionary learning

In this section, we recall various formulations of the dictionary learning problem, starting with an unsupervised method more suited to reconstruction and followed by the supervised method tailored around a specific task (here, classification).

Unsupervised dictionary learning

Dictionary learning has been widely used in reconstruction tasks. In its classical formulation, the goal is to learn a set of atoms directly from data. Let's consider a set of n signals Y = [y 1 , ..y n ]. A dictionary D can be learned by solving:

min D,x k n k=1 y k -Dx k 2 2 + λ x k 1 (1)
with D = (d ij ) i∈ [1,m],j∈ [1,K] being a dictionary, K the number of dictionary atoms d j , m the dimension of y k , and x k a sparse vector containing the coefficients to reconstruct y k . • 2 and • 1 denote 2 -norm and 1 -norm respectively. Once a sparse code x k is obtained, the original signal can be approximated by computing ŷk ≈ Dx k . This problem has been widely studied and many approaches exist in order to get both dictionary D and coefficients x k [START_REF] Aharon | K-svd: an algorithm for designing overcomplete dictionaries for sparse representation[END_REF], [START_REF] Olshausen | Emergence of simple-cell receptive field properties by learning a sparse code for natural images[END_REF], [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF].

In this formulation, the reconstruction error is minimized and the sparsity can be controlled with the value of the parameter λ (a higher λ increases sparsity). Using such unsupervised approach can yield good results in reconstruction problem and even in classification tasks [START_REF] Raina | Self-taught learning: Transfer learning from unlabeled data[END_REF], [START_REF] Wright | Robust face recognition via sparse representation[END_REF], because it can often find the underlying patterns in the data. However, it has been shown that better results could be obtained by tuning the dictionaries for a specific task [START_REF] Mairal | Task-driven dictionary learning[END_REF], [START_REF] Bradley | Differential sparse coding[END_REF].

Supervised dictionary learning

Supervised dictionary learning methods began to be investigated [START_REF] Mairal | Task-driven dictionary learning[END_REF], [START_REF] Yang | Supervised translation-invariant sparse coding[END_REF] in order to take advantage of parsimony in classification tasks. Encoding a datum using a dictionary can be seen as a projection into another coordinate system. The objective is to obtain projected features that are discriminative in the new space.

Let's assume we know a set of pairs (y k , l k ) where y k , k ∈ [1, n] is a set of signals (i.e images represented as column vectors) and l k , k ∈ [1, n] is the corresponding label for y k . We define L, a differentiable classification loss function and W, its set of parameters.

The supervised dictionary learning problem can be written as follows:

xk = argmin x y k -Dx 2 2 + λ x 1 (2) min W,D n k=1 L(l k , xk , W) (3)
The objective here, is to minimize the suitable cost function L with respect to its parameters W and a dictionary D (Fig. 1) using gradient descent for example. The main difference between this formulation and the previous one (Eq. 1) is that the goal now is to minimize the classification loss instead of a reconstruction error term. To minimize the cost function L with respect to the parameters D and W, it is possible to use a method similar to the backpropagation algorithm [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF] used in neural networks.

Computing the gradient of L with respect to the parameters W is usually simple. The main difficulty when solving (Eq. 3) is the minimization of the cost function L with respect to dictionary D because it does not appear explicitly and involves another optimization problem (solving for xk , (Eq. 2)). To overcome this problem, a way to compute the gradient of the cost function L(l k , xk , W) with respect to the dictionary D is needed. This problem has been tackled in [START_REF] Mairal | Task-driven dictionary learning[END_REF], [START_REF] Bradley | Differential sparse coding[END_REF].

In this paper, we follow the ideas of [START_REF] Mairal | Task-driven dictionary learning[END_REF] which show the differentiability of L and give the steps to compute its gradient with respect to the parameters W and the dictionary D. According to the paper, the desired gradient can be computed as follows :

∇ D L(l k , xk , W) = Dβ x k + (y -Dx k )β (4)
We define Λ = {i|x i = 0}), the set of non-zero coefficients of the considered code x k .

Let's consider a vector β ∈ R K . β Λ which is the vector β restricted to the indices in Λ is defined as follows :

β Λ = (D Λ D Λ ) -1 ∇ xkΛ L(l k , xk , W) (5)
where xkΛ corresponds to xk restricted to its non-zero coefficients and β j = 0, if j / ∈ Λ.

Patch decomposition and pooling

Dictionary learning methods have been used for reconstructing or classifying either full image or image patches. It means that in practice, a signal y k can be either an image (Section 2.2) or a patch reshaped as a column vector containing the pixel values (Section 3.2).

A supervised dictionary learning approach can successfully learn patterns for a classification task. However, in practice, some limitations can be observed: in particular, if we take the example of image classification, these methods are the most effective when the input images are relatively small and the object of interest homogeneously localized. Otherwise, the dictionary used would need a huge number of atoms in order to obtain an efficient sparse decomposition. Moreover, classifying a set of patches extracted from an image instead of the image itself is different, from the dictionary methods point of view. Indeed, when dealing with patches, a method to fuse the information of the set of patches is needed.

This particular problem has been studied by Yang et al. [START_REF] Yang | Supervised translation-invariant sparse coding[END_REF]. In the paper, the authors proposed to perform a single sparse coding step at patch level (with a patch size smaller than the input image) followed by numerous pooling steps in order to efficiently reduce the dimensions and obtain a multi-scale representation.

To be able to deal with large images, we recall the pooling function: the pooling operation is used to insert robustness and translation invariance to the features and are an effective way of reducing the dimensions. It is a mean of aggregating spatial local features in an image. Let's consider the classification of an image I. First, it can be decomposed into local overlapping patches y k which are encoded on dictionary by computing the coefficients xk . These patches can be spatially localized on a grid which gives the relative position of all patches. Then the codes xk obtained can be, for example, averaged over a small group of patches reducing the total number of patches.

3 Multi-layer supervised dictionary learning

Multi-layer framework

Intuitively, sparse coding can extract important characteristics for reconstruction in unsupervised frameworks and for classification in supervised methods.

The idea of the contribution is to reiterate the sparse coding layer in order to increase the discriminability of the features. The method is inspired by the convolutional networks [START_REF] Springenberg | Striving for simplicity: The all convolutional net[END_REF]: the convolution by a filter is replaced by a sparse coding step. The other goal of the proposed method is to control the dimension of the input patches by reducing the sparse coding of a large image, computationally intensive, to the sparse coding of small patches which can be processed more efficiently.

Encoding a vector on a dictionary is similar to projecting into a new space. The projection is non-linear and the resulting vector is sparse. The vector is then used as a new input for the following layer. So, adding another dictionary encoding step is akin to doing another projection in a new coordinate system. Each dictionary is learned in a supervised manner. The process can be repeated many times, with as much dictionaries as the number of layers in the architecture.

The proposed multi-layer can be described as follows (Fig. 2):

1. An input image is decomposed into a set of overlapping patches ordered to retain their spatial localization. 2. Each patch y k is encoded on a first dictionary D (1) . Since the spatial localization of the patches has been retained, the set of encoded sparse codes xk can be represented as a 3D volume X with a depth equal to the number of atoms in dictionary D (1) . 3. The resulting 3D volume is treated as a 3D image input for the next layer.

(1) and ( 2) can be repeated for the chosen number of layers.

To complement 3), for example, if we consider the q-th layer, we can write: Y (q) = X(q-1) , meaning that the stacked codes at layer q -1 are used as an input image Y (q) in layer q (see Fig. 2). The image Y (q) is then decomposed into 3D patches ŷ(q) k and x(q) k are obtained by encoding y (q) k with D (q) . More generally, we can replace Y (q) = X(q-1) by y (q) k = f ( X(q-1) ) where f can denote a transformation on the codes coupled with the patch decomposition.

In order to optimize the multi-layer architecture for classification, it is needed to find the optimal dictionaries at each layer. k which are encoded by dictionnary D (1) . The codes x

(1) k are processed and restructured into a 3D volume X (1) and then decomposed again into 3D patches y

(2) k in the second layer. These patches are encoded using the dictionary D (2) .

Formulation

Let's consider a set of input features Y = {y

(1) 1 , • • • , y (1)
p } associated to a label l. This set is obtained by decomposing an input image into p patches. In the section, to simplify the notations, each subscript k used for denoting the index of patches or codes is tied to a specific layer q (it can be read as k q ). The upper index (q) denotes the q-th layer. Let's consider the proposed algorithm with Q layers, its formulation is as follows:

For q ∈ [1, Q -1] (same as in (Eq. 2))

x(q) k = argmin

x y (q) k -D (q) x 2 2 + λ q x 1 (6) y (q+1) k
is obtained from the output of the previous layer x(q) k by applying a transformation f (see [START_REF] Yang | Supervised translation-invariant sparse coding[END_REF] for an example with f being the max-pooling function) followed by a new patch decomposition step. In this paper, we propose to use the average-pooling function.

To optimize the classification cost function with respect to the dictionaries at each layer, we use the backpropagation algorithm. Therefore, we need to compute the gradients with respect to the various dictionaries D (1)

, • • • , D (Q) .
By extending the formulation given in (Eq. 3), we obtain: min

W,D (1) ,••• ,D (Q) n k=1 L(l k , x(Q) k , W) (7)
We only use the output x(Q) k of the last layer as features for the classification (Fig. 3) and the cost function is minimized over the entire training set of n images. = f ( X( 1) ) as input to the second layer. Backpropagation is then used to update both dictionaries simultaneously.

Computation of the gradients

To use the backpropagation algorithm (i.e computing each gradient using the chain rule, the gradient is computed the same way as presented in Section 2.2 by replacing xk by x(q) k and y k by y (q)

k . For a pair (Y, l), the gradient of L with respect to dictionary D (Q) (the dictionary of the last layer) is computed using equations (Eq. 4) and (Eq. 6). If we introduce the notation using the layer number, the equation for the last layer becomes:

∇ D (Q) L(l k , x(Q) k , W) = D (Q) β x(Q) k + (y (Q) k -D (Q) x(Q) k )β (8)
with β defined as: for the indexes contained in the set Λ,

β Λ = (D (Q) Λ D (Q) Λ ) -1 ∇ x(Q) kΛ L(l k , x(Q) k , W) (9)
and β j = 0, if j / ∈ Λ. To compute the gradient of the layer q, the computations for the dictionary become:

∇ D (q) L(l k , x(Q) k , W) = D (q) β x(q) k + (y (q) k -D (q) x(q) k )β ( 10
)
β Λ = (D (q) Λ D (q) Λ ) -1 ∇ x(q) kΛ L(l k , x(Q) k , W) (11)
We underline that only the last layer Q intervenes in the classification step that is why the term is L(l k ,

x(Q) k , W) and not L(l k , x(q) k , W): by choice, the output of the last layer is a single code vector. To compute the gradient of the cost function L(l k ,

x(Q) k , W) with respect to the dictionary D (q) of the qth layer using backpropagation, we need to compute ∇ x(q) kΛ L(l k ,

x(Q) k , W). The computation of this gradient can be decomposed as follows:

∂L ∂ x(q) = ∂L ∂ x(q+1) ∂ x(q+1) ∂y (q+1) ∂y (q+1) ∂ x(q) ( 12
)
where ∂ x(q) ∂y (q) : ∂

x(q) Λ ∂y (q) = (D (q) Λ D (q) Λ ) -1 D (q) Λ (13)
and 0 elsewhere. We remind that the transformation f between x(q) and y (q+1) can be the identity, a pooling operation or a more general transformation (see 3.1) combined with a decomposition into patches, so the backpropagation includes a image "reconstruction" step to reverse the patch decomposition (Fig. 4).

Experimentations

We tested the proposed algorithm on the MNIST [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF] dataset and CIFAR-10 dataset [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF]. The first well-known MNIST dataset used for classification regroups a set of handwritten digits divided in 10 classes (i.e 0 -9) and contains 60000 (28 × 28) pixels images for training and 10000 for testing. These images have been rescaled to (32 × 32) pixels to be able to fit the CIFAR-10 dataset.

We have chosen to use the cross-entropy function (Eq. 14) for the classification loss as it has proven to give good results in multiclass classification problems. The chosen classifier is a linear classifier coupled with softmax for the output. If we consider a classification problem with C classes, the cross-entropy loss is computed as follows:

L(l k , x(Q) k , W) = - C i=1 l ik log(p ik ) (14)
where p ik is defined by:

p ik = exp(x (Q) k w i ) C j=1 exp(x (Q) k w j )
To demonstrate the proposed method, we choose to use an architecture with Q = 3 layers. For the first layer, we decompose the input image into patches of (5 × 5) pixels with a stride of 1 pixels. The second layer use patches of size (5 × 5). Then, a pooling step is done on a 2 × 2 region without overlap. We follow last layer with size (5 × 5). After the last layer, only one code remains for the whole image and this code is used as input for the classification.

The optimization method used is the stochastic gradient descent with a batch size of 10. The training set is shuffled randomly and the training samples for each batch are used in order. The learning step is initially fixed at 0.3 and divided by 2 every 3 passes on the full dataset.

For parameter λ (Eq. 1), we choose λ = 0.1 for all three layers. Empirically, this value leads to good reconstruction while giving very sparse codes. Increasing this value too much can lead to patches not being reconstructed (only zeros) while reducing the value (i.e by an order of magnitude) increases the number of non-zero coefficients and the computation costs without necessarily improving the performances.

We tested two configurations of architecture: we used K = 25, 25, 50 atoms and K = 50, 50, 100 atoms for the three layers. Increasing the number of atoms in the intermediate layers leads to very high dimensional input for the subsequent layer hindering the computational performances. The tests has been run on Matlab and we could not test very large dictionaries.

The results in Table 1 are obtained with the classifier directly learned by the algorithm. We used the original training and test sets with no data augmentation. Supervised learning greatly improves the perfomances for the tested configurations. Moreover, the gain in perfomances between the two choices of the number of atoms is small even though the number of atoms is doubled. It may be explained by the fact that the images in the MNIST dataset are not really complex so additional atoms are not needed to describe more features.

Table 2 regroups some performances of state of the art methods on the MNIST dataset. The results for the proposed method are obtained with no data augmentation as opposed to [START_REF] Mairal | Task-driven dictionary learning[END_REF] and [START_REF] Yang | Supervised translation-invariant sparse coding[END_REF].

Methods

Dictionary size K

Error rate 1 layer (unsupervised) 700 3.71% 3 layers (unsupervised) 25, 25, 50 4.93% (5 × 5), (5 × 5), (5 × 5) 3 layers (unsupervised) 25, 25, 50 2.2% (5 × 5), (5 × 5), (5 × 5) 3 layers (supervised) 25, 25, 50 0.46% (5 × 5), (5 × 5), (5 × 5) 3 layers (supervised) 50, 50, 100 0.41% (5 × 5), (5 × 5), (5 × 5) Table 1. Performance comparison between supervised and unsupervised learning with a linear classifier on the MNIST dataset.

Methods

Error rate Yang et al. [START_REF] Yang | Supervised translation-invariant sparse coding[END_REF] 0.84% Mairal et al. [START_REF] Mairal | Task-driven dictionary learning[END_REF] 0.54% Proposed method 0.41% We also tested our method on the CIFAR-10 dataset [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF] which is constructed with 60000 real images (50000 images for learning and 10000 images for testing) of size (32 × 32) pixels, separated into 10 classes. This dataset is more challenging than the MNIST dataset because the variance of view points, poses and localizations of the object of interest is much higher. In particular, the dictionary learning methods which process the whole image at once usually have some difficulties to learn the features efficiently. For this test, we used the same structure as the one used for the MNIST dataset (3 layers) with the same parameters (Table 3). Only a few works present classification results on the CIFAR-10 dataset using a dictionary learning method only. However, other methods exist (i.e Convolutional neural networks) [START_REF] Hinton | Improving neural networks by preventing co-adaptation of feature detector[END_REF] to deal with this kind of dataset.

Methods

Accuracy Fawzi et al. [START_REF] Fawzi | Dictionary learning for fast classification based on soft-thresholding[END_REF] 53.44% Coates et al. [START_REF] Coates | An analysis of single-layer networks in unsupervised feature learning[END_REF] 79.6% Coates et al. [START_REF] Coates | The importance of encoding versus training with sparse coding and vector quantization[END_REF] 81.5% Proposed method 83.03% Table 4. Performance comparison on the CIFAR-10 dataset.

Table 4 shows some results on the CIFAR-10 dataset without data augmentation. Fawzi et al. [START_REF] Fawzi | Dictionary learning for fast classification based on soft-thresholding[END_REF] use a single layer dictionary learning method for comparison. Coates et al. [START_REF] Coates | An analysis of single-layer networks in unsupervised feature learning[END_REF], [START_REF] Coates | The importance of encoding versus training with sparse coding and vector quantization[END_REF] used unsupervised multi-layer sparse coding with large dictionaries (up to 4k atoms). The proposed method performs better using a few number of atoms (undercomplete dictionaries) showing the capability of learning discriminative dictionaries. Better performances may be obtained by increasing the number of atoms in the layers, however at the moment, the MATLAB implementation used is too slow.

During our experiments, we noticed that the number of atoms in the first layer (image layer) is important, for example choosing K = 15, 25, 50 (15 instead of 25 in the first layer) could leave to a drop of about 10% in performances. The same can also be said of the number of atoms of the last layer (size of the features) but with a lesser extent.

Conclusion

In this paper, we have presented a multi-layer dictionary learning framework. It can potentially handle an image of any size as input and performs the learning of features at the patch level. Its goal is to allow the use of supervised dictionary learning methods on images while evading the computational issues that arise when dealing with large dictionary atoms.

We still have not fully investigated this method and we will continue to work on the proposed structure in order to study the effects of the choices of the different parameters (dictionary size, sparsity, number of layers). For future work, we will confront this method with more complex datasets, containing larger images, to challenge the limit of this approach.

Fig. 1 .

 1 Fig. 1. A signal y k associated to a label l k is encoded by a dictionary D, the resulting code xk is used as an input to a cost function with parameters W. The cost L(l k , xk , W) is computed. Then, the dictionary D and the parameters W are updated to fit the classification problem.

Fig. 2 .

 2 Fig. 2. Example of an architecture with 2 layers. An input image in presented to the first layer. The image is decomposed into patches y (1)

Fig. 3 .

 3 Fig. 3. Proposed structure with 2 layers. Input vectors go through a sparse coding step. We have y (2) k

Fig. 4 .

 4 Fig. 4. Example of the reconstruction step from 5 × 5 patches to 7 × 7 × K volume.

Table 3 .

 3 Performance comparison between supervised and unsupervised learning with a linear classifier on the CIFAR-10 dataset.

Table 2 .

 2 Performance comparison on the MNIST dataset.

Acknowledgement

This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01).


Yann Le Gal

Mangaiyarkarasi Rajkumar

Antoine Vacher

Vincent Dorcet

Thierry Roisnel

Marc Fourmigué

Frédéric Barrière

Thierry Guizouarn

Dominique Lorcy
email: dominique.lorcy@univ-rennes1.fr.

A sulfur-rich -electron acceptor derived from 5,5'-bithiazolidinylidene: charge-transfer complex vs. charge-transfer salt

Novel -electron acceptors are still highly desirable for the formation of conducting salts or as n-dopable semiconductors. We describe here two synthetic approaches to substitute a dicyanovinylidene group, C=C(CN) 2 for a thioketone (C=S) in the recently described DEBTTT acceptor where DEBTTT stands for (E)-3,3'-diethyl-5,5'-bithiazolidinylidene-2,4,2',4'-tetrathione. These electron withdrawing groups enhance the electron accepting ability as demonstrated through electrochemical investigations, without hindering the formation of short intra-and intermolecular S•••S contacts in the solid state. Association of this acceptor 1 with tetramethyltetrathiafulvalene (TMTTF) and decamethylferrocene (Fe(Cp*) 2 ) afforded 1:1 adducts which were analyzed by single crystal X-ray diffraction. Combined with vibrational and magnetic properties, it appears that [TMTTF][1] behaves as a neutral charge-transfer complex while [Fe(Cp*) 2 ][1] is an ionic salt. The concentration of the spin density on the exocyclic sulfur atoms in 1 -• , favors the setting of direct antiferromagnetic interactions in [Fe(Cp*) 2 ][1].

Introduction

Organic electron acceptors with a -conjugated framework have focused a lot of attention either for the elaboration of conducting materials, 1 or for the development of next generation of organic field-effect transistors (OFETs). 2,3,4,5 The elaboration of conducting materials can result from the redox reaction between a π-electron donor and a π-electron acceptor as in the prototypical TTF•TCNQ charge transfer (CT) salt. 1 These molecular materials are formed if the oxidizing agent, the acceptor, and the reducing agent, the donor, exhibit the appropriate electron affinity (EA) and ionization potential (IP) respectively to allow a partial or total CT between the donor and the acceptor. 6,7 The IP of the donor and the EA of the acceptor can be estimated electrochemically from their redox potential values in solution and the difference between the oxidation potential of the donor (E ox ) and the reduction potential of the acceptor (E red ) indicates if a CT is susceptible to occur. 6,7 Within this framework, we recently reported the synthesis of a sulfur-rich electron acceptor, the DEBTTT [(E)-3,3′-diethyl-5,5′-bithiazolidinylidene-2,4,2′,4′-tetrathione], which has proved to be a valuable acceptor in the preparation of CT salt with [Fe(Cp*) 2 ]. 8 It is worth mentioning that DEBTTT is also a valuable candidate in the elaboration of air-stable n-channel organic field-effect transistors, since the rare presence of sulfur atoms in such acceptors favor stronger intermolecular interactions for enhanced charge mobilities. 9 Nevertheless, the oxidizing ability of this acceptor is below those traditionally used to oxidize tetrathiafulvalene derivatives. Thus, there is an interest in extending this new family of acceptors, by increasing their electron acceptor ability. In this context, we have developed different approaches for the synthesis of DEBTTT and extended it to novel derivatives, either by replacing sp 2 sulfur atoms by more electronegative oxygen atoms or by modifying the nitrogen alkyl substituents. 10 However, these modifications didn't significantly enhance their electron acceptor ability. Herein, we wish to present another structural modification on the DEBTTT prototype, in order to increase the overall accepting ability. For instance, we substituted the thioketone (C=S) at the 2 position of the thiazole rings by a dicyanovinylidene group, C=C(CN) 2 . The presence of this well-known electron withdrawing group should increase the overall electron acceptor character of the molecule. Accordingly, we report the syntheses and characterizations of a novel electron acceptor the (E)-3,3′-diethyl-

Results and Discussion

The target acceptor skeleton is prepared starting from the thiazole ring bearing a protected dithiolene ligand 2 as reported in Scheme 1. 11 In order to graft a dicyanovinylidene group on the thiazole core, we first transformed the thioketone 2 into a thiazolium salt 3 by adding to a solution of 2 in CHCl 3 , trimethylorthoformate and borontrifluoride diethyletherate (Scheme 1). 12 The thiazolium salt 3 in the presence of NEt 3 reacts with malononitrile in refluxing dichloromethane to give the dicyanovinylidene 4 in 60% yield. As previously reported, 10 an efficient synthesis of the acceptor skeleton consists in transforming the protected dithiolene ligand with cyanoethyl groups into the protected dithiolene ligand within a dithiole-2-one cycle. Then, by simply heating the bicyclic structure into refluxing toluene, the acceptor skeleton is formed. Hence, we attempted to form first the bicyclic structure starting from 4 (Scheme 2). Thus, we deprotected the dithiolene ligand by treating 4 with LDA and then we added triphosgene in the medium. During the work up we noticed the evolution of the bicyclic structure 5 into a highly colored derivative. Assuming that the bicyclic structure 5 was transformed easily to the acceptor 1, due to the presence of the dicyanovinylidene group, we heated the reaction mixture into toluene without isolating 5.

Purification was achieved through column chromatography affording dark purple crystals after slow evaporation of the solution. The acceptor 1 was therefore obtained directly from 4 in 48% yield. We also attempted another approach for the synthesis of 1 starting from preformed DEBTTT. Indeed, it has been reported earlier that thiocarbonyl derivatives can react with tetracyanoethylene oxide to afford dicyanovinyl derivatives 14 and this reactivity has been used by Gompper et al. to convert dithiole-2-thione derivatives into the corresponding dicyanomethylene derivatives. 15 Applied to DEBTTT, this strategy also raises the question of selectivity as two different thiocarbonyl groups are present. The reaction of DEBTTT with tetracyanoethylene oxide (Scheme 3) was carried out in toluene at 0-20°C in the presence of an excess of tetracyanoethylene oxide. Using these conditions we successfully isolated 1 in 56% yield. With this new electron acceptor in hand, and before CT salt experiments, we first evaluated its electronic properties, as detailed below. The redox properties of 1 were analyzed by cyclic voltammetry in CH 2 Cl 2 using nBu 4 NPF 6 as supporting electrolyte. The cyclic voltammogram (CV) of this acceptor (Figure 3) exhibits two reversible monoelectronic reduction waves assigned to the reversible reduction of the neutral species to the anion radical and then to the dianion at E 1 = 0.06 V and E 2 = -0.34 V vs. SCE respectively. It is of interest to compare these redox potentials with those of the well-known -electron acceptor such as TCNQ (tetracyanoquinodimethane) (E 1 = 0.18 and E 2 = -0.37 V vs. SCE), as well as the sulfur analogue DEBTTT (E 1 = -0.05 V and E 2 = -0.44 V vs SCE in CH 2 Cl 2 ). 8 These redox potentials show that 1 exhibits a slightly lower acceptor character than TCNQ, but that it is a stronger acceptor than DEBTTT as the first reduction potential is anodically shifted by 110 mV. Thus, we succeeded in improving the accepting ability through the introduction of two dicyanovinylidene groups. UV-vis-NIR spectroelectrochemical investigations were carried out on a dichloromethane solution of 1 (c = 6  10 -4 M) with nBu 4 NPF 6 as supporting electrolyte (Figure 4). The neutral compound exhibits absorption bands in the UV-vis range from 250 nm to 600 nm. Gradual reduction from neutral to radical anion species, 1 -• , leads to a decrease of UV visible absorption bands at λ = 457 and 543 nm, concomitantly to the increase of the bands at λ = 404 and 582 nm and the growth of two new bands at λ = 665 and 800 nm, characteristic of the formation of the anion radical species. Further reduction to the dianion species, 1 2-, leads to the decrease of all the absorption bands centered at 404, 582, 665 and 800 nm and the increase of a strong band at λ = 466 nm. These results are consistent with our previous work on the DEBTTT where a similar behavior was observed upon reduction. 10 As two dicyanovinylidene groups are now present on the acceptor skeleton of 1, we wondered if the IR nitrile stretching absorption bands in 1 will be sensitive to its reduction state. Indeed, a linear correlation has been established between the  (C≡N) frequencies and the reduction state for numerous TCNQ complexes and salts. 16 Accordingly, we performed a FTIR spectroelectochemical investigation on 1, as well as on TCNQ itself for comparison purposes. This study was performed in dichloromethane with nBu 4 NPF 6 as supporting electrolyte. The spectra obtained for 1 and TCNQ under the three redox states are reported in Figure 5. In the neutral state, 1 exhibits a nitrile stretching vibration at ν C≡N = 2219 cm -1 . Upon gradual reduction to the radical anion, 1 -• , a new band appears at lower energy at ν C≡N = 2200 cm -1 concomitantly with the decrease of the band centered at 2219 cm -1 . Then, upon further reduction to the dianionic species, two new bands grow on the spectrum at even lower energy, 2186 and 2157 cm -1 , with a concomitant intensity decrease of the band localized at 2200 cm -1 . Concerning the TCNQ analyzed in the same experimental conditions, the neutral species exhibits the nitrile stretching vibration band at ν C≡N = 2222 cm -1 . This band is observed at a similar wavenumber than that observed for 1, indicating a similar degree of conjugation in both acceptors. Upon reduction to the monoreduced species, TCNQ -• , two new bands centered at 2180 and 2154 cm -1 increase. Reduction to the dianionic species TCNQ 2- induces an intensity increase of the band at 2154 cm -1 together with the observation of a new band at 2105 cm -1 . Altogether a low energy shift of 117 cm -1 is observed from TCNQ to the dianionic TCNQ 2-while a much smaller shift of 62 cm -1 is observed between 1 and 1 2-, a further indication of a decreased participation of the malononitrile moieties in the reduction of 1, when compared with TCNQ.

In order to get better insight into the reduction processes, DFT calculations [Gaussian09, B3LYP/6-311G**] were carried out on the neutral and the monoanionic species, that is 1 and 1 -• , together with TCNQ for comparison purposes. Full geometry optimizations led to the molecular structure depicted in Figure 6 and the calculated bond lengths for the neutral and reduced species of 1 are collected in Table 1. The optimized geometry of 1 is in very good agreement with the experimental one obtained from X-ray diffraction regarding bond angles and bond distances (Table 1). The LUMO of 1 (Figure 6) is delocalized over the whole molecule as observed for the LUMO of TCNQ. Its energy lies in-between those of TCNQ and DEBTTT, confirming the electrochemical investigations (see above). We note however that the contribution of the malononitrile moieties to the LUMO do not exceed that of the corresponding sulfur atoms in DEBTTT. In other words, these malononitrile moieties in 1 exert a stronger inductive effect that the sulfur atoms of the thione in DEBTTT, but do not allow for a particularly extended charge delocalization in the reduced species. This is further confirmed by the evolution of the calculated bond lengths, between 1 and the anion radical 1 -

•

. It essentially affects the central S=C-C=C-C=S fragment (Scheme 4), with the lengthening of the C=S and the C=C double bonds and concomitant shortening of the C-C bonds upon reduction, with also some minor variations on the other bonds of the acceptor skeleton.

1 in [TMTTF][1] at RT 1 in [TMTTF][1] at 150K 1 in [(Cp*) 2 Fc][1] at 150 K 1 -• calc

Charge transfer salts and complexes with electron donors

In order to form CT salts with this novel acceptor, we choose two electron donor molecules exhibiting different redox properties such as the tetramethyl tetrathiafulvalene (TMTTF), and the decamethylferrocene Fe(Cp*) 2 . Indeed, both donors can be reversibly oxidized to the monocharged species at E ox = +0.26 V for TMTTF and at E ox = -0.05 V for Fe(Cp*) 2 and, demonstrating that these values, Fe(Cp*) 2 is a stronger reducing agent than TMTTF. In solution, electron transfer is directly associated to the positive or negative value of the potential difference E = E red (Acceptor) -E ox (Donor). In the solid state, the extra stabilization brought to ionic structures can allow for a partial CT for negative E values, in the approximate range -0.30/-0.20 < E < 0. The degree of CT between the donor and the acceptor 1 can be assessed from the analysis of the intramolecular bond lengths of the acceptor core within [TMTTF][1]. In the presence of TMTTF, the acceptor exhibits bond lengths similar to those found in neutral 1, indicating the formation of a neutral charge-transfer complex, both in the room temperature and in the 150K X-ray structures. This is further confirmed by the structural characteristics of the TMTTF moiety itself. Indeed, the central C=C bond length of the TMTTF is also known to vary with the redox sate of the TTF core. Here in [TMTTF][1], this C=C bond length amounts to 1.353(5) Å, characteristic of a neutral TTF derivative. 18,19 This demonstrates that the redox difference mentioned above (E = -0.20 V) is here too negative to stabilize a CT in this neutral compound that we will therefore describe as a CT complex.

In [Fe(Cp*) 2 ][1], alternated chains of donor and acceptor running along c are observed (Figure 8a). The intramolecular bond lengths (Table 1) in the acceptor 1 in [Fe(Cp*) 2 ][1] are strongly modified from the structure of the neutral compound, with the central C=C bond as well as the C=S bonds significantly lengthened, while the C-C bonds of the thiazoline rings are significantly shortened. This confirms the presence of the reduced species 1 -• in [Fe(Cp*) 2 ][1], in accordance with the electrochemical E positive value at + 0.11 V (See above). A projection view of the unit cell along the stacking axis c (Figure 8a) demonstrates that these alternated columns interact laterally with each other, either along a or along (a+b). The lateral interactions along a can be described as in-registry (Figure 8b), with the shortest Fe…Fe distances found between an iron atom at (x,y,z) and a neighboring one at 8.659(1) Å (x+1, y, z) and 11.421( 16) Å (x+1, y, z+1). On the other hand, the interactions between columns along the (a+b) direction adopt an out-of-registry pattern (Figure 8c), with larger Fe…Fe distances at 11.731(4) Å (with Fe at 0.5-x, 0.5+y, 1.5-z) and 13.020(3) Å (0.5-x, 0.5+y, 0.5-z). Comparison with the all-sulfur DEBTTT analog described earlier shows that, despite the large size of 1 relative to DEBTTT, the shortest Fe…Fe distance is observed here at 8.659 Å while it amounts to the larger 9.338 Å value in Another possibility to determine the degree of CT between a donor and an acceptor exhibiting nitrile substituents is to investigate the nitrile stretching absorption band frequency ( CN ). This approach has been widely used in the literature on crystalline TCNQ salts as the strongest nitrile vibration band observed for a neutral TCNQ is found at  CN = 2227 cm -1 while it appears at lower energy value for TCNQ -1 ( CN = 2185 cm -1 ) in its Na + TCNQsalt. 16 We have shown above through spectroelectrochemical analyses that a sizeable shift of the  CN stretching frequency is observed upon reduction of 1 in solution. The IR spectra of solid acceptor 1 and both complexes, [TMTTF][1] and [Fe(Cp*) 2 ][1], show  CN stretching frequencies at respectively 2215 cm -1 , 2213 cm -1 and 2198 cm -1 , confirming that the acceptor is neutral in the presence of the TMTTF while it is reduced with Fe(Cp*) 2 .

The magnetic susceptibility of [TMTTF][1] and [Fe(Cp*) 2 ][1] were measured between 1.8 and 300 K on a SQUID magnetometer. No paramagnetism was found in the whole temperature range for [TMTTF][1], confirming that the complex is indeed neutral.

Contrariwise, as shown in Figure 9, the T product for [Fe(Cp*) 2 ][1] amounts to 1.14 cm 3 K mol -1 at room temperature, in accordance with the expected value for the sum of two S = ½ species with g values close to 2 for the radical anion 1 -• and close to 2.7-2.8 for the decamethyl ferricinium cation. 20 The decrease of the T product at lower temperatures indicates the presence of antiferromagnetic interactions. The T data have been fitted to a Curie-Weiss law to give a Weiss constant of -3.83(4) (C = 1.159 cm 3 K/mol). Such antiferromagnetic interactions are quite unusual in alternated D +• A -• chains built out of decamethylferricinium salts, 20 and should find their origin in a direct overlap between the radical anion species. A close inspection of the X-ray crystal structure of [Fe(Cp*) 2 ][1] reveals indeed a short S•••S contact (3.515 Å) between radical anions running along the a direction (Figure 10), to be compared to twice the sulfur van der Waals radius (2 x 1.80 = 3.60 Å). Tight binding calculations [START_REF] Ren | Crystal and Electronic Structure Analysis Using CAESAR[END_REF] of the  LUMO-LUMO interaction energy associated with this overlap interaction between the acceptor's LUMOs amounts to 27.8 meV, a sizeable value consistent with the observed antiferromagnetic response.

Conclusion

A novel electron acceptor bearing dicyanovinylidene substituents on a bithiazolylidene core has been prepared and characterized. This acceptor exhibits a planar skeleton together with several sulfur atoms which can respectively increase the accepting isolate other salts of reduced 1, with a specific interest for mixed valence ones as they can potentially lead to highly conduction materials. Besides, the involvement of neutral 1 in field effect transistors will be also considered as good charge mobilities can be anticipated from these favorable overlap interactions.

Experimental section

All commercial chemicals were used without further purification. The solvents were purified and dried by standard methods. All the NMR spectra were obtained in CDCl 3 unless indicated otherwise. Chemical shifts are reported in ppm and 1 H NMR spectra were referenced to residual CHCl 3 (7.26 ppm) or CD 3 CN (1.94 ppm) and 13 Synthesis of the protected ligand 4. To a suspension of 2 (500 mg, 1.58 mmol) in chloroform (30 mL) was added trimethylorthoformate (0.69 mL, 6.34 mmol) and borontrifluoride diethylether (1 mL, 7.9 mmol). The reaction mixture was stirred 5 hours at 75°C under nitrogen. The solvent was evaporated in vacuo, the oil was washed with diethylether affording 3 as a brown oil which was used in the next step without further purification. Yield: 80%; 1 H NMR (CD 3 CN , 300 MHz) δ (ppm) = 1.42 (t, 3H, 3 J = 7.3 Hz, CH 2 -CH 3 ), 2.80 (m, 4H, S-CH 2 ), 2.92 (s, 3H, S-CH 3 ), 3.27 (m, 4H, CH 2 -CN), 4.49 (q, 2H, 3 J = 7.3 Hz, CH 2 ); 13 To a suspension of 3 (1.83 g, 4.40 mmol) in dichloromethane, malononitrile (322 mg, 3.99 mmol) and triethylamine (654 µL, 0.76 mmol) were added. The mixture was stirred 5 hours at room temperature under nitrogen atmosphere then washed with 2N HCl (50 mL) and water (3 x 20mL). The concentrated solution was purified by chromatography on silica gel using ethyl acetate and petroleum ether (1:1) as eluent affording 4 as a brown powder. Crystals of sufficient quality were obtained by CH 2 Cl 2 slow evaporation. Yield: 32%; Mp: 112°C; 1 H NMR (300 MHz) δ 1.43 (t, 3H, 3 J = 7.1 Hz, CH 3 ), 2.78 (m, 4H, S-CH 2 ), 3.17 (m, 4H, CH 2 -CN), 4.49 (q, 3H, 3 J = 7.1 Hz, CH 2 ); 13

Synthesis of 1.

To a 0°C cooled solution of 4 (200 mg, 0.57 mmol) in dry THF (10 mL) was added a solution of LDA freshly prepared from n-BuLi (0.50 mL, 1.26 mmol) and diisopropylamine (0.17 mL, 1.26 mmol) in 10 mL of dry THF. After stirring for 20 min at -20°C, triphosgene (341 mg, 1.15 mmol) was added and the solution was stirred for 12 hours at room temperature. The solvent was evaporated in vacuo and 10 mL of toluene were added on the crude product under nitrogen atmosphere. The mixture was refluxed for 12 hours. The concentrated solution was purified by chromatography on silica gel using CH 2 Cl 2 as eluent affording 1 as purple crystals. Crystals of sufficient quality were obtained by slow concentration of a CH 2 Cl 2 solution of 1. Yield: 48%; Mp: >230°C; 1 H NMR (300 MHz) δ 1.45 (t, 6H, 3 J = 7.1 Hz, CH 3 ), 4.74 (q, 4H, 3 J = 7.1 Hz, CH 2 ); 13

Synthesis of acceptor 1 from DEBTTT.

To a 0°C cooled solution of DEBTTT (50 mg, 0.14 mmol) in toluene (15 mL) was added a drop of water and a solution tetracyanoethylene oxide (164 mg, 1.14 mmol) in 5 mL of acetonitrile. The reaction mixture was stirred 3 hours at 0°C and 12 hours at room temperature. The solvent was evaporated in vacuo and dichloromethane (80 mL) was added. The solution was washed with water (5×30 mL) and dried over MgSO 4 .

The concentrated solution was purified by chromatography on silica gel using CH 2 Cl 2 as eluent affording 1 as purple crystals. Yield: 56%. A novel sulfur rich -electron acceptor affords with electron donors 1:1 adducts: a neutral charge-transfer complex and a paramagnetic ionic salt.

13

 13

4 Scheme 1 . 4 .

 414 Scheme 1. Synthetic path to the malononitrile derivative 4.

Figure 1 .

 1 Figure 1. Molecular structure and numbering schemes for 4.

4  2 .

 42 Scheme 2. Synthetic path to the electron acceptor 1.

Scheme 3 .

 3 Scheme 3. Alternative synthesis of 1 from the preformed electron acceptor DEBTTT.

Figure 2 .

 2 Figure 2. Molecular structure and numbering schemes for 1 (top), and side view (bottom).

Figure 3 .

 3 Figure 3. Cyclic voltammogram of 1 (CH 2 Cl 2 , nBu 4 NPF 6 , V vs SCE, Pt, 100 mV s -1 )

Figure 4 .Scheme 4 .

 44 Figure 4. UV-vis-NIR absorption spectra monitored from 0.14 to 0V (left) and from -0.04 to -0.6 V (right) of the electrochemical reduction of 1 in dichloromethane with 0.2 M nBu 4 NPF 6 .

Figure 5 .

 5 Figure 5. FTIR spectroelectrochemical spectra of 1 (top) and TCNQ (bottom) in dichloromethane with 0.1M nBu 4 NPF 6 .

Figure 6 .

 6 Figure 6. HOMO and LUMO of DEBTTT (left), 1 (middle) and TCNQ (right) shown with the same cut-off at 0.02 [e bohr -3 ] 1/2 .

  6b,17 With the first reduction potential of 1 measured at +0.06 V, we obtain with Fe(Cp*) 2 a positive E value of +0.11 V, anticipating a CT to an ionic salt. On the other hand, with the less reducing TMTTF molecule, E amounts to -0.20 V, indicating a borderline situation.The CT complexes were prepared by mixing an acetonitrile solution of 1 with a dichloromethane solution of the donor. After a couple of days and slow concentration of the solutions, deep dark crystals were obtained. X-ray crystal structure determinations carried out on single crystals reveal a stoichiometry of one donor with one acceptor in both compounds, formulated as [TMTTF][1] and [Fe(Cp*) 2 ][1]. Selected bond lengths of the acceptor skeleton in both CT salts are collected in Table1. As shown in Figure7, [TMTTF][1] (triclinic system, space group P-1) is organized into alternated stacks running along a. Short intermolecular S•••S contacts between the donor and acceptor molecules are identified as well as short S•••S contacts between the acceptor molecules along the b axis.

Figure 7 .

 7 Figure 7. Solid state organization in [TMTTF][1]. The black dotted lines represent short (< 3.7 Å) intermolecular S•••S contacts. Hydrogen atoms were omitted for clarity.

[

  Fe(Cp*) 2 ][DEBTTT].

Figure 8 .

 8 Figure 8. Different views of [Fe(Cp*) 2 ][1] with: (a) projection view along the stacking axis c, (b) detail of the in-registry alternated columns and (c) detail of the out-of-registry alternated columns. Hydrogen atoms were omitted for clarity.

22

 22

Figure 9 .

 9 Figure 9. Temperature dependence of the T product for [Fe(Cp*) 2 ][1] at 10000 Oe (with  defined as molar magnetic susceptibility equal to M/H per [Fe(Cp*) 2 ][1] unit). Black circles indicate measured data; red line represents the best fit obtained with the Curie-Weiss model described in the text.

Figure 10 .

 10 Figure 10. View of the shortest S•••S contacts in the chain of radical anions running along a in [Fe(Cp*) 2 ][1].

  ability and enhances intermolecular contacts in the solid state. Association with two different donor molecules, namely tetramethyltetrathiafulvalene (TMTTF) and decamethylferrocene (Fe(Cp*) 2 ) afforded 1:1 adducts. Analysis of their structural, electrochemical, vibrational and magnetic properties show that [TMTTF][1] behaves as a neutral charge-transfer complex while [Fe(Cp*) 2 ][1] is an ionic salt, formulated as [Fe(Cp*) 2 + ][1 -• ]. The presence of the exocyclic sulfur atoms in 1 -• ; together with the concentration of the spin density on these atoms favors the setting of direct antiferromagnetic interactions. Work is now in progress to

  C NMR spectra were referenced to CHCl 3 (77.2 ppm) or CD 3 CN (118.3 and1.32 ppm). Melting points were measured on a Kofler hot-stage apparatus and are uncorrected. Mass spectra were recorded with Waters Q-Tof 2 instrument by the Centre Régional de Mesures Physiques de l'Ouest, Rennes. Elemental analyses were performed at the Centre Régional de Mesures Physiques de l'Ouest, Rennes. Column chromatography was performed using silica gel Merck 60 (70-260 mesh). DEBTTT was prepared according to literature procedures. 8 Cyclic voltammetry were carried out on a 10 -3 M solution of 1 in CH 2 Cl 2 , containing 0.1 M nBu 4 NPF 6 as supporting electrolyte. Voltammograms were recorded at 0.1 Vs -1 on a platinum electrode and the potentials were measured versus Saturated Calomel Electrode (SCE). The spectroelectrochemical setup was referenced versus SCE in CH 2 Cl 2 -[NBu 4 ][PF 6 ] 0.2 M. A 3600 plus SHIMADZU and FTIR varian 640 IR spectrophotometers were employed to record the UV-vis-NIR and IR spectra.

4 (

 4 C NMR (CD 3 CN, 75 MHz) δ (ppm) = 13.0 (CH 3 ), 18.6 (S-CH 3 ), 18.8 (-CH 2 -CN), 19.1 (-CH 2 -CN), 33.1 (S-CH 2 ), 49.7 (CH 2 ), 119.3 (CN), 138.1 (C=C), 143.C=C), 178.7 (C-S); HRMS (ESI) calcd for C + [C 12 H 16 N 3 S 4 ] : 330.02271 Found: 330.0224.

5 (

 5 C NMR (75 MHz) δ 14.CH 3 ), 44.0 (C=C(CN) 2 ), 48.0 (N-CH 2 ), 111.6 (=C(CN) 2 ), 112.6 (=C(CN) 2 ), 128.1 (C=C), 169.1 (C=C(CN) 2 ), 184.9 (C=S); UV-vis (CH 2 Cl 2 ) λ (nm) Ɛ (L.mol -1 cm -1 ) 366 (23460), 517 (3940); IR (CN) = 2215 cm -1 ; HRMS (ESI) calcd for M + .(C 16 H 10 N 6 S 4 ) : 413.98498 Found: 413.98553.

  C NMR (75 MHz) δ 15.2 (CH 3 ), 18.8 (CH 2 CN), 18.9 (CH 2 CN), 31.8 (S-CH 2 ), 32.4 (S-CH 2 ), 45.0 (C=C(CN) 2 ), 45.2 (N-CH 2 ), 117.0 (CN), 117.1 (CN), 124.3 (C=C), 138.1 (C=C), 168.5 (C=C(CN) 2 ); IR (CN) = 2172, 2200 and 2253 cm -1 ; HRMS (ESI) calcd for [M + Na] + [C 14 H 13 N 5 NaS 3 ] : 370.02308 Found: 370.0234; Anal.calcd for C 14 H 13 N 5 S 4 ; C: 48.39, H: 3.77, S: 27.68. Found : C: 48.11, H:3.78, S:27.51.

Table 2

 2 Crystallographic data

	Compound		4	1		[TMTTF][1] at RT	[TMTTF][1] at 150 K	[Fe(Cp*) 2 ][1]
	Formula		C 14 H 13 N 5 S 3	C 16 H 10 N 6 S 4	C 26 H 22 N 6 S 8	C 26 H 22 N 6 S 8	C 36 H 40 FeN 6 S 4
	FW (g•mol -1 )	347.47	414.54	674.98	674.98	740.83
	Crystal system	monoclinic	triclinic	triclinic	triclinic	monoclinic
	Space group	P 2 1 /c	P	1	P	1	P	1	P2 1 /n
	a (Å)		9.9664(7)	5.3638(2)	7.2959(11)	7.2379(10)	8.6586(5)
	b (Å)		9.6751(6)	7.2562(3)	8.7231(12)	8.6275(12)	20.4937(12)
	c (Å)		16.8053(13)	12.7329(5)	12.7388(18)	12.664(2)	10.9221(6)
	 (°)		90	77.773(2)	102.719(8)	102.003(8)	90
	 (°)		99.922(5)	80.204(1)	101.011(7)	100.361(8)	109.724(2)
	 (°)		90	69.110(1)	91.481(7)	91.428(9)	90
	V (Å 3 )		1596.23(19)	450.06(3)	774.33(19)	759.30(19)	1824.38(18)
	T (K)		150(2)	294(2)	293(2)	150(2)	150(2)
	Z		4	1		1		1		2
	D calc (g•cm -3 )	1.446	1.529	1.447	1.476	1.349
	 (mm -1 )		0.467	0.541	0.605	0.617	0.677
	Total refls.		8303	7308	12634	7173	14104
	Uniq.	refls.	3611(0.0503)	2051 (0.0292)	3442 (0.0368)	3446 (0.0584)	4121(0.0474)
	(R int )
	Unique		2960	1872	2363	2604	3551
	refls.(I>2s(I))
	R 1 , wR 2		0.0423, 0.099 0.0261, 0.0711 0.0417, 0.1016	0.061, 0.1361	0.0937, 0.2136
	R 1 , wR 2 (all	0.0527, 0.106 0.0292, 0.0747 0.0685, 0.1140	0.0813, 0.1464 0.1067, 0.2203
	data)

Elaboration of [TMTTF][1].

To a hot solution of TMTTF (6 mg, 0.024 mmol) in dichloromethane (10 mL) was added a hot solution of 1 (10 mg, 0.024 mmol) in acetonitrile.

The mixture was refluxed for 10 minutes and the solution was allowed to reach room temperature slowly. Slow evaporation of the solution afford deep purple crystals. diffractometers operate with graphite-monochromated Mo-Kα radiation (λ = 0.71073 Å). The structures were solved by direct methods using the SIR97 program, 23 and then refined with full-matrix least-square methods based on F 2 (SHELXL-97) 24 with the aid of the WINGX program. 25 All non-hydrogen atoms were refined with anisotropic atomic displacement parameters. H atoms were finally included in their calculated positions. Crystallographic data on X-ray data collection and structure refinements are given in Table 2 for