Benjamin Elie

Yves Laprie

Pierre-André Vuissoz

Freddy Odille

High spatiotemporal cineMRI films using compressed sensing for acquiring articulatory data

Keywords: Dynamic speech MRI, Compressed Sensing, Articulatory data

The paper presents a method to acquire articulatory data from a sequence of MRI images at a high framerate. The acquisition rate is enhanced by partially collecting data in the kt-space. The combination of compressed sensing technique, along with homodyne reconstruction, enables the missing data to be recovered. The good reconstruction is guaranteed by an appropriate design of the sampling pattern. It is based on a pseudo-random Cartesian scheme, where each line is partially acquired for use of the homodyne reconstruction, and where the lines are pseudo-randomly sampled: central lines are constantly acquired and the sampling density decreases as the lines are far from the center. Application on real speech data show that the framework enables dynamic sequences of vocal tract images to be recovered at a framerate higher than 30 frames per second and with a spatial resolution of 1 mm. A method to extract articulatory data from contour identification is presented. It is intended, in fine, to be used for the creation of a large database of articulatory data.

I. INTRODUCTION

Studies on speech production require the knowledge of both audio recordings and articulatory data in order to define the existing relationships between the articulatory gestures and the acoustic clues. Due to several constraints (health hazard, access to internal tissues. . . ), the acquisition of articulatory data is challenging and raises several major issues.

Speech production involves two levels of coordination. Speech articulators, which change the acoustic properties of speech, have to move according to the sequence of phonemes to articulate. This mainly corresponds to the coarticulation phenomenon. In addition, the realization of supraglottal constrictions and cavities has to be coordinated with the glottis setup so the the aerodynamic properties of the vocal tract are compatible with the nature of the excitation source. These two coordination mechanisms have different temporal time scales. At the segmental level, i.e. that of speech sounds, the order of magnitude of the coordination between the supraglottal cavities and the vocal folds is the millisecond, while that between supraglottal articulators is the hundred of milliseconds. This means that the study of coarticulation, which is our main objective, requires a sampling rate well above 10 Hz, at least 30 Hz to get articulatory data which can be exploited. Indeed, this gives approximately one or two images at least for each speech sound.

Recently, Magnetic Resonance Imagery techniques (MRI) have been of particular interest for articulatory data acquisitions because of the numerous advantages they present [START_REF] Narayanan | An approach to real-time magnetic resonance imaging for speech production[END_REF]- [START_REF] Sajan | A fast and flexible mri system for the study of dynamic vocal tract shaping[END_REF]. Unlike X-ray images, they are harmless for the subject, and they enable a single slice to be selected, or a set of slices for a 3D representation. However, the major drawback is the required acquisition time. Acquiring the whole k-space corresponding to a single slice may take several hundred of milliseconds, and consequently, limit the temporal resolution to be no less than this value, which is not sufficiently low to correctly observe fast articulatory movements.

Historically, speech MRI have used spiral sampling schemes to enhance the acquisition rate [START_REF] Narayanan | An approach to real-time magnetic resonance imaging for speech production[END_REF]. This sampling scheme offers good image quality given the high acquisition rate. However, it may generate strong undesired artifacts, such as unrealistic tongue tip and lip elongations, that disturb the articulator contours estimation from these images. To prevent these artifacts, the proposed framework uses a Cartesian-based sampling scheme. The choice of a Cartesian-based sampling scheme is also motivated by its adaptability to be used with compressed sensing [START_REF] Lustig | Compressed sensing mri[END_REF] and with homodyne reconstruction [START_REF] Youla | Generalized image restoration by the method of alternating orthogonal projections[END_REF]. These mathematical frameworks allow images to be recovered from partial Fourier information. This paper presents a method to simultaneously integrate several acceleration techniques for MRI to be used in articulatory data acquisition. Sec. II details the theoretical background of compressed sensing and homodyne reconstruction applied in this paper, as well as the choice of the sampling scheme. Simulations and experimentations are presented in Sec. III, and the results in Sec. ??. The method for exploiting articulatory data is presented in Sec. IV.

II. THEORY

A. Compressed sensing

Compressed sensing (CS) is a mathematical framework enabling a subsampled signal, i.e. a signal that is sampled at a rate that does not fulfill the Shannon theory of signal sampling [START_REF] Lustig | Compressed sensing mri[END_REF], to be recovered. The accuracy of the CS recovery lies on the assumption that the signal x ∈ C n to be recovered is K-sparse in a certain base, i.e. it exists a sparse-transform Ψ so that Ψx ∈ C n contains only K < n non-zero elements. In that case, in the presence of noise, only m ≥ K observations of x are sufficient to find a solution by solving the convex problem

x = argmin x ||Ψx|| 1 s.t. ||Φx -b|| 2 ≤ , (1)
where Φ ∈ R m×n is the CS encoding matrix that contains only 0 and 1, so that Φx = b, and b ∈ C m is the observed signal, namely the subsampled version of x, and is a tolerance value to account for noise in the observation data. The 1 -norm ||x|| 1 of the vector x is defined as the sum of the modulus of the complex elements of x. Another important condition for accurate recovery is the incoherence of the encoding matrix Φ with the sparsifying transform Ψ.

When dealing with MRI, the desired signal is usually the image intensity, namely the inverse spatial Fourier transform of x, denoted F -1 sp x. In our case, the sparsifying transform Ψ has been chosen as the temporal Fourier of the image intensity signal, namely Ψ = F t F -1 sp x, where F t is the temporal Fourier transfrom operator. This is justified by the fact that most of the image, namely the head, does not move, and that only a small portion makes relatively slow movements. As a result, most of the pixels exhibit a highly sparse temporal Fourier transform. In comparison with wavelet-based transforms, it has been shown to guarantee better contrasts at the vocal tract boundary walls. In practice, Eq. 1 is then

ρ = argmin ρ ||F t ρ|| 1 s.t. ||ΦF sp ρ -b|| 2 ≤ , (2)
where ρ is the set of images to be recovered.

B. Distributed compressed sensing

In practice, MRI techniques use multichannel coils. The availability of the multichannel coils is commonly used to increase the acquisition speed by parallel imaging techniques, such as SENSE (SENSitivity Encoding) [START_REF] Klaas P Pruessmann | Sense: sensitivity encoding for fast mri[END_REF], or GRAPPA (Generalized Autocalibrating Partially Paralleled Acquisitions) [START_REF] Mark A Griswold | Generalized autocalibrating partially parallel acquisitions (grappa)[END_REF]. It can also be used in CS techniques by exploiting the fact that the signals observed by the different coils are strongly correlated. Basically, their sparsity under the chosen sparse representation should be similar. Consequently, it is possible to introduce such joint sparsity as an a priori in the inverse problem by simultaneously minimizing the 1 -norm of the sparse representation of the signals in each coils, and also the number of non-zero rows of the signal matrix (the matrix whose columns are the signal in each coil). This last constraint comes from the assumption that the non-zero coefficients of the sparse representation of the coil signals share the same location. This problem, called Distributed Compressed Sensing (DCS) [START_REF] Liang | Accelerating sense using distributed compressed sensing[END_REF], writes

P = argmin P ||F t P|| 1,2 s.t. ||ΦF sp P -B|| 2,2 ≤ , (3)
where P ∈ C n×l is the matrix whose columns contain the reconstructed image vectors of the l coils, and B ∈ C m×l is the observation matrix. The mixed 1,2 -norm of the matrix P is defined as the 1 -norm of the 2 -norm of each row of P, hence

||P|| 1,2 = n i=1 ||P i || 2 , (4)
where P i is the ith row of P.

C. Homodyne reconstruction

Homodyne reconstruction techniques are based on the assumption that k-space should exhibit an hermitian symmetry because of the real-valued nature of the images to be recovered. Consequently, the knowledge of only half of the Fourier space is necessary. In practice, this property is never totally valid because of phase errors in the images. However, it is still possible to perform phase correction from a partial kspace sampling, where the portion of the sampled k-space is slightly greater than 1/2.

Homodyne reconstructions [START_REF] Douglas C Noll | Homodyne detection in magnetic resonance imaging[END_REF] use a phase correction image defined as p * (x, y) = e -j∠ρ lr (x,y) ,

where ρ lr is a low-resolution version of the image ρ. It is the inverse spatial-Fourier transform of the observed k-space to which a rectangular weight function, centered around the very first central lines, has been applied to conserve only the low spatial frequency domain. This phase correction image is then multiplied to the image obtained by inverse spatial-Fourier transform of the zero-filled observed k-space. The reconstructed image is the the real part of the phase corrected image. Among homodyne reconstruction techniques, POCS (Projection Onto Convex Sets) [START_REF] Youla | Generalized image restoration by the method of alternating orthogonal projections[END_REF] has been shown to perform well with small k-space fractions. It consists in iteratively applying the phase correction to the reconstructed data until modifications from an iteration to the next goes under the desired threshold.

In application with CS, because of large amount of missing data, the low-resolution image ρ lr is computed from the temporal mean value of the acquired kt-space.

D. Choice of the subsampling pattern

Recent studies about dynamic speech MRI have focused on spiral sampling schemes [START_REF] Fu | Highresolution dynamic speech imaging with joint low-rank and sparsity constraints[END_REF], [START_REF] Sajan | A fast and flexible mri system for the study of dynamic vocal tract shaping[END_REF]. This sampling trajectory is very efficient for accelerating the acquisition process, but is less robust to undesired motion artifacts. Because of its simplicity and its wide use in practice, a pseudo-random Cartesian sampling scheme has been preferred for the study. Variable density sampling used with Cartesian sampling scheme, where central lines (or low frequency) are privileged has been proven to give more accurate recovery than uniformly random sampling schemes [START_REF] Krahmer | Stable and robust sampling strategies for compressive imaging[END_REF].

Given these considerations, the sampling trajectory is designed as follows. First, a number of encoded lines per temporal frame, called n lpf , is set to an optimal value. It is a trade-off between good temporal resolution (low n lpf ) and high reconstruction quality (high n lpf ). Then, the number of fully encoded center lines, called n cl , with n cl ≤ n lpf , is chosen. The n cl central lines are constantly encoded at each temporal frame. Finally, the remaining n lpf -n cl lines to be encoded at temporal frame t are randomly chosen given the following probability function

p(k y , t) = 1 [1 -(k y -n y /2)] r(t) , (6)
where k y is the line number of the k-space, n y /2 is the center line (null frequency) of the k-space, and r(t) ∈ [0, 0.5] is a number chosen randomly at each temporal frame according to a uniform distribution. This distribution leads to a variable sampling density that decreases as one digresses from the center lines of the k-space. An example of sampling trajectory is given in Fig. 1. It shows the sequential pattern of the acquired line as a function of time. The first 4 subsampled k-spaces are also represented.

E. Proposed reconstruction method

To summarize, the experimental protocol to acquire articulatory data at high spatiotemporal resolution may be expressed as follows 1) choice of the slice to be recovered (usually midsagittal slice), 2) choice of acquisition parameters (n y , n x , n lpf , and n cl ) according to the desired spatiotemporal resolution, 3) compute the pseudo-random Cartesian sampling scheme according to Eq. ( 6), 4) recover missing sampled line information with POCS algorithm [START_REF] Youla | Generalized image restoration by the method of alternating orthogonal projections[END_REF], where the phase correction is computed from the temporal mean value of the acquired kt-space, 5) distributed compressed sensing reconstruction using Eq. ( 4) and SPGL1 solver [START_REF] Van Den | SPGL1: A solver for large-scale sparse reconstruction[END_REF], [START_REF] Van Den | Probing the pareto frontier for basis pursuit solutions[END_REF], 6) denoising processing to enhance image quality, using appropriate denoising algorithm [START_REF] Maggioni | A nonlocal transform-domain filter for volumetric data denoising and reconstruction[END_REF], 7) coil combination to merge data into a single combined coil [START_REF] Zhang | Coil compression for accelerated imaging with cartesian sampling[END_REF].

III. RESULTS

A. Simulations

Simulations with numeric phantoms are performed to assess the recovery quality of the method as a function of the number of lines per frame n lpf , which is related to the obtained framerate. Numeric phantoms are created from recorded images of speech MRI obtained with an independent technique [START_REF] Vuissoz | Sound synchronization and motion compensated reconstruction for speech cine mri[END_REF]. The image sequences are then interpolated on a random time vector, where each time step corresponds to a single line acquisition time, to simulate acceleration and speed decrease of articulatory movements. The corresponding kt-space is then subsampled in regards with the different tested sampling trajectories. Sparse reconstruction is then applied to the simulated data. Images are 256×256 pixels of size 1.016×1.016 mm 2 . Each sequence contains 128 images acquired with 16 coils. The time step for the simulation is set to the value of the used repetition time in practice, which is 3.5 ms, and n cl is set to 5.

Fig. 2 shows the NRMSE (Normalized Root Mean Square Error) value of recovered images from the simulations as a function of n lpf . To compute the NRMSE value, only pixels that are included in the region of interest defined by a rectangle bounded by the vocal tract are considered. As expected, the reconstruction quality is enhanced as the number of lines per frame increases. However, for n lpf > 10, there is no significant increase of the reconstruction quality. Consequently, in practice, setting n lpf to 10 is a good compromise between high framerate and good image quality.

B. MRI experiments

MRI experiments were performed on a 3T Signa HDxt MR system (General Electric Healthcare, Milwaukee, WI). Speech MRI data were obtained from 2 healthy volunteers with written informed consent and approval of local ethics committee. The data were collected with an 16-channel neurovascular coil array. The protocol consisted in a sagittal slice through the middle of the vocal track acquired with a custom modified Spoiled Fast Gradient Echo (Fast SPGR, TR 3.5ms, TE 1.1ms, line BW 83.33 kHz, flip angle 30 degrees, half echo in frequency direction, matrix 256×256, 512 temporal frames). For each temporal frame the modification consisted to acquire only a randomized fixed size sample (n lpf lines) of the 256 phase lines of the k-space that gives an acquisition time per temporal frame of 3.5×n lpf milliseconds and a total acquisition time for the vocal production of 512×3.5×n lpf seconds.

In Fig. 3, the evolution of the lip opening as a function of time, as well as the movement of the back of the tongue during the utterance "J'ai pigé la phrase" (/Ze.pi.Ze.la.fKAz/), are plotted. Boxes in right plots indicate the production of the word "phrase" (/fKAz/). The temporal resolution and the image quality are sufficient to visualize the articulator movements. For instance, the lip opening increases to pronounce /fKAz/: it goes from a narrow constriction for the voiceless labiodental fricative /f/, to a relatively wide opening for the open vowel /A/, and then decreases again for the alveolar voiced fricative /z/. At the same time, as shown in the bottom strip plot in Fig. 3, the back of the tongue rises to create an uvular constriction to make the voiced uvular fricative /K/.

Fig. 4 shows strip plots of repeated utterance /ara/, containing the alveolar trill /r/. The recovered film has a framerate of 48 frames per second, and the spatial resolution is 1.016×1.016 mm 2 . Moments of the production of the alveolar trill /r/ are indicated by boxes on the strip plots.

The high spatiotemporal resolution enables the oscillations of the tongue tip to be seen. It is also interesting to note the coarticulation of the tongue tip and its back: at each production of /r/, they both make simultaneous constrictions. The uvular constriction made with the back of the tongue probably enables the oral pressure upstream the alveolar constriction to raise, so that aerodynamic conditions around the tongue tip cause it to auto-oscillate.

IV. EXPLOITING ARTICULATORY DATA

Although the acoustics of the vocal tract is completely determined by the vocal geometry and the source conditions, outlining the contour from the glottis up to the lips as a whole does not suffice for modeling coarticulation. Indeed, articulators do not move as a whole from one position to the next. Articulators are organized in groups, each group realizing a particular gesture. Speech production involves overlapping gestures which have to be modeled separately. It is therefore important to outline each articulator independently, i.e. the mandible, the tongue, both lips, the epiglottis, the larynx and the velum.

It should be noted that bones (and particularly the mandible in the case of the vocal tract) which clearly appear on Xrays images become invisible on MR images. The mandible which gives crucial information about mouth opening has to be derived indirectly from the contour of the incisor root, thus with some imprecision due to the limited extent of this tooth on the image. Conversely, other contours and primarily the tongue, are more visible on MR images than X-ray images. We have developed specific software, called Xarticulators, for outlining articulator contours and we explored two directions for outlining contours. The first consists of delineating the tongue contour by hand. This is feasible for a small number of images in order to get good quality contours and check that acoustic simulations produce the expected acoustic signal. For larger databases it is possible to take advantage of the fact that, unlike X-ray, MR imaging produce contours which do not overlap between each other. This feature is very interesting in terms of contour tracking and we exploited it by adapting the semi-automatic tracking algorithm developed by Fontecave and Berthommier [START_REF] Fontecave Jallon | A semi-automatic method for extracting vocal-tract movements from x-ray films[END_REF]. The main principle is to delineate the target contour in key images randomly selected in the cineMRI, and then to index images where the contour has to be tracked with respect to the key images by using a DCT (Discrete Cosine Transform) distance. The image used for indexing images is limited to the region (not necessarily a rectangle) where the contour to be tracked is located. The resulting contour is obtained by averaging contours of the closest images. This semi-automatic tracking requires superfrom the user who can add key images to correct errors corresponding to an image too far from the existing key images. Fig. 5 shows example of articulator contours tracked by the presented method. It corresponds to the utterance "Des abat-jours" (/de.za.bA.ZuK/), and shows the lip opening (upper plot) and the alveolar region opening (middle plot) as a function of time. The recorded audio signal, to which MRI noise has been removed thanks to a source separation technique [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF], is shown at the bottom, along with the phonetic segmentation. The temporal evolution of the articulator contours agrees with the phonetic segmentation: an alveolar constriction is created during the pronunciation of /z/, lips are closed during /b/ and suddenly opens for the following /A/, and lips are very close each other for pronouncing the diphone /Zu/.

V. CONCLUSION

The presented framework for acquiring 2D images of the vocal tract enables the temporal evolution of articulator contours to be extracted with a good temporal resolution. Simulations with numeric phantoms have shown that images are still accurately even with a very high decimation factor. In practice, this may degrade the image quality, and consequently, make the contour extraction more difficult, however, it could be used to analyze particular very fast movements, such as alveolar trills. In the context of natural speech, setting the framerate at around 30 frames per second is a good trade-off between image quality and acquisition rate. Movements of articulators, such as lips or tongue tip can be finely analyzed thanks to a semi-automatic tracking algorithm. An example has been presented in this paper, where tracked articulator movements agree with the phonetic segmentation of the recorded audio signal. A large articulatory database is currently being fed, following the presented framework1 . It is intended to create accurate articulatory model from statistical methods, representing realistic anatomic data.

Figure 1 .

 1 Figure 1. Sampling trajectory used for the study. Left column: sequential sampling pattern of the acquired line as a function of time: (top) detail line after line, (bottom) global pattern, frame per frame. Right : first 4 acquired k-spaces. Note that each line is not fully sampled. Missing data are recovered using the homodyne reconstruction algorithm POCS [5].

Figure 2 .

 2 Figure 2. Median NRMSE value of recovered images from the simulations as a function of n lpf . Error bars indicate upper and lower median absolute deviations, each of them computed from a set of 5 realizations.

Figure 3 .

 3 Figure 3. Strip plots of the repeated utterance "J'ai pigé la phrase" (/Ze.pi.Ze.la.fKAz/). Top: strip plot of the lip opening. Bottom: strip plot of the posterior constriction. Slice positions are indicated by a vertical dotted line on the midsagittal slice at the left. Boxes on right plots indicate the productions of the word "phrase" (/fKAz/). Time resolution is 35 ms. Spatial resolution is 1.016 × 1.016 mm 2 , and slice thickness is 5 mm.

Figure 4 .

 4 Figure 4. Strip plot of the repeated utterance /ara/. Top: strip plot of the tongue tip movement. Bottom: strip plot of the uvular constriction. Slice positions are indicated by a vertical dotted line on the midsagittal slice at the left. Boxes on right plots indicate the productions of the alveolar trill /r/. Time resolution is 20.8 ms. Spatial resolution is 1.016 × 1.016 mm 2 , and slice thickness is 5 mm.

Figure 5 .

 5 Figure 5. Strip plots of the articulator contours extracted from the utterance "Des abat-jours" (/de.za.bA.ZuK/). Top plot: lip opening. Middle plot: alveolar region opening. Bottom plot: audio signal. Phonetic segmentation is indicated by vertical dashed lines. Left plot: midsagittal view of the vocal tract with articulator contours and position of the strip plot (vertical dotted line).

Video files are available at:www.loria.fr/ ∼ belie/pages/csmri.html

ACKNOWLEDGEMENT

The authors thank FEDER and Région Lorraine for their financial support.


Patrik Feth
email: patrik.feth@iese.fraunhofer.de

Rasmus Adler
email: rasmus.adler@iese.fraunhofer.de

Service-based Modeling of Cyber-Physical Automotive Systems: A Classification of Services

Keywords: automotive, Car-to-X, ConSerts, cyber-physical systems, safety, services

Systems of systems such as Smart Ecosystems, Cyber-Physical Systems, and the Internet of Things support flexible collaborations among heterogeneous participants with open interfaces. To assure safety in all possible collaboration scenarios, we introduced in previous work the ConSerts approach. This approach assumes that system interactions are captured via required and provided services. Considering the interaction between vehicles and infrastructure, this fundamental assumption is problematic as there is no commonly accepted approach for modeling these future Car-to-X scenarios. Existing modeling approaches in the automotive domain consider the realization of the functionality of one vehicle and typically have a functional, dataflow-oriented character. In this paper, we will derive a classification for services in order to contribute to the definition of our required service-based modeling of cyberphysical automotive systems.

I. INTRODUCTION

As a result of progress in technology and immense investments by companies and governments, we are currently witnessing the transformation of traditional vehicles into autonomously driving vehicles. The functionality of driving autonomously requires an extensive amount of data to capture current and possible future driving situations. It is not likely that all the necessary information can be gathered by the autonomously driving vehicle itself; rather, different sources of information need to contribute to the vehicle's perception of the environment. Technologies such as the Tactile Internet allow these sources to be potentially distributed across the entire globe. In reality, it will be nearby infrastructure systems, other vehicles, or a remote control center that will communicate with the autonomously driving vehicle. As driving usually implies a change of a vehicle's physical location, these systems will change during runtime. This orchestration of adapting system collaborations motivates the need to not only consider a single vehicle during development but to introduce a new abstraction layer on the Car-to-X level. From an outside, naive view, the functionality of a vehicle has not changed: it is still driving and thus performing a genuinely safety-critical task. Now that there are other systems contributing to this task, this functionality is no longer realized by the vehicle itself but rather by an automotive cyberphysical system comprising infrastructural devices, other vehicles, and even remote cloud-based systems. This implies that it is no longer sufficient to consider safety on the vehicle level: the analysis needs to be complemented by safety activities on a higher abstraction layer. To this end, we have developed in previous work the ConSerts approach [START_REF] Schneider | Conditional Safety Certification of Open Adaptive Systems[END_REF]. The ConSerts approach assumes that every system in a system of systems scenario comprises a set of reconfigurable components and that each configuration provides some services by using some required services. ConSerts are "conditional safety certificates" describing what can be guaranteed for a provided service depending on some demands on required services. Considering the static composition of components in a system, ConSerts can be used to complement traditional means for assuring safety such as integration testing. Considering a system of systems scenario and the dynamic interaction of components of different systems, ConSerts enable runtime checks to ensure safe collaboration. A fundamental prerequisite for applying the ConSerts approach is a service view on the system of systems level. However, relevant standards in the development of automotive systems like AUTOSAR1 or ISO 26262 [2] do not consider the Car-to-X level and no guidance can be found to derive such a view. In the context of the EMC² project, an extensive state-of-the-art analysis of service-oriented architectures for embedded systems has been conducted [START_REF] Eckel | State of the art and SoA architecture requirements[END_REF]. The analysis concludes that a trend towards such architectures can be found but an accepted approach in the automotive context is not presented in the report. In order to support the definition of our required service-view on the Car-to-X level, we will classify services that are relevant for the automotive domain. The remainder of the paper is structured as follows: Section 2 illustrates our understanding of a service view, refining a general service definition as given in [START_REF] Avizienis | Basic Concepts and Taxonomy of Dependable and Secure Computing[END_REF] by means of an example, introduces ConSerts-related terms, and motivates a classification of services. Section 3 presents our proposal for a classification of automotive services. Section 4 concludes the paper and suggests future research directions.

II. EXAMPLE

The following example introduces the ConSerts-related vocabulary in italic letters and illustrates the challenge of defining a service view. The example is derived from a case study we are conducting in the context of the European Truck Platooning Challenge 2 . In this challenge, two trucks shall drive semi-automatically in a convoy in order to maximize the use of the slipstream. The leader truck is controlled normally by a driver. The follower truck is controlled automatically, as the long reaction time of humans requires a large safe distance to the leader tuck. While driving at the desired short distance, safety needs to be guaranteed by the components that realize the automation. Depending on the implementation of the platoon driving, these components might be scattered over the different trucks. For instance, the component that determines the safe distance might be deployed on the follower truck and receive speed information about the leader truck from a component in the leader truck. Under this assumption, there is a safety dependency between the two trucks: the follower truck needs to obtain the speed values at a sufficiently high level of quality to safely control its own acceleration with respect to the safe distance. Figure 1 illustrates this and other assumptions in a service view. We do not claim here to have derived a perfect service view for platoon driving. The example is only intended to clarify the terminology used. The user of the platoon driving system on the follower truck is provided with the application service Platoon Driving. To realize this service, a set of basic services are required. For this example and throughout the remainder of the paper, we assume that these services are provided by non-reconfigurable components as this simplification does not affect our intended service classification. As shown in figure 1, we assume that the application service Platoon Driving is provided by a component "Platoon Drive Planning", which uses three basic services. The first is the 2 https://www.eutruckplatooning.com/default.aspx service "Determine Safe Distance", which defines which distance is currently necessary to ensure safety. The second is the service "Determine Acceleration to Reach Safe Distance", which delivers an acceleration plan for achieving a certain safe distance. The third service is the "Set Follower Truck Acceleration", which controls the acceleration of the follower with respect to a certain plan. The service "Determine Safe Distance" is provided by a component "Safe Distance Planning", which requires two services from the component "Leader Truck Status Perception". One delivers the current braking capability of the leader truck and the other delivers the current speed of the leader truck. This component is located in the leader truck, whereas the component Safe Distance Determination is located in the follower truck. Thus both components require a service "WiFi Comm", which enables communication between the trucks. Based on such a service view, we could define ConSerts for each component in order to perform runtime checks that assure safe collaboration between the components of different trucks. The ConSerts formalize the relation between guarantees for provided services and demands on required services. Guarantees and demands have a safety property type and an integrity level. Safety property types define "what" is assured in the form of a certain service failure mode, such as "too low" for the service "Determine Safe Distance". The integrity level defines "how much" it is assured and complies with the commonly used concept in safety standards like ISO 26262 [2] in order to capture the level of confidence. An additional concept in the ConSert approach are runtime evidences for assuring some context assumptions made during the definition of ConSerts but which are not related to a single required service (e.g., the independence of services with respect to common cause failures). The modeling of ConSerts is formal enough to perform automated analyses that check if all demands and guarantees are fulfilled. Depending on the concrete composition scenario, these analyses are performed at development time, at the point in time when components start collaborating at runtime, or permanently during the runtime collaboration. However, for modeling and analyzing ConSerts we require a service view answering the question which basic services contribute to an application service and in which way. So far, we have not found any sophisticated modeling method, but we have been able to identify different types of basic services. As different types of services have specific failure modes, a classification of services enables us to introduce failure mode classification and corresponding guidelines for deriving safety properties. In the following, we will thus discuss the different types of services a service modeling approach for Car-to-X communication should have.

III. DISCUSSION

Car-to-X communication (also known as Vehicle-to-Vehicle and Vehicle-to-Infrastructure communication) refers to the exchange of information between traffic participants and the infrastructure to offer application services to the driver. Considering the basic services that support such application services, we initially distinguish between communication infrastructure services and information exchange services. Communication infrastructure services are services that enable the exchange of information, e.g., our WiFi service in the example. Information exchange services are services that use communication infrastructure services to provide different kinds of information. In order to refine the class of information exchange services and introduce related subclasses, we consider how information is generated and processed in a vehicle. As shown in figure 2, the initial information is generated by sensors. Afterwards, this information is evaluated in a perception layer in order to create information about the current environmental state. This information is then used to plan the behavior of the vehicle. The information generated during the planning is then broken down into information that can be processed by the actuators. The most common scenario is obviously that one vehicle offers some of its perception information to other vehicles. This is the case, for instance, in our Platooning example, as the leader truck its vehicle speed with the follower truck. However, a vehicle might also share information from other layers. For the agriculture domain, it makes sense, for instance, for the implement to control the acceleration of a tractor. In this scenario, the implement delivers the intended acceleration to the tractor and the tractor breaks this information down into information for its engine. Theoretically, it can even happen that vehicles share information from the lowest layers. However, we can think of no use case where one vehicle would evaluate the raw sensor data of another vehicle or where one vehicle would directly control the raw data of another vehicle. For this reason, we see three types of information that might be exchanged: perception information, planning information, and arbitration information. According to these three classes of information, we divide the class of information exchange services into three sub-classes: perception services, planning services, and arbitration services. In the following, we will discuss these classes in more detail.

A. Application services and basic services

The existing ConSerts methodology already differentiates between application services and basic services. Application services are a natural starting point for a first hazard and risk analysis as they describe what is delivered to humans and thus state the overall goal of collaboration. A hazard and risk analysis should analyze which failures and risks are inherently related to this goal and under which conditions the provision of the service is safe. These safety conditions should be captured by the guarantees of the application service. Following the ConSerts approach, each of these guarantees is to be broken down into some demands on basic services that realize the application service. Considering the application service Platoon Driving, the goal is, for instance, to drive as closely as possible in order to save fuel. A related service failure is to drive too closely as this might cause a collision. The Platoon Driving service should thus guarantee that no collision will occur. Considering the service view in figure 1, this guarantee has to be broken down into the three required services of the component "Platoon Driving Follower Truck".

B. Planning services

Planning services are the basic services that are mostly related to application services. They determine behavior that is needed in order to realize application services. Considering the Platooning example, the services "Determine Safe Distance" and "Determine Acceleration to Reach Safe Distance" are such planning services. Planning services require perception services that provide them with the necessary (environmental) information in order to plan the behavior and some arbitration services that realize their planned behavior. The complexity of planning services is increasing all the time, as application services replace more and more very complex human tasks and the requirements on the performance of the tasks often become higher. The Platoon Driving service, for example, has performance requirements that cannot be fulfilled by a human driver. This increasing complexity leads to a discussion about the "safety of the intended functionality", as currently emerging in the creation of a new version of the ISO 26262 standard. To define a safe specification of a decision-making algorithm that should take full responsibility in almost every possible situation is definitely a task that requires special attention.

C. Perception services

Perception services provide planning services with the information they need. As this information is typically on a higher abstraction level than what is actually measured by sensors, components providing perception services generally require other perception services. Due to the increasing complexity of planning services, more and more information is required that can hardly be determined by one vehicle. Thus, future automotive vehicles will require perception services provided by other vehicles or by some external infrastructure.

Interpretation of sensor data should be performed by the system that has most of the context knowledge. Only in a limited number of cases will this be the vehicle under consideration. For some information, multiple sources might be available at a certain point in time. In such a case, the vehicle has to select one of many available perception services. The guarantees specified in the ConSerts approach shall act as a driver for this selection from the safety point of view.

D. Arbitration services

Arbitration services achieve a certain vehicle behavior by controlling actuators. As the realization of a vehicle behavior that is required by a planning service is very complex and far away from what can actually be controlled with actuators, components providing arbitration services generally require other arbitration services. For instance, the service "Set Follower Truck Acceleration" might be provided by a component that requires a service "Set Moment of the Rear Axle" and a service "Set Moment of the Front Axle". Arbitration services are obviously inherently safety-critical. Nevertheless, it is possible that these services will cross the border of a vehicle, e.g. to reduce the consequences of an inevitable crash or to stop a vehicle in case of a chase by the police. A failure of an arbitration service is the wrong stimulation of an actuator based on a correct set value. Depending on the level of detail provided by the planning services, we do not expect a lot of new challenges for autonomous vehicles compared to traditional vehicles. An exception is the impact of security on safety. The correlation between security and safety is very obvious in this setting, but also needs to be considered for the other service classes.

E. Communication infrastructure services

Infrastructure services require particular consideration as they are needed to maintain safe functionality throughout the system's operation. As part of context modeling in the system engineering lifecycle, the relevant infrastructure is known early in the project. If, for example, no wireless connection is possible in a Car-to-X configuration, only very limited functionality can be offered to the user. A failure of such a service means that it cannot be provided at the anticipated level of quality. The quality level of the required infrastructure services needs to be monitored permanently by the mechanism of runtime evidences. Adequate concepts for reacting to any degradation need to be defined. In the example, the service called WiFi Comm is an infrastructure service: In the case of a low-quality WiFi connection, e.g. message delay beyond a defined threshold, the system needs to increase the distance to the follower truck and hand over the control to the driver.

We motivated our classification with the dynamic collaboration of components of different vehicles. However, considering novel automotive architectures such as those presented in [START_REF] Küfen | Ein erweitertes Konzept zur zentralen Informationsplattform für perzeptive, intelligente Fahrzeugsysteme und seine Möglichkeiten[END_REF] and [START_REF] Hudecek | A Novel Approach Supplementing Implementation of Future Scenarios for Fully Automated Driving[END_REF], the components within a vehicle follow a similar dynamic way of interaction that is unpredictable at design time. We thus also see great potential for our ConSerts approach and the related service classification for safety assurance with respect to the next generation of automotive architectures. In this context, we consider the use of ontologies and approaches such as Semantic Service Provisioning [START_REF] Kuropka | Semantic Service Provisioning[END_REF] of major importance. Existing methods model services in ontologies but not in a safety-critical context. Combining the two concepts and model services as part of an ontology enables the performance of algorithms known from search engines for service matching in future automotive applications. To use this potential in a safety-critical domain, it needs to be investigated how such an ontology needs to be enriched with safety-relevant information and how reliable safety analysis can be conducted on them.

IV. CONCLUSION AND FUTURE WORK

To implement future applications such as autonomous driving, vehicles will evolve from single systems to heavily connected automotive cyber-physical systems. As safety-critical functionality will be implemented by collaborations on the Car-to-X level, safety considerations will become important for this high abstraction level. Previous work introduced ConSerts as a safety measure on the system of systems level. To support the creation of the service view required to define ConSerts, we have given a classification of services in the automotive domain that we derived from the fundamental architecture of information processing in automotive systems and from our experience using the ConSerts approach. We have presented initial thoughts on these service classes and related failures. As future work, we plan to refine our service classification and enhance it with a related service failure classification that supports the definition of safety properties for ConSerts. Furthermore, we will investigate how to use ontologies to capture the different types of exchanged information and perform automated runtime safety-analyses to assure that every component always receives the intended safety-relevant information.

Fig. 1 .

 1 Fig. 1. Service view of the platoon driving system

Fig. 2 .

 2 Fig. 2.: Classification of services Vehicle-to-vehicle communication enables vehicles to exchange the information they generate in the different layers.The most common scenario is obviously that one vehicle offers some of its perception information to other vehicles. This is the case, for instance, in our Platooning example, as the leader truck its vehicle speed with the follower truck. However, a vehicle might also share information from other layers. For the agriculture domain, it makes sense, for instance, for the implement to control the acceleration of a tractor. In this scenario, the implement delivers the intended acceleration to the tractor and the tractor breaks this information down into information for its engine. Theoretically, it can even happen that vehicles share information from the lowest layers. However, we can think of no use case where one vehicle would evaluate the raw sensor data of another vehicle or where one vehicle would directly control the raw data of another vehicle. For this reason, we see three types of information that might be exchanged: perception information, planning information, and arbitration information. According to these three classes of information, we divide the class of information exchange services into three sub-classes: perception services, planning services, and arbitration services. In the following, we will discuss these classes in more detail.

AUTomotive Open System Architecture: www.autosar.org

ACKNOWLEDGMENT

The work presented in this paper was created in the context of the Fraunhofer Digital Commercial Vehicle Technology Cluster, which is funded by the State of Rhineland-Palatinate and the Fraunhofer-Gesellschaft.


Daniel Barlet

Note on rational 1-dimensional compact cycles. second version

Keywords:

Note on rational 1-dimensional compact cycles.

The aim of this short note is to prove the following result.

Theorem 0.0.1 Let M be a complex space. Let R be the subset of the space C 1 (M ) consisting of compact rational 1-dimensional cycles in M . Then R is a closed analytic subset in C 1 (M ).

By a rational 1-dimensional cycle we mean that each irreducible component of such a cycle is a rational curve (may be singular).

Note that this result is classical in the projective context.

The proof of the theorem uses the following proposition.

Proposition 0.0.2 Let π : U → V be a geometrically flat map between reduced complex spaces with one dimensional fibres. Assume that for a point v 0 ∈ V the fibre π -1 (v 0 ) has an irreducible component γ 0 which has genus ≥ 1. Then there exists an open neighbourhood V 1 of v 0 in V such that for any v ∈ V 1 the fibre π -1 (v) has an irreducible component of genus ≥ 1.

1

In the previous statement, the genus of an irreducible compact curve is, by definition, the genus of its normalization. Recall also that the geometric flatness assumption means that there exists a holomorphic map ϕ : V → C 1 (U ) such that for each v ∈ V we have π -1 (v) = |ϕ(v)| and such that for v generic in V the 1-cycle ϕ(v) is reduced (that is to say that all multiplicities are equal to 1). We begin by some general results in order to show that it is enough to prove the proposition and the theorem in the case when V is normal and when the fibres of π are connected.

Lemma 0.0.3 Let U → V be a geometrically flat map between reduced complex spaces with holomorphic fibre map ϕ : V → C n (U ). Let τ : Ṽ → V be a proper modification of V and denote π : Ũ → Ṽ the strict transform of π by τ . Then π is geometrically flat and the fibre at a point ṽ ∈ Ṽ is the n-cycle ϕ(τ (ṽ)) × {ṽ} in Ũ .

proof. Let S ⊂ V be the center of the modification τ . Then for ṽ ∈ τ -1 (S) the fibre of π is ϕ(τ (ṽ)) × {ṽ} as a cycle in Ũ ⊂ U × V Ṽ . As Ũ is a closed analytic subset in U × V Ṽ and as the family of cycles ṽ → ϕ(τ (ṽ)) × {ṽ} is an analytic family of compact cycles in U × V Ṽ such that, for ṽ generic, they are contained in Ũ , this is an analytic family of cycles in Ũ and this gives the holomorphic fibre map for π.

Remarks.

1. If n = 1 and if R is the set of points in Ṽ such that the fibre of π is rational then τ (R) is the subset of V where the fibre of π is rational.

2. This lemma allows to assume that V is a normal complex space in the proofs of the proposition 0.0.2 and the theorem 0.0.1 Lemma 0.0.4 Let U → V be a geometrically flat map between irreducible complex spaces and assume that V is normal. Let π : U → W and g : W → V be a Stein factorization of π. Let ν : W → W be the normalization of W and let π : Ũ → W be the strict transform of π by ν. Put g : W → V be the composition g := ν • g. As π is equi-dimensional proper and with a normal basis, it is geometrically flat with connected fibres. Also g is proper, finite and surjective with a normal basis so it is geometrically flat with 0-dimensional fibres. Let θ : W → C n ( Ũ ) and f : V → Sym k ( W ) the corresponding holomorphic fibre maps. Then the fibre map for the geometrically flat map π • g is given by

f • Sym k (θ) • Add : V → C n ( Ũ ), where Add : Sym k (C n ( Ũ )) → C n ( Ũ ) is the addition map of n-cycles in Ũ .
proof. Remember that, by definition of a Stein reduction, the fibre of g of a point v ∈ V is the set of connected components of π -1 (v) and that the fibre of π at a point w ∈ g -1 (v) is the connected component of π -1 (v) given by w. So π and also π have connected fibres. As g is proper (and finite) and as W is irreducible the image by g of non normal points in W is a closed analytic subset in V with no interior point. So it is clear that the holomorphic map f • Sym k (θ) • Add is a fibre map for π • g at the generic points of V . This is enough to conclude.

Corollary 0.0.5 In the situation of the previous lemma with n = 1, let R be the subset of W of points such the fibre of π is rational. Then the subset of point in V such that the fibre of π is rational is equal to

R := {v ∈ V / f (v) ∈ Sym k ( R)}. Then, if R is closed (resp. analytic), so is R.
proof. This corollary is clear because a compact curve is rational if and only if each of its connected component is rational.

remark. With the previous results, it is enough to prove the proposition 0.0.2 and the theorem 0.0.1 with the following extra assumptions : V is normal and all fibres of π are connected.

proof of the proposition. We shall use the following result (see [B.80]): Let C be a reduced compact curve in a complex space M , and let L be a holomorphic line bundle on C. Then there exists an open neighbourhood M of C in M and a holomorphic line bundle L on M inducing L on C. Moreover, if L is topologically trivial on C we may choose L topologically trivial on M . Note that the last point is not stated in loc. cit but is a trivial consequence of the proof given there.

Consider now C 0 = |ϕ(v 0 )| = π -1 (v 0 ) and let ν : C0 → C 0 be the normalization of C 0 . Define the coherent sheaf F := ν * (O C0 ) on C 0 . We have an exact sequence of coherent sheaves 0 → O C 0 → F → Q → 0
where Q has support in a finite set. Then H 1 (C 0 , Q) vanishes and we have a surjective map

H 1 (C 0 , O C 0 ) → H 1 (C 0 , F) → 0.
Using the surjectivity above, we can find a topologically trivial line bundle L on C 0 which is not holomorphically trivial on each non rational component of C 0 . Thanks to the result quoted above and to the properness of π we can find an open neighbourhood V 0 of v 0 in V an a line bundle L on U 0 := π -1 (V 0 ) which is topologically trivial on U 0 and induces L on C 0 . Now let Z be the subset of the space C 1 (L) of

connected 1 compact 1-cycles in L such that their direct image on U 0 is contained in ϕ(V 0 ) ⊂ C 1 (U 0 ). This subset Z is a closed analytic subset in C 1 (L) because ϕ(V 0 ) is a closed analytic subset in C 1 (U 0 ) as ϕ : V 0 → C 1 (U 0
) is a proper holomorphic map and as the direct image by the projection p : L → U 0 is holomorphic. Remark that for each v ∈ V 0 the set Z contains the cycle ϕ(v) of L which is the zero section of the restriction of L to |ϕ(v)| with suitable multiplicities, in order that its direct image on U 0 is equal to ϕ(v). This defines a closed holomorphic embedding of ϕ(V 0 ) in Z ⊂ C 1 (L).

We shall show now that the direct image map f * : Z → ϕ(V 0 ) for compact 1-cycles induced by the projection C 1 (L) → C 1 (U 0 ) has positive dimensional fiber at ϕ(v) ∈ Z when ϕ(v) is rational.

Assume that for some v ∈ V 0 the 1-cycle C := ϕ(v) is rational. Then the restriction of the line bundle L on C is holomorphically trivial and any compact 1-dimensional cycle in L |C can be move (by vertical translation). So any point in Z in the fibre of f * over C = ϕ(v) is not isolated. This proves our assertion. Now C 0 has at least one irreducible component, say γ, which is not rational; so the corresponding point of ϕ(v 0 ) ∈ Z is isolated in its fibre for f * . Indeed, the zero section is the only reduced compact 1-dimensional cycle in L |γ as L |γ is topologically trivial but not holomorphically trivial by construction. So any connected compact 1-cycle near-by ϕ(v 0 ) in Z ∩ f -1 * (C 0 ) must have support in the zero section of L on each non rational component of C 0 . As ϕ(v 0 ) is connected this implies that on a rational component of ϕ(v 0 ) which meets an irrational component, the corresponding component of a near-by cycle to the cycle ϕ(v 0 ) in f -1 * (C 0 ) has to vanish at some point (the intersection with some non rational component). Then the corresponding component of such a cycle is the zero section over this rational component (we have only constant sections on rational components). As we assume C 0 connected and as the cycles in Z are connected, we conclude that ϕ(v 0 ) ∈ Z is an isolated point in its fibre of f * . Now the subset T of points t ∈ Z such that the dimension at t of the fibre of f * is at least equal to 1 is a closed analytic set in Z. The intersection of T with the closed embedding of ϕ(V 0 ) is Z defines a closed analytic subset in ϕ(V 0 ) and then also of V 0 which contains the subset of rational fibers of π in V 0 . As we have shown that v 0 is not in this closed analytic subset, we obtain an open neighourhood

V 1 of v 0 in V 0 such that for any v ∈ V 1 the cycle ϕ(v) is not rational.
Corollary 0.0.6 Let M := U × P 1 be a reduced complex space and let p : M → P 1 be the projection. Let p * : C 1 (M ) → C 1 (P 1 )

N.[P 1 ] the direct image for compact 1-cycles. Define X := p -1 * (1.[P 1 ]). As p * is holomorphic, this is a closed analytic subset in C 1 (M ). Let S 0 be the subset of C 1 (M ) of irreducible cycles which are in X. This is a Zariski open subset in X (see [B-M] prop. IV 7.1.2). Then the closure S 1 of S 0 in X contains only rational cycles.

proof. Let π : U → V := S 1 be the projection of the graph of the tautological family of 1-cycles parametrized by S 1 . As the generic cycle in this family is irreducible, by definition of S 0 , all fibres of π are connected. Also the generic fibres are rational because for s ∈ S 0 it is reduced and isomorphic to P 1 . If there is a non rational cycle in S 1 , then there exists, thanks to the previous proposition, a non empty open set of non rational cycles in this family. But S 0 is open and dense in S 1 this gives a contradiction.

proof of the theorem. Let V be an irreducible component of C 1 (M ). As we may normalize V thanks to the lemma 0.0.3, we can assume that the generic cycle in V is reduced. Let W be a relatively compact open set in V . Then there exists, thanks to the proposition IV 7.1.2 in [B-M 1], an integer k ≥ 1 such that for any v ∈ W the cycle v has at most k irreducible components. Denote π : U → W the projection of the graph of the tautological family of compact curves in M parametrized by W . Define the subset S l ⊂ C 1 (U × P 1 ) as the image by the addition map of cycles of (S 1 ) l in C 1 (U × P 1 ). As the addition map is proper and finite, S l is a closed analytic subset of C 1 (U × P 1 ) for each integer l ≥ 1. Let q : U × P 1 → U the projection. We shall prove that the subset of rational cycles in the family (v) v∈W is exactly given by the subset

R := ∪ k l=1 q * (S l ) ∩ W
which is a closed analytic subset in W because the map q is proper and the subset S l ⊂ C 1 (U × P 1 ) is a closed analytic subset. First remark that each cycle in R is rational as the direct image of a rational cycle is rational and we proved that each cycle in S 1 is rational in corollary 0.0.6. Conversely, let v ∈ W such that v is rational. Then there exists an integer l ∈ [1, k] and l holomorphic generically injective2 maps (distinct or not) f 1 , . . . , f l from P 1 to U such that the sum of there images is v and with graphs G 1 , . . . , G l which are points in S 0 . Then v = l i=1 q * (G i ) and so v is in R. To conclude the proof it is enough to say that a subset which is closed and analytic on any open relatively compact subset in V is a closed analytic subset in V .

  . 78] Barlet, D. Majoration du volume ... in Lecture Notes n 0 822 (1980) (Sem. Lelong-Skoda 78-79), Springer. • [B-M 1] Barlet, D. et Magnusson, J. Cycles analytiques complexes I : théorèmes de préparation des cycles, Cours spécialisés n 0 22 Société Mathématique de France (2014).

see th. IV 7.2.1 for the analyticity of this condition on compact cycles.

the normalization maps.


Pierre Guiraud

Etienne Tanré

Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size

Keywords: 2010 AMS subject classifications:92B25, 92B20, 60F10, 60J75 Network of LIF neurons, Synchronization, Stochastic stability, Interacting Ornstein Uhlenbeck processes, Excitatory synapses

, namely, fully-connected and totally excitatory integrate and fire neural network with Gaussian white noises. Using a large deviation principle, we prove the stability of the synchronized state under stochastic perturbations. Then, we give a lower bound on the probability of synchronization for networks which are not initially synchronized. This bound shows the robustness of the emergence of synchronization in presence of small stochastic perturbations.

Introduction

Neurons are the cells of the nervous system which are able to generate transmissible electrical signals called action potentials, or spikes for short, and to encode information in the sequences of time intervals separating them (inter-spikes intervals (ISI)). The information encoded in spikes sequences is transmitted and processed in the network formed by the neurons, by mean of synaptic interactions triggered by the spikes. These interactions may facilitate (excitatory interactions) or prevent (inhibitory interactions) the emission of spikes in postsynaptic neurons. Among the collective behaviors emerging from this coupling, the synchronization is one of those that has attracted much attention in experimental and theoretical works: a network is fully synchronized when all its neurons spike simultaneously.

Mathematical studies of synchronization have been carried out first in deterministic neural models. For instance, in [START_REF] Mirollo | Synchronization of pulse-coupled biological oscillators[END_REF], the authors prove that in fully-connected and totally excitatory leaky integrate and fire neural network models the synchronization occurs for almost all initial state. When weak interactions are assumed, a large class of models can be reduced to canonical systems of phase coupled oscillators [START_REF] Hoppensteadt | Weakly connected neural networks[END_REF]Izhikevich, 1999a). This formalism allows to study the existence and the stability of synchronized solutions in weakly coupled general networks (Izhikevich, 1999b), including networks containing inhibitory synapses and considering synaptic delays [START_REF] Van Vreeswijk | Partial synchronization in populations of pulse-coupled oscillators[END_REF][START_REF] Van Vreeswijk | When inhibition not excitation synchronizes neural firing[END_REF]. Synchronized solutions and more generally phase-locked solutions in integrate and fire networks with delays have also been analyzed in the case of strong or arbitrary intensity coupling, see for instance Bressloff andCoombes (1999, 2000).

Successive in vivo recordings of the membrane potential of a neuron show the existence of a large variability in its ISI. Although this variability could be the result of the complex encoding of information in the neural network, it nevertheless suggests the presence of noise in the neural activity. As a matter of fact, the sources of noise are multiple and we can mention thermal noise, channel noise, synaptic noise, and the noise resulting of background activity of the whole network. Noise is commonly modeled by a Gaussian stochastic process: an additive white noise is considered in the dynamics of the membrane potential. Gaussian noise models a weak interaction between the neurons of a sub-network and those of the sequel of the network. This can be derived rigorously by assuming balanced excitation and inhibition outside of the sub-network, and a Poisson statistics for the times of the incoming spikes [START_REF] Gerstner | Spiking neuron models: Single neurons, populations, plasticity[END_REF].

The mathematical analysis and the problem of efficient numerical simulations of neural networks with Gaussian noise can be tackled when considering simplified models such as (leaky) integrate and fire models. To this class of models it is possible to associate a discrete time Markov chain containing all the informations to study the time evolution of the spiking times [START_REF] Touboul | A markovian event-based framework for stochastic spiking neural networks[END_REF]. This approach allows to deduce qualitative properties of the network and also to build efficient algorithm to simulate exactly the ISI. Nevertheless, the dimension of the Markov chain depends on the characteristics of the network and in most cases a good knowledge of the transition distribution of the chain is needed.

For instance, for Perfect Integrate and Fire (PIF) or Leaky Integrate and Fire (LIF) network models with inhibitory interactions, the countdown sequence, which after any spiking time gives for each neuron the remaining time until it emits a spike supposing no interaction meanwhile, is a Markov chain. For networks with excitatory interactions, to keep the Markov property of the countdown process it is necessary to change the interaction rules [START_REF] Turova | Synchronization of firing times in a stochastic neural network model with excitatory connections[END_REF] (taking into account the last presynaptic spike and forgetting the previous ones). Instead of changing the interaction rules, another way to have the Markov property is to consider a chain of higher dimension composed by the pair of the countdown process and the membrane potential of each neuron at the spiking times [START_REF] Touboul | A markovian event-based framework for stochastic spiking neural networks[END_REF]. Unfortunately, there is no explicit expressions for the transition probabilities of this chain, even for simple models as LIF networks. The inefficiency of the Markov chain approach makes difficult the study and simulation of neural networks with excitatory synapses.

The mean-field limit of such networks when the number of neurons goes to infinity has also been studied (Delarue et al., 2015b,a). It exhibits another kind of synchronization, defined by the fact that the number of neurons which spike simultaneously is a macroscopic proportion of the full network (i.e ηN, where the networks is consti-tuted by N neurons and 0 < η ≤ 1). In this context, the size of the interactions has to decrease with the number of neurons: they are of order 1/N.

In the present paper, we study the complete synchronization for finite networks, which can be considered as noisy versions of the model of [START_REF] Mirollo | Synchronization of pulse-coupled biological oscillators[END_REF], that is, fully-connected and totally excitatory integrate and fire neural network with noise. Using a large deviation principle, we show that, if the network is synchronized, it remains synchronized with large probability, provided the random perturbations are small enough. In a second time, we study the emergence of synchronization in a network which is not initially synchronized. The proof developed in [START_REF] Mirollo | Synchronization of pulse-coupled biological oscillators[END_REF] cannot be easily adapted, since it relies on the preservation of an order which is destroyed in presence of noise. Therefore, to obtain an estimation of the probability that the synchronization occurs before the n-th firing of the network, we propose an alternative strategy to those of [START_REF] Mirollo | Synchronization of pulse-coupled biological oscillators[END_REF]; [START_REF] Turova | Synchronization of firing times in a stochastic neural network model with excitatory connections[END_REF], which use some arguments developed for deterministic networks in [START_REF] Catsigeras | Integrate and fire neural networks, piecewise contractive maps and limit cycles[END_REF]. This estimation is obtained supposing a specific relation between intensity of interaction and the number of neurons. These results show that both the emergence of synchronization proved in a deterministic setting and the synchronized state itself are robust under small stochastic perturbations.

The paper is organized as follows. In Section 2 we introduced the model. Section 3 is devoted to the main results which are proved in Section 5. The paper also contains numerical results in Section 4 and an annex about large deviation principle for Ornstein-Uhlenbeck process.

Model

We consider a leaky integrate and fire neural network with Brownian noise. We suppose the network contains N neurons which are labelled by an index in the set I := 1, N . For each i ∈ I, the quantity V i,ε (t) represents the membrane potential of the neuron i at time t ≥ 0, in presence of a noise whose intensity is parametrized by ε ≥ 0.

As usual in integrate and fire neural networks, the dynamics of the model considered here has two regimes: a sub-threshold regime and a firing regime. We say that the network is in the sub-threshold regime if the potential of all the neurons is smaller than a fixed threshold potential θ . When the potential of one neuron in the network reaches the threshold, it emits a spike and we say that the network is in the firing regime. Then, the potential of the spiking neurons is reset to a potential that we will suppose equal to zero (and less than θ ) and the neurons that do not spike, but are connected with a spiking neuron, suffer a jump in their potential depending on the synaptic weights. Now we detail the two regimes of the network. For all n ≥ 0, we denote by τ ε n the n-th instant when the network is in the firing regime. More precisely, by convention τ ε 0 = 0 and for all n ≥ 0 the instant τ ε n+1 is the first instant after τ ε n when a spike is emitted in the network. We suppose that the sequence (τ ε n ) n∈N obeys the following induction:

τ ε n+1 := min i∈I inf t > τ ε n : Ṽ i,ε n (t) ≥ θ ∀n ≥ 0, (1)
where, Ṽ i,ε n is the solution of the stochastic differential equation

Ṽ i,ε n (t) = V i,ε (τ ε n ) + ˆt τ ε n -γ Ṽ i,ε n (s) -β ds + √ ε (W i t -W i τ ε n ) ∀t ≥ τ ε n , (2)
where γ > 0, β > θ and (W 1 t ) t≥0 , (W 2 t ) t≥0 , . . . (W N t ) t≥0 are independent Brownian motions. Note that for n ≥ 1, the quantity V i,ε (τ ε n ) is the potential of the neuron i just after the n-th firing regime (we will describe it soon). At time τ ε 0 = 0, we suppose that the potentials V i,ε (0) are included in [α, θ ), where α < 0.

It follows that for each n ∈ N and i ∈ I, the process

( Ṽ i,ε n (t),t ≥ τ ε n
) is an Ornstein-Uhlenbeck process. So, it can also be written as

Ṽ i,ε n (t) = φ t-τ ε n (V i,ε (τ ε n )) + √ εe -γt ˆt τ ε n e γs dW i s ∀t ≥ τ ε n , (3)
where

φ t (x) = (x -β )e -γt + β .
In the sub-threshold regime, that is for t ∈ (τ ε n , τ ε n+1 ) with n ≥ 0, we suppose that the membrane potential of each neuron satisfies

V i,ε (t) = Ṽ i,ε n (t). (4)
Now we describe the firing regime. We denote by V i,ε (τ ε n -) the value of the potential of neuron i, in the sub-threshold regime just before the instant τ ε n . It is a shorthand notation for:

V i,ε (τ ε n -) := lim t τ ε n V i,ε (t) = φ τ ε n -τ ε n-1 (V i,ε (τ ε n-1 )) + √ εe -γτ ε n ˆτε n τ ε n-1
e γs dW i s .

We now define step by step the set J(n) of neurons which spike at time

τ ε n . The first component of J(n) is J 0 (n) = {i ∈ I, V i,ε (τ ε n -) ≥ θ }.
The set J 0 (n) corresponds to the neurons which spike spontaneously (i.e. without any interaction with the other neurons in the network). In practice, in our model, the set J 0 (n) is almost surely a singleton. We then define

J 1 (n) = {i ∈ I \ J 0 (n), V i,ε (τ ε n -) + ∑ j∈J 0 (n) H ji ≥ θ },
where the constants H ji are the synaptic weights. They represent the effects on the neuron i of a spike emitted by the neuron j. We assume the network fully connected and exclusively composed of excitatory connections i.e.

m := min i, j∈I H ji > 0. ( 5
)
Note that a neuron i of J 1 (n) has a membrane potential smaller than θ at time τ ε nbut larger than θ after receiving at time τ ε n the kicks H ji of the spiking neurons j of J 0 (n). The set J 1 (n) is therefore the set of the neurons which spike at time τ ε n by interaction with the neurons of J 0 (n). In the same way, we define by induction the sets

J p+1 (n) = {i ∈ I \ ∪ 0≤q≤p J q (n), V i,ε (τ ε n -) + ∑ j∈∪ 0≤q≤p J q (n) H ji ≥ θ }.
The set

J(n) := ∪ p∈N J p (n)
is thus the set of the labels of the neurons which emit a spike at time τ ε n . Note that this union is finite, since our definition implies that J p+1 (n) ⊂ (∪ 0≤q≤p J q (n)) C . Therefore, a neuron can not spikes twice (or more) at time τ ε n . Another way to obtain this property could be to introduce a refractory period in the previous model, but it would add technical difficulties. Nevertheless, the model considered in this paper can be interpreted as the limit of the similar model with a refractory period, when the refractory period goes to zero. Now we can define the state of the network in the firing regime as

V i,ε (τ ε n ) =    0 if i ∈ J(n) V i,ε (τ ε n -) + ∑ j∈J(n) H ji if i / ∈ J(n). (6)
The definition of

J(n) ensures that V i,ε (τ ε n ) < θ forall i ∈ I.
This ends the n-th firing regime and the network starts again to evolve according to the equation ( 4) of the subthreshold regime.

Results

Recall that the network is synchronized at time t if all the neurons spike simultaneously at time t. We have already introduced the sequence of spiking times

τ ε 1 , • • • , τ ε n , • •
• at which at least one neuron emits a spike. We are here interested in events of the form

S ε k := { the network is synchronized at time τ ε k } (7) BS ε n := { the network has been synchronized at least once before time τ ε n }. (8)
Obviously, we have

BS ε n = ∪ n k=1 S ε k S ε k = {J(k) = I}.
In a deterministic framework, once the network is synchronized it remains synchronized for ever. But when stochastic perturbations are introduced the potentials of the neurons do not remain equal after synchronization. Therefore, it is not guarantied that the network get synchronized again at the next spiking time. Nevertheless, an estimation of the probability of this event is provided by the following theorem:

Theorem 3.1 There exists ε 0 > 0, such that for every ε ∈ (0, ε 0 ] the probability that the network stays synchronized at time τ ε n+1 given that for some integer n it is synchronized at time τ ε n satisfies,

P S ε n+1 S ε n ≥ 1 -exp -γ m 2 4ε N .
This result principally shows that the synchronization is not destabilized by small stochastic perturbations. More precisely, if the intensity ε of the noise goes to 0, then the probability that the network stays synchronized goes to 1, and for sufficiently small intensity of the noise the network stays synchronized with high probability. The proof of Theorem 3.1 is given in Section 5.2. The main idea is to use a large deviations principle to obtain a lower bound on the probability that the potential of each neuron remains close to the deterministic synchronized solution until some neurons reach the threshold potential. In this case, the potential of the neurons are sufficiently close to each other at the firing time, and the interactions between neurons induce the synchronization. Now we study the probability that the networks get synchronized.

Theorem 3.2 Assume (V i,ε (t), i ∈ 1, N ,t ≥ 0) evolves according to (4) and (6) and that the number N of neurons in the network satisfies

N ≥ θ -α m θ -α m + 2 . ( 9
)
Then, for all n ∈ [ θ -α m , mN θ -α ], the probability that the network synchronizes during the first n spiking times verifies

P (BS ε n ) ≥ 1 -NΦ -m 2γ ε inf n - θ -α m , β -θ m + × 1 -Φ -m 2γ ε N n - θ -α m -nΦ -m 2γ ε N + , (10)
where Φ denotes the cumulative distribution function of a standard Gaussian random variable and (x) + := sup(x, 0) denotes the positive part of a real number x.

Note that the condition (9) ensures that the integer

n 0 = θ -α m + 1 always belongs to the interval [ θ -α m , mN θ -α ].
The sequence of events (BS ε n ) n is obviously increasing. So, we deduce from Theorem 3.2 that for every n ≥ n 0 (without any upper bound constraint on n)

P (BS ε n ) ≥P BS ε n 0 ≥ (1 -NΦ (-p ε 1 inf{1 + p 2 -p 2 , p 3 })) + × 1 -Φ -p ε 1 N -p 2 p 2 + 1 p 2 + 1 -p 2 + 1 Φ (-p ε 1 ) N + ≥ (1 -NΦ(-p ε 1 inf{1, p 3 })) + (1 -Φ(- N p ε 1 p 2 +1 + p ε 1 p 2 ) -(p 2 + 2)Φ (-p ε 1 )) N + , (11)
where p ε 1 , p 2 and p 3 are the dimensionless parameters defined by

p ε 1 := m 2γ ε p 2 := θ -α m and p 3 := β -θ m .
Note also that condition (9) can be written as N ≥ p 2 (p 2 + 2). An analysis of (11) shows that the lower bound on P(BS ε n ) is an increasing function of p ε 1 and a decreasing function of p 2 . Therefore, if n ≥ n 0 the lower bound of P(BS ε n ) increases when the intensity ε of noise decreases, and goes to 1 when ε goes to 0. This result shows that the phenomenon of synchronization observed in deterministic networks, such as those of [START_REF] Mirollo | Synchronization of pulse-coupled biological oscillators[END_REF], is stable under stochastic perturbations. In other words, in presence of small stochastic perturbations, the synchronization occurs in a finite time (smaller than τ ε n ) with a high probability which goes to 1 in the deterministic limit. Moreover as expected in excitatory networks, strong interaction between neurons facilitates the synchronization. Indeed, our lower bound on the probability of synchronization is an increasing function of the interaction parameter m. Furthermore, with larger m our estimation is valid for smaller number of neurons (see ( 9)) and smaller number of spiking times (n θ -α m ). Remark 3.3 In the proof of Theorem 3.1, we obtain control on the probability that Card(J 1 (n + 1)) = N -1 given J(n) = I which is not a necessary condition for synchronization. Indeed, order the membrane potential of the neurons at time τ ε n+1 -

θ = V κ(1),ε (τ ε n+1 -) > V κ(2),ε (τ ε n+1 -) > • • • > V κ(N),ε (τ ε n+1 -).
A sufficient condition for synchronization is

V κ(i),ε (τ ε n+1 -) ≥ θ -(κ(i) -1)m, ∀i ∈ 1, N .
This last condition is weaker than the bound we use in the proof of Theorem 3.1 but, unfortunately, even the law of V κ(i),ε (τ ε n+1 -) is out of reach: it is related to the order statistic of a family of N Ornstein-Uhlenbeck processes at the first time at which one of them hits θ . This law is unknown. [START_REF] Sacerdote | Joint distribution of first exit times of a two dimensional Wiener process with jumps with application to a pair of coupled neurons[END_REF] give first results in this direction in a very simplified setting with N = 2 neurons and γ = 0 (Perfect Integrate and Fire model). See also a recent review for the LIF model with N = 1 in [START_REF] Sacerdote | Stochastic integrate and fire models: a review on mathematical methods and their applications[END_REF]. Similar results in our context should permit to accurately evaluate the probability of synchronization.

Numerical simulations

In this section, we illustrate our theorems with figures obtained by numerical simulations of the model.

In Figure 1, we illustrate Theorem 3.1: we plot as function of the intensity of the noise ε the probability of the network to be synchronized at the (n + 1)-th firing time, given that it was synchronized at the n-th firing time. For this figure, the simulation has been done with N = 1599 neurons, γ = 1, θ = 1, m = 0.05 and three values of β : β = 1.2 (red), β = 1.02 (green) and β = 0.98 (blue). The first two values of β satisfy our hypothesis β > θ , but not the last one.

As predicted by our theorem, for β > θ the plots show that once the network is synchronized, it stays synchronized with very large probability, provided the noise is sufficiently small. In other words, the synchronized state is stable under small stochastic perturbations.

When β < θ , no spike are produced in the deterministic case (ε = 0). However, in presence of noise (ε > 0) spikes are emited even if β < θ . We observe no significative difference in the behavior of P S ε n+1 S ε n as a function of ε for β = 0.98 < θ and θ < β = 1.02.

Figure 2 shows in the upper panel the probability P (BS ε n ) of synchronization before the n-th firing of the network as a function of n. In the lower panel, the figure shows the probability P BS ε n \ BS ε n-1 of first synchronization at the n-th firing time versus n. The plots are done for several values of the intensity of the interactions. For the deterministic model (ε = 0), synchronization occurs in a finite time [START_REF] Mirollo | Synchronization of pulse-coupled biological oscillators[END_REF][START_REF] Catsigeras | Integrate and fire neural networks, piecewise contractive maps and limit cycles[END_REF]. In Figure 2, we see that even in presence of noise the synchronization occurs in a short time with a high probability. We also observe the same monotony properties as the upper bound (11): the higher the intensity of the interactions, the higher is the probability that the network synchronizes fast. The proof consists in estimating

P sup ∈ 1,n Card(J( )) = N sup ∈ 1,n Card(J( )) ≥ A ( 12
)
and P sup

∈ 1,n Card(J( )) ≥ A , (13)
P(BS ε n ) = P sup ∈ 1,n
Card(J( )) = N using Bayes formula. A lower bound of ( 13) is obtained in Lemma 5.2 for A = N n . The control of the quantity ( 12) is given in Lemma 5.4.

We first prove a technical lemma for the explicit expression of the potential V i,ε before the first spiking time of neuron i.

Lemma 5.1 For all n ≥ 1 and i ∈ I we have

1 {i/ ∈∪ n =1 J( )} V i,ε (τ ε n ) =1 {i/ ∈∪ n =1 J( )} φ τ ε n (V i,ε (0)) + n ∑ =1 e -γ(τ ε n -τ ε ) ∑ j∈J( ) H ji + √ εe -γτ ε n ˆτε n 0 e γs dW i s , (14)
where 1 S denotes the indicatrice function of a set S, that is 1 S (ω) = 1 for ω ∈ S and 1 S (ω) = 0 for ω / ∈ S.

Proof: By formula ( 6)

1 {i / ∈J(1)} V i,ε (τ ε 1 ) = 1 {i / ∈J(1)} φ τ ε 1 (V i,ε (0)) + ∑ j∈J(1) H ji + √ εe -γτ ε 1 ˆτε 1 0
e γs dW i s and the lemma is true for n = 1. From the same formula (6) at time τ ε n+1 we have

1 {i/ ∈∪ n+1 =1 J( )} V i,ε (τ ε n+1 ) =1 {i/ ∈∪ n+1 =1 J( )} φ τ ε n+1 -τ ε n (V i,ε (τ ε n )) + ∑ j∈J(n+1) H ji + √ εe -γτ ε n+1 ˆτε n+1 τ ε n e γs dW i s . (15)
Supposing the lemma is true at rank n and using φ t (x + y) = φ t (x) + ye -γt we obtain

φ τ ε n+1 -τ ε n 1 {i/ ∈∪ n =1 J( )} V i,ε (τ ε n ) =φ τ ε n+1 -τ ε n 1 {i/ ∈∪ n =1 J( )} φ τ ε n V i,ε (0) + n ∑ =1 e -γ(τ ε n+1 -τ ε ) ∑ j∈J( ) H ji + √ εe -γτ ε n+1 ˆτε n 0 e γs dW i s 1 {i/ ∈∪ n =1 J( )} . (16)
Using

1 {i/ ∈∪ n+1 =1 J( )} = 1 {i/ ∈∪ n+1 =1 J( )} 1 {i/ ∈∪ n =1 J( )}
and 15) and ( 16), we deduce

1 {i/ ∈∪ n+1 =1 J( )} φ t (x) = 1 {i/ ∈∪ n+1 =1 J( )} φ t 1 {i/ ∈∪ n =1 J( )} x , from equations (
1 {i/ ∈∪ n+1 =1 J( )} V i,ε (τ ε n+1 ) =1 {i/ ∈∪ n+1 =1 J( )} φ τ ε n+1 -τ ε n φ τ ε n V i,ε (0) + ∑ j∈J(n+1) H ji + √ εe -γτ ε n+1 ˆτε n+1 τ ε n e γs dW i s + n ∑ =1 e -γ(τ ε n+1 -τ ε ) ∑ j∈J( ) H ji + √ εe -γτ ε n+1 ˆτε n 0 e γs dW i s ,
and it immediately follows that the property is true at rank n + 1 2 Lemma 5.2 Assume the support of the initial condition is included in (α, θ ). Let n ≥ θ -α m then,

P max ∈ 1,n Card(J( )) > N n ≥ 1 -NΦ -m 2γ ε inf n - θ -α m , β -θ m Proof: Since, n =1 J( ) = I =⇒ ∃ ∈ 1, n s.t. Card(J( )) ≥ N n ,
we have

P max ∈ 1,n Card(J( )) > N n ≥ P n j=1 J( ) = I . (17)
From Lemma 5.1 we deduce that for any i ∈ I

P i / ∈ n =1 J( ) ≤ P φ τ ε n (V i,ε (0)) + n ∑ =1 e -γ(τ ε n -τ ε ) ∑ j∈J( ) H ji + √ εe -γτ ε n ˆτε n 0 e γs dW i s < θ ≤ P φ τ ε n (V i,ε (0)) + mne -γτ ε n + √ εe -γτ ε n ˆτε n 0 e γs dW i s < θ ≤ P φ τ ε n (V i,ε (0) + mn) + √ εe -γτ ε n ˆτε n 0 e γs dW i s < θ ≤ P inf{V i,ε (0) + mn, β } + √ εe -γτ ε n ˆτε n 0
e γs dW i s < θ .

Now, remind that

L √ εe -γt ˆt 0 e γs dW s = N 0, ε 1 -e -2γt 2γ ∀t ∈ R.
On the other hand, by monotonicity of the cumulative distribution function Φ, for all a ≥ a 0 and σ 2 ≤ σ 2 0 , we have

∀x ≤ a, P N (a, σ 2 ) < x ≤ Φ x -a 0 σ 0 . ( 18
)
Then, with

a 0 = inf{α + mn, β }, a = inf{V i,ε (0) + mn, β }, x = θ , σ 0 = ε 2γ and σ = 1-e -2γt
2γ , if inf{α + mn, β } ≥ θ , we have

P i / ∈ n =1 J( ) ≤ Φ   θ -inf{α + mn, β } ε 2γ   = Φ -m 2γ ε inf n - θ -α m , β -θ m .
The result follows then from (17) and

P n =1 J( ) = I ≤ N ∑ i=1 P i / ∈ n =1 J( ) . 2 Lemma 5.3 For all n ≥ 0, t ≥ τ n , i ∈ 1, N , ∀C ≥ θ -α, P Ṽ i,ε n (t) +C < θ ≤ Φ (θ -α -C) 2γ ε + nΦ -m 2γ ε . (19)
Proof: We start with n = 0. For all t, conditionally to the initial value V i,ε (0), Ṽ i,ε 0 (t) has a gaussian distribution with mean

φ t (V i,ε (0)) and variance ε(1 -exp(-2γt)) 2γ .
The proof of ( 19) for n = 0 is ended by using ( 18) and an integration with respect to the initial law V i,ε (0).

P Ṽ i,ε 1 (t) +C < θ =P Ṽ i,ε 1 (t) +C < θ , Ṽ i,ε 1 (τ 1 ) ≥ α + P Ṽ i,ε 1 (t) +C < θ , Ṽ i,ε 1 (τ 1 ) < α .
Using conditional probability, an upper bound for the first term is

Φ (θ -α -C) 2γ ε .
Furthermore,

P Ṽ i,ε 1 (t) +C < θ , Ṽ i,ε 1 (τ 1 ) < α ≤P Ṽ i,ε 1 (τ 1 ) < α ≤P Ṽ i,ε 1 (τ 1 ) < α, i ∈ J(1) + P Ṽ i,ε 1 (τ 1 ) < α, i / ∈ J(1) ≤P Ṽ i,ε 0 (τ 1 ) + m < α, i / ∈ J(1) ≤P Ṽ i,ε 0 (τ 1 ) + m < α ≤Φ -m 2γ ε .
So, for n = 1,

P Ṽ i,ε 1 (t) +C < θ ≤ Φ (θ -α -C) 2γ ε + Φ -m 2γ ε .
To conclude, we use a straightforward induction on n, splitting cases according to the position of Ṽ i,ε n (τ n ) with respect to α.

2 Lemma 5.4 Let A be in [ θ -α m , N -1]. ∀ ≥ 1, P (Card(J( )) = N|Card(J( )) ≥ A) ≥ 1 -Φ (θ -α -mA) 2γ ε -Φ -m 2γ ε N (20) Proof: Given Card(J( )) ≥ A, a sufficient condition to have Card(J( )) = N is Ṽ i,ε -1 (τ ) + mA ≥ θ , ∀i ∈ 1, N .
On the one hand, assuming A ≥ θ -α m , we apply Lemma 5.3 with C = Am. On the other hand, using the conditional independence of ( Ṽ

i,ε -1 (t) -Ṽ i,ε -1 (τ -1 )) t≥τ -1 , given Ṽ i,ε -1 (τ -1 ), we obtain (20). 2 Since N ≥ θ -α m ( θ -α m + 2), there are integers in [ θ -α m , mN θ -α ].
For such an integer n, we can apply Lemma 5.4 with A = N n and Lemma 5.2 to obtain

P sup ∈ 1,n Card(J( )) = N ≥ 1 -NΦ -m 2γ ε inf n - θ -α m , β -θ m × 1 -Φ -m 2γ ε N n - θ -α m -nΦ -m 2γ ε N .
5.2 Proof of the Result of Stability (Th. 3.1)

Firstly, we know that the system has the homogeneous Markov property, so

V i,ε (τ ε n+1 ) = 0, ∀ i ∈ I conditionned by V i,ε (τ ε n ) = 0, ∀ i ∈ I
have the same probability for any n. Thus, we prove the result of Theorem 3.1 for n = 0, that is we consider a system of N neurons such that ∀i ∈ 1, N , V i,ε (0) = 0, i.e. the network is initially synchronized. We estimate in this section a lower bound for the probability to stay synchronized at the next spiking time of the network τ ε 1 (see ( 1) and ( 2)).

The main tool of the proof of Theorem 3.1 is the Large Deviation Principle for an Ornstein Uhlenbeck process. We give an explicit expression of the good rate function in the following Lemma.

Lemma 5.5 Let x be the solution of the linear equation and X ε the solution of the associated noisy dynamic.

X ε (t) = x 0 + ˆt 0 K -γX ε (s)ds + √ ε W t , (21)
x(t) = x 0 + ˆt 0 K -γx(s)ds. (22)
A Annex: Large Deviation Principle for Ornstein Uhlenbeck Process

We just recall results of Freidlin-Wentzell theory and apply it to the particular case of Ornstein-Uhlenbeck process.

We consider a one dimensional deterministic function (x(t),t ≥ 0) evolving according to

x(t) = x 0 + ˆt 0 b(x(s))ds, (26)
where b : R → R is a uniformly Lipschitz continuous function. The main question is related to the distance between x and its stochastic perturbation by an additive noise:

X ε (t) = x 0 + ˆt 0 b(X ε (s))ds + √ ε W t . (27)
Precisely, we fix a final time T and a distance δ and we aim to estimate

P sup 0≤t≤T |X ε (t) -x(t)| ≥ δ .
Answers to this classical question can be obtained with Freidlin-Wentzell theory. We introduce the functional set

H x 0 1 := f : f (t) = x 0 + ˆt 0 φ (s)ds, φ ∈ L 2 ([0, T ]),t ∈ [0, T ] .
Theorem A.1 The family (X ε (t), 0 ≤ t ≤ T ) satisfies a large deviation principle with good rate function I x 0

I x 0 ( f ) =    1 2 ˆT 0 f (t) -b( f (t))
2 dt for f ∈ H

x 0 1 +∞ otherwise.

The proof can be found e.g. in Dembo and Zeitouni (2010, Sect. 5.6). We apply with the set

Γ = f ∈ C 0 [0, T ], sup 0≤t≤T | f (t) -x(t)| ≥ δ .
The LDP writes

-inf f ∈ • Γ I x 0 ( f ) ≤ lim inf ε ε log P sup 0≤t≤T |X ε (t) -x(t)| ≥ δ lim sup ε ε log P sup 0≤t≤T |X ε (t) -x(t)| ≥ δ ≤ -inf f ∈ Γ I x 0 ( f ),
where • Γ and Γ denote respectively the interior and the closure of Γ. In our case the first and last quantities are equal, so it now remains to evaluate inf

f ∈Γ I x 0 ( f ) = inf f ∈Γ∩H x 0 1 1 2 ˆT 0 f (t) -b( f (t)) 2 dt.
In our case, b(x) = -γx+K and we find an explicit expression for the previous quantity. Let us introduce

Ψ(ϕ) = ˆT1 0 ϕ (t) 2 + γ 2 ϕ 2 (t)dt for ϕ ∈ H 0 1 , |ϕ(T 1 )| = δ .
The optimal infimum of Ψ is reached by

ϕ T 1 (t) = δ sinh γt sinh γT 1 for t ∈ [0, T 1 ].
since for all Λ ∈ H 0 1 such that Λ(T 1 ) = 0,

Ψ(ϕ T 1 + Λ) = ˆT1 0 ϕ T 1 (t) + Λ (t) 2 + γ 2

Figure 1 :

 1 Figure 1: P S ε n+1 S ε n versus ε for N = 1599, m = 0.05, γ = 1 and θ = 1.0. Red plot β = 1.2, green plot β = 1.02 and blue plot β = 0.98. In the upper panel ε ∈ [0, 0.1] and the lower panel is a zoom in the range ε ∈ [0, 0.01].

Figure 2 :

 2 Figure 2: P (BS ε n ) versus n (upper panel) and P BS ε n \ BS ε n-1 probability of first synchronization at the n-th firing time versus n (lower pannel). The values of the parameters are: ε = 0.01, β = 1.2, α = 1, N = 1599 and θ = 1. Magenta m = 0.001, green m = 0.0015, blue m = 0.0018 and yellow m = 0.002.

  (ϕ T 1 (t) + Λ(t)) 2 dt = Ψ(ϕ T 1 ) + Ψ(Λ) (-ϕ T 1 (t) + γ 2 ϕ T 1 (t))dt = Ψ(ϕ T 1 ) + Ψ(Λ) ≥ Ψ(ϕ T 1 ). ) dt = δ 2 γ cosh (γT 1 ) sinh (γT 1 ) . Therefore the function Ψ(ϕ T 1 ) is a decreasing function of T sup 0≤t≤T |X ε (t)x(t)| ≥ δ = -γδ 2 2 1 + cosh (γT ) sinh (γT ) < -γδ 2 .

  Consider ϕ ∈ H 0 1 , ϕ ∞ ≥ δ and introduce T 1 = inf {t ∈ [0, T ], |ϕ(t)| = δ }. We define a modification φ ∈ H 0 1 such that | φ(T 1 )| = δ : φ(t) = ϕ(t) for t ∈ [0, T 1 ] δ exp (-γ(t -T 1 )) for t ∈ [T 1 , T ].

	Denote ϕ(t) = f (t) -x(t), we have
		ˆT 0	f (t) -b( f (t))	2 dt =	ˆT 0	ϕ (t) + γϕ(t))	2 dt.
	So, it remains to evaluate:
					inf 1 , ϕ ∞ ≥δ ˆT 0 ϕ∈H 0	ϕ (t) + γϕ(t)	2 dt.
	We have
	ˆT 0	ϕ (t) + γϕ(t))	2 dt ≥	ˆT 0	φ (t) + γ φ(t)	2 dt =	ˆT1 0	φ (t) + γ φ(t)	2 dt.
	So, we have to evaluate
								ˆT1
		inf 0≤T 1 ≤T	inf 1 ,|ϕ(T 1 )|=δ ϕ∈H 0	0	ϕ (t) + γϕ(t)	2 dt.
	ˆT1						ˆT1		ˆT1
		ϕ (t) + γϕ(t)	2 dt =	ϕ	2 (t) + γ 2 ϕ 2 (t)dt + γ	2ϕ(t)ϕ (t)dt
	0						0 ˆT1		0
						=	ϕ	2 (t) + γ 2 ϕ 2 (t)dt + γδ 2
							0

Centro de Investigación y Modelamiento de Fenómenos Aleatorios -Valparaíso, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso, Chile pierre.guiraud@uv.cl

Université Côte d'Azur, Inria, France. 2004 route des Lucioles, BP93, 06902 Sophia-Antipolis Cedex, France Etienne.Tanre@inria.fr

Acknowledgement PG and ET have been supported by Mathamsud Project 13MATH-04 -SIN -. PG thanks University of Valparaiso project DIUV 45/2013. ET has also been supported by the Inria Equipe Associée Anestoc-Tosca. ET is grateful to Inria Sophia Antipolis -Méditerranée "Nef" computation cluster for providing resources and support.

For any fixed level δ > 0 and finite time T , let

We have

Proof: See Annexe 2 Now we finish the proof of the Theorem 3.1. A suffisant condition for the network to stay synchronized at τ ε 1 is

Let us denote

the solution to the associated deterministic equation (i.e. ε = 0) starting from 0. We fix δ ∈ (0, min{βθ , θ }), then there exist 0 25) is satisfied for δ ≤ m/2, then all the neurons spike at time τ ε 1 and so the network stays synchronized. As the potential Ṽ i,ε 0 are independent before τ ε 1 , condition ( 25) is fulfilled with a probability greater than

and the probability of synchronization satisfies

From Lemma 5.5 it follows that

N for all ε small enough.