Fatiha Hamma¹

Daniel Philip Venmani¹

Kamal Singh²

Bruno Jahan¹

Synchronization in Industrial IoT: Impact of propagation delay on time error

Keywords: Time synchronization, propagating delay compensation, time error, 5G/B5G, Industry 4.0/5.0

The growing needs for wireless connectivity in smart industries raises opportunities for mobile service providers as well as industrial players, also known as verticals. This inevitably has increased challenges for both the parties involved. This is increasingly true for timing and synchronization requirements. Within this context, in this article, we present a detailed analysis of 'time synchronization errors'. These errors are commonly known to affect the delivery of accurate time needed to synchronize the robots and machines present inside the Industrial Internet of Things (IIoT) domain. To eliminate such errors, we propose a novel compensation technique to compensate for the random errors which occur due to the propagation delay in the wireless medium. The article concludes by providing numerous results based on simulations with discussions proving the efficacy of our proposal.

I. INTRODUCTION

Time synchronization is increasingly vital for wired and wireless communication, especially in time-critical industrial applications. This necessity is driven by the demands of Industry 4.0 and the forthcoming Industry 5.0, emphasizing the need for precise time synchronization.

In particular the end-to-end (starting from the time source -GNSS-based or IEEE 1588-based, until the end robot) time synchronization requirements are defined to be no more than 1 microsecond for end-to-end time sensitive communications, in the context of 3GPP Release 16 specifications defined for 5G [1]. This involves transporting timing packets over wired networks as well as wireless networks. Transporting timing packets over wired networks is relatively simple, due to the fact that the errors which accumulate over a wired medium (fibre, copper cables, etc.) could be estimated in advance and could be eliminated using compensation techniques. One of the most known compensation techniques for wired medium is 'Calibration' [START_REF] Rost | Time transfer through optical fibres over a distance of 73 km with an uncertainty below 100 ps[END_REF]. On the other hand, transporting timing over wireless medium is more difficult, due to the nature of the wireless medium itself. The air interface, also known as the radio interface is known to encounter several types of errors which are mostly random. Some of known factors to influence the errors over the air interface are directly related to channel conditions, such as multi-path, fading, mobility, etc. Our focus in this paper is on error compensation for the wireless medium, with particular attention paid to the study and analysis of various errors which occur in the wireless medium while trying to establish a synchronisation over-theair between the User Equipment (UE) and the Base Station (BS). Our contributions in this paper are as follows:

• Time Compensation Technique: We propose a time compensation technique specifically designed to address the time error caused by the wireless medium. We also provide a comprehensive analysis and understanding of the impact of time error on the synchronization accuracy, using detailed simulations. This proposed technique effectively minimizes the synchronization errors.

• Signal Synchronization based only on Downlink: In our research, we propose an approach to synchronize the clocks of User Equipment (UEs) with static locations, using only a downlink signal for the case of smart factories. This method significantly reduces the overall time error thereby increasing synchronising accuracy, ensuring highly accurate and precise synchronization. By eliminating the need for additional signals, such as uplink signals, or complex synchronization mechanisms, our approach simplifies the synchronization process while maintaining good accuracy levels. In the context of our paper, the UE could be a 'relay' placed between the Base Station (BS) and the robots of the factory or the UE may represent the robots themselves directly. We assume that this device will be confined to a given factory location. Our paper is organised as follows. Section II provides a detailed analysis on the various factors that affect the Propagation Delay (PD). Section III proposes our novel technique to compensate for the random errors which occur due to the propagation delay in the wireless medium. Our error compensation technique leverages on the existing methods of error calculation standardised by 3GPP and therefore does not require any changes to existing implementations of the mobile BSs or the UEs. Our results in Section IV show that our compensation technique satisfies the emerging demands of 5G, as defined in [1]. Section V concludes this work.

II. BACKGROUND AND RELATED WORKS

Similar to 4G, 5G uses Synchronization Signal Block (SSB) to synchronize the UEs 'over-the-air'. However, the frame structure of 5G SSB is quite different than that of 4G. The frame structure used for over-the-air synchronization is defined in the 3GPP specifications [START_REF]G Base Station (BS) radio transmission and reception[END_REF]. The SSB consists of 4 OFDM symbols, the first for Primary Synchronization Signal (PSS), the second for Secondary Synchronization Signal (SSS), and the rest for Physical Broadcast Channel (PBCH). Synchronization is established through an exchange of these signals between the BS and the UE, in frequency domain as well as in time domain. This enables synchronisation in both frequency as well as in time. More details on this initial attach process could be found in [START_REF] Omri | Synchronization procedure in 5G NR systems[END_REF].

Propagation delay (PD), represented from here onwards mathematically as P DL , is defined as the amount of time that a signal needs to be received by the UE from the BS; the greater the distance between the UE and the BS, the higher the P DL would be [START_REF] Schüngel | Optimized Propagation Delay Compensation for an Improved 5G RAN Synchronization[END_REF]. This will cause the local clock present inside the UE, T U E , to lag behind the reference BS local clock, T BS . The most commonly known methods to estimate propagation delay are based on Timing Advance (TA) and Round Trip Time (RTT) methods. These are legacy techniques and for in-depth details, please refer [START_REF] Raitoharju | Estimation of base station position using timing advance measurements[END_REF]. These two methods date back to 3G. Although both these methods are able to fulfil the existing 3GPP requirements for time synchronization, we argue that they may not satisfy the emerging synchronization requirements of the future Industrial Internet of Things (IIoT) domain. For instance, 3GPP TS 23.501 [1] cites that the most demanding synchronisation requirement in the context of smart industries is 900 ns, which has been defined for motionto-motion control application within a smart grid.

In order to improve the time synchronization accuracy for 5G, new approaches have been introduced. It is worth mentioning here that propagation delay estimation and compensation, while remaining a very important operational subject, remains an obscure field of research. While the number of related works directly implicating this subject is very limited, there are a handful of very interesting related works. To point a few, the authors of [START_REF] Hadi | 5G Time Synchronization: Performance Analysis and Enhancements for Multipath Scenarios[END_REF] proposed to use Channel State Information Reference Signal (CSI-RS) instead of SSB to benefit from its higher bandwidth to cover larger distance between the base station and User Equipment (UE). The results show that the proposed signals cover more distance between UE and gNB compared to the usual synchronization signals. However, their results were limited only for subcarrier spacing of 15 kHz for 3.6 MHz carrier frequency bandwidth for the SSB signals. In [START_REF] Schüngel | Optimized Propagation Delay Compensation for an Improved 5G RAN Synchronization[END_REF] the authors chose another different method for time synchronization. They used multiple TA measurements to estimate the position of UE in order to determine the PD. Their technique causes significant overhead over the wireless medium and is not suitable to satisfy the synchronisation requirements of 5G, notably the 900 ns [1]. In [START_REF] Xu | RAN enhancement to support propagation delay compensation of TSN[END_REF], the authors introduced a propagation delay estimation and compensation technique which combines both TA and RTT methods. They talked about the downlink/uplink time synchronization budget from a RAN perspective. This solution, as they have pointed-out themselves, has the disadvantages of complexity and wasted resources. Following this in [START_REF] Shi | Evaluating the performance of over-the-air time synchronization for 5g and tsn integration[END_REF], the authors investigate frequency and scalability aspects of overthe-air time synchronization. Their performance evaluation reveals the conditions under which 1 microsecond or less requirement for time synchronization can be achieved. They adopt an approach where the accuracy of time synchronization is achieved through the use of 'special equipment' in the presence of clock drift and different air-interface timing errors related to reference time indication. The complexity lies within the fact that the network requires an upgrade for the use of such 'special equipment'.

Unlike the above mentioned works, through our work, we show that our proposal is efficient in order to satisfy the emerging demands of 5G while satisfying the many different criteria such as different Sub Carrier Spacing (SCS), different carrier frequency bandwidth, different channel models, etc., without creating any additional impact to the existing implementations or network operations. This means network operators could leverage on their already deployed 5G base stations (gNBs) in order to fulfil even future emerging synchronisation demands from verticals. Furthermore, our technique uses only downlink synchronization signals between UE and 5G base station without the need for uplink transmissions. This may simplify compensation calculation, reduce system latency, and improve the performance of the network since there will be fewer transmissions (no uplink transmissions), less consumed power, and fewer resource allocations.

III. ON THE IMPACT OF PROPAGATION DELAY ESTIMATION AND COMPENSATION

The main challenge in estimating the total synchronization error is finding the propagation delay error and compensating for it. While doing this, there are different types of errors that come into picture. They are:

Base station timing error: on the base station side, 3GPP defines the maximum Time Alignment Error (TAE) as a requirement [START_REF] Schüngel | Heterogeneous synchronization in converged wired and wireless time-sensitive networks[END_REF], and it is caused due to the mismatch between the frames transmitted from different radio units (or antenna connectors). This mismatch can be caused by the clock drift or due to the delays incurred by the synchronization protocols. Table I summarizes the timing values of TAE specified by 3GPP for different scenarios . Two types of base station errors are involved in this exchange: the base station Transmission timing error listed in Table I and the base station detection error, which is a constant value of 100 ns, shown in [START_REF] Schüngel | Heterogeneous synchronization in converged wired and wireless time-sensitive networks[END_REF]. User Equipment (UE) error: Uplink and downlink exchanges between the UE and BS lead to two main errors related to the UE: the initial transmit timing error, denoted in this paper as E U E,U L,T x and referred by 3GPP as T e , is listed in Table II. The second error is the downlink reception error observed in the UE, denoted as E U E,DL,Rx . The downlink frame time on UE side represents the time of arrival (ToA) of the downlink signal. The downlink frame timing detection error E U E,DL,Rx depends on SCS of the uplink signals. b) Channel: The wireless channel's characteristics can have an impact on the timing synchronization procedure due to multi-path propagation and interference. 3GPP defines two channel models for link level simulations, namely TDL and CDL [12]. TDL is the model of taps with different delays (therefore the name Tapped Delay Line); each tap is modeled as a random variable. Clustered Delay Line (CDL) models a received signal consisting of multiple delayed clusters. Multipath fading is simulated by adding attenuated and phase shifted copies to the transmitted signal.

c) Shadowing: The existence of obstacles between the transmitter and the receiver causes the shadowing effect. For this study, we additionally assess the situation for line-of-sight (LOS) and non-line-of-sight (NLOS).

A. Proposal to estimate PD for devices

We propose our method for estimating the propagation delay for a known UE with a fixed TX-RX distance, in order to compensate for the estimated PD. We then investigate the effect of the channel conditions (such as path-loss, multipath, and shadowing) between UE and BS. We assume a use case with stationary devices, considering control-to-control communication for industrial controllers with a service area of 100 m 2 , which requires a total time error budget of 900 ns [1]. In this use case, majority of IoT devices are immobile or have limited mobility (mobility confined to a space and does not exceed 5km/h) in small service areas [START_REF] Tan | Enabling indoor mobile millimeter-wave networks based on smart reflect-arrays[END_REF]. Therefore we base our hypothesis on the assumption that the distances between the base station and the UE do not vary randomly. This allows us to establish fixed values for propagation delay estimation and compensate for it relatively 'easily' rather than using TA or RTT methods. By assuming a fixed distance between the TX and the RX, the remaining error gets limited to channel conditions such as multipath or fading. Only variations due to parameters like atmospheric conditions could have a severe impact on the PD.

In our case, only a downlink signal from the base station to the UE is used, without the need to use an uplink signal as in the case of TA or RTT methods. This enables very little overhead over the dedicated wireless medium. In addition this decreases the total time error in the synchronization procedure, because the errors E U E,U L,T x or E BS,U L,Rx become absent. Our proposal considers a stationary UE or a UE with mobility in limited area, such that the approximate location is known. It is realistic to assume that we know the location of the smart factory. In the case of higher mobility, we can use a fixed relay UE, placed inside the smart factory, to redistribute the time synchronization received from the base station.

Thus, calculation of propagation delay compensation error assumes knowledge of the approximate distance between the base station and the UE, as from above we consider the UE confined to a factory location. With a Non-Disclosure Agreement (NDA), it is possible to obtain the approximate positions of the Factory or UEs present either inside or around the premises of the factory for a first use. This will introduce some error, but this error stays minimal as specified in the next paragraph. In addition, we would need only a one-way downlink signal instead of the two-way signals (downlink and uplink as in TA or RTT). This enables lower overhead over the dedicated wireless medium, in addition to decreasing the time to synchronize the UEs with less time error. With this, we deduce the timing of UE, T U E as:

T U E = T BS + P DL + E T otal (1)
The total remaining error, E T otal , is traditionally estimated by adding the different downlink and uplink errors as E U E,U L,T x , E BS,U L,Rx , E U E,DL,Rx , E U E,U L,T x , the channel transmission errors and errors related to the calculation of the time advance or RTT. However in our proposal, the total remaining error is derived from only using the downlink errors E U E,DL,Rx and E gN B,DL,T x , in addition to the error of the estimated distance, E d,est , and the error of the transmission channel, Err chan , which is shown in eq. ( 2) and Figure 2.

E T otal = E d,est + Err chan + E gN B,DL,T x + E U E,DL,Rx (2)
Fig. 2: Illustration of proposed propagation delay estimation Our proposal involves the reference time to be sent by the mobile network clock (base station). For low mobility devices in the IIoT domain, the clock on the device ( equation 1) is adjusted by using the propagation delay compensation P DL , estimated using the distance between UE or the smart factory, which is fixed in our use case, and gNB. The proposed total propagation delay will be the sum of propagation delay compensation P DL and the calculated error E T otal as explained in Figure 2. E d,est is low in our case, since a localisation error of 10 m will lead to 33.33 ns of error considering the speed of light, and we assume that the location of smart factory is known with such error margin. It should be mentioned that for a distance below 200 m, there is no need for propagation delay estimation and compensation [START_REF] Schüngel | Heterogeneous synchronization in converged wired and wireless time-sensitive networks[END_REF]. The timing errors due to transmitters and receivers of gNB and UE would be constant, since they remain to be the same physical components. The other part of error Err chan will be due to multipath and synchronisation error losses. We study this error through simulations in the following section.

IV. NUMERICAL RESULTS AND ANALYSIS

In this section, we outline the simulations that were used to estimate the only unknown variable in our equation 2, in order to better calculate the total error E T otal .

A. Simulation setup and parameters

Numerous simulations were conducted using the 5G toolbox of MATLAB. We simulated a signal of 6 GHz. By carefully simulating several scenarios, we analyse the occurrence of remaining error and investigate whether we respect the 3GPP specification, i.e. the budget of 900 ns for control-to-control communication for industrial controllers with a service area of 100 m² [1].

B. Step-by-step process of our simulations

We simulated a TDL-C channel. This created multiple delayed versions of the input signal. By doing this, we were able to simulate a realistic multipath channel. We then added a path loss that is compliant to the 3GPP specification [12]. We generated the synchronization signal block (SSB) mentioned in [11]. We then used FFT in order to transform the generated SSB from frequency domain to time domain. The transformed signal is 'delayed' before transmitted. This 'external delay' is added in order to inter-correlate the received signal and the local reference signal. This inter-correlation allows us to compensate the propagation delay value using the distance and the speed of light. The peak of the correlation represents the estimated propagation delay with channel effects. We analyse our link-level simulations here. We analyse Err chan and results show that a lot of error budget (for a target E total < 900 ns) is still remaining to account for E d,est (33.33 ns for up to 10m of localisation error), E gN B,DL,T x (around 65 ns) and E U E,DL,Rx (around 100 ns). Next, we add a detailed analysis for our simulation results, and discuss different parameters which affect synchronization error.

C. Analysis of the proposed method for delay compensation

a) Distance vs Time error: The results show the impact of the maximum distance. Several simulations are run with UE randomly placed up to a given maximum distance. We will examine cumulative distribution function (CDF) which shows the probability (Y-axis) of achieving a value lower than or equal to a given point on the X-axis. Figures 3,4,5, show CDF of Err chan over a maximum range of distances up to 1 km, 3 km, and 6 km respectively. These ranges are reasonable as we are considered rural environment for IIoT factories. On the X axis, we plot the time error values Err chanl up to 1 µs. We also include zoomed figures to show the details. First we fixed the delay spread to 30ns. For SCS 15 kHz, we observe that for 88% of times Err chan error is low for a maximum distance of 6 km. At other times the synchronisation is lost due to reception errors. For maximum distance of 3km or less, we have low error for almost 100% of times. Similar simulations were conducted for delay spread of 300 ns (as delay spread can be high with such high distances). Results are shown in Figures 6,7,8. For SCS 15 kHz, the error now is limited to 200 ns approx. for 70% of times with max. distance of 6 km and almost 100% of times with max distance of 3 km or less. We can also observe that synchronisation loss occurs more often when the delay spread is increased. This is explained by the increase in Inter Symbol Interference (ISI). For example for same maximum distance of 6 km, CDF shows that, when compared side by side for different DSs, synchronisation loss occurs more often when delay spread is higher i.e. 300 ns as compared to 30 ns. b) Sub Carrier Spacing vs Time error: In general, we can see that sub-carrier spacing of 15 kHz is relatively more robust as compared to other SCS values. For maximum delay spread of 300 ns, it can satisfy the requirements of error < 900 ns in 70% of cases when maximum distance is 6 km and close to 100% when maximum distance is 3 km or less. Other SCS values perform relatively poorer with worse values for highest SCS. For example for SCS 120 kHz, the synchronisation requirement is satisfied only for 30% and 55% with maximum distance of 6 km and 3 km respectively. This is due to decreasing cyclic prefix interval leading to increased Inter Symbol Interference (ISI). Nevertheless, all SCS satisfy the synchronisation requirements almost 100% of times when the maximum distance is 1 km. c) Delay spread vs Time error: We may also observe how different SCS are affected by a random value delay spread (DS) in Figure 9. DS refers to the difference of the time of arrival of multiple visions of the same signal at the receiver due to the multipaths. We vary DS for analysis because DS can vary from environment to environment. A higher DS indicates a less robust channel with a significant time difference between the earliest and latest signal arrivals, which leads to more ISI.

Figure 9 shows how DS affects time synchronization error and illustrates the CDF of Err chan over a range of DS values between 32 ns and 1148 ns, as they are the minimum and maximum values of DS suggested by 3GPP [12] for different scenarios of Urban Macro (UMa), Rural Macro (RMa) and RMa Outdoor-to-Indoor (RMa O2I). With SCS of 15 kHz, 77% of transmitted signals had a Err chan of less than 700 ns ( while rest of the time the synchronisation was lost), compared to only 10% with SCS of 120 kHz. Higher sub-carrier spacing can result in a broader channel response, which lengthens the channel's delay spread. Thus, SCS should be chosen carefully depending on application needs and the traits of the wireless channel, including the delay spread.

d) Frequency vs Time error: We used a 6 GHz carrier frequency for the conducted simulation. Adopting high carrier frequencies opens the possibility of using new sub-carrier spacing of 60 kHz and 120 kHz, which may increase sensitivity to frequency and phase noise, but decrease resistance to Doppler shift. The transmitted symbols have a shorter duration, using higher sub-carrier spacing also reduces the effect of ISI. V. DISCUSSIONS AND CONCLUSION This paper proposed a new method for estimating and compensating propagation delay (PD). The idea is to correct the time error caused by the wireless medium when time synchronisation is delivered directly from the MNO's BSs. Only one-way downlink signal is used (instead of the classical two-way uplink-downlink) in order to reduce the signalling overhead. This allowed to improve the synchronisation accuracy for IIoT devices inside factories. Different scenarios, notably Distance vs Time Error, Sub-carrier Spacing vs Time Error, Delay Spread vs Time Error were evaluated. The best possible criteria which resulted in the least time error value were analyzed in order to obtain the best time synchronisation values. Our results reveal the impact of the transmission channel. Overall conclusions show that the distance between the BS and the UE has the maximum impact on the time error regardless of the application, while sub carrier spacing and delay spread should be chosen carefully depending on the application needs and the traits of the wireless channel. Future works would include advanced and precise models such as ray tracing models considering Refractive Index Surface (RIS) that would allow for more fine tuning of the time errors leading to better time synchronization values. Practical prototype in actual industrial IoT environments may be foreseen to confirm the effectiveness of the proposed solution.

  Figure1shows various errors due to base station and UE.

Fig. 1 :

 1 Fig. 1: Time errors inside BS (5G gNB here) and UE

Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 :Fig. 7 :Fig. 8 :

 345678 Fig.3: CDF plot for Err chan for variable distances ranging from 0 to 1km and delay spread of 30 ns

Fig. 9 :

 9 Fig.9: CDF plot for Err chan for a fixed distance of 3 km and variable random delay spread.

TABLE I :

 I TAE for BS specified by 3GPP

	Scenarios	Max TAE
	MIMO transmission	65 ns
	Intra-band contiguous carrier aggregation	260ns
	Intra-band non-contiguous carrier aggregation, 3µs
	Inter-band carrier aggregation,	3µs

TABLE II :

 II T e Timing Error Limit

	Frequency	SCS of SSB sig-	SCS of uplink sig-	Te
	Range	nals (KHZ)	nals (kHz)

Table I presents

 I E gN B,DL,T x , and Table II presents E U E,DL,Rx .


Josephine Simonnot
email: josephine.simonnot@cnrs.fr

Christine Esclapez
email: christine.esclapez@prism.cnrs.fr

Caroline Boë

How can we think about the links between tradition and creation in a sub-island territory? Music, Bridges, Passages: Towards a Mediterranean paradigm?

How can we think about the links between tradition and creation in a sub-island territory?

Archival appraisal of the Repertorium in order to establish a medium-and long-term heritage conservation and enhancement policy.

Axis 3 Archive as common territory

Local action with village users. Immersive listening point proposed in front of each craftsman's shop. « I think that for tourists, this will give them an even more cultural image and a deeper connection to the village, rather than just a commercial one. In a very short space of time, it can give a social and cultural dimension, even without explanations. »

Le Chant du Geste

Valorization through research: Active listening to the territory

Cultural

House of listening « At first glance, there are several houses for listening: the sounds of nature; the sounds of cities; language; music..., but, in fact, the way listening inhabits them forces us to change the correlations between the elements that make up each house. The house of listening is not a solid building, but a dynamic relationship between listening and its environment.

Historic Géographic

Le chant du geste, bridge-labyrinth

What bridges?

« Acquatic » relationship

Another relationship to time

Oscillation

The archive of the future: a rhythmic & cyclical bridge « The collective believed that the archive lies between two shores, between the fixed and the moving. In this way, it would enable us to construct a different relationship with time. No longer a linear relationship, but a logic of elsewhere, of space. In the end, we agreed that we needed to build an almost aquatic relationship with the archive: it is source, movement, irrigation. (...) The archive is not just a document, but something that can be considered as a space, where elements circulate and intersect, something that is perpetually repeated, like an eternal return. »

  workshop • Sound recordings • Discussions on heritage music • Discussions on the perception of the sound of artisanal gestures Acousmatic composition • Concrete sounds linked to handcrafting • Early music Artisan•es dans leurs ateliers : Toni Casalonga, Marie-Claire Darnéal, Ugo Casalonga, Cyril Quilichini, Hervé Quilichini, Virginie Torrelli. Photos : C. Boë © CC BY.

Interwiews on Canal U, Télé AMU :

https://www.canal-u.tv/chaines/tele-amu/videos-realisees-dans-le-cadre-du-projet-corsica-sud-archives-cosa Teaser of the movie : https://www.canal-u.tv/chaines/tele-amu/videos-realisees-dans-le-cadre-du-projet-corsica-sud-archives/bande-annonce-du

https://cosa.hypotheses.org


Edwin Friedmann
email: edwin.friedmann@cea.fr

Jean-Philippe Poli
email: jean-philippe.poli@cea.fr

Olivier Hotel
email: olivier.hotel@cea.fr

Fuzzy Classifiers for Chemical Compound Recognition from SAW Sensors Signals

Keywords: Fuzzy rule-based system, volatile chemical compounds, surface acoustic wave sensors, FURIA, fuzzy decision trees

Chemical vapor analysis devices are booming, thanks to a growing need in areas such as security and quality control. These control devices are based on various technologies that are the subject of important researches in an ever-growing community of physicists and electronics. However, the data from these sensors are often processed by conventional algorithms poorly configured for the purpose of automatically recognizing target chemical compounds. These algorithms are often based on statistical models that are not always adapted to a limited number of learning data and demonstrated reproducibility problems for these kind of sensors. In this article, we propose to train fuzzy models and compare their performances with the classical methods of the state of the art, to show how practical they can be for such applications. Three different uses cases will be studied: toxic chemicals recognition, detection of counterfeit coffee capsules and detection of a chemical weapon among everyday products.

I. INTRODUCTION

For decades, the recognition of volatile chemical compounds was done using classic olfactometry approaches: for instance, using a panel of experts or exploiting the developed olfactory capabilities of certain animals. However, physiological differences and concentration problems led to imprecise, even non-reproducible identifications. For example, it is estimated that a trained dog can only stay focused between 15 and 30 minutes. To overcome these shortcomings, alternative methods have been developed: gas chromatography, mass spectrometry, optical spectrometry and chemical sensors [START_REF] Li | Overview of odor detection instrumentation and the potential for human odor detection in air matrices[END_REF].

The problem of classification of "odours" has been successfully treated, in particular with fuzzy logic [START_REF] Guillaume | Fuzzy inference systems to model sensory evaluation[END_REF]- [START_REF] Szulczyński | Application of fuzzy logic to determine the odour intensity of model gas mixtures using electronic nose[END_REF]. To carry out this task, different descriptors are used for classification, such as the physico-chemical quantities of the molecules [START_REF] Kissi | Pattern recognition system based on empirical knowledge: Sandalwood and camphoraceous odours application[END_REF], [START_REF] Marsala | Fuzzy Decision Trees to Extract Features of Odorous Molecules[END_REF], the characteristics resulting from the responses of sensory sensors [START_REF] Llobet | Fuzzy artmap based electronic nose data analysis[END_REF], [START_REF] Szulczyński | Application of fuzzy logic to determine the odour intensity of model gas mixtures using electronic nose[END_REF], or even sensory analyses carried out by experts [START_REF] Guillaume | Fuzzy inference systems to model sensory evaluation[END_REF]. Although they all rely on the formalism of fuzzy logic, the classification methods used are also very varied, ranging from neural networks like Fuzzy ARTMAP [START_REF] Llobet | Fuzzy artmap based electronic nose data analysis[END_REF], to methods of fuzzy rules induction [START_REF] Guillaume | Fuzzy inference systems to model sensory evaluation[END_REF], [START_REF] Marsala | Fuzzy Decision Trees to Extract Features of Odorous Molecules[END_REF] and modeling by experts [START_REF] Kissi | Pattern recognition system based on empirical knowledge: Sandalwood and camphoraceous odours application[END_REF], [START_REF] Szulczyński | Application of fuzzy logic to determine the odour intensity of model gas mixtures using electronic nose[END_REF].

This article focuses on chemical sensors, which are booming thanks, among other things, to their transportability and their low cost. Their general principle is based on the physicochemical interaction between a volatile compound and a layer of sensitive material. In particular, this work deals with SAW (Surface Acoustic Wave) sensors that will be introduced in section II.

In [START_REF] Hotel | A review of algorithms for saw sensors e-nose based volatile compound identification[END_REF], the recognition of chemical compounds has been studied with classical approaches of machine learning (neural networks, SVP, kNN, etc.). However, the reactions of SAW sensors are not exactly reproducible, which leads to the need of a large number of training examples.

This article addresses the same problem but this time using fuzzy rule-based classifiers. They have the advantage to return a rule-based model that can be evaluated by experts of the domain afterwards. Furthermore, one can have access to the details of rules activated during a process of classification, making the decision more transparent.

As experts cannot formulate a rule base, we use machine learning. We will therefore compare two approaches: FURIA and fuzzy decision trees. These two algorithms represent two different strategies to induce rules from data.

The article is composed as follows. Section II explains the principles of SAW sensors and the feature extraction. Section III reminds the principles of the two methods of rule induction, which are used in this work. We then describe the experimental protocol in section IV and we present the results (section V) and discuss them (section VI) before concluding.

II. SAW SENSORS FEATURE EXTRACTION

The particularity of this work is to be applied on a more recent generation of chemical sensors based on Surface Acoustic Wave (SAW). Figure 1 presents an illustration of such sensors. When molecules interact with the sensitive surface of a SAW sensor, the physical properties of the latter change. These changes are converted into a signal. In our case, modified diamond nanoparticles are used as sensitive coatings that can be mixed with other nanoparticles to improve their sensitivity to chemical compounds. It is thus common to use a matrix of such sensors with different coatings, and so to manipulate a multi-valued signal.

Input transducer Sensing film

Surface Acoustic Wave

This differs from most of the previous works cited in the introduction since we do not have a characterization of the molecules but only signals from which we will extract features.

Figure 2 represents the response of the sensor when exposed to a coffee capsule. Each curve is in fact the output of a SAW sensor whose functionalization (the reactive layer) is specified in legend. This figure shows that the multi-valued signal is composed of two parts: a phase called transient regime and a phase called stationary regime. In practice, there is undoubtedly information in the transient mode but the sampling rate of these sensors being rather low (of the Fig. 2: Example of output of the sensor for a coffee capsule exposure. order of 10Hz), it often happens that this phase is not seen. The descriptors thus use the stationary phase.

The most used features in the literature [START_REF] Hotel | A review of algorithms for saw sensors e-nose based volatile compound identification[END_REF], [START_REF] Yan | Electronic nose feature extraction methods: A review[END_REF] are the amplitudes of the different signals during this stationary phase.

However, Hotel et al [START_REF] Hotel | Saw sensor's frequency shift characterization for odor recognition and concentration estimation[END_REF] proposed new interpretable descriptors based on the mathematical equations, which model the vibrations of the reactive layer. Indeed, it was established that the frequency shift of SAW sensors is the superposition of two main contributions (the electro-acoustic one can be neglected) [START_REF] Raj | Investigations on the origin of mass and elastic loading in the time varying distinct response of zno saw ammonia sensor[END_REF]:

1) a viscoelastic contribution due to changes in the coating's Young modulus; and 2) a mass loading effect due to changes of the coating film's mass. [START_REF] Raj | Investigations on the origin of mass and elastic loading in the time varying distinct response of zno saw ammonia sensor[END_REF] modeled these equations by first order linear differential equations. Since these contributions constitute a signature of the odour, [START_REF] Hotel | Saw sensor's frequency shift characterization for odor recognition and concentration estimation[END_REF] suggests solving the equations with the information contained in the signals. The features are their solutions. Unfortunately, the lack of available information prevent from solving them analytically. [START_REF] Hotel | Saw sensor's frequency shift characterization for odor recognition and concentration estimation[END_REF] uses metaheuristics to approximate the solutions.

These features show a real improvement of the performances in most applications [START_REF] Hotel | Saw sensor's frequency shift characterization for odor recognition and concentration estimation[END_REF]. However, the features represent uncertain data. This motivates the use of fuzzy logic, in particular the two methods we present in the next section.

III. FUZZY RULE-BASED MODEL INDUCTION

A. Learning Fuzzy Rules with FURIA

FURIA [START_REF] Hühn | Furia: an algorithm for unordered fuzzy rule induction[END_REF], standing for Fuzzy Unordered Rule Induction Algorithm, is an extension of the RIPPER [START_REF] Cohen | Fast effective rule induction[END_REF] nonfuzzy rule learning algorithm. Briefly, RIPPER is a multilabel classification algorithm that learns rules having as a premise a conjunction of inequalities on numeric attributes The first consists in applying a one-vs-rest decomposition of the data to obtain an unordered rule base, instead of eliminating the classes one after the other and get a decision list like previous rule base learning algorithms. The latter approach can cause a bias in favor of the default class, which is corrected in FURIA by treating each class equivalently.

Then, as in RIPPER, the rules for each class are learned by starting with an empty rule and adding the predicates, which maximize the gain of information from FOIL [START_REF] Hühn | Furia: an algorithm for unordered fuzzy rule induction[END_REF] until the rule only covers target data. The rule set stops growing when all the target instances are covered or if the total description length of the rule base is 64 bits higher than the smallest previously calculated.

After growth, the previous set of rules is optimized to avoid over-fitting. Each rule R i is compared to two newly constructed rules: a replacement rule R ′ i , empty before growth, and a revision rule R ′′ i , equal to R i before growth. They are both temporarily placed in the i th position of the rule set, developed as above according to FOIL's information gain, and pruned to minimize the error of the total rule base.

Between R i , R ′ i and R ′′ i , the rule that generates the smallest minimum description length (MDL) is kept. The remaining target examples are covered as in the growth phase and this optimization can be repeated several times (2 times in general).

The second major modification occurs at the end of the algorithm: the inequalities initially learned are transformed into fuzzy sets with a trapezoidal membership function, as illustrated on Figure 3. For a rule, we determine the best fuzzification of each predicate: the original interval defined at the core of the trapezoid and the data not covered by the predicate are evaluated as the support boundary of the trapezoid, via a purity measurement [START_REF] Hühn | Furia: an algorithm for unordered fuzzy rule induction[END_REF]. The purest of the fuzzy intervals thus constructed is kept and this procedure is repeated until all the inequalities are fuzzified.

Finally, we obtain a set of unordered fuzzy rules. This type of model involves two possible classification problems: examples covered identically by two rules with different

B. Inducing Fuzzy Decision Trees

Fuzzy decision trees (FDT) are a generalization of decision trees [START_REF] Dubois | Fuzzy Sets in Approximate Reasoning and Information Systems[END_REF]. They appear in the early 90s with fuzzified versions of ID3 [START_REF] Cios | Continuous id3 algorithm with fuzzy entropy measures[END_REF], [START_REF] Weber | Fuzzy-ID3: A class of methods for automatic knowledge acquisition[END_REF]. To the authors of [START_REF] Yuan | Induction of fuzzy decision trees[END_REF], fuzzy decision trees are classifiers closer to human thinking and that are more robust to imprecision, conflict and missing information.

The most used FDTs are derived from the ID3 algorithm [START_REF] Janikow | Fuzzy decision trees: issues and methods[END_REF]- [START_REF] Qin | Decision tree learning with fuzzy labels[END_REF]. They consist in determining strong fuzzy partitioning of the input variables before the learning algorithm in tree structure. The fuzzy sets obtained via clustering (Figure 4) are then used as categorical variables in the original ID3 algorithm, each node dividing a variable into its different fuzzy sets. An instance x is distributed among the children of the node according to its membership degree µ A j i (x) to each fuzzy subset A j i of the variable i split by the node. The measure of discrimination, i.e. the entropy for ID3, has been adapted to consider the different membership degrees for each branch.

Another category of fuzzy decision trees groups the algorithms that fuzzify the thresholds used in the nodes of trees handling numerical data [START_REF] Olaru | A complete fuzzy decision tree technique[END_REF], [START_REF] Chandra | Fuzzifying gini index based decision trees[END_REF]. Instead of performing a crisp binary division of a variable, the boundary is fuzzified to obtain a fuzzy transition between the left and right children. In fuzzy SLIQ [START_REF] Chandra | Fuzzifying gini index based decision trees[END_REF], the width of the linear part of the splitting function depends on the standard deviation of the variable, which is split. In [START_REF] Olaru | A complete fuzzy decision tree technique[END_REF], the authors perform a Fibonacci search to optimize the width of the slope after that the optimal threshold has been found.

More recently, Intuitionistic Decision Trees [START_REF] Bujnowski | An approach to intuitionistic fuzzy decision trees[END_REF], based on Fuzzy ID3, are showing good performances on several datasets.

In this work, we focus on the fuzzy variants of ID3 and C4.5 decision trees, with the following workflow:

1) Clustering for each input variables in order to get modalities. 2) Strong fuzzy partitioning with triangular membership functions whose tops are the centroids of the clusters. 3) FDT growth: the variable minimizing the fuzzy entropy is selected at each step. 4) Classification: for each leaf, the majority class is selected.

IV. EXPERIMENTAL PROTOCOL

The number of sensors can be different from one experiment to another. Each sensor is coated with a diamond nanolayer and has a fundamental frequency of 433.9 MHz [START_REF] Chevallier | New sensitive coating based on modified diamond nanoparticles for chemical saw sensors[END_REF]- [START_REF] Girard | Electrostatic grafting of diamond nanoparticles: a versatile route to nanocrystalline diamond thin films[END_REF]. The signals are recorded and the characteristics on these signals are extracted to constitute the learning and validation sets.

The learning algorithms will be evaluated on three databases consisting of the experiments presented below, and on two types of descriptors presented in the following subsection. The first data set consists of acquisitions of 5 different toxic gases, the second of experiments on authentic and counterfeit coffee capsules, and the third one of acquisitions of DMMP, simulating Sarin gas, everyday products (e.g., soap) and mixtures of compounds.

A. Use Cases

We select three real-world use cases. They are relevant since the experimental environment is less and less controlled: for instance, the toxic chemicals experiment has been conducted in a lab, while the DMMP experiment has been conducted with a backpack and the ambient air. We will now describe the data sets for the three use cases.

1) Toxic chemicals recognition:

The device used is based on a network of eight SAW sensors. The sensors were exposed to ammonia (N H 3 ), sulfur dioxide (SO 2 ), hydrogen sulfide (H 2 S), methanol (CH 3 OH) and toluene (C 7 H 8 ) at a concentration of 10 ppm, 8 ppm, 6 ppm, 4 ppm and 2 ppm. The gases were generated from the dilution of a nitrogen calibrated gas. The temperature of the sensors (22 o ) and the flow rate (200 ml / min) above them were kept constant. The sensors were exposed to gas for 15 seconds, then the gas cell containing the sensors was purged for 30 seconds. 17 exposure-purge cycles were performed for each gas at each concentration.

2) Detection of counterfeit coffee capsules: This time, the selected device is based on a network of four SAW sensors. The sensors were exposed to 21 types of authentic commercially available coffee capsules and 7 types of counterfeit capsules. The contents of the capsules were emptied into a sealed beaker; the volatile compounds of each sample were transported through the gas cell containing the sensors using a pump. The temperature of the sensors was kept constant ( 22 o C). An exposure of 20 cycles (30 seconds) -concentration (30 seconds) was performed for each sample. The measurement process was repeated for 2 different capsules of the same coffee.

3) Detection of DMMP among interferers: The selected device is based on a network of eight SAW sensors. The sensors were exposed to DMMP and interferents: water, ethanol, shower gel and fertilizers, and DMMP-interfering and interfering-interfering mixtures. After diffusion of the gases in a backpack for 10 min, the vapors are inhaled in through the sensors via a pump. The temperature of the sensors was kept constant (22 o C). Several acquisitions were made, over several days in order to obtain a sufficiently large database.

V. RESULTS

We compare here the performances in terms of classification, obtained by applying statistical learning methods and fuzzy classifiers presented in section III on the three databases and the two types of features: tables I, II and III.

For each data set and each feature set, the performances of the best statistical method are presented [START_REF] Hotel | A review of algorithms for saw sensors e-nose based volatile compound identification[END_REF] (called "baseline" in the tables). As a reminder, they were obtained by a grid search of the best hyperparameters of the SVM, LMNN, BaggedTree and XGBoost methods. In the case of toxic chemicals recognition, the correct classification rate was used as a performance measure, while for the detection of counterfeit coffee capsules and the detection of DMMP among interferers, sensitivity has been calculated.

Similarly, the search for optimal parameters for FURIA and fuzzy decision trees is done via a grid search on the number of rule base optimization for the first (from 0 to 4 iterations), and a grid search on the number of terms learned per input variables for the second (from 3 to 7 terms).

The evaluation is carried out with a 5-fold cross-validation process. The database is divided into 5 subsets of similar size. The first serves as a test base first and the other four as a learning base, and so on, from the second to the fifth. Each table collects the mean and the standard deviation of the evaluation metric. In the case of fuzzy classifiers, the number of average rules is also indicated in order to judge the complexity of the models learned. The duration of the model induction is also shown.

Overall, the cross-validation performance of fuzzy classifiers on the three datasets gives at least results close to those obtained with statistical models, or even better in some cases. Numerically, there is often less than 1 % difference between the two types of methods. This is particularly the case for the sensitivities obtained on the DMMP basis (table III).

On the coffee and DMMP bases, we also find an improvement in the results, tables II and III, when we consider the mass and viscoelastic contributions compared to the results obtained with the amplitudes as descriptors. The gain is most significant on the basis of coffee capsules: +13 % for the FDT algorithm and +16.9 % for FURIA.

VI. DISCUSSION

A. FURIA

The FURIA algorithm gives scores close to statistical learning methods for DMMP, table III, but lower in the case of toxic chemicals, table I. For the recognition of counterfeit coffees, table II, the score is 10% lower with the amplitudes as descriptors but goes back to the level of the statistical methods with the contributions. Apart from the number of rule base optimizations, the scores were obtained with standard values for hyperparameters: the minimum number of instances covered by a premise is fixed to 2 and the number of folds for optimization to 3. If these hyperparameters are varied, there is no significant change in the performances.

In addition to classification performance, we note that the rule bases learned by FURIA are small, from 7 to 32 rules in average. Regarding the number of training examples for each base (600, 1600 and 160 respectively), there is at least one rule for 20 examples in average. Thus, there is no over-fitting with this method and the models obtained are compact.

From the confusion matrices given in table IV, we verify that there is no bias in favor of the most frequent class CH 3 OH like it could appear with RIPPER algorithm.

B. Fuzzy Decision Trees

The fuzzy decision trees give in the three cases quasiequivalent scores to the statistical methods: the difference is at most 1.4 %, except with the contributions on the coffee data set where there is a gain of 4.4 %, table II.

These performances were obtained by testing a few combinations on the hyperparameters of the induction algorithm, namely, the minimum degree of membership of an example in a branch, the termination criteria on the gain of entropy, the sum of the degrees of belonging to a node, and the minimum number of examples in a node. Compared to FURIA, performance varies more strongly by changing the hyperparameters of the induction algorithm.

The strong partitioning of the input variables facilitates the interpretability of the model. With, for example, a partition with three fuzzy sets, we can assign the terms Low, Medium and High to the learned sets, making the rule base accessible to the final users. There is however a greater number of rules learned than for FURIA. We note that for the coffee data set, 938 rules for 1600 examples are induced in average, i.e. less than two examples per rule. The model suffers from overfitting and the accuracy (or sensitivity) as the only performance criterion is not sufficient. It is then a question of finding a compromise between performance at evaluation, number of rules and computation time. We can also use rule base simplification algorithms to return reduced size models.

VII. CONCLUSION AND PERSPECTIVES

In this article, we have successfully applied fuzzy rule-based systems to the recognition of chemical compounds by an array of SAW sensors. In order to get the rule bases, we chose to use two popular methods: FURIA and fuzzy decision trees.

We collected data in three different use cases: the detection of toxic compounds, the recognition of counterfeit coffee capsules and the detection of the presence of DMMP in a backpack. We also retained two types of descriptors in the bibliography, based on one hand on the amplitudes, and on the other hand on the mass and the visco-elastic contributions.

With a few exceptions, fuzzy classifiers lag behind or even exceed statistical classifiers, with an accuracy of almost 99% on toxic chemicals. In addition, we can note that the standard deviations remain relatively low with the fuzzy methods, less than 5%, which shows a good robustness of the models in these applications. From all of these results, we conclude in the interest to add these fuzzy classifiers in our scope of search for the model learning process.

The use case of coffee capsules is delicate and the results are no doubt due to the limitations of the sensors used: in fact, coffee is made up of around a thousand different molecules and the functionalization of the sensors were the same for all three use cases.

We intend to continue this work by making better use of the quality the descriptors based on the contributions. Indeed, coming from an approximation based on a meta-heuristic optimization method, it is possible to fall into local extrema and have degraded solutions. We will study the quality of these features and use it during learning and recognition.

Fig. 1 :

 1 Fig. 1: Principles of SAW sensors

Fig. 3 :

 3 Fig. 3: Example of trapezoidal membership functions used in FURIA

Fig. 4 :

 4 Fig. 4: Strong partition of the domain of an input variable, automatically obtained with a clustering approach

TABLE I :

 I Accuracy on toxic chemicals

		Eight amplitudes
		Score	Rules	Duration
	Baseline	98.1 ± 0.7%	-	< 1s
	FURIA	96.4 ± 1.7%	21	1.2s
	FDT	98.8 ± 0.9%	203.6	9.7s
		Sixteen contributions
		Score	Rules	Duration
	Baseline	98.5 ± 0.8%	-	< 1s
	FURIA	95.0 ± 1.8%	23.4	1.0s
	FDT	98.0 ± 0.9%	219	15s

TABLE II :

 II Sensitivity on coffee data set

		Four amplitudes
		Score	Rules	Duration
	Baseline	62.6 ± 8.7%	-	< 1s
	FURIA	52.6 ± 1.9%	32.4	2.9s
	FDT	61.2 ± 4.0%	197.2	35s
		Eight contributions
		Score	Rules	Duration
	Baseline	69.8 ± 7.9%	-	< 1s
	FURIA	69.5 ± 4.3%	18.8	5.6s
	FDT	74.2 ± 3.3%	938.6	110s

TABLE III :

 III Sensitivity on DMMP data set

		Eight amplitudes
		Score	Rules	Duration
	Baseline	96.6 ± 2.6%	-	< 1s
	FURIA	96.8 ± 4.4%	8	0.2s
	FDT	96.7 ± 5.0%	57.6	4.6s
		Sixteen contributions
		Score	Rules	Duration
	Baseline	98.3 ± 1.1%	-	< 1s
	FURIA	97.7 ± 3.1%	7.6	0.2s
	FDT	97.7 ± 3.2%	58.2	2.0s

TABLE IV :

 IV Confusion matrices on toxic chemicals Eight amplitudes

					Predicted
			CH 3 OH	N H 3	SO 2	8 C 7 H	H 2
		CH 3 OH	170	0	0	0	0
	Actual	N H 3 SO 2 C 7 H 8	0 0 0	133 0 1	1 158 8	4 0 141	0 0 0
		H 2	0	0	0	0	140
		(a) Confusion matrix for baseline
					Predicted
			CH 3 OH	N H 3	SO 2	8 C 7 H	H 2
		CH 3 OH	168	0	0	2	0
	Actual	N H 3 SO 2 C 7 H 8	0 0 1	129 0 7	3 155 1	5 1 140	1 2 1
		H 2	0	2	1	0	137
		(b) Confusion matrix for FURIA
					Predicted
			CH 3 OH	N H 3	SO 2	8 C 7 H	H 2
		CH 3 OH	170	0	0	0	0
	Actual	N H 3 SO 2 C 7 H 8	0 0 1	134 0 3	0 158 0	4 0 146	0 0 0
		H 2	0	0	1	0	139
		(c) Confusion matrix for FDT


N Collin

B Wattrisse

S Le Floc'h

D Ambard

J Sigüenza

NUMERICAL DETERMINATION OF AN ANATOMICAL STRUCTURE'S UNLOADED STATE FROM IN VIVO MEDICAL IMAGING

Keywords:

Context

• Implantable Medical Device (IMD) deployment's computation in patient specific anatomical structures.

•

Objectives

Inspect impact of the choice of stress-free configuration of anatomical structures in the result of IMD-anatomical structure interaction's computation.

Shell modelling

Stress-free configuration determination algorithm already exist in the litterature 1,2 , but are adapted to volumic mesh modelization of structure.

Medical imaging, with segmentaton, provides geometrical inputs of anatomical structures as their median surfaces. To avoid a step of volumic reconstruction and thickness discretization that would results in more degree of freedom and longer computation time, one could model those structures with shells. Our contribution is to adapt such algorithm to shell modelling.

Initialization:

x m → X k=0

Apply loading to X k , get loaded configuration x k , strain U and rotation R

Next Iteration:

U -1 R -1 x m → X k+1

Perspectives

• Implement more complex boundary conditions, heterogeneous loading or contact with other structures, to adress more cases. • Assign a different material to the arterial wall than the aneurysm.

• Use Aitken type relaxation to improve convergence.

•

  Inputs: loaded geomety x m & mechanical assumptions X k is an unloaded configuration of x m Does x k fit to x m ? to find stress-free configuration to implement computations with based of hyperelastic model and patient specific geometry. The convergence of such algorithm depends of the compatibility between the measured geometry, the boundary conditions and the assumed material of the structure. References 1 Bols et al. A computational method to assess the in vivo stresses and unloaded configuration of patient-specific blood vessels. 2013 2 Raghavan et al. Non-Invasive Determination of Zero-Pressure Geometry of Arterial Aneurysms. 2006 3 Costalat et al. Soft hyperelastic material from Biomechanical wall properties of human intracranial aneurysms resected following surgical clipping. 2011 4 Steiner et al. Imposed homegeneous pressure, with values obtained from Monitoring the injured brain: ICP and CBFdiscretized in thickness • 3 dofs per nodes: displacements • only median surface discretized • 6 dofs per nodes: displacements & rotations Σ = ∂ W ∂ C Application to in vivo neural aneurysms Hypothesis: • Soft hyperelastic material from in vivo aneurysm wall 3 • Homogeneous blood pressure 4 P i • No displacement on contours of the


Ruben Hermann

Camille Robert

Vincent Lagadec

Mathieu Dupre

Denis Pelisson

Caroline Froment Tilikete

Catch-Up Saccades in Vestibular Hypofunction: A Contribution of the Cerebellum?

Keywords: Vestibular areflexia, Bilateral vestibulopathy, Head-impulse test, Catch-up saccades, Cerebellar dysfunction, CANVAS

Long-term deficits of the vestibulo-ocular reflex (VOR) elicited by head rotation can be partially compensated by catchup saccades (CuS). These saccades are initially visually guided, but their latency can greatly decrease resulting in short latency CuS (SL-CuS). It is still unclear what triggers these CuS and what are the underlying neural circuits. In this study, we aimed at evaluating the impact of cerebellar pathology on CuS by comparing their characteristics between two groups of patients with bilateral vestibular hypofunction, with or without additional cerebellar dysfunction. We recruited 12 patients with both bilateral vestibular hypofunction and cerebellar dysfunction (BVH-CD group) and 12 patients with isolated bilateral vestibular hypofunction (BVH group). Both groups were matched for age and residual VOR gain. Subjects underwent video head impulse test recording of the horizontal semicircular canals responses as well as recording of visually guided saccades in the step, gap, and overlap paradigms. Latency and gain of the different saccades were calculated. The mean age for BVH-CD and BVH was, respectively, 67.8 and 67.2 years, and the mean residual VOR gain was, respectively, 0.24 and 0.26. The mean latency of the first catch-up saccade was significantly longer for the BVH-CD group than that for the BVH group (204 ms vs 145 ms, p < 0.05). There was no significant difference in the latency of visually guided saccades between the two groups, for none of the three paradigms. The gain of covert saccades tended to be lower in the BVH-CD group than in BVH group (t test; p = 0.06). The mean gain of the 12° or 20° visually guided saccades were not different in both groups. Our results suggest that the cerebellum plays a role in the generation of compensatory SL-CuS observed in BVH patients.

Introduction

During high velocity head rotations, gaze stabilization is mainly achieved through the vestibulo-ocular reflex (VOR). In case of vestibular loss, this slow eye movement reflex can be compensated by catch-up saccades (CuS) that redirect gaze toward the target [START_REF] Halmagyi | A clinical sign of canal paresis[END_REF]. In the early stages of a vestibular deficit, patients mostly produce visually guided catch-up saccades to bring the image of the target back onto their fovea after the head movement (hence also called overt saccades). When vestibular function does not recover, some of these catch up saccades show a reduced latency to the extent that they unfold during the head movement, hence also called covert saccade or short-latency catch up saccades (SL-CuS) [START_REF] Manzari | Vestibular function after vestibular neuritis[END_REF]. Both the short latency and the regularity of these SL-CuS have been shown to be linked to better dynamic visual acuity and to have positive functional impact [START_REF] Hermann | Are covert saccade functionally relevant in vestibular hypofunction?[END_REF][START_REF] Batuecas-Caletrio | The vestibulo-ocular reflex and subjective balance after vestibular schwannoma surgery[END_REF]. 1 3 These SL-CuS can occur as early as 70 ms after the beginning of the head movement [START_REF] Tian | Vestibular catch-up saccades in labyrinthine deficiency[END_REF]. As the shortest latency of visually guided saccades is around 100 ms (express saccades) [START_REF] Fischer | Human express saccades: extremely short reaction times of goal directed eye movements[END_REF], this raises the questions regarding what actually triggers these SL-CuS. Different sources of information participating in the triggering of SL-CuS could include the proprioceptive cervical receptors, residual vestibular function (in particular from the contralateral ear), cognitive function, visual information, and multisensory integration [START_REF] Tian | Vestibular catch-up saccades in labyrinthine deficiency[END_REF][START_REF] Colagiorgio | New insights into vestibular-saccade interaction based on covert corrective saccades in patients with unilateral vestibular deficits[END_REF][START_REF] Van Nechel | Visual input is the main trigger and parametric determinant for catch-up saccades during video head impulse test in bilateral vestibular loss[END_REF][START_REF] Kasai | Eye-head coordination in labyrinthine-defective human beings[END_REF][START_REF] Black | The active head-impulse test in unilateral peripheral vestibulopathy[END_REF]. Beside the triggering signals, the neural circuits which underlie these quick compensatory catch-up saccades are also still unknown.

The cerebellum is crucially involved in general motor control, including eye movements control, and particularly in plastic modifications of oculomotor responses such as adaptation, learning, and compensation [START_REF] Manto | Consensus paper: roles of the cerebellum in motor control-the diversity of ideas on cerebellar involvement in movement[END_REF]. Save for some exceptions [START_REF] Takagi | Effects of lesions of the oculomotor vermis on eye movements in primate: saccades[END_REF][START_REF] Piu | The cerebellum improves the precision of antisaccades by a latency-duration trade-off[END_REF], the cerebellum does not play a major role when it comes to saccade latency. It is nevertheless crucial for vestibular and saccadic adaptation [START_REF] Pélisson | Sensorimotor adaptation of saccadic eye movements[END_REF]. Additionally, it is strongly connected to the vestibular nuclei. Therefore, beyond the demonstrated contribution of the cerebellum to theVOR adaptation mechanisms in case of chronic vestibular deficit, this structure could also participate in this critical gaze stabilization function by contributing to the generation of SL-CuS.

One way to demonstrate the role of the cerebellum in this compensatory mechanism is to study gaze stabilization in patients presenting with both vestibular and cerebellar deficits, like patients with cerebellar ataxia neuropathy vestibular areflexia syndrome (CANVAS). Our hypothesis is that cerebellar dysfunction in these patients could hinder the shortening of CuS latency as compared to patients with vestibular deficits only.

The main objective of this study was to evaluate this hypothesis by measuring the impact of cerebellar dysfunction on CuS in patients with cerebellar ataxia and vestibular hypofunction. In the first experiment, we recorded VOR and CuS during head impulse test in two groups of patients with bilateral vestibular hypofunction of which one had additional cerebellar dysfunction.

In a second experiment, performed in two similar groups of patients, we aimed to better understand the specificity of cerebellar contribution to CuS by comparing the latency of purely visually guided saccades.

Material and Methods

This prospective study was performed in the neuro-ophthalmology unit of the Lyon University Hospital between November 2019 and March 2022.

We recruited patients with bilateral vestibular hypofunction and cerebellar dysfunction (BVH-CD group) and patients with isolated bilateral vestibular hypofunction (BVH group). Data of 6 previously recorded patients (2016) were used for the BVH group [START_REF] Hermann | Are covert saccade functionally relevant in vestibular hypofunction?[END_REF].

All subjects met the clinical BARANY criteria A, B, C, and D. For criteria C, we only used vHIT as our study focused on compensation during high velocity head movements.

In order to reduce biases while comparing the latency of the first catch up saccade between the group of BVH patients and the group of BVH-CD patients, we aimed for these two groups to be as comparable as possible. As age has been shown to affect SL-CuS in patients with bilateral vestibular hypofunction [START_REF] Hermann | Are covert saccade functionally relevant in vestibular hypofunction?[END_REF], we matched mean age across our BVH-CD group and our BVH group. Even though it has not been shown that residual vestibulo-ocular reflex (VOR) impacts CuS latencies, we also chose to match residual vestibular function across both groups, based on the mean VOR gain measured during vHIT of the lateral semicircular canals.

BVH-CD Group

Inclusion criteria were:

• Age between 18 and 90 • Bilateral vestibular hypofunction (BVH) as defined by the Barany Society criteria [START_REF] Strupp | Bilateral vestibulopathy: diagnostic criteria consensus document of the classification committee of the Bárány Society[END_REF] Cerebellar dysfunction:

-Obvious cerebellar atrophy on MRI -At least one of the following clinical manifestations: postural and segmental cerebellar dysmetria, cerebellar dysarthria, specific cerebellar oculomotor disorders (at least one of the following: saccade hypermetria, downbeat nystagmus, obvious deficit of the visual inhibition of the visuo-vestibulo-ocular reflex)

Subjects were not included if they had other underlying central neurological condition, otological disorder other than vestibular hypofunction, corrected standard visual acuity lower than 5/10 Snellen equivalent, ocular motor palsy, ocular instability in primary gaze position, instability of the cervical spine, or if they were taking drugs interfering with eye movements.

A total of 12 subjects were included in the BVH-CD group. Ten patients had cerebellar ataxia neuronopathy and vestibular areflexia syndrome (CANVAS) with positive RFC1 testing. Two patients had chronic progressive cerebellar dysfunction associated with bilateral vestibular hypofunction, of unknown origin. Clinical manifestation of cerebellar oculomotor disorders as well as other clinical cerebellar manifestation are shown in Table 1.

3

BVH Group

Inclusion criteria were:

-Age between 18 and 90 -Bilateral vestibular hypofunction (BVH) as defined by the Barany Society criteria [START_REF] Strupp | Bilateral vestibulopathy: diagnostic criteria consensus document of the classification committee of the Bárány Society[END_REF] Exclusion criteria were similar to the other group with the exception of the exclusion in case of any underlying neurologic disorder.

A total of 12 subjects were included in the BVH group. Etiologies were the following: 5 idiopathic, 4 iatrogenic (aminoglycoside); 1 Meniere's disease, 1 post infectious, 1 bilateral vestibular neuritis.

Material

Video Head Impulse Test (vHIT)

Head and eye movements during head impulse test were recorded using a lightweight portable vHIT device (Hardware: ICS Impulse, GN Otometrics, Taastrup, Denmark; Software: Otosuite Vestibular software). Head movements were recorded with a nine-axis motion sensor. Movements of the right eye were recorded with a high-velocity infrared camera. Both the head motion-sensor and eye camera were mounted on a lightweight frame and run at a 250 Hz sampling rate. The infrared camera was calibrated by having the subject gaze toward two light dots projected at eye level on the wall from built-in lasers.

Eye tracker

Visually guided saccades were recorded using high-velocity infrared cameras (EyeBRAIN® tracker now distributed by SURICOG Company, Paris, France). Movements of both eyes were recorded with a sampling rate of 300 Hz and an accuracy of 0.5°. Eye-tracker was calibrated with a build in 9-point calibration system.

Method

Head Impulse Test (vHIT)

This evaluation procedure has been described in details in a previous study [START_REF] Hermann | Are covert saccade functionally relevant in vestibular hypofunction?[END_REF]. In short, a single experienced examiner standing behind the subject performed outward horizontal impulses. Subjects had to gaze toward a target located 2 m in front of them. Their head was tilted forward to align the plane of their horizontal semicircular canals with the horizontal plane. A minimum of 10 valid horizontal head impulses (head speed > 200°/s) was achieved in each direction. Head and eye velocity data were then exported in CSS format for off line analysis.

Visually Guided Saccades (VGS)

Horizontal visually guided saccades were recorded in 3 different conditions: STEP, GAP, and OVERLAP. In all conditions, participants were seated in a dimly lit room facing a computer screen 60 cm in front of them. Their head was stabilized using a head and chin rest. For each trial, subjects first had to look at a dot in the center of the screen. Then they fixated a second dot as soon as it appeared at 12° or 20° from the central dot, either to the left or the right. After 1000 ms, the second dot disappeared, and the central dot appeared again for 2400 to 3600 ms until the next trial started.

For the STEP condition, the central dot disappeared at the same time as the second (lateral) dot appeared. For the GAP condition, the central dot disappeared 200 ms before the lateral dot appeared. For the OVERLAP condition, the central dot never disappeared. In each paradigm, twelve

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randomized outward saccades (6 to the left and 6 to the right) were recorded at 12° and at 20°. Thus, a total of 72 visually guided saccades were recorded.

Analysis of Head and Eye Movements

Data analysis for both vHIT and VGS was done in a program developed in our lab and running on MATLAB v.8.1 (Math-Works, MA, USA). Details of the vHIT analysis have already been published [START_REF] Hermann | Are covert saccade functionally relevant in vestibular hypofunction?[END_REF]. In short, movements were first identified automatically and differentiated from artefacts by using a 5°/s head or eye velocity threshold. Then for each identified movement, five cursors defined, respectively, the starting and ending positions, the starting and ending times, and the time of maximum velocity. Each automatically detected movement was checked and cursors could be manually adjusted. In line with the previously described protocol, we chose to identify a maximum of three eye movements per head impulse (vestibulo-ocular reflex and up to 2 catch-up saccades). Covert saccades (CS) were defined as saccades occurring before the end of the head movement and overt saccades (OS) after the end of the head movement. The first catch-up saccades were defined as the first corrective saccade and could correspond to covert or overt saccades. For each head impulse, VOR gain was calculated as the amplitude ratio between eye movement and head movement at the end of VOR sequence. The end of the VOR sequence was defined as either the start of the covert saccade or the maximum point of VOR amplitude in case of absence of covert saccades. Saccadic gain was calculated as the ratio of eye movement amplitude during CS or OS and of total head movement amplitude. Latencies were calculated as the onset time of each eye movement (VOR, CS, OS) relative to the beginning of head movement. For each subject, the occurrence of CS (frequency in percent) was determined as the total number of CS relative to the total number of head impulses. The consistency of CS initiation was determined by the mean of individual standard deviation of the latency.

For visually guided saccades, analysis was done on both eyes. Eye movements were identified automatically by using an eye velocity threshold of 30°/s to differentiate movements from artefacts. Cursors were manually adjusted if necessary. We analyzed the latency, amplitude, and gain of primary saccades and only the amplitude and gain of secondary saccades by using the average between the values of both eyes. Gain was defined as the ratio between the actual saccade amplitude and the amplitude required to capture the target. Latency was defined as the difference in time between the appearance of the second dot and the beginning of the eye movement. Primary saccades measured in STEP condition were considered accurate if their amplitude was 12° + / -1 for a target amplitude of twelve degrees or 20° + / -2 for a target amplitude of twenty degrees; they were considered hypometric or hypermetric if their amplitude was below or above these thresholds, respectively.

Statistical Analysis

All data were stored and analyzed using JASP (JASP Team, Version 0.16.2, 2022). Statistical analyses were done using independent samples Student t test, Mann-Whitney test, or ANOVA depending on the normality of the distribution, as established by Shapiro-Wilk test, and the number of variables tested. All tests were two-tailed, and a statistical threshold p value < 0.05 was used.

Results

Experiment 1: Video Head Impulse Test

Twelve subjects were included in each group with respectively 9 and 8 males for the bilateral vestibular hypofunction and cerebellar dysfunction group (BVH-CD) and the bilateral vestibular hypofunction group (BVH). The mean age in the BVH-CD and BVH groups was, respectively, 67.8 years (SD 5.6) and 67.2 years (SD 6.9) (t test; p = 0.80). The mean residual VOR gain in the BVH-CD and BVH groups was, respectively, 0.24 (SD 0.15) and 0.26 (SD 0.15) (t test p = 0.78).

The mean latency of the first catch-up saccade differed significantly between the BVH-CD group (204 ms, SD 75 ms) and the BVH group (145 ms, SD 26 ms) (Mann-Whitney; p = 0.04). Details on latency of the first catch-up saccade (CuS) are shown in Fig. 1.

extreme, 1st quartile, mean (cross), median (horizontal line), 3rd quartile, and upper extreme. *: statistically significant difference between the two groups (p < 0.05).

Furthermore, the standard deviation of the latency of the first CuS was significantly greater in the BVH-CD group (49.6 ms) than in the BVH group (31.2 ms) (Mann-Withney; p = 0.043).

The mean latency of covert saccades did not differ between the BVH-CD group (132 ms) and the BVH group (124 ms) (t test; p = 0.43). Note that the gain of covert saccades tended to be lower in the BVH-CD group (0.33; SD 0.1) than in BVH group (0.45, SD 0.14) (t test; p = 0.06). The covert saccades did not differ in mean frequency between the two groups (t test; p = 0.39), being present in 52% of trials in BVH-CD patients and 63% of trials in BVH patients.

Finally, the mean latency of overt saccades was also similar in the BVH-CD group (264 ms) and the BVH group (257 ms) (t test; p = 0.73).

Experiment 2: Visually Guided Saccades

Primary Saccade Latency

Ten subjects underwent testing of visually guided saccades from each group. The mean age was, respectively, 69.8 (SD 9) and 70.0 (SD 6) for these BVH and BVH-CD sub-groups (t(18) = 0,000; p = 1,000). Eight men and 2 women participated in each group.

A three-way ANOVA (factors group x paradigm x eccentricity) did not reveal any significant effect of group on the latency of the primary visually guided saccades ((F(1,18) = 0.006; p = 0.941) and no significant interaction between group and paradigm ((F(2,36) = 0.720; 0.494) and between group, paradigm, and eccentricity ((F(2,36) = 0.246; 0.783). As expected, there was a significant effect of both eccentricity (F(1,18) = 8.029; p = 0.011) and paradigm (F(1,18) = 67.841; p < 0.001) on the latency of the first primary visually guided saccade.

Detailed results of the latency data are shown in Fig. 2.

The mean gain of the visually guided saccade to the 12° target was, respectively, 0.88 (SD 0.14) and 0.96 (SD 0.05) for BVH-CD and BVH (t test p = 0.20). Saccades to the 20° target reached a mean gain of, respectively, 0.92 (SD 0.10) and 0.96 (SD 0.03) for BVH-CD and BVH (Mann-Whitney p = 0.43). For the 12° target, hypometric saccades were more frequent in the BVH-CD group than in the BVH group with, respectively, 47% and 20% (Mann-Whitney p = 0.04). Conversely for the 20° target, hypermetric saccades were more frequent for the BVH-CD group than for the BVH group with, respectively, 8% and 0% (Mann-Whitney p = 0.04).

Discussion

The main goal of this study was to evaluate the impact of cerebellar dysfunction on the latency of compensatory catch-up saccades (CuS) during head impulse testing in patients with bilateral vestibular hypofunction (BVH). We found that the first catch up saccade had a significantly longer latency in patients with BVH and cerebellar dysfunction (BVH-CD) than in patients with only BVH. Additionally, the triggering of these first saccades was disorganized in BVH-CD patients as shown by an increased standard deviation of the latency when compared to BVH patients. In contrast, the latency of visually guided saccades in STEP, GAP, and OVERLAP conditions was similar between both groups of patients. Together, these results suggest that the cerebellum contributes to the compensatory response which is observed in BVH patients and which consists of a reduced latency of catch-up saccades and thus of the emergence of short-latency CuS (SL-CuS).

Mean residual gain of the lateral canals VOR in the BVH-CD population was 0.24 (SD 0.15) and in the BVH population 0.26 (SD 0.15). These values are close to the 0.28 mean VOR gain we had published in a previous study on 20 patients with BVH [START_REF] Hermann | Bilateral Vestibulopathy: vestibular function, dynamic visual acuity and functional impact[END_REF].

When looking at all covert saccades together, i.e., compensatory saccades occurring before the end of head movement, no significant difference of latency has been observed between the BVH and BVH-CD patient's groups. This is to be expected as, according to their definition according to the head movement, covert saccades do not represent a neurophysiological entity but rather constitute a heterogeneous category of saccades. For example, a saccade occurring at 180 ms could potentially be called covert saccade if the duration of the head movement is 185 ms or overt saccade if the duration of the head movement is 175 ms. For this reason, we emphasized on analyses of the first compensatory catch-up saccade (CuS), which could be covert or overt. The threshold for discriminating between the different types of CuS (short or long latency) is yet to be determined but should probably vary with age.

Thus, when focusing on the first CuS, we found that their mean latency (204 ms, SD 75 ms) significantly increased in BVH-CD patients as compared to BVH patients (145 ms (SD 26 ms)). This suggests that the cerebellum plays a role in triggering CuS.

One hypothesis is that dysfunctions associated to cerebellar ataxia in our BVH-CD patient's population would affect the saccadic network and result in an increased latency of all saccadic responses. Indeed, saccade anomalies are a typical sign of cerebellar lesions, particularly those invading the dorsal cerebellar vermis and posterior fastigial nucleus. These anomalies are mostly related to inaccuracy of saccadic amplitude. Lesions of the dorsal cerebellar vermis impair saccade adaptation, thus disrupting the maintenance or restoration of saccadic accuracy [START_REF] Takagi | Effects of lesions of the oculomotor vermis on eye movements in primate: saccades[END_REF][START_REF] Barash | Saccadic dysmetria and adaptation after lesions of the cerebellar cortex[END_REF][START_REF] Jenkinson | Disruption of saccadic adaptation with repetitive transcranial magnetic stimulation of the posterior cerebellum in humans[END_REF][START_REF] Colnaghi | θ-burst stimulation of the cerebellum interferes with internal representations of sensory-motor information related to eye movements in humans[END_REF]. Consistent with this literature, our BVH-CD patients produced a significantly higher percentage of inaccurate (hypometric and/or hypermetric) saccades when compared to the BVH group. Although increased saccadic latency is not a typical sign of cerebellar dysfunction, it has been documented for visually guided saccades after the ablation of the dorsal cerebellar vermis in primates [START_REF] Takagi | Effects of lesions of the oculomotor vermis on eye movements in primate: saccades[END_REF] and for anti-saccades in idiopathic cerebellar atrophy [START_REF] Piu | The cerebellum improves the precision of antisaccades by a latency-duration trade-off[END_REF]. To test the hypothesis that cerebellar dysfunction on its own would be responsible for increased saccadic latency, we analyzed the latency of visually guided saccades in BVH-CD and BVH groups. We did not find any difference of VGS latency between these groups, either for the STEP, GAP, or overlap condition or for the 12° or 20° target eccentricity. The increase in latency of the first CuS during the head impulse test is therefore not due to a general increase in saccade latency linked to an alteration of the saccadic system following cerebellar dysfunction.

Another hypothesis is that BVH associated to cerebellar dysfunction in our patients' population is due to cerebellar and/ or other central dysfunction rather than a peripheral vestibular dysfunction. In that case, BVH in both populations could not be truly compared. Lesions of the cerebellum can profoundly alter slow eye movements in a manner that depends on the lesion location. Typical manifestations of lesions of the flocculus and paraflocculus are impaired smooth pursuit, spontaneous nystagmus, and gaze holding deficit [START_REF] Rambold | Partial ablations of the flocculus and ventral paraflocculus in monkeys cause linked deficits in smooth pursuit eye movements and adaptive modification of the VOR[END_REF][START_REF] Zee | Effects of ablation of flocculus and paraflocculus of eye movements in primate[END_REF] which can also be seen in BVH-CD such as CANVAS [START_REF] Dupré | Update on cerebellar ataxia with neuropathy and bilateral vestibular areflexia syndrome (CANVAS)[END_REF]. Lesions of the flocculus/paraflocculus region also impair adaptation of the VOR [START_REF] Rambold | Partial ablations of the flocculus and ventral paraflocculus in monkeys cause linked deficits in smooth pursuit eye movements and adaptive modification of the VOR[END_REF][START_REF] Zee | Effects of ablation of flocculus and paraflocculus of eye movements in primate[END_REF][START_REF] Miles | Plasticity in the vestibulo-ocular reflex: a new hypothesis[END_REF][START_REF] Lisberger | Signals used to compute errors in monkey vestibuloocular reflex: possible role of flocculus[END_REF] but do not usually lead to loss of vestibular function. In case of CANVAS, vestibular impairment is due to a ganglionopathy of the vestibular nerve [START_REF] Szmulewicz | Cerebellar ataxia with neuropathy and bilateral vestibular areflexia syndrome: a histopathologic case report[END_REF]. Therefore, the vestibular impairment is probably independent of the cerebellar dysfunction in BVH-CD and more likely corresponds to peripheral vestibular dysfunction comparable to that in BVH patients.

The last hypothesis is that the cerebellum, given its wellknown contribution to adaptive changes of motor or sensorimotor functions, drives or helps in the establishment of a SL-CuS within a network involved in gaze stability when vestibular dysfunction remains over the long term. Indeed, the cerebellum plays a fundamental role in motor learning. Following physiological or pathological changes, it participates in the recalibration of various motor or sensorimotor responses and in the preservation of their performance. This cerebellar role has been studied in depth in the case of visuomotor adaptation [START_REF] Tzvi | Mini-review: the role of the cerebellum in visuomotor adaptation[END_REF]. In the BVH-CD group, the mean latency of the first CuS during head impulse tests (204 ms) was very similar to the mean latency of VGS in the gap paradigm (206 ms and 212 ms for targets at 12° and 20°, respectively). Based on this similarity, we argue that BVH-CD patients can use the classical visually guided saccades network to generate CuS, but not the saccade network which develops progressively and specifically triggers SL-CuS in BVH patients with preserved cerebellar function. Organization of CuS in vestibular hypofunction probably reflects efficient central compensation and has been shown to be correlated to lower Dizziness Handicap Inventory Scores [START_REF] Matiñó-Soler | A new method to improve the imbalance in chronic unilateral vestibular loss: the organization of refixation saccades[END_REF]. We suggest that within the saccade network involved in the establishment of SL-Cus, the cerebellum is probably also responsible for decreasing the variability of the latency of the first CuS as reflected by the difference in standard deviation of this parameter between both groups.

Conclusion

Patients with added cerebellar dysfunction show an increased latency of catch-up saccades when compared to patients with isolated vestibular deficit, despite the fact that visually guided saccades latency remains similar in both groups. Our results suggest that the cerebellum plays a role in the generation of compensatory short latency catch-up saccades observed during head impulse testing in patients with bilateral vestibular hypofunction. The information of the role of the cerebellum in the neural network of compensatory catch-up saccade is one important step in understanding the occurrence of short-latency catch up saccades. Other studies analyzing the role of proprioceptive cervical receptors, residual vestibular function, cognitive function, visual information, and multisensory integration will help to desiccate this network.

Fig. 1 Fig. 2

 12 Fig. 1 Boxplots of the latency of the first catch-up saccade (CuS) during vHIT showing lower

Table 1

 1

	Clinical features of cerebellar manifestations of	Subject	J_SP	J_VVOR	GEN	DBN	HS	SWJ	L_DyM	WG	DyA	Ad
	the 12 subjects of the BVH-CD group: J_SP, jerky smooth	1
	pursuit; J_VVOR, jerky visuo-
	vestibular reflex; GEN, gaze
	evoked nystagmus; DBN,
	down beat nystagmus; HS,
	hypermetric saccades; SWJ,
	square wave jerks; L_DyM,
	limb dysmetria; WG, wide
	gait; DyA, dysarthria; Ad,
	adiadochokinesia

Acknowledgements This study was supported by the Hospices Civils de Lyon (69HCL15_0746). We thank, E. Ionescu, O. Dumas, C. Becaud, Y. Sornay ,and G. Bolot. We thank E. Koun for his help in data management.

Data Availability

The datasets used in this study can be accessed if needed by contacting the corresponding author.

Author Contribution All authors contributed to the study conception and design was done by CFT, RH, and DP. Material preparation, data collection, and analysis were performed by RH, CR, VL, and MD. The first draft of the manuscript was written by RH and CFT. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Declarations

Ethical Approval The study was undertaken in agreement with French law (March 4, 2002) and the Declaration of Helsinki. All subjects were informed about the design and purpose of the study, and all gave their informed, written consent to the protocol. Approval was received from the local ethical committee on human experimentation (Comité d'éthique de Lyon autorisation no. . The previous study during which the data of six of the BHV patients was obtained was approved by the National French ethical committee on human experimentation (no. 160165B-32).

Competing Interests The authors declare no competing interest.

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