Ehsan Enferad

Christophe Giraud-Audine

Frédéric Giraud

Michel Amberg

Betty Lemaire-Semail

Differentiated Haptic Stimulation by Modal Synthesis of Vibration Field

To date, several focusing techniques have been proposed to realize localized stimulation on haptic interfaces. The localized stimulation can be achieved using arrays of actuators with delayed excitation, or by time reversal techniques which require a preliminary learning phase. The stimulation is thus limited to prerecorded cases. Additionally, these techniques are sensitive to parameters variation and disturbances. On the other hand, modal decomposition is another technique which allows to realize arbitrary vibration fields congruent with the boundary conditions without any preliminary learning phase. In this paper, we propose to use modal decomposition to realize a desired vibration velocity field in order to have differentiated stimuli. The experimental results validate the ability of the method to approximate satisfactorily a desired reference form. The psychophysic evaluations show that users can differentiate and localize the stimulation while exploring a surface with two fingers.

I. INTRODUCTION

Currently, there is a growing effort to implement haptic feedback on devices such as smartphones or tablets to enhance the user's experience. As an alternative to unbalanced rotor of rotating actuators, which offer poor controllability and cause a global vibration of the device, many proposal can be found in literature such as friction modulation [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF], or focused elastic deformation [START_REF] Woo | Vibration rendering on a thin plate with actuator array at the periphery[END_REF]. The first approach is based on the vibration of the whole active surface (usually the screen). This results in a modulation of the apparent friction coefficient, which can be explained by the non linear behavior of the air trapped between the interface and the finger (squeeze effect), or by intermittent contact of the vibrating surface with the finger [START_REF] Vezzoli | Physical and Perceptual Independence of Ultrasonic Vibration and Electrovibration for Friction Modulation[END_REF], [START_REF] Wiertlewski | Partial squeeze film levitation modulates fingertip friction[END_REF]. The vibration can then be switched on and off according to a measurement of the user's finger position to lure the user and render texturized surfaces [START_REF] Ben Messaoud | Amplitude control of an ultrasonic vibration for a tactile stimulator[END_REF]. However, since the whole surface is vibrated, it is difficult to offer a multi finger haptic interaction although some approaches could address this problem [START_REF] Ghenna | Control and evaluation of a 2-d multimodal controlled-friction display[END_REF]. The second category, i.e focusing techniques, is realized using phase array or time reversal. Localized deformation of the interface can then be realized which can be used to create differentiated stimulation. In phase array techniques, a large number of transducers generate vibrations which are emitted with a time delay proportional to the distance separating a transducer and the focus area [START_REF] Woo | Vibration rendering on a thin plate with actuator array at the periphery[END_REF]. The signals *This work was realized in the frame of the MINT Team of the Institut de Recherche sur les Composants logiciels et matériels pour lInformation et la Communication Avancée IRCICA are simple, usually a few periods of sine waves, but the result can be somewhat coarse, and since the energy is spread on a large area the deformation is limited. Time reversal on the other hand uses more complicated signals which are recording by transducers of the propagated vibration due to an initial deformation at some location. Each transducer then plays backward the recorded signal, in a synchronous manner, to "replay" the original deformation [START_REF] Hudin | Localized Tactile Feedback on a Transparent Surface through Time-Reversal Wave Focusing[END_REF]. Therefore, an identification procedure, is required. The main drawback is that the tests conditions may differ from the operational condition in a real environment. To address this issue, closed loop is devised, but to our knowledge, not implemented yet. This paper is a first step toward generating arbitrary vibration field based on control in the modal space with an evaluation of its potential for multifinger tactile exploration. The main idea of the paper is to use the property of the modal shapes of an elastic structure which, under some assumptions, form an orthogonal basis. In the first part we recall the main properties, and the protocol to realize a given velocity field thanks to its projection on a reduced set of modal shapes. Imposing velocity is motivated by the result found in [START_REF] Vezzoli | Physical and Perceptual Independence of Ultrasonic Vibration and Electrovibration for Friction Modulation[END_REF]. In addition, we propose a way to calculate the voltages waveform applied to the transducers to control the transient in open-loop. The second part is devoted to experimental validations of the proposed method. A simple velocity field is successfully approximated over a limited number of identified modes measured on a simple thin beam equipped with piezoelectric ceramics. Finally, the third part discusses psychophysic tests that demonstrate the ability of the proposed technique to suggest different sensations on two fingers of the user. The results show that the users are able to recognize the zone with the largest velocity.

II. THEORETICAL BACKGROUND

We consider a device constituted by an elastic media equipped with piezoelectric ceramic glued on one side. The geometry considered here is flat, prismatic and thin compared to other dimensions, similar to a screen. The material is linear, the dissipation is small, i.e the structure is lightly damped. In such case, the theory recalled below holds [8] [9].

A. Modal decomposition

The solution to vibration problem of an elastic medium can be addressed using modal decomposition. The mode shapes displacement vectors ϕ k , which depends on the geometry of the structure and the boundary conditions (BC), actually define a complete orthogonal basis with respect to the inner product :

ϕ j , ϕ k = D ϕ j • ϕ k dΩ (1)
where D denotes the integration domain, and dΩ is an elementary volume. For the scope of the paper, only plane geometries submitted to flexing deformation will be considered. As a consequence, the only relevant component of the displacement vector ϕ k is the out of plane displacement component, that will be denoted ϕ k in the following. To obtain a normalized basis, one defines the normalization by setting the amplitude of the mode shape such that :

ρϕ j , ϕ k = δ jk ( 2
)
where ρ is the density of the medium, ϕ j are the normalized modal shapes, and δ jk = 1 if k = j and 0 otherwise . Hence, any time varying shape w(x, y,t) compatible with the BC can finally be expressed as :

w(x, y,t) = ∞ ∑ k=1 ϕ k (x, y)η k (t) (3)
where η k (t) are the time coefficients relative to mode k. In order to find these functions, the dynamic equation of the plate which is written :

ρh ẅ + D∇ 2 w + p(x, y,t) = 0 ( 4
)
where D is the flexing rigidity of the plate, h is its thickness, and ∇ 2 is the double Laplacian operator. Moreover, p(x, y,t) is the normal loading of the plate. It is either due to external perturbations, or, in our case, by actuators such as piezoelectric patches. Applying the inner product 1 to Eq 4 and a given mode ϕ k , and considering the property (2) yields the decoupled dynamic equations governing each mode :

ηk (t) + 2ξ k ω k ηk (t) + ω 2 k η k (t) = ψ k (t) ψ k (t) = -p(t), ϕ k (5)
ω k is the resonant frequency of mode k, and ξ k is a damping coefficient introduced empirically to account for the damping of the mode. ψ k is the modal force. Since p accounts for the forces applied by the piezoelectric actuators, the definition given in 5 shows that any mode can be excited by an actuator as long as the geometry of the actuator is chosen to ensure that ψ k = 0. Hence, an actuator will be able to excite several modes as long as the previous condition is fulfilled and that the time variation of p is chosen adequately with regards to the frequency response of the selected modes. Moreover, p is proportional to the voltage v(t) applied to the ceramics, and therefore it is also the case for ψ k (t) :

ψ k (t) = G k v(t),
with G k the gain. This property will be helpful to define the voltages.

B. Modes selection

As discussed above, in theory, an infinite number of modes is required to realize a given deformation shape. While, due to the physical constrains in the actuators such as their placement, the available geometries and more importantly calculation time limitation in the case of closed loop control, a finite number of modes should be excited practically. Consequently, the first issue is to simplify the modal projection of the desired velocity field. This will of course result in an approximation. Therefore, the included modes, which are called modelled modes, should be selected properly to yield an acceptable trade-off between complexity and fidelity to the initial desired velocity field. In this paper, as an example, the objective is to generate at the given time T F the velocity map depicted on Fig. 1 (black meshed surface). The desired deformation velocity field at focusing point ẇ(x, y, T F ) has been decomposed to the first fifty modes and the modal velocity coefficients of the modes have been calculated for nine modelled modes as shown in Fig. 2 in black color. According to this figure, six modes have a significantly higher contribution in ẇ(x, y, T F ), and were selected as a possible simplification. The selected modes were ϕ 1 , ϕ 3 , ϕ 5 , ϕ 7 , ϕ 11 and ϕ 13 which are shown in blue color in Fig. 2 and the resulting velocity is shown in color surface in Fig. 1. As it is seen, an acceptable approximation of desired velocity field can be achieved utilizing six selected modes. Using modal decomposition, the initial value of the modes at focusing moment is known and their time dependence is described by eq. ( 5). As a result, a comprehensive knowledge of the excitation signal is available and in the perspective of closed loop control, an at resonance control of the modes is required, which already exists in literature [START_REF] Giraud | Vector control method applied to a traveling wave in a finite beam[END_REF] [START_REF] Ben Messaoud | Amplitude control of an ultrasonic vibration for a tactile stimulator[END_REF].

C. Open loop voltages calculation

The choice of the modes and their amplitudes as discussed above only concerns the instant when the focusing occurs that is the procedure above has yield the values ηk (t F ). Therefore, it remains to define the transient of the modes i.e calculate the time law ηk (t) while ensuring their calculated values at t = t F . To do so, we base the choice of these functions on the following consideration : in order to limit the driving voltages that will result from the procedure, the frequency used to excite a mode should be close to its resonant frequency, and we set its raise time. Therefore, the time law (reference trajectory) during the raising phase is an increasing modulated exponential defined by :

ηk (t) = A k e ct sin (dt -α k ) ( 6
)
where A k is an amplitude that must be determined, c = ξ k ω k and d = 1ξ 2 k ω k . ξ k is the transient damping that will be imposed with the relevant voltage. The corresponding displacement field η k (t) and acceleration field ηk can be deduced by integrating and differentiating with respect to time respectively. Moreover, the initial condition η k (0) = 0 is imposed. In this paper, we chose c = a and d = b, that is the modes should evolve close to their natural (free) motion. This is motivated by the fact that at resonance, the forces -hence the voltages-needed for the vibration are minimal. Considering now the dynamic equation of one mode [START_REF] Ben Messaoud | Amplitude control of an ultrasonic vibration for a tactile stimulator[END_REF], and replacing the displacement, velocity and acceleration fields, one gets after some algebra :

ψ k (t) = A k 4a 2 e a(t-t F ) sin (bt -α k ) b (7)
To ensure the required modal velocities, it remains to calculate A k from the condition A k e at F sin (bt F ) = ηk (t F ). To ensure that the modes are in phase at the focusing instant t F , one must set the phase α k such that bt Fα k = ± π 2 + 2n k π where n k is an arbitrary integer. Finally, the voltages are deduced thank to the velocity gain of the considered mode. This is done through an identification procedure described in the next section.

III. EXPERIMENTAL SET UP

In order to investigate the feasibility of generating desired vibration field with modal excitation method, an experimental setup was designed to test a simple case of modal vibration generation. For this purpose, a beam equipped with piezoelectric actuators was realized and is presented in subsection III-A. The excitation voltage signals of actuator have been generated by a Textronix AFG 3022B arbitrary function generator. The generated signal has been connected to actuator after amplification with the power amplifier, as shown in Fig. 3(a). Velocity is measured a single point laser vibrometer (Polytec OFV 505). The measured vibration velocity is acquired by a Picoscope 3404 oscilloscope. All measurements are synchronized and processed using dedicated MATLAB routines that we developed.

A. Design of beam

An aluminum beam with dimensions 100×6×2 mm 3 with free-free boundary condition has been designed (Fig. 3(b)). Six piezoelectric actuators were glued on the beam and can excite the fifteen identified modes although as discussed only six of them will be used in this study. The two actuators at the center and the two outermost are respectively electrically connected, while the last ones are not used in the experiment, but provide some homogeneity of the geometry in order to avoid localized deformation resulting from the section variation.

B. Identification of mode shapes and modal parameters

The characteristic modal parameters (ω k ,ξ k and G k ) must be identified on the actual mechanical structure. To do so, a frequency domain identification test has been realized on the beam. The piezoelectric actuators were excited with a chirp signal while the vibration of the beam was measured using a laser vibrometer to construct the Frequency Response Function (FRF) at regularly spaced location of the beam. Using peak-picking method, the resonance frequencies ( f k = ω k 2π ) were measured. By measurement of the half-power bandwidth around each resonance frequency, the damping factors (ξ k ) were identified. Finally, since at resonance ηk = G k v(t) 2ξ k ω 2 k , the modal gains (G k ) have been identified. The identified parameters are presented in table I for nine identified modes and their mode shapes (ϕ k (x, y)) are depicted on Fig. 4.

IV. EXPERIMENTAL VALIDATION OF MODAL FOCUSING

A. Validation of the proposed method

Here, the method to implement the multi-modal excitation in the beam is validated against the desired final velocity field. As was discussed, the six more significant modes were selected, and were excited using two sets of actuators. Indeed, to avoid saturation of the amplifiers, the voltage resulting of the superimposition of the "modal" voltages defined by eq. ( 7) were split between the center actuator set (ϕ 1 ,ϕ 3 , ϕ 11 , ϕ 13 ) and the side actuator set (ϕ 5 , ϕ 7 ). The assignment of a particular mode excitation to an actuator set is done by selecting the best modal gain values between the two sets. On fig. 5, the amplitude vs position on the beam (horizontal axis) and time (vertical axis) is presented near the focusing which was set to be at 25 ms and in the middle of the beam. It is clearly visible that the velocity field in this area is dominating on a spot measuring roughly 1.5 cm, the speed measured being 200 mm s -1 . This speed is maintained to that level for roughly 20 µs. The overall process lasts 50 ms.

B. Evaluation of the velocity field at focusing time

To validate the velocity field obtained at t = 25 ms, fig. 7 presents its modal projection. The first two modes have been very well excited, but then some discrepancies are observed especially for modes ϕ 5 and ϕ 11 with a maximum value of the error of 6 mm s -1 . Considering the fact that this is obtained in open loop from experimental estimates of the damping, these results are considered satisfactory. This is confirmed by the experimental velocity field at the focusing instant, which is depicted on fig. 7, and is similar to the reference on fig . 1. However, it can be noticed that in addition to the bending, a torsion movement is present, probably due to the imperfection in the gluing of the actuator or of the boundary conditions. This could also explain the Sprectrogram of the vibration velocity at the center during a focusing differences between the desired and the experimental modal projections. However, the experimental velocity field is close enough to the reference to consider that the effect on the tactile stimulation should be similar.

V. PSYCHO PHYSIC EVALUATION

A. Scope of the study

The idea is to evaluate the impact of the gradient of the velocity field for multi-finger interaction by contrast with a located interaction with a single finger as in [START_REF] Hudin | Localized Tactile Feedback on a Transparent Surface through Time-Reversal Wave Focusing[END_REF]. The point is that, at this stage, the rise time of the two lowest frequencies modes are prominent as can be seen from the spectrogram of fig. 8. Thus, using one finger to explore the beam lead to no specific stimulation apart from shocks detectable everywhere. By contrast, the effect is striking when two fingers or more are in contact with the vibrating surface. In this case, the vibrating sensation ceases in the finger in contact with the lowest level of velocity, while the second finger still detects repeated "shocks". To confirm this, the following experiment was set.

B. Protocol

The surface of the beam has been divided into five equal zones, named 'A', 'B', 'C', 'D' and 'E' as shown in Fig. 9. During the psychophysic test, the users were asked to use two fingers simultaneously, each of them being located on two different zones. The combination imposed were (A,C), (B,D) and (C,E), in order to reduce the mechanical cross talk between two fingers too close. The users were asked to test one of the combination at random, while the voltage were set to obtain maximum speeds 80 mm s -1 , 100 mm s -1 , 150 mm s -1 and 200 mm s -1 in random order. The stimuli for a given setting and location were repeated at a frequency of 7 Hz. For each run, the participants were asked to situate where they detected an event. The answers were categorized as "right", "left" or "both". Ten participants, 3 women and 7 men aged between 22 and 38 years old, have been repeating the above mentioned processes for two sets of psycho physic experiments. In one experiment, the participants were asked to put index and middle finger of one hand on the sets of locations (A,C), (B,D) and (C,E) and compare the sensations under the index and middle fingers of the same hand (test 1). In the second experiment, participants put the index fingers of both hands simultaneously on the same sets of location and answered the same question (test 2).

C. Results and discussion

Fig. 10 sums up the result of test 1. In this figure, the dark blue bars with labels of 'C', 'A' and 'C & A' represents the answers of participants when comparing vibration in spot 'A' and 'C' on the beam (as shown in Fig. 9), and clearly show that the participants dominantly chose spot 'C' regardless of the vibration amplitude. They declared to have no feeling at 'A' and to detect a vibration at 'C'. Only one participant had a similar feeling at 150 mm s -1 . The yellow bars in Fig. 10, illustrates response of participant for comparing the haptic feeling between spot 'C' and 'E', this time the 'C' column is dominantly selected by the ten participants for all vibration amplitudes. Only one participant felt that the stimulations in 'C' and 'E' were the same when vibration the amplitude was at the lowest levels 80 mm s -1 and 100 mm s -1 . In the third case, comparing between spot 'B' and 'D' represented as light blue and light green bars on the Fig. 10 (column 'B', 'D' and 'B and D'), the sensation is different. Some participants felt a stronger vibration on the spot 'B', while only one chose 'D'. Actually, most participants had the impression that both fingers were stimulated hence the majority of 'B & D'. In fact, although the instantaneous speed in 'C' on vibration velocity field when focusing occurs ( fig. 7) dominates, looking at the temporal evolution of the vibration on Fig. 5 shows that during transient spots 'B' and 'C' (located in the vicinity of -0.025 mm and 0.025 mm on the horizontal 'x' axis) present symmetrical speed extrema before and after the focusing (roughly at 24.8 ms and 25.2 ms) although the amplitudes represent roughly half of the central speed at focusing. That is the whole focusing process influences the haptic feeling experience of the users. Actually, it seems that the user "filters out" the vibration as if, in the process, the dominant speed was the only one perceived. Indeed, the 'A & C' and 'C & E' combination present the highest speed contrast, as opposed to 'B & D' where the speeds are almost equal. In the two first cases, the participants decide for the correct location, while for the last they cannot decide, and actually describe the experience as "shocks between the fingers". The second experience was designed to confirm that this ability to feel only the dominant side was not induced by some mechanical coupling induced by the hand : if the whole hand is vibrated at the lowest vibration amplitude, it makes sense that we feel only the difference of speed. Hence, a similar psycho physic experiment was repeated by the participants using both index as it is very unlikely that the path between the hands through the body can propagate the vibration. The results are repeated and follow the pattern observed in the previous test (depicted in Fig. 11).

VI. CONCLUSIONS

In this paper, the concept of the modal projection and modal excitation of vibration was proposed for haptic applications and applied to realize a simple haptic interface. The voltages were calculated to obtain the focusing at a specified time. The resulting velocity field presented the high speed contrast as expected and a limited number of the modes was used. According to the psycho physic experiment results, the participants experience a differentiated haptic stimulation when using two fingers of the same or two different hands. Moreover, they can localize correctly the peak of the velocity field at the center of beam ('C' spot). The first observations indicate that the sensation results from the difference of the vibration speed under each finger so that only the finger exposed to the largest amplitude feels the stimulation.

Fig. 1 .

 1 Fig. 1. Ideal reference velocity field (black meshed surface) and practical reference velocity field obtained after modal superimposition on the truncated experimental modal basis (colored surface).

Fig. 2 .

 2 Fig. 2. Modal coefficient of nine modelled modes of the beam (black bars) for the desired velocity field and the selected modal coefficient to excite in experiment (blue bars)

Fig. 3 .

 3 Fig. 3. (a) The photo of experimental setup with the beam on the xy-plotter, measurement and excitation equipments. (b) The geometry of beam which is fixed to a plastic chassis with double sided tape glue, the six actuator are visible in bottom side of the beam.

Fig. 4 .

 4 Fig. 4. Identified mode shapes (ϕ k (x, y)) for nine modelled modes of beam.

Fig. 5 .Fig. 6 .

 56 Fig. 5. spatial-temporal evolution of multi-modal vibration. Vertically : time from 24 ms 26 ms, horizontally : position along the central line of beam. Relative vibration velocity in mm/s

Fig. 7 .

 7 Fig. 7. Experimental carthography of spatial form generation in t = T F

Fig. 9 .

 9 Fig. 9. The scheme of beam which is divided to five spots of ('A', 'B', 'C', 'D' and 'E'), the participant is touching spot 'A' by index finger and spot 'C' by middle finger in photo.

Fig. 10 .

 10 Fig. 10. The perception results of index and middle fingers of ten participants, dark blue bars are number of perception for comparison of (A,C) spots, light blue bars are for (B,D) and yellow bars are for (C,E) spots.

TABLE I IDENTIFIED

 I

	ϕ 1	229	0.0437	+114
	ϕ 3	2553	0.0165	+654
	ϕ 5	7137	0.0265	+6500
	ϕ 7	12701	0.0168	-8346
	ϕ 9	22810	0.0112	+10414
	ϕ 11	29390	0.0166	-6285
	ϕ 13	58303	0.0101	-41448
	ϕ 15	68543	0.0176	-166030
	ϕ 17	87855	0.0147	+134615

MODAL PARAMETERS Mode f k (Hz) ξ k G k (mm/(sV ))

The authors are with Univ. Lille, Centrale Lille, Arts et Métiers Paris Tech, HEI, EA

-L2EP -Laboratoire d'Electrotechnique et d'Electronique de Puissance, F-59000 Lille, France ehsan.enferad@ed.univ-lille1.fr


EA Joan Duprez
email: duprez.joan@gmail.com

Jean-François Houvenaghel
email: jeanfrancois.houvenaghel@chu-rennes.fr

Sophie Drapier
email: sophie.drapier@chu-rennes.fr

Manon Auffret
email: auffret.manon@gmail.com

Dominique Drapier
email: d.drapier@ch-guillaumeregnier.fr

Gabriel Robert
email: gabriel.hadrien.robert@gmail.com

Marc Vérin
email: marc.verin@chu-rennes.fr

Paul Sauleau
email: paul.sauleau@univ-rennes1.fr

Continuous subcutaneous apomorphine infusion does not impair the dynamics of cognitive action control in mild to moderate Parkinson's disease

Keywords: Cognitive action control, Parkinson's disease, CSAI, Activation-suppression, Simon task

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Introduction

Patients with Parkinson's disease (PD) whose motor fluctuations can no longer be satisfactorily controlled by oral medication can be proposed continuous subcutaneous apomorphine infusion (CSAI). This treatment has repeatedly been reported to have a beneficial effect, with reductions in motor fluctuations, as well as in nonmotor symptoms [1]. By contrast with previous reports [2], recent studies have suggested that the beneficial motor effect of CSAI comes without the deleterious cognitive side effects associated with other treatments such as deep brain stimulation of the subthalamic nucleus [3,[START_REF] Gaspari | Clinical and neuropsychological follow up at 12 months in patients with complicated Parkinson's disease treated with subcutaneous apomorphine infusion or deep brain stimulation of the subthalamic nucleus[END_REF][START_REF] Drapier | Apomorphine infusion in advanced Parkinson's patients with subthalamic stimulation contraindications[END_REF]. So far, however, few studies have examined the effect of CSAI on the cognitive functions thought to be affected by other antiparkinsonian treatments.

One of the most important of these functions is cognitive control, as it allows us to adapt to our environment according to our intentions, ignoring any irrelevant information [START_REF] Ridderinkhof | Neurocognitive mechanisms of action control: resisting the call of the Sirens[END_REF]. Among the different processes participating in an efficient cognitive control, cognitive action control refers to the online control allowing to suppress unwanted and automatic action tendencies in favor of an intention-driven behavior [START_REF] Ridderinkhof | Neurocognitive mechanisms of action control: resisting the call of the Sirens[END_REF]. This cognitive process is often assessed using experimental conflict tasks such as the Simon task [START_REF] Simon | Reactions toward the source of stimulation[END_REF]. This task requires participants to produce a "left" or "right" answer according to the color of a stimulus displayed on the left-or righthand side of a screen, regardless of its position.

When the position and color of the stimulus are incongruent and indicate different responses, a conflict arises and the participant has to suppress the automatic response activated by the position of the stimulus. Incongruent situations lead to more errors, and responding accurately has a cognitive cost expressed as an increase in reaction time (RT). This is commonly called the congruence effect.

During the past decade, most studies investigating cognitive action control have focused on its dynamics, using the activation-suppression model and distributional analyses [START_REF] Ridderinkhof | Activation and suppression in conflict tasks: Empirical clarification through distributional analyses[END_REF]. This model postulates that fast responses in incongruent situations are more error-prone, reflecting a process called impulsive action selection, while slow responses indicate a smaller congruence effect, reflecting the efficacy of a suppression mechanism. The conditional accuracy function (CAF) can be used to investigate impulsive action selection, as it represents accuracy as a function of congruence plotted against RT distribution, divided into several bins (usually 5-7). Lower accuracy for the first incongruent bin indicates stronger impulsive action selection. The suppression mechanism is shown by the delta plot representing the congruence effect (incongruent RT minus congruent RT for correct responses) as a function of RT distribution, as with the CAF. The steeper the negative slope between the two last bins, the greater the suppression (see [START_REF] Van Den Wildenberg | To head or to heed? Beyond the surface of selective action inhibition: a review[END_REF] for a review). The effect of PD and its treatment on these dynamics has been thoroughly studied in recent years. For instance, patients with PD have been shown to display stronger impulsive action selection and weaker suppression, compared with healthy controls [START_REF] Duprez | Impulsive oculomotor action selection in Parkinson's disease[END_REF][START_REF] Wylie | The effect of Parkinson's disease on interference control during action selection[END_REF][START_REF] Wylie | The effect of Parkinson's disease on the dynamics of on-line and proactive cognitive control during action selection[END_REF]. Antiparkinsonian treatment also has an effect on these dynamics, as van Wouwe et al. [START_REF] Van Wouwe | Dissociable Effects of Dopamine on the Initial Capture and the Reactive Inhibition of Impulsive Actions 0in Parkinson's Disease[END_REF] found that patients with PD displayed greater suppression when they were on their dopaminergic medication than when they were off medication. Furthermore, a study by Wylie et al. [START_REF] Wylie | Dopamine agonists and the suppression of impulsive motor actions in Parkinson disease[END_REF] revealed that dopamine agonists weaken the suppression of irrelevant activations, while the number of impulsive errors is unaffected. In this study, the dopamine agonist dose was correlated with the strength of suppression, with higher doses being associated with weaker suppression. Deep brain stimulation of the subthalamic nucleus impacts both impulsive action selection and suppression. Wylie et al. [15], for example, showed that patients undergoing active stimulation make more impulsive errors than when they are off stimulation, but are more efficient at suppressing irrelevant activations, providing they take enough time to respond.

Although some studies have suggested that CSAI has no deleterious cognitive effects, its potential effect on impulsive action selection and suppression in conflict situations has never been examined. CSAI does not seem to modify performances in standard neuropsychological assessments performed during routine clinical care [3,[START_REF] Gaspari | Clinical and neuropsychological follow up at 12 months in patients with complicated Parkinson's disease treated with subcutaneous apomorphine infusion or deep brain stimulation of the subthalamic nucleus[END_REF][START_REF] Drapier | Apomorphine infusion in advanced Parkinson's patients with subthalamic stimulation contraindications[END_REF][START_REF] Borgemeester R W K | Continuous subcutaneous apomorphine infusion in Parkinson's disease patients with cognitive dysfunction : A retrospective long-term follow-up study[END_REF], but an effect on the dynamics of cognitive action control cannot be ruled out. Although the impact of classic oral agonists has already been studied [START_REF] Wylie | Dopamine agonists and the suppression of impulsive motor actions in Parkinson disease[END_REF], apomorphine differs from these dopaminergic agonists in terms of its pharmacodynamics and continuous administration modality [START_REF] Ribarič | The pharmacological properties and therapeutic use of apomorphine[END_REF]. The goal of the present study was to verify that add-on CSAI has no negative impact on cognitive action control and its dynamics. To this end, we used an oculomotor version of the Simon task that has proved to reveal subtle changes in cognitive action control in PD or healthy older participants [START_REF] Duprez | Enhanced Impulsive Action Selection in Middle-Aged Adults-Insights From an Oculomotor Simon Task[END_REF][START_REF] Duprez | Impulsive oculomotor action selection in Parkinson's disease[END_REF]. We compared the classic congruence effect, as well as the key features of impulsive action selection and suppression, measured before the introduction of CSAI and six months later in patients with mild to moderate PD. for inclusion in this study (Table 1). They had mild to moderate disease severity (Hoehn and Yahr stages I-III in the onmedication state [START_REF] Hoehn | Parkinsonism onset, progression, and mortality[END_REF]) and had been hospitalized in order to introduce apomorphine hydrochloride treatment (Apokinon®; Aguettant, Lyon, France) using an infusion pump (Microjet Crono PAR; Cane Medical Technology, Italy) as part of their routine clinical care. The CSAI treatment had been proposed to these patients due to motor fluctuations occurring despite an optimized oral treatment. Before CSAI, these patients displayed an averaged fluctuation score (calculated as the sum of items 36 to 39 of the UPDRS IV) of 3.2 ± 1.2 and an average off-state score of 1.15 ± 0.5. This score represents a proportion of 26 to 50 % of off-state during the day. CSAI has been reported to reduce these offstate periods known to have a strong negative impact on patients' daily life [START_REF] Drapier | Quality of life in Parkinson's disease improved by apomorphine pump: the OPTIPUMP cohort study[END_REF][START_REF] Martinez-Martin | Chronic subcutaneous infusion therapy with apomorphine in advanced Parkinson's disease compared to conventional therapy: a real life study of non motor effect[END_REF]. Patients were free of severe cognitive deterioration (Mattis Dementia Rating Scale score > 130; [START_REF] Mattis | Dementia rating scale professional manual[END_REF]), and had no other neurological or psychiatric pathologies. The severity of their motor symptoms was assessed using the third part of the Unified Parkinson's Disease Rating Scale (UPDRS). All patients were assessed under their usual medication at baseline (before the introduction of CSAI) and after 6 months of apomorphine treatment (M6; mean time since baseline: 6 ± 0.3 months). At baseline, medication included both dopamine agonists and levodopa therapy in 19 patients, and levodopa alone in one patient.

Methods

The study was approved by the ethics committee of Rennes University Hospital and conducted in accordance with the Declaration of Helsinki and current French legislation (Huriet Act). All the patients gave their informed consent to take part in the study. CSAI was used during the day for a mean duration of 13.6 ± 4 hours (range: 6-20) at a mean hourly rate of 4.9 ± 0.8 mg (range: 3.8-7) and a mean bolus number of 2.6 ± 1.5 per day (range: 0-5) with a mean dose of 3.4 ± 1.3 mg (range: 0-5) per bolus.

CSAI treatment

Task design and procedure

We used an oculomotor version of the Simon task in which participants were instructed to make a left or right eye movement according to the color (blue or yellow) of a stimulus, while ignoring its location (left or right), as fast and accurately as possible. Participants performed a total of 300 trials, divided into five blocks with self-paced breaks between these blocks. Two types of situation happened pseudorandomly with an equal probability: a congruent one where the location of the stimulus corresponded to the response side indicated by its color; and an incongruent one where the location and color of the stimulus indicated opposite responses. Color mapping was pseudorandomly assigned to each patient, and the reverse mapping was used at M6. Eye movements were recorded using an EyeBrain T2® head-mounted eyetracker (e(ye)BRAIN, Ivry-sur-Seine, France) with a 300 Hz sampling rate and an angular resolution of 0.5°. For more details, see [START_REF] Duprez | Enhanced Impulsive Action Selection in Middle-Aged Adults-Insights From an Oculomotor Simon Task[END_REF].

Analysis

All data analyses were performed using R© software (Version 3.1.0). The first eye movement after stimulus presentation was deemed to be the participant's response, and was considered correct if it corresponded to the side indicated by the color of the stimulus. Eye movements with an amplitude below 2° and a latency below 100 ms or above 1000 ms were removed (for more details, see [START_REF] Duprez | Enhanced Impulsive Action Selection in Middle-Aged Adults-Insights From an Oculomotor Simon Task[END_REF]). Outlier latencies of more than three standard deviations from mean RT were also discarded. Overall, 2.03% of the data were removed. For each participant, we calculated the mean RT for correct trials and mean accuracy for each congruence condition, to evaluate the congruence effect. We also used distributional analyses to assess impulsive action selection and suppression [START_REF] Ridderinkhof | Activation and suppression in conflict tasks: Empirical clarification through distributional analyses[END_REF]. We computed CAFs and delta plots showing accuracy and the congruence effect (incongruent RT minus congruent RT of correct trials) as a function of RT distribution divided into five bins (see [START_REF] Duprez | Enhanced Impulsive Action Selection in Middle-Aged Adults-Insights From an Oculomotor Simon Task[END_REF]). The first bin of each CAF in the incongruent situation was used to assess impulsive action selection, and the value of the final slope of each delta plot was used to assess suppression [START_REF] Van Den Wildenberg | To head or to heed? Beyond the surface of selective action inhibition: a review[END_REF].

As accuracy was not normally distributed, it was arcsine transformed prior to the statistical analysis. The congruence effect, accuracy for the first bin of the incongruent CAF, and last slope value of the delta plot were compared between baseline and M6 using repeated-measures analyses of variance (ANOVAs). For each test, we checked the assumptions of normality and homoscedasticity of the residuals. Clinical variables were compared using paired Wilcoxon signedrank tests. The significance threshold was set at p = 0.05.

Results

Clinical effect of CSAI

We found a significant decrease in motor fluctuations, as assessed with the sum of Items 36-39 of the UPDRS-IV at M6, compared with baseline (Table 1; V = 59, p = 0.02). Oral treatment was significantly reduced at M6, with a significant decrease in the levodopa equivalent daily dose (LEDD; V = 145, p = 0.001), as well as in the total oral medication daily dose (V = 150, p = 0.0005). The oral agonist dose tended to decrease (V = 64, p = 0.05) whereas overall LEDD increased, owing to the introduction of apomorphine treatment (V = 13, p = 0.001).

Congruence effect

We found no difference between baseline and M6 in overall RT, suggesting that patients responded at the same speed before and after the introduction of CSAI (F(1, 19) < 1, p = 0.4). A strong congruence effect was found, with a significantly shorter RT in the congruent situation than in the incongruent one reflecting the cognitive cost of conflict resolution (Fig. 1; F(1, 19) = 34.9, p < 0.0001). No difference was found in the size of the congruence effect between baseline and M6, as revealed by the absence of a significant interaction effect (mean congruence effect = 89 ± 62 ms at baseline and 111 ± 72 ms at M6; F(1, 18) < 1, p = 0.5). This result suggests that the cost of conflict resolution was unaffected by the introduction of CSAI.

Overall accuracy did not differ between baseline and M6, suggesting that CSAI had no impact on patients' precision (F(1, 19) < 1, p = 0.9). As with RT, we found a strong congruence effect, with a significant decrease in accuracy in the incongruent situation, compared with the congruent one (Fig. 1; F(1, 19) = 138.2, p < 0.0001). The size of the congruence effect on accuracy was unaffected by CSAI, as suggested by the absence of a significant interaction effect (mean congruence effect = 39% ± 28 at baseline and 46% ± 28 at M6; F(1, 18) = 1.4, p = 0.24).

Impulsive action selection

We found no difference between baseline and M6 in accuracy for the first bin of the incongruent situation (Fig. 2;F(1, 19) = 0.19, p = 0.66). This suggests that patients made just as many impulsive errors 6 months after the introduction of CSAI as they did before. Impulsive action selection is regarded as a key component of the dynamics of cognitive action control, and was our primary focus in this study, along with selective suppression. As the absence of an effect cannot be ruled out solely on the basis of the absence of a significant difference, we computed the 95% confidence interval of the effect between baseline and M6 for impulsive action selection. The mean difference between baseline and M6 was -0.03 (95% CI [-0.09, 0.04]), indicating a negligible decrease in accuracy between baseline and M6. Group differences reported for PD in the Simon task literature are about 0.1 [START_REF] Wylie | The effect of Parkinson's disease on interference control during action selection[END_REF]15,23]. As the 95% confidence interval for the differences in our study fell below what is usually reported as a meaningful difference, we can safely assume that there were no differences between baseline and M6 on impulsive action selection in our study.

Suppression of the irrelevant activation

There was no significant difference between baseline and M6 in the slope value of the final bin of the delta plot (Fig. 3; F(1, 19) < 1, p = 0.88). This suggests that CSAI did not weaken the suppression process, as it remained at its baseline level. As for impulsive action selection, we computed the 95% confidence interval for the mean difference between baseline and M6 to see whether there was a potentially meaningful effect in our data despite the absence of a significant difference. The mean difference between baseline and M6 was -0.01 (95% CI [-0.18, 0.15]). In the literature on conflict control in PD, significant differences are usually about 0.2 [15, [START_REF] Wylie | Dopamine agonists and the suppression of impulsive motor actions in Parkinson disease[END_REF][START_REF] Van Wouwe | Dissociable Effects of Dopamine on the Initial Capture and the Reactive Inhibition of Impulsive Actions 0in Parkinson's Disease[END_REF], which is outside the range of the 95% confidence interval for our data. This strongly supports the absence of an effect between baseline and M6 in our study.

Discussion

In the current study, we wanted to check that add-on CSAI has no impact on a key cognitive process (i.e., cognitive action control), and more specifically on the dynamics of this process, as reflected by impulsive action selection and suppression.

Regarding motor improvement, motor fluctuations decreased and oral medication was reduced accordingly, in line with the literature [3, [START_REF] Katzenschlager | Double-blind, randomized, placebo-controlled, Phase III study (TOLEDO) to evaluate[END_REF][START_REF] Drapier | Quality of life in Parkinson's disease improved by apomorphine pump: the OPTIPUMP cohort study[END_REF]. Regarding cognitive action control, as expected, our results revealed no difference in the congruence effect between baseline and 6 months after the introduction of CSAI. In other words, the cost of conflict resolution was unaffected by the introduction of apomorphine. The primary focus of our study was the effect of CSAI on impulsive action selection (reflected by accuracy for the first CAF bin in the incongruent situation) and suppression (reflected by the last delta plot slope value). We found that patients displayed similar accuracy at baseline and M6 for fast incongruent responses, and exerted equally strong suppression of automatic activations. The 95% confidence intervals for the differences between baseline and M6 on these two measures were not within the range of the meaningful effects that are usually reported in the literature using the Simon task, arguing in favor of an absence of effect. Thus, our results strongly suggest that CSAI has no impact on impulsive action selection and suppression in patients with mild to moderate PD, in accordance with previous reports of the absence of neuropsychological impairment after CSAI [3,[START_REF] Gaspari | Clinical and neuropsychological follow up at 12 months in patients with complicated Parkinson's disease treated with subcutaneous apomorphine infusion or deep brain stimulation of the subthalamic nucleus[END_REF][START_REF] Drapier | Apomorphine infusion in advanced Parkinson's patients with subthalamic stimulation contraindications[END_REF].

The absence of a deterioration in cognitive action control may arise from two aspects of CSAI treatment. First, the overall action of the apomorphine molecule may differ from that of other dopamine agonists. Apomorphine is a nonselective dopamine agonist that can interact with both D1-like and D2-like receptors [START_REF] Ribarič | The pharmacological properties and therapeutic use of apomorphine[END_REF], unlike most other antiparkinsonian dopamine agonists, which mainly bind to D2-like receptors. Wylie et al. [START_REF] Wylie | Dopamine agonists and the suppression of impulsive motor actions in Parkinson disease[END_REF] reported that weaker suppression of automatic activation correlated with higher doses of dopamine agonists (agonists binding to D2-like receptors). As apomorphine differs from the usual agonists, it may well have a different effect on cognitive action control. Another, nonexclusive, hypothesis is that the continuous nature of CSAI treatment, as opposed to the pulsatile doses of standard oral medication, results in greater physiological dopaminergic stimulation without any fluctuation in dose effects. This may stem either from the continuous administration modality of the apomorphine itself or from the overall reduction in oral treatment. In turn, this may maintain a certain level of cognitive performance, such as cognitive action control in the current study. Previous studies investigating the cognitive effect of antiparkinsonian treatments compared the performances of patients between their best (on medication) and worst (off medication) motor conditions. It is therefore difficult to compare the results of other studies with our own, and further investigations are needed to test the hypothesis that continuous versus pulsatile treatment has differential effects on cognitive action control.

Our study had several limitations. The patients in our group were free from cognitive deterioration at baseline and their disease was generally less advanced than that of participants included in previous studies [3,[START_REF] Gaspari | Clinical and neuropsychological follow up at 12 months in patients with complicated Parkinson's disease treated with subcutaneous apomorphine infusion or deep brain stimulation of the subthalamic nucleus[END_REF][START_REF] Drapier | Apomorphine infusion in advanced Parkinson's patients with subthalamic stimulation contraindications[END_REF]. This raises the question of a potential differential effect of CSAI on cognitive action control in patients with or without cognitive impairment. Thus, although we can hypothesize that CSAI is safe regarding cognitive action control in patients with mild to moderate disease severity, the generalizability of its effect to patients at a more advanced stage of the disease needs to be investigated further. However, it should be noted that the recent study of Borgemeester et al. [START_REF] Borgemeester R W K | Continuous subcutaneous apomorphine infusion in Parkinson's disease patients with cognitive dysfunction : A retrospective long-term follow-up study[END_REF] has shown that CSAI had no negative impact on executive functioning in PD patients with cognitive dysfunction. Another point is that the absence of an effect on cognitive action control in our study has to be offset against the size of the patient sample and the specificity of the task. Confirmation of our results requires similar studies using other cognitive tasks and a larger group of patients. Lastly, it would be very interesting to directly compare the effects of apomorphine and other dopamine agonists on cognitive action control.

Conclusion

The present study was the first to investigate the effect of CSAI on the dynamics of cognitive action control. Our results strongly suggest that CSAI has no negative impact on impulsive action selection and suppression in patients with mild to moderate PD. These results are important, as the nonmotor and cognitive effects of antiparkinsonian treatments have a major impact on patients' quality of life (see [1] for a review). Although further studies are needed to confirm the absence of cognitive deterioration with CSAI, our results argue strongly in favor of its safety for cognitive action control.

Ethical standards

This study was approved by the ethics committee of Rennes University Hospital and conducted in accordance with the Declaration of Helsinki and current French legislation (Huriet Act).

All the patients gave their informed consent to take part in the study.

Source of funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Figure 1 :

 1 Figure 1: Mean RT (A) and accuracy (B) according to congruence and treatment phase. Error bars represent the standard error of the mean.

Figure 2 :

 2 Figure 2: CAF for the incongruent situation, plotted as a function of RT distribution divided into five bins. Accuracy for the first bin indicates the strength of impulsive action selection. Error bars represent the standard error of the mean.

Figure 3 :

 3 Figure 3: Delta plots showing the congruence effect as a function of RT distribution, divided into five bins. The value of the final slope indicates the strength of selective suppression. Error bars represent the standard error of the mean.

  M, Qamar MA, Evans A, Chaudhuri KR (2016) The efficacy of apomorphine -A non-motor perspective. Parkinsonism Relat Disord 33:S28-S35. doi: 10.1016/j.parkreldis.2016.11.020 2. Růzicka E, Roth J, Spacková N, et al (1994) Apomorphine induced cognitive changes in Parkinson's disease. J Neurol Neurosurg Psychiatry 57:998-1001. 3. Alegret M, Valldeoriola F, Martí M, et al (2004) Comparative cognitive effects of bilateral subthalamic stimulation and subcutaneous continuous infusion of apomorphine in Parkinson's disease. Mov Disord 19:1463-1469.

  apomorphine subcutaneous infusion in reducing OFF time in Parkinson's disease patients with motor fluctuations not well controlled on optimized conventional treatment [abstract]. Mov Disord. 32 (suppl 2)

Table 1

 1 We recruited 20 patients with idiopathic PD from the Neurology Department of Rennes University Hospital (France)

	: Clinical data (mean ± standard deviation) before (baseline) and after (M6) the start of CSAI. Statistical
	comparisons were carried out using the Wilcoxon paired test. Dyskinesia and fluctuation scores were calculated as the
	sum of items 32 to 35 and 36 to 39 (respectively) of the part IV of the UPDRS.
		N	Baseline	M6	p-value	V statistic
	Sample size		20	20
	Age (years)	20/20	58.9 ± 10.5	-
	Education (years)	20/20	19.2 ± 8.7	-
	Sex (M:F)	20/20	9:11	-
	Disease duration	20/20	11.7 ± 4.8	-

Acknowledgements

The authors would like to thank all the volunteers who took part in this study, as well as Elizabeth Wiles-Portier who revised the English style.

Dr Sophie Drapier received speech honorarium from Teva and Medtronic and served on scientific advisory boards for Aguettant and Britannia.

Dr Marc Vérin has served on the Scientific Advisory Board for Aguettant and Orkyn and received speech honorarium from Teva and Medtronic.

Conflict of interest statement


Roman A Motiyenko

Laurent Margulès

Didier Despois

Jean-Claude Guillemin

Ch

Laboratory spectroscopy of methoxymethanol in the millimeterwave range †

Methoxymethanol,

Introduction

To select new target compounds for the interstellar medium (ISM), the most usual approach is probably the analogy with the compounds already detected in this medium. ‡ However, considering some plausible chemical reactions between the detected compounds can lead to more original targets. These approaches are strongly supported when such species have been found in lab simulations of the chemistry of the ISM in the gas phase or on the grains.

Methanol has been detected a long time ago in the ISM. 1 Its formation in this medium probably occurs on the grains 2,3 by the reaction of hydrogen radicals on solid carbon monoxide leading to methanol via the formaldehyde. 4,5 The reaction of hydrogen on formaldehyde can give two products, the hydroxymethyl CH 2 OH and methoxy CH 3 O radicals but the former can also be produced by reaction of methanol on hydrogen radical. The methoxy radical has been recently detected in the ISM 6 while the study of the millimeter spectrum of the hydroxymethyl radical has been only recently performed but was not followed by its detection in the ISM up to date. 7 In the case where it is formed in this medium, a possible explanation could be a rapid radical recombination in the grains to form a dimer prior to the desorption of the monomer. 5 The dimer, ethyleneglycol, has already been observed in the ISM. 8 If this hypothesis is correct and because the methoxy radical has been unambiguously detected, the recombination of both radicals (CH 3 O + CH 2 OH) could lead to the formation of another product, the methoxymethanol CH 3 OCH 2 OH. Note that the association of two CH 2 OH radicals to form (CH 2 OH) 2 , although appealing by its simplicity, seems to be not favoured as the CH 2 OH radical has a low mobility due to OH bonds. 9 Methoxymethanol could also be produced by addition of CH 3 O radical on formaldehyde followed by hydrogen abstraction if such reactions are barrier-less:

CH 3 O + H 2 CO --→ CH 3 OCH 2 O CH 3 OCH 2 O + H --→ CH 3 OCH 2 OH
At the opposite of the ethylene glycol isomer, few studies have focused on methoxymethanol. Nevertheless, it has been demonstrated that low-energy (1-20 eV) electron-induced reactions in condensed thin films of methanol and post-irradiation temperature-programmed desorption of CH 2 OH and CH 3 O produce ethylene glycol and methoxymethanol. 10,11 Such reactions could play a dominant role in ionizing radiation-induced chemical syna Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523 -Université de Lille 1 Sciences et Technologies, Villeneuve d'Ascq Cedex, France. E-mail: roman.motienko@univ-lille1.fr b Laboratoire d'Astrophysique de Bordeaux, Université de Bordeaux, CNRS, Pessac, France. c Institut des Sciences Chimiques de Rennes, UMR 6226 CNRS -ENSCR, Rennes, France. † Electronic Supplementary Information (ESI) available: spectral assignment and fits. See DOI: 10.1039/C7CP05932A/ ‡ http://www.astro.uni-koeln.de/cdms/molecules Fig. 1 In blue: the recorded spectrum between 235 and 265 GHz dominated by Q-type Ka = 3 ← 2 transitions of methanol. The methanol lines were saturated at analog-to-digital converter to obtain higher dynamic range for weaker lines of methoxymethanol. In red the spectra of conformers I and III of methoxymethanol simulated on the basis of the results obtained in this study. The amplitude of the synthetic spectra is scaled on assigned lines of the methoxymethanol.

thesis of the ISM and, it should be noted that most of the products identified in these studies starting from methanol have been detected in this medium. 12 In addition, in these studies it was suggested that CH 3 OCH 2 OH could serve as a tracer molecule for electron-induced reactions in the interstellar medium. 13 Consequently, on the basis of all these observations, we have considered methoxymethanol as a good candidate for the ISM since its detection or non-detection would improve the current knowledge in the interstellar chemistry.

The detection of new compounds in the ISM needs to have in hand the spectra of the target compound for comparison with spectra coming from interstellar clouds. For over more than sixty years, rotational spectroscopy has allowed the detection of many species in the ISM but only compounds with a permanent dipolar moment can be concerned. The rotational spectrum of methoxymethanol has never been studied before. Previous publications include high-level theoretical calculations that yielded rotational constants, harmonic force field, and the information on the conformational stability and some transitions states. 14 These calculations were performed in view of subsequent experimental observations of the rotational spectrum of methoxymethanol via O( 1 D) insertion reaction with dimethyl ether since such reaction was successfully applied for methane and ethylene to produce respectively methanol and vinyl alcohol. 15 In the present study, we report the first assignment and analysis of the rotational spectrum of methoxymethanol that was synthesized via chemical approach described below. We also report the results of quantum chemical calculations performed to aid in the interpretation of the observed spectra.

Experiment

Synthesis

The synthesis of Priede et al. 16 has been modified to obtain methoxymethanol on a preparative scale. Paraformaldehyde (6.60 g, equivalent to 0.22 mol of formaldehyde) and anhydrous methanol (128 g, 4.0 mol) were introduced in a 250 mL cell equipped with a stirring bar and a stopcock. The suspension was cooled at 77 K and the cell degassed before to close the stopcock. The suspension was heated at 95 • C for 3 h, whereupon clear solution formed. After cooling at room temperature, the methoxymethanol was purified by trap-to-trap distillation in vacuo (0.1 mbar). Three slow distillations with partially selective condensation in a trap immersed in a bath cooled at -55 • C led to methoxymethanol containing less than 5% of methanol. Yield: 7.6 g (56%). 1 H NMR (CDCl 3 ) δ 3.37(s, 3H, CH 3 ); 4.38 (t, 1H, 3 J HH = 7.9 Hz, OH); 4.67 (d, 2H, 3 J HH = 7.9 Hz, CH 2 ). 13 C NMR (CDCl 3 ) δ 55.1 ( 1 J HH = 142.3 Hz (q), CH 3 ); 90.7 ( 1 J HH = 162.1 Hz (t), CH 2 ).

Conventional Absorption Spectroscopy

We recorded the rotational spectrum of methoxymethanol using fast-scan terahertz spectrometer in Lille. The details of the spectrometer except of the fast-scan feature are described by Zakharenko et al. 17 . As a radiation source in the spectrometer, we use commercially available VDI frequency multiplication chain that is driven by home-made fast sweep frequency synthesizer. The fast sweep system is based on the up-conversion of AD9915 direct digital synthesizer (DDS) operating between 320 and 420 MHz into Ku band by mixing the signals from AD9915 and Agilent E8257 synthesizers with subsequent sideband filtering. The DDS provides rapid frequency scan with up to 50 µs/point frequency switching rate. In the present study, the spectrum was recorded in the frequency ranges 150 -200 GHz, 220 -330 GHz and 400 -460 GHz. Because of the methoxymethanol line weakness the spectrum was scanned with slower rate of 1 ms/point, and to additionally increase the signalto-noise ratio, each spectrum was averaged 8 times. Owing to kinetic instability of methoxymethanol the measurements were performed in the so-called "flow mode". During the experiment, the absorption cell, which consisted of 1.2 m Pyrex glass tube with Teflon windows, was kept at room temperature, whereas the sample of methoxymethanol was cooled at a temperature of about -50 • C and evaporated outside of the cell. Then, the sample was continuously injected through a side opening at one end of the cell and pumped out through another side opening at the other end. The optimum gas pressure in the cell was kept close to 25 Pa. The recorded spectra are very dense mostly owing to same order of magnitude intensities of the rotational lines of all three stable conformations of methoxymethanol as described below. The spectra are also dominated by the lines of methanol and formaldehyde that may originate either from impurities in the sample or from decomposition products. An example of the recorded spectrum between 235 and 265 GHz is shown on Fig 1 . To obtain higher dynamic range with respect to weak lines of methoxymethanol, we saturated the intensities of strong lines in the analog-to-digital converter at the detection. Therefore, the strongest lines presented on Fig. 1 are cut off and do not follow characteristic at 2f detection over/below baseline intensities ratio of 3/1. All the methanol and formaldehyde lines were excluded from the spectrum, as they hampered the analysis owing to their saturated intensities.

Theoretical Calculations

The present quantum chemical calculations were performed using the Gaussian 09 suite of programs. 18 First, we performed structural optimization and harmonic force field calculations at MP2 19 level of theory and using Peterson and Dunning's correlation-consistent aug-cc-pVTZ basis set, which is of triple-ζ quality Table 1 The energies of transition states of methoxymethanol with respect to the conformer I. The transition states are defined on Fig. 3 Transition β ( and includes both diffuse and polarized functions. 20 The geometries were fully optimized using the "tight" convergence option. As it was found in the previous study, methoxymethanol has three stable conformations denoted here as I, II and III. Their structures, atom numbering, and calculated dipole moment values are shown on Fig. 2. The conformer I is the most stable one. The relative energies of the conformers II and III resulting from structural optimization are respectively 717 and 925 cm -1 . The relative energies with zero-point energy correction from harmonic force field calculations are correspondingly 672 and 813 cm -1 . Two most stable conformations I and II have molecular skeleton formed by heavy atoms in gauche configuration, and they differ only by the orientation of the hydroxyl group hydrogen. Consequently, conformations I and II have rather similar rotational constants (see Table 3) but quite different components of the permanent dipole moment. The conformation III is characterized by planar heavy atoms skeleton, and by gauche configuration of the OH group.

Second, we calculated two-dimensional potential energy surface (PES) of methoxymethanol. Owing to large computational volume, the density functional theory (DFT) calculations were performed with Becke's threeparameter hybrid functional 21 employing the Lee, Yang, and Parr 22 correlation functional (B3LYP), and using 6-311++G(3df, 2pd) basis set. In methoxymethanol, the heavy atoms connected by three single bonds are the origins of three large amplitude motions (LAM). The LAMs may be parametrized by three dihedral angles: α(C6-O5-C1-H3, methyl torsion), β(O9-C6-O5-C1, skeletal torsion) and γ(H10-O9-C6-O5, OH torsion). For each conformation, the methyl torsion may be described by typical three equivalent minima on one-dimensional PES which is approximated using the following expansion formula:

V (α) = V 3 2 (1 -cos 3α) + V 6 2 (1 -cos 6α) + ... (1)
In the previous study, 14 the barrier to internal rotation of the methyl top for conformation I was estimated to be 669 cm -1 . The other two LAMs are two-dimensional problems as two bond angles need to be changed along the pathway between equivalent minima. We calculated the PES as function of two dihedral angles and that is presented on Fig. 3. The calculations were performed by varying angles β and γ in the range from 0 • to 360 • by a step of 5 • . The geometry was fully optimized at each step. Consequently, the energy of each point on PES corresponds to the optimized structure. As it may be seen from Fig. 3, each conformation is characterized by two energetically equivalent local minima on PES denoted with letters a and b. The minima are connected by pathways across five different transitions states TS1 to TS5. Owing to symmetry properties, the transition states from TS1 to TS4 have two equivalent configurations. The information on different barrier heights may be obtained directly from the calculated PES. To provide more precision and a point of comparison with the calculations of stable conformers, the transition states were further optimized using "tight" convergence criteria at MP2 level and with aug-cc-pVTZ basis set. As expected, each optimization produced one negative vibrational frequency that corresponds to the first order saddle point. The energies of all five transitions states with respect to the conformer I are presented in Table 1. From these optimizations, the most interesting result is obtained for TS5 that separates two equivalent configurations of the conformer III. Taking the relative energy of the conformer III into account, the barrier to interconversion between two gauche configurations of the conformer III is about 492 cm -1 . Consequently, tunneling splittings owing to OH torsional motion are expected in the rotational spectrum of the conformer III. The splittings owing to tunneling between two equivalent configurations of conformers I, and II are also possible. However, higher barrier heights and also much larger barrier widths should make tunneling probabilities and thus tunneling splittings much smaller compared to those of conformer III.

Finally, the barriers to internal rotation of the methyl group were calculated for all three conformations. In these calculations, we retained MP2/aug-cc-pVTZ as well as DFT M06-2X/6-311++G(3df, 3pd) method/basis set combinations. 23 The latter method has previously shown to provide relatively high precision of the results as compared to experimental values 24,25 demanding at the same time much less computational effort. In the present case, the computational time required for calculations was smaller by a factor of seven for M06-2X/6-311++G(3df, 3pd) than for MP2/aug-cc-pVTZ. For each conformer, to estimate the height of the barrier to methyl torsion, we calculated one-dimensional PES by varying the angle alpha from 0 • to 360 • by a step of 5 • . Then, the resultant PES was fitted to Eq.1 to get the values of the V 3 and V 6 terms. The results of the fits are presented in Table 2. For the conformation I, the comparison of the theoretical values of V 3 with the experimental one is provided in the 'Analysis' section, and in Table 3.

The values of the barrier heights for all three conformations are influenced mostly by steric effects. In this view, the conformation III may be compared to dimethyl ether, CH 3 OCH 3 , 26 in which the main contribution to relatively high barrier of 925 cm -1 is due to the steric interaction between two pairs of out-of-plane hydrogen atoms. For the conformations I and II, the interaction is reduced as one of the hydrogens is replaced by hydroxyl group oxygen. In the absence of steric repulsion between hydrogen atoms, the barrier to internal rotation is determined by other effects. Here one may give examples of the V 3 value of 372 cm -1 for methyl formate, CH 3 OCHO, 27 or 399 cm -1 for methyl cyanate, CH 3 OCN. 28 In case of the conformer I, the interaction between a pair of hydrogen atoms increases the barrier height, as compared to the two examples cited above. The conformer II represents an intermediate case between I and III, where in addition to the interaction between a pair of hydrogens, an interaction between hydrogens of methyl and hydroxy groups may take place leading to higher barrier to internal rotation in comparison with conformer I.

Analysis

According to ab initio calculations, the most stable conformer of methoxymethanol has the lowest dipole moment, while two others conformations are significantly less stable, but they have much higher dipole moments. By combining Boltzmann factors at room temperature and line strength proportional to dipole moment squared one gets a situation where the intensities of a-type transitions of all three conformations are comparable. Our calculations show that normalized relative intensities of a-type transitions of three conformations are I:II:III = 0.51:0.20:0.29. The experimental spectrum of methoxymethanol qualitatively confirms this estimation.

Conformer I. The conformer I was assigned on the basis of the spectral predictions obtained using the results of quantum chemical calculations. To account for the internal rotation of the methyl top, we used the Hamiltonian model based on combined internal and principal axes approaches which is realized in the XIAM code. 29 In XIAM, pure rotational part of the Hamiltonian operator is defined in the principal axis system, whereas the part that describes the internal rotation is defined in the internal axis system chosen to reduce the effect of the angular momentum generated by the torsional motion. After the diagonalization of the torsional part of the Hamiltonian, the resulting eigenvalues are rotated (transformed) to the principal axis system, and a global solution of the torsion-rotation problem is thus obtained. The advantage of XIAM code is that the main parameters of the Hamiltonian are directly linked to molecular structure. In addition to usual rotational constants, the rotation of the internal axis system with respect to the principal axis system is defined via two angles and δ that may be adjusted in fitting of experimental data. Both angles describe the orientation of the methyl top, δ is the angle between the internal rotation axis and the principal axis z, and is the angle between the projection of the internal rotation axis onto xy-plane and the principal axis x. Another important adjustable parameter F 0 is the inverse of the inertia moment of the methyl top.

The initial predictions of the rotational spectrum of the conformation I were calculated using XIAM code with rotational and centrifugal distortion constants as well as δ, , V 3 and F 0 parameters resulted from optimized ab initio molecular structure and harmonic force field. Such calculations were very important, as the initial assignment of the conformation I lines in very dense spectrum was a difficult task. Indeed, due to extremely crowded spectra we were not able to recognize typical high-K a patterns of a R 0,1 series transitions that were predicted to have the highest intensity in the frequency range of the measurements. Instead of searching for such patterns, we focused on low-K a transitions that were finally assigned owing to particular multiplet structure produced by torsional splittings. Typically, the tunneling through the torsional barrier splits each rotational transition into two components of equal intensity labeled according to symmetry species A and E of C 3v symmetry point group that describes the methyl top. In case of strong torsion-rotational interaction, the relative intensities of the components may vary, and additional splittings may arise producing thus more complicated pattern than simple doublet. According to initial predictions, such interactions occur for low-K a transitions. An example of such multiplet is shown on Fig. 4. In the spectra, we found multiplets similar to predicted ones but shifted in frequency by several hundreds of megahertz. The first assignment and fitting of such multiplets to the Hamiltonian model used in XIAM code allowed the determination of several parameters and significantly reduced the shift between predicted and observed multiplets. The following assignment was performed in a usual "bootstrap" manner, when the results of the least-squares fit of transitions with lower values of quantum numbers were used to calculate the predictions of transitions with higher values of quantum numbers.

In total, we assigned 1176 rotational transitions of the conformer I in the ground vibrational state. All the transitions were fitted to the model containing 19 parameters with overall root-mean-square deviation of 0.04 MHz. The fitted parameters are presented in Table 3 and compared to the results of ab initio calculations. One may notice generally good agreement between theoretical calculations and experimental values. In Table 3, the rotational constants of the conformer II are also given. As it may be seen, the rotational constants of the two most stable conformations are rather close, and simple comparison between theoretical and experimental values is not enough to determine which conformation was assigned and analyzed. To distinguish between conformations, we used dipole moment components. Theoretical calculations show that for the conformation I, the a-type transitions should be much stronger than b-type, as µ a dipole moment component is at least two times higher that µ b . Whereas for the conformation II, both the a-and b-type transitions should have approximately the same intensities. The multiplet presented on Fig. 4 is composed by strong a-type transitions and weak b-type transitions, and consequently, it may be attributed to the conformation I.

The determined value of the barrier to internal rotation compares well with the values obtained by two different calculations presented above. The V 3 value may also be used as additional evidence of the conformer I assignment. Taking previous results into account, one may validate the choice of DFT M06-2X method as a lower computational cost alternative for the evaluation of torsional barrier heights. Based on the present analysis of the conformer I, reliable frequency predictions of the rotational spectrum in the ground vibrational state are obtained in the frequency range up to 500 GHz and for all transitions involving quantum numbers 0 < J < 55 and 0 < K a < 20.

Conformer II. The rotational constants of the conformer II listed in Table 3 were estimated from MP2/augcc-pVTZ calculations by applying scaling coefficients. The latter were obtained via direct comparison between theoretical and experimental constants of the conformer I. The spectral predictions calculated using scaled rotational constants of the conformer II allowed the assignment of a series of K a = 0 and K a = 1 a R 0,1 transitions. However, the following assignment process was difficult as the assigned lines could not be fitted within experimental accuracy and using newly calculated prediction one could hardly assign higher K a lines. The fit produced unreasonably high values of centrifugal distortion constants that deviated rather significantly from those predicted from quantum chemical calculations. In this way, we were able to assign the transitions up to K a = 2. We were also able to assign the origins of high-K a c Q and c R type bands, as the µ c dipole moment component is the highest for the conformer II. However, compared to calculated predictions c Q series contained much more lines indicating some additional splittings of rotational lines. The problem of centrifugal distortion constants may be explained by quite asymmetric character of the potential well of the conformer II (see Fig. 2) that deviates significantly from harmonic oscillator approximation in which the theoretical constants were obtained. The problem of additional splittings may arise from coupling between methyl torsion and two other LAMs. Unfortunately, at present, there is no model that is able to treat the problem of methyl top internal rotation and two tunneling motions within the accuracy of high-resolution terahertz experiment. Therefore, we provide only the list of assigned lines of the conformer II in the ESI † .

Conformer III. The conformer III is a prolate asymmetric top close to symmetric top limit κ = 0.98. Owing to this feature, the high K a a R 0,1 bands represent a congested series of lines spaced on frequency scale approximately by B +C and thus easily assignable in broadband spectra. Close inspection of each series revealed that each transition in series has a double structure. The doublet splittings originate from tunneling between two equivalent gauche configurations of the conformer III. The tunneling splits each rotational energy level into two components usually labeled as 0 -and 0 + . To treat the tunneling splittings, we applied the method based on reduced-axis-system (RAS) approach proposed by Pickett. 30 It is well suited for molecules with a double minimum-potential. In matrix form, in the basis of individual wavefunctions of each tunneling substate |0 + and |0 -, the RAS Hamiltonian has the following form:

H = H rot -H ∆ H I H I H rot + H ∆ (2)
In Eq.2, H rot is the standard Watson S-reduction Hamiltonian in the I r coordinate representation, H ∆ is the part of the Hamiltonian that allows fitting averaged rotational constants for both tunneling substates: ) with the energy difference between two substates ∆E = 2E * . H I is a perturbation Hamiltonian containing F xy (P x P y + P y P x ), and F yz (P y P z + P z P y ), terms and their centrifugal distortion corrections. The non-diagonal terms determine the orientation of the reduced axis system with respect to the principal axis system. In total, we assigned more than 700 rotational transitions of the conformer III. All the assigned transitions are of a-type. Owing to structural features, the parameter ρ that describes the coupling between internal rotation of the methyl top and overall rotation of the molecule is much higher for the conformer III (ρ III = 0.19) compared to two other conformers (ρ I,II = 0.07). For the conformer III, in combination with relatively high barrier to internal rotation, it leads to unresolvable tunneling splittings for the a-type transitions and rather large splittings of 1 to 15 MHz for b-and c-type transitions. This fact may explain why we were able to assign only a-type transitions of conformer the III, as the model of the RAS Hamiltonian does not take the methyl top internal rotation into account. In addition, the c-type transitions connect two tunneling substates |0 + and |0 -and, therefore, their frequencies strongly depend on ∆E parameter.

H ∆ = E * + E * J P 2 + E * K P 2 z + E * 2 (P 2 + + P 2 -) + ... ( 3
The assigned transitions frequencies were fitted within experimental accuracy using the Hamiltonian in Eq.2 with overall rms deviation of 0.048 MHz. The results of the fit are presented in Table 4. The energy difference between two tunneling substates follows the general tendency observed for similar molecules exhibiting OH torsion. In the case of 2-hydroxyacetonitrile HOCH 2 CN, 31 theoretical calculations yield the barrier height of 427 cm -1 , while the energy difference ∆E determined from the rotational spectra analysis is 3.76 cm -1 . For a similar molecule, 4-hydroxy-2-butynenitrile HOCH 2 C 3 N, [START_REF] Motiyenko | 70th International Symposium on Molecular Spectroscopy[END_REF] theoretically estimated barrier height of 365 cm -1 corresponds to the observed tunneling splitting of 4.55 cm -1 . In the present study, the barrier to OH torsion is predicted to be higher than in two previous cases, 492 cm -1 , which is consistent with the ∆E value from the least-squares fit of 3.02 cm -1 .

Conclusions

The results of the current study provide a reliable basis for the detection of methoxymethanol in the ISM. The most stable conformation I was assigned and analyzed. However, low dipole moment value of the conformation I may be rather discouraging for astrophysical observations. Compared to the CH 3 O and CH 2 OH radicals, the dipole moment of the conformer I of methoxymethanol is lower by factor of 7.5 and 5.8 respectively. The intensity of rotational lines scales as a square of dipole moment. At a characteristic temperature of 10 K, assuming equal abundance of the two radicals and CH 3 OCH 2 OH, and taking the rotational partition functions into account one gets roughly relative intensities factors of 250 in case of CH 3 O/CH 3 OCH 2 OH, and 73 in case of CH 2 OH/CH 3 OCH 2 OH. The hydroxymethyl radical has not yet been detected in the ISM. The CH 3 O radical was detected in different cold sources 6,[START_REF] Bacmann | [END_REF] but the mechanisms of its formation are not clearly understood. It is generally believed that in cold prestellar cores, methanol is formed on grains via successive CO hydrogenation, with CH 3 O as one of the intermediates of such reaction. The experimental simulations at 10 K confirm such possibility of the methoxy radical formation. 5 In addition, these experiments also suggest that the dominant formation pathway of CH 2 OH is the hydrogen abstraction of CH 3 OH. Theoretical calculations show that at low temperature CH 3 O radical may isomerise into CH 2 OH 34 providing a possibility of the methoxy radical sink. If such isomerization is indeed effective in the ISM conditions, then subsequent recombination of CH 3 O and CH 2 OH radicals would lead to higher abundance of methoxymethanol making its detection more probable. However, at present, there are no reliable experimental data on isomerization of CH 3 O at low temperatures, and on the CH 3 O + CH 2 OH reaction rates. In this view, the results of the present study would also be useful for exploring these two problems by means of high resolution rotational spectroscopy.

Fig. 2

 2 Fig. 2 Structure, atom numbering, relative energies, and dipole moment components of three stable conformations of methoxymethanol. The energies and dipole moments are obtained from MP2/aug-cc-pVTZ calculations.

Fig. 3

 3 Fig.3The calculated potential energy surface of methoxymethanol as a function of two dihedral angles β(O9-C6-O5-C1) and γ(H10-O9-C6-O5). See Fig.2for the definition of the coordinates.

Fig. 4

 4 Fig. 4 Example of low-Ka A -E symmetry multiplet due to internal rotation splittings in the conformation I of methoxymethanol. Relative intensities of individual components are indicated by red sticks.

Table 2

 2 Parameters of the potential function V (α) (in cm -1 , Eq. 1) describing the methyl group torsion for three stable conformations of methoxymethanol determined by quantum chemical calculations

	Conformer Parameter Value a	Value b
	I	V 3 V 6	567.3(34) 552.6(42) -16.2(30) -14.8(37)
	II	V 3 V 6	708.2(38) 699.4(35) -6.5(21) -4.3(12)
	III	V 3 V 6	798.8(24) 772.7(29) -15.6(21) -17.4(26)
	a MP2/aug-cc-pVTZ.
	b M06-2X/6-311++G(3df,3pd).

Table 3

 3 Rotational and torsional parameters of conformer I of methoxymethanol Numbers in parentheses are two standard deviations in the same units as the last digit.

	Rotation

a b MP2/aug-cc-pVTZ calculations. c

Table 4

 4 Rotational parameters of the conformer III of methoxymethanol MP2/aug-cc-pVTZ calculations. b Numbers in parentheses are two standard deviations in the same units as the last digit.

	Parameters	Experiment theory a
	A (MHz)	32328.14(10) b 32534.76
	B (MHz)	4350.32898(47) 4366.71
	C (MHz)	4070.97615(46) 4086.61
	D J (kHz)	1.073687(51)	1.07
	D JK (kHz)	-3.3717(26)	-3.46
	D K (kHz)	[121.2] c	121.2
	d 1 (kHz)	-0.10047(11)	-0.102
	d 2 (kHz)	-0.001644(55) -0.00128
	H KJ (Hz)	-0.406(15)
	∆E (MHz)	90678.0(11)
	E * J (MHz) E * K (MHz) E * 2 (MHz) E * JJ (kHz) E * 2J (kHz)	0.40917(10) -3.085(82) 0.18311(20) -0.001921(51) -0.00150(10)
	F yz (MHz)	36.7781(57)
	F xy (MHz)	-0.1715(42)
	F yzK (MHz)	0.00583(15)

a c Fixed to ab initio value.

Acknowledgements

This work was supported by the French Programme National "Physique et Chimie du Milieu Interstellaire", and by ANR-13-BS05-0008 IMOLABS of the French Agence Nationale de la Recherche.


Laurent Daou

Mathieu Luglia

Claude Perissol

Virgile Calvert

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