Smail Ahmim
email: smail.ahmim@gmail.com

Morgan Almanza

Vincent Loyau

Frédéric Mazaleyrat

Alexandre Pasko

Fabien Parrain

Martino Lobue

Self-oscillation and heat management in a LaFeSi based thermomagnetic generator

Keywords: thermomagnetic generator, thermal energy harvesting, magnetocaloric materials, self-oscillation, piezoelectric materials

Harvesting and directly converting the largely untapped and omnipresent lowgrade heat into electricity has huge technological potential for supplying internet of things (IoT) devices, and wireless sensors in smart buildings applications. To this end, we present a self-actuating, autonomous, thermomagnetic generator (TMG) showing state-of-the-art net power output 4.2 µW (240 µW cm -3 ), working over a 35.1 °C temperature difference. The core of our design is made of cheap and easily available piezoelectric buzzers and of commercial LaFeSi magnetocaloric materials from Vacumschmelze. The device self oscillating cycle is studied as a dynamical system with particular attention to the interplay between mechanical and thermal dynamics.

Introduction

The number of Internet of Things (IoT) connected devices is expected to increase two-fold over the next five years [START_REF] Newell | Review of power conversion and energy management for low-power, low-voltage energy harvesting powered wireless sensors[END_REF]; this will prompt a rising demand for autonomous power supply generators to sustain low-power wireless sensor nodes. Besides, the pressure to develop small, reliable, and compact energy scavenging devices to power up implantable medical electronics (IME) has been constantly rising for the past twenty years [START_REF] Shi | Implantable energy-harvesting devices[END_REF]. Considering the essential role played by wireless sensors nodes in the development of smart low carbon fingerprint buildings, and the growing relevance of IME to tackle endemic diseases and large scale health crisis, small scale energy harvesting can be counted between the key technological challenges of the next couple of decades.

Due to its ubiquity in buildings and urban environment, and in human body as well, low grade heat looks as one of the straightforward energy sources to target for applications whose aimed power-output falls in a range between some hundreds and tens of µW [START_REF] Newell | Review of power conversion and energy management for low-power, low-voltage energy harvesting powered wireless sensors[END_REF][START_REF] Yi | A battery and leadless heart-worn pacemaker strategy[END_REF].

Thermoelectric (TE) devices offer a direct conversion tool from a heat flux to electricity. Nevertheless keeping a constant temperature gradient, and avoiding detrimental heat leakages over small length scales, can be a difficult goal. This, jointly with the low factor of merit shown by available TE, represents a major shortcoming of this solution at the millimeter scale and for low grade heat sources [START_REF] Gueltig | High-performance thermomagnetic generators based on heusler alloy films[END_REF].

For the last couple of decades, since the discovery of Giant Magnetocaloric effect [START_REF] Pecharsky | Giant magnetocaloric effect in Gd 5 (Si 2 Ge 2 )[END_REF], we witnessed the rising of a pretty large scientific community focused on caloric materials [START_REF] Moya | Caloric materials near ferroic phase transitions[END_REF], and on their application to energy conversion [START_REF] Kitanovski | Present and future caloric refrigeration and heat-pump technologies[END_REF], cooling, heat pumping, and power generation. A caloric material is a substance where the action of a field (viz. a thermodynamic intensive variable) induces a relevant response in one or both thermal state variables (i.e entropy and temperature). In general, the effect is magnified when the substance is on the verge of a ferroïc transition [START_REF] Moya | Caloric materials near ferroic phase transitions[END_REF]. Caloric materials are named with a prefix pointing at the field triggering the effect so that magnetocaloric, electrocaloric, barocaloric, elastocaloric, refer to magnetic, electric, pressure, and stress fields respectively. In addition, multicaloric behavior may appear due to cross effects associated with coupling between order parameters.

Our focus on autonomy (i.e. battery-less devices) makes magntocaloric materials (MCM) the more suitable choice due to the possibility to use magnets as field sources. Moreover magnetocaloric effect has been studied for a longer time, and we dispose of an unprecedented choice between families of materials that have been mostly optimized to be deployed in room temperature magnetic cooling devices [START_REF] Lyubina | Magnetocaloric materials for energy efficient cooling[END_REF]. An opportunity that can not be missed to give the old thermomagnetic generation (TMG) concept [START_REF] Tesla | Thermo-magnetic motor[END_REF][START_REF] Edison | Pyromagnetic motor[END_REF][START_REF] Brillouin | Thermomagnetic generator[END_REF] a new chance.

Modifying the magnetic order through a temperature change is the core of TMG. The conversion to electrical energy is either done directly via Faraday law, or through an intermediate step through mechanical energy which is eventually converted by a transducer. Waste heat harvesting often imply an imposed temperature difference ∆T res between the hot source and the heat sink, so that the power output as a function of ∆T res is commonly used as a figure of merit. Some promising results have been obtained at ∆T res ≈ 100 K. Recently, Gueltig and co-workers [START_REF] Gueltig | High-performance thermomagnetic generators based on heusler alloy films[END_REF] realized a TMG showing an average electric power density of 118 mWcm -3 for ∆T res = 140 K. Before that, Chun et al. presented a TMG [START_REF] Chun | Thermo-magneto-electric generator arrays for active heat recovery system[END_REF] with 38 mWcm -3 for ∆T res = 80 K.

Nonetheless, most of the waste heat available reservoirs are in the so called low grade form, involving a ∆T res of at most few tens of degrees above room temperature. This motivates an increasing attention towards temperature spans within this interval. Working over ∆T res = 50 K, Ujihara et al. [START_REF] Ujihara | Thermal energy harvesting device using ferromagnetic materials[END_REF] estimated a power of 3.6 mW/cm 2 (power per surface area of the active material). This power is indirectly extrapolated using the efficiency at resonance of a piezoelectric transducer. Guelting et al. [START_REF] Gueltig | High frequency thermal energy harvesting using magnetic shape memory films[END_REF] realized a TMG that shows an average power density of 1.6 µWcm -3 for ∆T res ≈ 25 K. More recently, Waske et al. [START_REF] Waske | Energy harvesting near room temperature using a thermomagnetic generator with a pretzel-like magnetic flux topology[END_REF] show a maximum electric power density of 118 µWcm -3 with ∆T res = 30 K.

It is worth noting that the latter value, one of the higher published to date, represents the power output without considering the energy required to actuate the thermal switch (i.e. the pump keeping the flow of the fluid used to exchange heat).

Once thermal energy has been converted into magnetic one, conversion into electricity can be achieved through different methods (see [START_REF] Kitanovski | Energy applications of magnetocaloric materials[END_REF] for a recent and exhaustive review). Direct conversion through Faraday law represents the most straightforward road [START_REF] Waske | Energy harvesting near room temperature using a thermomagnetic generator with a pretzel-like magnetic flux topology[END_REF]; in some cases an intermediate mechanical energy conversion step has been successfully used, jointly with Faraday law, in order to enhance conversion efficiency [START_REF] Gueltig | High frequency thermal energy harvesting using magnetic shape memory films[END_REF][START_REF] Gueltig | High-performance thermomagnetic generators based on heusler alloy films[END_REF][START_REF] Joseph | Upscaling of Thermomagnetic Generators Based on Heusler Alloy Films[END_REF]. On the other hand the mechanical energy can also be harvested through other transducers as electroactive materials [START_REF] Chun | Thermo-magneto-electric generator arrays for active heat recovery system[END_REF], and piezoelectric materials [START_REF] Ujihara | Thermal energy harvesting device using ferromagnetic materials[END_REF][START_REF] Carlioz | Générateur piézoélectrique à déclenchement thermomagnétique[END_REF].

Recently, we presented a piezoelectric based TMG [START_REF] Ahmim | A thermal energy harvester using LaFeSi magnetocaloric materials[END_REF] using a commercial first order MCM (CALORIVAC ® from Vacumschmelze) as active substance. The adopted design is similar to the one presented in [START_REF] Ujihara | Thermal energy harvesting device using ferromagnetic materials[END_REF] with two major differences: the magnetic field source is a Halbach array in order to achieve high field gradients keeping demagnetizing fields as low as possible; the active substance undergoes a first order magnetic transition allowing higher efficiency at maximum power as shown in a recent study [START_REF] Almanza | First-versus secondorder magnetocaloric material for thermomagnetic energy conversion[END_REF]. The thermal management is assured by alternatively moving the MCM from the hot source to the heat sink and vice versa, while the mechanical energy is recovered from the deformation of the beam where the material is fixed. This design, generally referred as linear [START_REF] Kitanovski | Energy applications of magnetocaloric materials[END_REF], is particularly well suited to work over a well controlled thermodynamic four-strokes cycle (viz. two quick adiabats, and two heat exchange transformations) [START_REF] Ahmim | Thermal energy harvesting system based on magnetocaloric materials[END_REF], and to reach a self-oscillating regime where no external actuation is needed to drive the device [START_REF] Jenkins | Self-oscillation[END_REF][START_REF] Gueltig | High-performance thermomagnetic generators based on heusler alloy films[END_REF][START_REF] Ahmim | A thermal energy harvester using LaFeSi magnetocaloric materials[END_REF]. The device reaches up to 0.12 µW electric power-output with a density of 6.8 µW.cm -3 per volume of the MCM [START_REF] Ahmim | A thermal energy harvester using LaFeSi magnetocaloric materials[END_REF]. It works over a temperature difference ∆T res = 38 K, with a 0.8 mm linear displacement, at a frequency f = 0.12 Hz. In spite of its state-of-the-art performance the main drawback of this design, and the key obstacle limiting its power output, is its relatively low working frequency.

Here we present a new prototype largely overcoming the performance of the previous one, with 4.2 µW net power-output, 240 µW cm -3 (i.e. over 30 times the output of the previous device), working over a similar temperature span, ∆T res = 35 K, at an improved frequency, f = 0.41 Hz.

In the following section we shall present and discuss the new design and its main advantages. We shall focus on the key objective of achieving a bistable dynamical cycle. The design and characterization of the new moving part will be described in detail and the directly measured output under optimal working conditions presented.

The third section is devoted to the study of the stability of the self-oscillating behavior, studied as a dynamical system driven by a conservative mechanical equation of motion coupled to a diffusive thermal equation. The dynamical system phase-space is sampled by varying an external parameter (i.e. the distance d between the heat reservoirs), and its phase portrait discussed.

The physical mechanisms underlying the observed dynamics are discussed in the fourth section. The relevance of this study is two-fold. On the one hand, it assesses the reliability of the device over different working conditions, a pretty relevant issue for an actual device. On the other, it allows to investigate the subtle interplay between the thermal and the mechanical cycles, showing that this is the key physical mechanism underlying the behavior and the performance of a TMG self-actuating device. A section devoted to discuss the main results and to the perspectives opened by the presented achievements will close the paper.

Basic working principle and design

A schematic representation of the prototype cross section and a set of pictures are shown in Fig. 1 and2 respectively. The TMG is composed of a MCM plate, a disc spring, a field source, a hot source, and a heat sink. The MCM is a La(Fe,Si) 13 H plate (0.25 × 7 × 10 mm 3 ) with a Curie temperature T C = 30 °C. The disc spring is made of two piezoelectric buzzers (see Fig. 2). They are inversely glued to each others through a thin CuBe ring. Each piezoelectric buzzer is cut, using a water-jet, in the shape of a cross in order to get the researched stiffness. The latter has been directly measured and is equal to 8.2 N.mm -1 . The total displacement of the MCM when switching from direct contact with one reservoir to the other is d = 0.48 mm. The field source is an Halbach structure, built using NdFeB, 5×(1.5 × 1.5 × 10 mm 3 ) parallelepiped magnets. This configuration allows a stronger force with a steeper change over small displacements compared to the field generated by a single magnet [START_REF] Ahmim | Thermal energy harvesting system based on magnetocaloric materials[END_REF]. The hot source temperature is kept constant through an electrical resistance while the cold reservoir is made with a water exchanger. An opening in the piezoelectric buzzer ensures the MCM direct thermal contact with the heat sink. The temperatures of the heat source and the of the heat sink are T h = 51.5 °C and T c = 16.4 °C respectively, giving a temperature difference between reservoirs ∆T res = T h -T c = 35.1 °C. The device works over a four strokes cycle driven by the competition between the magnetic and the elastic forces, F m and F e respectively. The latter has been measured, and depends linearly on the MCM position x (viz. on the beam deformation) as shown in Fig. 4 (i.e. the green line passing trough points A and B). The former depends on the field gradient (i.e. on the MCM position x), and on the MCM magnetization M (T, H(x)) where T is the temperature of the material and H(x) the magnetic field. Magnetic force isotherms computed with 3D Ansys MAXWELL, using as an input a set of M (T, H) (see Fig. 3) curves measured with a VSM magnetometer are shown in Fig. 3 (see [START_REF] Ahmim | A thermal energy harvester using LaFeSi magnetocaloric materials[END_REF] for more details). Defects in the assembly with the magnets (misalignment) or in the magnets themselves lead to a lack-off flatness. This issue is addressed by introducing an equivalent air gap of 0.1 mm when the MCM is on the hot side.. The four steps composing the cycle can be described as follows:

Heat sink

Heat source

Step 1 : the MCM is kept in contact with the heat sink by the elastic force. It cools down until the ferromagnetic volume fraction makes F m overcoming F e , namely when the material temperature corresponds to the magnetic force isotherm crossing the elastic force in the point x 1 = 0.58 mm (i.e point A in Fig. 4 matching the MCM position). This takes place at T = 18 °C (i.e. the yellow continuous line if Fig. 4).

Step 2 : the MCM switches towards the hot source; if this step corresponds to a mechanical instability, the movement is expected to be fast enough to be considered an adiabatic transformation. It is worth noting that during the movement, due to the field change, the temperature will slightly increase due to the adiabatic temperature change.

Step 3 : the MCM is in contact with the hot source, receiving heat and therefore becoming paramagnetic. During this process the magnetic force decreases till reaching the F m isotherm crossing F e at x 0 = 0.1 mm (i.e. the MCM position). This takes place at T = 45 °C (point B in Fig. 4).

Step 4 : once reached the mechanical instability corresponding to point B in Fig. 4 the MCM moves back to the heat sink. During this displacement the field decreases, triggering a negative adiabatic temperature change. This transformation too is expected to take place in adiabatic conditions.

Maximizing power by increasing frequency can be achieved through a reduction of the time associated with the four strokes. While the time spent to carry out steps 1 and 3 is related to the heat exchange process, steps 2 and 4 depend on the displacement velocity. On the one hand thermal contact is the

F m -- F m -- F m -- F m -- F --k moy =8,25 N/mm °C °C °C °C e
Step 3

B

Step 2

Step 4

A

Step 1 key to get fast heat exchange, on the other hand fast switching depends on a wise balance between forces in order to reach a bistable regime. In short, steps 2 and 4 must take place as instabilities with the driving force (magnetic one in step 2 and elastic one in step 4) largely overcoming the restoring one all along the displacement. Given the magnetic force isotherms, the working cycle is determined by the slope of F e (x), where a fine beam-stiffness tuning is of the utmost relevance to get a bistable dynamical cycle. The disc spring has been designed to work within the elastic regime so that Von Mises stresses remain below 90 MPa for copper and 50 MPa for piezoelectrics.

After studying different configurations and shapes through FEM calculations with large deflection, we opted for a cross shaped beam made of two piezoelectric buzzers. A buzzer consists of a piezoelectric disk glued on the top of a metallic disk (a CuBe metallic alloy in this case) of larger diameter, as shown in frame A of Fig. 5. The stiffness of the CuBe disk has been tuned by cutting it in the cross-shape shown in frames B and C of Fig. 5. The final setup presented here is obtained by assembling two buzzers in series (i.e. keeping the two piezoelectric disks on the outward sides). The double disc spring reduces the device area, increasing the quantity of piezoelectric material and easing the assembly thanks to a mechanical response less sensitive to its fixation. Its stiffness as a function of the displacement has been directly measured, the result being the green line, shown in Fig. 4. FEM simulations, shown in Fig. 6, confirm that the deformation is never getting beyond the elastic limit under a d = 1 mm displacement, more than twice the working point of the actual device. Considering the small working ∆T res , and the fact that the beam is never directly in contact with the reservoirs, a small but negligible temperature gradient through the bimorph is expected due to heat leakages from the MCM. Hereafter, no thermal effect is expected in the disc spring response.

The MCM displacement is measured using a laser sensor ILD1320 MICRO-EPSILON, the results are shown in Fig. 7. Steps 1 and 3 (i.e. MCM thermal contacts with the cold and hot end) last, t c = 1.37 s and t h = 0.9 s respectively. Steps 2 and 4 are steep, a signature of the researched bistable behavior. Working over a total displacement d = x 1 -x 0 = 0.48 mm the device reach a frequency f = 0.41 Hz which is three times higher than the frequency of our previous prototype [START_REF] Ahmim | A thermal energy harvester using LaFeSi magnetocaloric materials[END_REF]. The voltage induced on the piezoelectrics by the deformation is measured using a load resistor which maximizes the output power. Fig. 8 shows the voltage measured on the upper piezoelectric plate. The TMG net output electric power is P out = 4.2 µW (i.e. 240 µWcm -3 per volume of the active material) which corresponds to an electrical energy per cycle of 10.6 µJ.

The power entering from the hot side is 1.7 W, given the power output the efficiency is 2 10 -6 . The sources of efficiency reduction in our deign are manyfold. On the one hand only a small fraction of the incoming heat is actuallly passing through the TMG due to relevant heat leakages into air. Using the cited thermal model we estimate the global heat flux traversing the TMG to be ∼ 0.18 W (i.e. one order of magnitude lower than the actual incoming heat). Eventually, from our previous work, we estimate the final conversion efficiency into electric power to be reduced of a factor 1000. Indeed, harvesting low-grade thermal energy is, from the start, a matter of low efficiency, even Carnot's one being close to zero. Thence, here we put the main focus on autonomy (i.e. selfoscillation), and on reaching the power output needed by small MEMS scale systems.

Study of working cycle as a dynamical system

As discussed in the previous section, the device dynamical cycle is the result of the balance between the elastic force term F e (x), determined by the deformation (i.e. by the MCM position x), and the magnetic force F m (T, x). The latter depends on x through the field gradient, and on both variables, x, T through the magnetization M (T, H(x)). To be sure, a pretty complex functional dependence the finite elements code has to deal with. A mayor issue here is that, beside the mechanical cycle whose variables are x(t) and ẋ(t), the material undergoes a thermodynamic (i.e. a thermal) cycle with a single dynamical variable T (t) driven by a non-conservative equation of motion. This makes the global behavior a rather complex dynamical system operating on a three-dimensional phase-space whose points x(t) ≡ {x, ẋ, T }, with x ∈ R 3 , move along phase curves stemming from the coupling between the mechanical and the thermal dynamics. The phase velocity vector field is v(t) ≡ { ẋ, ẍ, Ṫ }.

In what follows we shall see that this minimal model allows grasping, at least qualitatively, most of the relevant features we experimentally observed. A further refinement taking into account the piezoelectric material effect on the buzzer stiffens, associated with transducers electric tension, will be needed to explain some details of the MCM movement when the dynamics goes beyond the pure bistable regime. This introduces an additional functional dependence in the elastic force term, F e ≡ F m (x, V (x)), and a new variable, V (t), making the phase-space a four dimensional one, with phase points x(t) ≡ {x, ẋ, T, V }, and x ∈ R 4 .

The dynamical system is constrained by two external parameters: the distance between the thermal baths d = x 1 -x 0 , limiting the displacement to the interval x 0 ≤ x ≤ x 1 ; the temperature difference between the reservoirs, ∆T res = T h -T c . The latter drives the temperature dynamics through a Fourier thermal law,

250 T dS(T, H(x)) dt = -K h (x)(T -T h ) -K c (x)(T -T c ) (1)
where S is the entropy of the MCM, T its temperature, K h (x), and K c (x) are the thermal conductance between the MCM and the hot and cold reservoir respectively, and, for the sake of simplicity, we neglect heat leakage through the beam. The dependence of K h and K c on the position x describes heat conduction when the material is in contact with the bath (e.g. K h (x 0 ) represents the thermal conductivity between the hot reservoir and the MCM when they are in direct contact), as well as conduction through an air layer when the material lose contact with the heat baths (e.g. K c (x) drives the heat flux between the MCM and the cold bath through an air gap of thickness x 1 -x, and K h (x) describe heat conduction between material and hot end through an air layer of thickness x -x 0 ).

In order to experimentally investigate the phase portrait of the TMG described as a dynamical system, we shall discuss its measured behavior by varying the distance d between the reservoirs. This will allow to appreciate the complex interplay between mechanical and thermal variables in a self-oscillating TMG device based on an otherwise pretty simple design.

Hereafter, we shall study the device responses when changing the distance d = x 1 -x 0 between reservoirs. The hot bath position is kept fixed at x 0 = 0.1 while the position of the cold reservoir, originally at x 1 = 0.58 mm is changed either towards the positive or the negative direction increasing or decreasing d respectively. More precisely, increasing or reducing d will correspond to x 1 > 0.48 mm or x 1 < 0.48 mm respectively. Measured displacement integral curves (i.e. x(t) plot) corresponding to the latter case are shown in Fig. 9. Curves acquired after increasing d are shown in Fig. 10.

The first issue apparent from the integral curves shown in Fig. 9 is that when the distance d is reduced the system dynamics is still bistable (viz. switches between reservoirs are sharp, taking place over a negligible time scale), the total period increases (viz. frequency decreases), and the longer periods are mostly to be ascribed to the increase of the time the MCM stays in contact with the cold bath t c . Table 1 allows a closer examination of the integral curves showing that while t h always increases when the gap between reservoirs is made thinner, the time spent in contact with the hot end t h undergoes an initial increase (i.e. till x 1 = 0.46 mm), followed by a systematic reduction. On the other hand, Fig. 10 shows the integral curves when x 1 is increased (i.e. x 1 > 0.58 mm), and Tab 2 summarizes the relevant data relative to these curves. Comparison between Fig. 9 and 10 makes clear that, decreasing the distance between baths, x 1 = 0.58 mm represents the onset of the bistable dynamical regime. Actually, when x 1 > 0.58 mm the system shows a complex behavior where the MCM no more attains contact with the cold end, and shows a damped oscillation over an intermediate equilibrium point followed by a transitory slow displacement ending when the material eventually switches back to the hot reservoir.

To understand this change in behavior as a function of x 1 , it is worth using a phase portrait of the dynamical system. For the sake of simplicity we shall start from the minimal model where we neglect the role of the tension on the piezoelectric material so that, as stated above, phase points evolve in a three dimensional space, x ∈ R 3 , with x(t) = {x, ẋ, T }. In this case the phase portrait of the dynamical system in the plane (x, T ) is defined by the force-balance line (FBL) where the total force acting on the MCM is zero,

F (x, T ) = F e (x) -F m (T, x) = 0. (2)
The line, representing the sub-set of phase space where ẍ = 0 (i.e. with phase velocity v F BL = { ẋ, 0, Ṫ }), splits the plane (x, T ) into two regions: one, below it, where the magnetic overwhelms the elastic force resulting in a total negative acceleration, dragging the MCM towards the hot end; the other, above it, where the elastic force is dominant and a positive force accelerates the material towards the heat sink. The point C, shown in Fig. 11, divides the FBL into a mechanical stable, and an mechanical unstable equilibrium branch. The hot and cold end are shown as vertical continuous thick lines, red and blue respectively. This is to remind that they represent an external constraint to the dynamical system, 2), divides the (x, T ) plane into two regions. One above the line where the elastic force is predominant. The other, below, where the magnetic force overwhelms the elastic one. The FBL line is divided into a stable (continuous black line), and an unstable (dotted black line) branch by the point C in the form of two rigid impenetrable walls (i.e. a constraint fully described by the parameter d = x 1 -x 0 ) limiting the phase space on the x axis to the interval

x 0 ≤ x ≤ x 1 .
Let us now discuss the typical phase trajectories associated with the integral displacement curves shown in Fig. 9 and10.

Purely bistable cycles are described by the integral curves where x 1 < 0.58 mm shown in Fig. 9. As an example we can take the curve corresponding to x 1 = 0.32 mm (the yellow line in Fig. 9) where the MCM stays in direct contact with the reservoirs to heat up or cool down alternatively, and switches abruptly between them when reaching the temperature laying on the unstable branch of the force-balance line (corresponding on Fig. 11 to point B and the yellow point A). If we neglect the adiabatic temperature change taking place when the material moves, corresponding to less than 2 °C given the material properties and the field change it is submitted to, the related phase trajectory can be represented as a rectangle in in (x, T ) plane. At B , where the system get in contact with the hot reservoir, the MCM can't move. However its temperature is lower than the temperature of the bath, the consequent heat exchange resulting in a temperature increase. More precisely, ẋ = ẍ = 0, and Ṫ > 0, with a phase-point velocity v = {0, 0, Ṫ } (represented in Fig. 11 with an upward yellow arrow). Getting to point B, lying on the FBL, ẍ ≥ 0 and the MCM accelerates toward the heat sink. Neglecting heat exchanges with air (viz. adiabatic limit), and the adiabatic temperature change, the phase point moves along a horizontal line where ẋ, and ẍ, are both positive, and Ṫ = 0, corresponding to a phase velocity vector v = { ẋ, ẍ, 0} (horizontal leftward yellow arrow in Fig. 11). The displacement stops at A where the material reaches the cold end (i.e. ẋ = ẍ = 0), and exchanges heat with the cold bath, here Ṫ < 0 as material temperature is greater than the cold bath one. Displacement form A to A, where the vertical line crosses the FBL, takes place with Ṫ < 0 (the downward vertical yellow arrow in Fig. 11) with phase velocity v = {0, 0, Ṫ }. When the phase point reaches the force-balance line, at A, the MCM switch back to the hot end (horizontal leftward yellow arrow in Fig. 11).

Let us now examine the integral curve corresponding to x 1 = 0.58 mm (the green lines in Fig. 9 and10). While the switch from the hot to the cold bath is still sharp (i.e. fast material movement), the switch in the opposite sense takes place at a slower rate, at least down to x ∼ 0.42 mm. This behavior can be explained tracing the corresponding phase trajectory on the (x, T ) plane as shown in Fig. 11. In this case the phase curve behaves in a pretty similar way to the purely bistable cycle along the B BA A lines (green arrows in Fig. 11). The main difference here is that the point A lays on the stable branch of the FBL. This makes the system moving along the force-balance line, between A and C, through a path of stable points. As far as the MCM loses direct contact with the cold reservoir, heat exchange continues, even if at a lower rate (i.e. K c (x) < K c (x 1 ) when x = x 1 ), inducing a temperature decrease (i.e. Ṫ < 0), and a consequent displacement of the material (i.e. ẋ < 0) depending on the FBL slope. This state of affairs suddenly changes when reaching C, where the system get unstable and switches towards the hot bath (represented by the green rightward arrow along CB segment in Fig. 11).

The third and more complex phase trajectory we shall consider corresponds to the behavior observed when x 1 ≥ 0.58 mm. Red lines and arrows in Fig. 11 describe the phase trajectory corresponding to the x 1 = 0.78 mm. In this case the MCM crosses the stable branch of the force-balance line at x ∼ 0.7 mm, before getting to the heat sink. A damped oscillation lasting approximately 10 -1 s takes place across the FBL till the phase point reaches the line at C . Afterwards the MCM is expected to moves along the FBL line exchanging heat with the heat sink through the air layer until reaching the point C where the system becomes unstable and switches back to B . The damped oscillation can be explained in the frame of the minimal dynamical model by the change in potential energy associated with the material cooling down, and possibly by adding to the description energy losses associated with air viscosity and eddy currents. On the other hand, while the trajectory C C shown in Fig. 11 describes a slow MCM displacement towards the hot reservoir, quite similar to the one associated with the above described AC path, it can hardly account of the MCM slow displacement towards the heat sink, just after the end of the oscillation, shown in Fig. 10 between points C andC .

In order to explain this behavior, characterizing all the x 1 > 0.58 mm integral curves, some substantial model improvements are required. The point here is that the minimal model neglects the effect of the piezoelectric material electric tension V on the elastic force. Actually, when the MCM moves, deformation under charge of the piezoelectric materials imposes an additional recall force (i.e. always opposed to the displacement). This force relaxes over a scale time of seconds when the tension is discharged on a resistance (viz. the final energy conversion step). To address this effect the elastic force must be described as a function of the position of the MCM x, of the tension on the piezoelectric transducers V , and of the sense of the deformation, namely F e ≡ F e (x, V ), the dynamical system phase space becoming a four dimensional one with x ≡ {x, ẋ, T, V }, and x ∈ R 4 . Now the force-balance line in the plane (x, T ) changes its shape depending on different piezoelectric deformations and states of charge. FBL corresponding to different values of x and V are qualitatively represented in Fig. 12 showing the way the C C path of Fig. 10 is associated to the forcebalance line shape change during the piezoelectric tension discharge.

Discussion

This section shall address the different physical mechanism underlying the behaviors described above, and their role on the overall working performance of the device.

As maintaining a stable self oscillating regime, and getting high power output through frequency improvement are the main goals of this work, we shall focus on these two features.

Starting from self-oscillation, it is worth noting that in spite of its complexity, and of the non-linear, and non-conservative nature of most of its features, the dynamical system does not show chaotic behaviour when varying the parameter x 1 over the interval between 0.17 mm and 2.18 mm, a pretty wide range indeed, spanning more than one order of magnitude.

Moreover, the self-oscillation is robust against variation of d. Namely, over the explored range of distances between reservoirs, the dynamical system is never getting into a fixed point stopping its oscillation. This second feature can be discussed more in detail through the projection of the phase portrait on the (x, T ) plane. A fixed point is a point of the phase space where all the components of the velocity vector are zero [START_REF] Arnold | Ordinary differential equations[END_REF], within the minimal model this corresponds to v = { ẋ, ẍ, Ṫ } = 0. While ẍ = 0 is the mechanical equilibrium condition, Ṫ = 0 corresponds to thermal equilibrium. Equation (1) imposes:

K h (x) (T h -T ) -K c (x) (T -T c ) = 0. (3)
This expression defines a boundary in Fig. 13), over the (x, T ) plane, getting from the point β ≡ (0, T h ) to the point α ≡ (x 1 , T c ), we shall refer to as the heat-balance line (HBL) (green line in Fig. 13).

Intersections points between the FBL and the HBL, are fixed points whenever they lay on the FBL stable branch (square points in Fig. 13). This case can be reached in two situations:

• when the MCM is in thermal equilibrium with one heat reservoir, a condition that can take place only if the material reaches the reservoir temperature (i.e. points α and β) before losing contact with its surface (i.e. before reaching the points A and B respectively), as shown in Fig. 13-(A);

• when the MCM is moving slowly enough to exchange heat with air (i.e. below the threshold speed where the displacement can be considered adiabatic), and the inward heat flux from the hot end equals the outward one towards the heat sink, as shown in Fig. 13-(B) on the A to C stable branch.

As the self-oscillation never gets jammed in the d interval explored, we can infer that fixed points are absent from the phase space subset we sampled.

Summarizing, self-oscillation is rather robust against changes of the distance between reservoirs, and the dynamical systems never gets on the onset of chaotic behavior. Both these results can be ascribed to the stabilizing effect the heat flux through the air-gap holds over the dynamics. As a matter of fact, while non-linearity, damping, and forced oscillations are generally the causes of chaotic dynamics, a non-equilibrium heat flux, as the one described by Eq.( 1), due to its trend towards a stationary maximum entropy production regime, stabilizes the dynamics making it less sensitive to changes in external parameters. Moreover it is noteworthy that in absence of the heat flux through air (viz. in vacuum), there wouldn't exist a HBL line and all the path along the FBL stable branch would be like a fixed points, this way, a huge part of the phase space unavailable to self-oscillating trajectories.

Focusing now on frequency, Table 1 and 2 show that the highest frequency matches with x 1 = 0.58 mm. On the one hand longer periods, more precisely, longer heat exchange times at the cold end t c , are observed when x 1 is increased, as reported in Table 2. This is an expected result related to the fact that the C C path of Fig. 11 takes place farther away from the heat sink with the consequent K c (x) reduction. On the other hand, the frequency reduction when x 1 is reduced and the device is working in the purely bistable regime looks, at first sight, quite unreasonable. A further refinement of the thermal management model may help Here we show the case where β lays below the FBL (below B) and α above the FBL (i.e. above A), so that both are fixed points of the dynamics (full green square point); (B) in this case the intersections between HBL and FBL with the reservoir surfaces are situated outside the phase-space subset accessible to the dynamics, on the contrary γ (full green square point), representing the intersection between the HBL and the FBL stable branch, is a fixed point.

to gain an insight into this behavior. Actually, identifying the surface of the hot and cold end where the MCM gets in direct thermal contact with the reservoirs is somewhat oversimplifying the real situation. Any actual energy harvester is kept in thermal contact with reservoirs (e.g. the interior of a building and the outside atmosphere) through some thermal component showing a finite thermal conductance, say K r c and K r h , referring to the cold and hot reservoir respectively. These heat exchange surfaces must exhibit a thermal capacity higher than the one of the active substance (i.e. the MCM in the present case) in order to keep their temperature constant all along the working cycle. However, the presence of a stationary heat flux through the device tends to reduce the temperature difference between the reservoir surfaces, the amount of the reduction depending on K r c , K r h , and on the flux intensity. From this point of view reducing the air gap between reservoir surfaces increases the heat flux intensity challenging the capacity of the heater, and of the cooler (i.e. the artificial heat reservoirs used in this setup) to keep T h and T c constant. Reducing T h and increasing T c means approaching the fixed point β to point B from above, and α to A from below (see Fig. 13). This is slowing down heat transfer till stopping oscillation when α, or β eventually passes the points where the FBL intersect the reservoir surfaces. The difficulty to keep the temperature gradient against heat leakages is a rather general feature of thermal harvesting devices, particularly when working over small lengths scales, and with tiny temperature gradients (e.g. this is frequently addressed in the literature on thermo-electric generators [START_REF] Apertet | Optimal working conditions for thermoelectric generators with realistic thermal coupling[END_REF]). Besides, the fact that the heat flux affects more the cold end than the hot one, as apparent from the disparity between the exchange time t h and t c apparent in Table 1 is something specific of our setup where the reduction of d increases the distance between the cold surface and the water cooler so that while K r h is unaffected, K r c is proportional to d. This explanation can be easily checked by increasing the heat flux through ∆T res change at fixed d. Within the above mentioned heat flux picture increasing ∆T res is expected to modify the reservoirs surface temperature slowing down the oscillation and stopping it when a fixed point appears on one reservoir. This is what has been observed fixing the cold sink at x 1 = 0.5 mm (viz. in the bistable region) and reducing the cooler temperature down to T c = 11.1 °C (i.e. with ∆T res = 40.4 °C) where the MCM eventually get stacked due to the appearance of a fixed point on the hot reservoir surface (i.e. the point β temperature getting lower than point B one as in Fig. 13-(A)).

Conclusions

Table 3 shows a summary of the data from the literature on the performance of existing TMG prototypes. Here we focus on two features, the power density normalized to the active material volume, and the working temperature difference between reservoirs ∆T res . Power densities span over four orders of magnitude, from tens of micro-Watts per cubic centimeter, up to more than one hundred milli-Watts per cubic centimeter.

However, ordering the data with respect to the working ∆T res two distinct groups are apparent. Devices working over a temperature span of the order of 100 °C or more [START_REF] Chun | Thermo-magneto-electric generator arrays for active heat recovery system[END_REF][START_REF] Gueltig | High frequency thermal energy harvesting using magnetic shape memory films[END_REF][START_REF] Gueltig | High-performance thermomagnetic generators based on heusler alloy films[END_REF][START_REF] Joseph | Upscaling of Thermomagnetic Generators Based on Heusler Alloy Films[END_REF] that generally get powers in the milli-Watts per cubic centimeter range, and prototypes working over a few tens degrees (i.e. ≤ 50 °C) where the power density is in the range of tens or hundreds of micro-Watts per cubic centimeter.

In the first group the works on the prototype realized at the Karlsruhe Institute of Technology [START_REF] Gueltig | High frequency thermal energy harvesting using magnetic shape memory films[END_REF][START_REF] Gueltig | High-performance thermomagnetic generators based on heusler alloy films[END_REF][START_REF] Joseph | Upscaling of Thermomagnetic Generators Based on Heusler Alloy Films[END_REF] deserve some special remarks. Firstly ∆T res does not properly matches with the design presented there as, while the hot reservoir temperature is fixed, the heat sink one is varying during the fast oscillating cycle where the MCM exchanges heat with the disc spring and with air. Actually, besides reporting the best throughput published so far, these studies have the great merit to show that frequency, and power improvements can be achieved in 500 a system working under far-from-equilibrium thermodynamic conditions getting rid, with no regrets, of the comparison with Carnot cycle and its adiabats and isotherms. The main drawback here is that the optimal performance is reached at the price of using a rather hot heat reservoir, with T h ∼ 170 °C. The heat sink changes its temperature during the cycle but remains in the proximity of room temperature, this is the reason of the ∆T res 100 °C reported in Table 3.

In the second group, with ∆T res below 50 °C, a part for the theoretical power density upper bound reported in [START_REF] Ujihara | Thermal energy harvesting device using ferromagnetic materials[END_REF] (i.e. the actual output has not been published), the highest powers are shown by the TMG presented in [START_REF] Waske | Energy harvesting near room temperature using a thermomagnetic generator with a pretzel-like magnetic flux topology[END_REF] and by the prototype presented here, both working on the same temperature interval of about 30 °C. Nonetheless it is worth noting that the device presented in [START_REF] Waske | Energy harvesting near room temperature using a thermomagnetic generator with a pretzel-like magnetic flux topology[END_REF] is not autonomous, needing a pump to actuate the heat-exchange fluid whose power consumption has not been reported, so that the net throughput, if any, is not known.

Summarizing, the thermo-magnetic harvester presented here show a state-ofthe-art performance and is, at the best of our knowledge, the sole autonomous Piezoelectric measured [START_REF] Ujihara | Thermal energy harvesting device using ferromagnetic materials[END_REF] 28.34 10 3 50

None estimated [START_REF] Chun | Thermo-magneto-electric generator arrays for active heat recovery system[END_REF] 38 10 3 80

PVDF measured [START_REF] Gueltig | High frequency thermal energy harvesting using magnetic shape memory films[END_REF] 1.6 100

Coils measured [START_REF] Gueltig | High-performance thermomagnetic generators based on heusler alloy films[END_REF] 118 10 3 100

Coils measured [START_REF] Joseph | Upscaling of Thermomagnetic Generators Based on Heusler Alloy Films[END_REF] 33 10 3 100

Coils measured

TMG prototype reported to show a net power output in the hundreds of micro-Watts per cubic centimeter, working in the vicinity of room temperature, and scavenging low grade heat over a range of few tens of degrees. Furthermore, as a recent review on the application of caloric materials to energy conversion [START_REF] Kitanovski | Energy applications of magnetocaloric materials[END_REF] puts it, the future of these technologies relies on frequency improvements that can hardly be achieved with devices needing a fluid to exchange heat with the reservoirs. In the design described here the MCM is the active material and the heat exchange substance at the same time, and the field source plays the two-fold role of triggering the caloric effect and switching the thermal management moving the material. In this context, the above presented study on the mechanical and thermal dynamics of the TMG, based on the phase portrait of an equivalent dynamical system, allows to grasp some key features of the device that would be otherwise difficult to appreciate.

On the one hand it does show how to take advantage of non-equilibrium fluxes, and further improvement towards a design recovering, through a mixed piezoelectric, and Faraday law transduction system, the energy associated with the damped oscillations observed in the x 1 ≥ 0.58 mm case are envisaged. This would allow to avoid the impact with one or both reservoirs reducing undesired aging of the material, and recovering part of the kinetic energy, otherwise dissipated, whence an improvement of the overall efficiency.

On the other hand, the study makes clear that, in order to reduce the size of the system, getting towards a MEMS scale harvester, working under reduced air density (low grade vacuum) is an option to be seriously considered, not only because of the increasing role of air viscosity when the MCM thickens scales down, but because of the aforementioned effect of the heat leakage through air on the exchange surfaces.

Both options represent worthwhile opportunities in the research of novel solid-state based energy conversion technologies.

Figure 1 :

 1 Figure 1: Schematic cross section of the TMG prototype.

Figure 2 :

 2 Figure 2: The beam (piezoelectric buzzer AB6505B from PUI Audio, Inc) and the MCM.

Figure 3 :

 3 Figure 3: Left: M (T, H) curves measured with a VSM magnetometer. Right: magnetic force isotherms computed with Ansys MAXWELL function of the air gap between the magnet and the MCM (i.e. the position of the MCM)

Figure 4 :

 4 Figure 4: Magnetic force for different temperatures indicated in the legend and the elastic force of the disc spring function of the position. The MCM is in contact with the heat source (red) when the MCM is in 0.1 mm and with the cold source (blue) when the MCM is in 0.48 mm. Points A and B are the cold and the hot switching points.

Figure 5 :

 5 Figure 5: Assembly of the disc spring. A) a single buzzer with the larger CuBe disk and the piezoelectric disk; B) sizes and shape of the final cross cut; C) final setup with two cross-shape cut buzzers assembled in series.

Figure 6 :

 6 Figure 6: Finite element simulations of Von Mises stress on the different parts of the buzzer (copper and piezoelectric) for a displacement of the MCM d = 1 mm.

  Position of the MCM (mm)

Figure 7 :

 7 Figure 7: The position of the MCM as a function of time with a heat source and heat sink temperatures of 51.5 °C and 16.4 °C respectively (i.e. ∆Tres = 35.1 °C). The heat sink position x 1 = 0.58 mm, the heat source is x 0 = 0.1 mm.

Figure 8 :

 8 Figure 8: The voltage produced by the upper piezoelectric as a function of time.

Figure 9 :Figure 10 :

 910 Figure 9: The displacement of the MCM (i.e. the x(t) integral curves) for different values of the distance d = x 1 -x 0 : x 1 = 0.58, 0.46, 0.32, 0.22, 0.17 mm, and x 0 = 0.1 mm.

Figure 11 :

 11 Figure 11: The FBL line, in black, defined by Eq.(2), divides the (x, T ) plane into two regions. One above the line where the elastic force is predominant. The other, below, where the magnetic force overwhelms the elastic one. The FBL line is divided into a stable (continuous black line), and an unstable (dotted black line) branch by the point C

Figure 12 :

 12 Figure 12: The force-balance line including the effect of piezoelectric tension on the elastic force. The path C C accounts of the observed slow displacement of the MCM towards the heat sink apparent in Fig.10.

Figure 13 :

 13 Figure13: (A) the HBL is represented as a green line joining point β to point α. Here we show the case where β lays below the FBL (below B) and α above the FBL (i.e. above A), so that both are fixed points of the dynamics (full green square point); (B) in this case the intersections between HBL and FBL with the reservoir surfaces are situated outside the phase-space subset accessible to the dynamics, on the contrary γ (full green square point), representing the intersection between the HBL and the FBL stable branch, is a fixed point.

Table 1 :

 1 Data derived from Fig.9.

	x 1 (mm) Frequency (Hz) Period (s) t h (s) t c (s)
	0.58	0.41	2.44	0.9	1.37
	0.46	0.27	3.6	1.51	2
	0.32	0.28	3.53	1.27	2.24
	0.22	0.28	3.55	0.86	2.68
	0.17	0.23	4.25	0.7	3.54

Table 2 :

 2 Data derived from Fig.10.

	x 1 (mm) Frequency (Hz) Period (s) t h (s) t c (s)
	0.58	0.41	2.44	0.9	1.37
	0.78	0.31	3.24	0.61	2.41
	1.58	0.27	3.68	0.59	3.07
	2.18	0.25	4.04	0.56	3.46

Table 3 :

 3 Comparison with different TMG prototypes

		Net power	∆T res
		(µW cm -3 )	(°C)	Transducers Comments
	[15]	320	30	Coils	non-autonomous
	This
	work	240	35.1	Piezoelectric measured
	[19]	6.8	38

Acknowledgment

This work benefited from the financial support of the project HiPerTherMag (ANR-18-CE05-0019) managed by the French National Research Agency. We would like to thank Dr. Alex Barcza from Vacuumschmelze for his kind support. Many thanks also to Eugene William from ENS Paris-Saclay and Nicolas Mercadié from IUT de Cachan for their priceless technical help.


PhD Cédrick T Bonnet
email: cedrick.bonnet@univ-lille.fr

MD Arnaud Delval

PhD Tarkeshwar Singh

MD Luc Defebvre

Parkinson's-disease-related changes in the behavioural synergy between eye movements and postural movements

Keywords: Parkinson's disease-related impairments, synergic model, eye and postural movements, automatic

come

INTRODUCTION

Parkinson's disease (PD) leads to major motor impairments, such as resting tremor, rigidity, hypokinesia, and postural instability [START_REF] Magrinelli | Pathophysiology of Motor Dysfunction in Parkinson's Disease as the Rationale for Drug Treatment and Rehabilitation[END_REF][START_REF] Yang | Parkinson's Disease and Cognitive Impairment[END_REF]. PD patients also have visual impairments (e.g. altered contrast sensitivity and double vision [START_REF] Ekker | Ocular and visual disorders in Parkinson's disease : Common but frequently overlooked[END_REF][START_REF] Weil | Visual dysfunction in Parkinson's disease[END_REF]) and cognitive impairments [START_REF] Yang | Parkinson's Disease and Cognitive Impairment[END_REF]. Here, we sought to determine how synergic interactions between vision and posture are affected in PD patients. Previously, [START_REF] Latash | Neural control of movement stability : Lessons from studies of neurological patients[END_REF] explained that analyses of impairments in synergy are relevant because they show more fundamental impairments than singular impairments such as postural instability. [START_REF] Latash | Neural control of movement stability : Lessons from studies of neurological patients[END_REF] explained that impaired synergy may be the cause of motor impairments, while postural instability may only be the result of these impairments. Recently, synergy analyses have shown that PD is associated with abnormally low indices of multimuscle synergy in postural control [START_REF] Falaki | Impaired synergic control of posture in Parkinson's patients without postural instability[END_REF], 2017b).Furthermore, the basal ganglia (affected in PD) are involved in the control of both eye movements and posture. More precisely, the caudate and putamen nuclei are interconnected structures that control saccadic eye movements and posture [START_REF] Hikosaka | Role of the Basal Ganglia in the Control of Purposive Saccadic Eye Movements[END_REF][START_REF] Visser | Role of the Basal Ganglia in Balance Control[END_REF]. Both nuclei are affected in PD [START_REF] Pasquini | Clinical implications of early caudate dysfunction in Parkinson's disease[END_REF][START_REF] Wang | Consistent decreased activity in the putamen in Parkinson's disease : A meta-analysis and an independent validation of resting-state fMRI[END_REF]. The neural circuits for saccades and postural control even overlap (see Figure 4 on p. 7 in [START_REF] French | A Review of the Pedunculopontine Nucleus in Parkinson's Disease[END_REF], which thus provided a mechanistic basis of PD-related impairments in the relationships between the two functions.

It is still not clear whether PD-related impairments affect the synergic interactions between vision and posture [START_REF] Devos | Driving and off-road impairments underlying failure on road testing in Parkinson's disease : Driving Skill Impairments in PD[END_REF][START_REF] Ewenczyk | Antisaccades in Parkinson disease : A new marker of postural control[END_REF]. Our research group has developed a quantitative model of how eye movements and limb movements synergically interact for success in ecological, precise, visual search tasks performed in the upright stance [START_REF] Bonnet | A functional synergistic model to explain postural control during precise visual tasks[END_REF]. We therefore used this model to test PD-related impairments in behavioural synergy between eye and postural movement in a precise visual search task. Precise tasks refer to tasks in which participants search for specific targets in a goal-directed fashion, whereas unprecise free-viewing tasks (used here as control tasks) involve the participant merely glancing at the environment in the absence of any goals. We expected to find that PD would impair the synergic interactions between eye movements and postural movements during a precise visual search task. The results presented below confirmed our expectations and suggested that PD patients may need to perform exercises combining eye and postural movements to recover goal-directed postural control. These combined exercises may benefit PD patients more than exercises involving eye movements alone or postural movements alone would.

MATERIALS AND METHODS

Participants

Twenty PD patients (12 males and 8 females) and 20 age-and sex-matched healthy older adults (12 males and 8 females) were included. The mean ± standard deviation (SD) age, body mass and height were respectively 60 ± 8 years, 78 ± 17 kg and 1.71 ± 0.06 m for the patients and 61 ± 7 years, 78 ± 13 kg and 1.71 ± 0.08 m for the age-matched healthy controls. The two groups did not differ on their physical characteristics (Fs(1,38)<0.29, p>0.05). To calculate of the sample size with sufficient statistical power, we used the results of our previous study of young adults [START_REF] Bonnet | Relations between Eye Movement, Postural Sway and Cognitive Involvement in Unprecise and Precise Visual Tasks[END_REF] and a bivariate normal model in G*power [START_REF] Faul | Statistical power analyses using G*Power 3.1 : Tests for correlation and regression analyses[END_REF]. Based on our previous study, the estimated effect size f was 0.624. With a two-tailed test and alpha risk of 0.05, a power of 0.8, and a phi correlation (H0) of 0, the required sample size was 9. We decided to recruit at least twice this number of patients (n=20) and then the same number of control participants as there were patients.

The PD patients were diagnosed by one of the two physicians in the investigating team (LD or AD). The diagnosis was confirmed immediately before the experimental session. The PD patients had a mean ± SD time since disease onset of 5 ± 2 years (range: 2-11). Their mean motor Movement Disorder Society Unified Parkinson's Disease Rating Score (MDS-UPDRS) motor score on-medication (part III; [START_REF] Hughes | Accuracy of clinical diagnosis of idiopathic Parkinson's disease : A clinico-pathological study of 100 cases[END_REF]) was 21 ± 8 (range: 10-39). The patients' mean Hoehn and Yahr stage was 2.0 ± 0.5 (range: 1-3) and the mean axial score out of 24 (computed by summing the scores for MDS-UPDRS III items 18 (speech), 22 (axial rigidity), 27 (rising from a chair), 28 (posture), 29 (postural stability) and 30 (gait); [START_REF] Bejjani | Axial parkinsonian symptoms can be improved : The role of levodopa and bilateral subthalamic stimulation[END_REF]) was 3.3±2.4 (range: 0-11).

Participants were excluded if they had a history or signs of vestibular, musculoskeletal or neurological disorders or diseases (other than PD in the patient group), recurrent dizziness, dementia (according to Diagnostic and Statistical Manual of Mental Disorders, 4 th Edition), cognitive decline (a Montreal Cognitive Assessment (MoCA) score below 27 [START_REF] Nasreddine | The Montreal Cognitive Assessment, MoCA : A brief screening tool for mild cognitive impairment[END_REF]), severe motor fluctuations, or dyskinesia. Participants were also excluded if they had fallen in the previous six months (based on their reply to a simple question) and if they were taking a medication known to affect postural control. The patients were on their usual dopaminergic medication (i.e. "on-medication") during this study. The mean total daily levodopa equivalent dose was 659±339 mg. The participants were only included if they had normal or corrected-to-normal vision, i.e. if they could clearly see images. The study was approved by the local independent ethics committee (reference: 2014-74). All the participants gave their written, informed consent to participation.

Apparatus

Three video projectors (Optoma HD83, London, England) were used to project the images onto a full semi-circular panoramic display (radius: 2.04 m; height: 2 m; Figures 1A and1C). The participants could see images projected onto the panoramic display with a maximum visual angle of 100° for left-right and 20° for up-down (Figure 1A and1C). The participants stood with their feet placed in a standardized position (14 cm, 17°;[START_REF] Mcilroy | Preferred placement of the feet during quiet stance : Development of a standardized foot placement for balance testing[END_REF]. Head, upper back and lower back movements were measured with a Polhemus magnetic tracking system (Polhemus Liberty 240/8-8 System, Colchester, VT) at 240 Hz. The markers were placed on (i) a helmet worn by the participant (the head marker), (ii) the upper back over the seventh cervical vertebra (the upper back marker), and (iii) a chest belt over the fifth lumbar vertebra (the lower back marker) (Figure 1C). Eye movements were recorded with iViewX software and eye tracking goggles (SensoMotoric Instruments, Teltow, Germany) attached to the helmet worn by the participants (Figure 1C). The iViewX system recorded the pupil position at a frequency of 50 Hz. Begaze software (SensoMotoric Instruments, Teltow, Germany) was used to characterize fixations in each trial. It defined a fixation as less than 13 pixels of eye movement for 80 ms. A MATLAB (MathWorks, Natick, MA, USA) script was used to project the image onto the wall and to synchronize the data from the magnetic tracking system, and the eye tracker.

The 12 images displayed to the participants were all images of rooms inside a house (the kitchen, the living room, etc.; Figure 1B). Six images were shown in the precise task, and six other images were shown in the free-viewing control task (Figure 1B). The order of the images shown in the tasks was counterbalanced so that all images were projected the same number of times to all participants. The order in which the six images per task were shown was randomized across participants. The order in which the two tasks were performed was also randomized. The images had been used in previously reported study [START_REF] Thibaut | Impact of age-related macular degeneration on object searches in realistic panoramic scenes[END_REF].

Tasks

The study was performed with two tasks (a search task and a free-viewing task) and six trials per task. Each trial lasted for 50 seconds. In each trial, the participants had to stare at a black cross (2°) projected in front of them for the first five seconds. The black cross then disappeared, and the participants were free to look wherever they wished in the image.

In the free-viewing task, the participants were instructed to look freely at the image and focus on its content. In the search task, the participants were instructed to detect the location of five objects in each image. The names of these five objects were displayed at the top of the image throughout the trial (Figure 1B, top panel). Before the tasks, the participants were shown a document with the names and pictures of the objects to be searched for. During each trial in the search task, the participants knew that they had to find as many target objects as possible. They could use the strategy of their choice, e.g. searching for one target at a time or search for them all at once. Once a target object had been found, the participant had to confirm the successful outcome by fixing the object for approximately 5 sec. They could look at the names of the five objects as many times as they liked. After each trial, the participants had to report how many objects they had found and to rate their level of confidence concerning their task performance (from 1 (low performance), to 5 (high performance)). If the participant found all five objects before the end of the trial, they had to continue looking for the same five objects again and again until the trial ended.

During each task, the participants were told to relax and stand in a comfortable position. They were allowed to move their head and other body parts to look at the images in the most comfortable way. However, they were not allowed to make voluntarily movements not related to the task (e.g. hand movements or deep breaths). The arms had to stay in the same position during each trial (in their trouser pockets, for example). The investigator checked that the participants performed the task as per the instructions.

Procedure

Once the participants arrived in the experimental room, they were familiarized with the equipment, the procedure, and the visual tasks. Next, the participants were set up with the markers and eye-tracker. The devices were then calibrated with the MATLAB script, and the participants performed the tasks. All the participants sat down and rested for approximately five minutes between tasks. The participants were allowed to stop the study at any time and for any reason.

Dependent variables

To analyse linear postural movements (lower back, upper back, and head), the range (R), SD, and mean velocity (V) in the mediolateral (ML) and anteroposterior (AP) axes were calculated. Two additional postural control variables were calculated: the general path length of postural movements and the ellipse area [START_REF] Paillard | Techniques and Methods for Testing the Postural Function in Healthy and Pathological Subjects[END_REF]. The ellipse area captures 85% of the postural sway [START_REF] Kinsella-Shaw | Effects of Visual Environment on Quiet Standing by Young and Old Adults[END_REF][START_REF] Latash | Movement sway : Changes in postural sway during voluntary shifts of the center of pressure[END_REF]. To analyse the eye movement time series, we used the same variables: R, SD, and V in both left/right and up/down directions, the path length, and the ellipse area. To analyse the characteristics of fixation, all saccades were excluded before calculating the SD and R. The latter analysis allowed us to determine whether the participants looked more carefully at objects located further away in one task than in another task.

In the search task, several performance variables were evaluated: the number of correct objects found, the percentage of incorrect objects reported, the time spent looking at the correct objects found, the eccentricity and size of correct objects found, and the self-reported confidence score. The time spent looking at the correct objects was computed with a MATLAB script by considering the characteristics of fixation (defined with the Begaze software). The head, upper back and lower back time series were all resampled at 50 Hz, i.e. the frequency of the eye movement time series. Only the last 45 seconds of data from each trial were analyzed. The five-second-long fixations on each object found in the search task could have confounded our results. To control for this, we deleted the longest fixations corresponding to the number of objects found in the eye time series.

Differences between our model of synergy and other models of synergy

The behavioural synergy model is concerned with how postural sway and eye movements are mutually controlled for success in precise visual search tasks performed in the upright stance [START_REF] Bonnet | A functional synergistic model to explain postural control during precise visual tasks[END_REF]. It is important to note that our behavioural synergic model does not encompass the coordination between the eyes and the rest of the body included in other synergy models(e.g., [START_REF] Anastasopoulos | Altered eye-to-foot coordination in standing parkinsonian patients during large gaze and whole-body reorientations[END_REF]. Furthermore, the behavioural synergy model does not encompass any type of muscle activity or any reflex (such as the vestibulo-ocular reflex or compensatory eye and head movements) [START_REF] Einhäuser | Human eye-head co-ordination in natural exploration[END_REF]. The model covers relationships between various behaviours (eye movements and body posture) and has already been validated in healthy young adults (Bonnet, Davin, & Baudry, 2019;[START_REF] Bonnet | Relations between Eye Movement, Postural Sway and Cognitive Involvement in Unprecise and Precise Visual Tasks[END_REF][START_REF] Bonnet | Functional Synergy Between Postural and Visual Behaviors When Performing a Difficult Precise Visual Task in Upright Stance[END_REF].

Tests to identify stabilizing relationships between eye and postural movements

Outliers (more than 2 SD outside the quartiles) were identified and deleted [START_REF] Tabachnik | Using Multivariate Statistics[END_REF]. We also tested the normality of the data distribution (using a Shapiro-Wilk test) and the homogeneity of the variance (using the Mauchly test) before our analyses.

To test the behavioural synergy model, we calculated the correlations between the characteristics of eye movements (angular variables) and postural movements (linear variables). We tested our main hypothesis by selecting results from the correlations matrices for the characteristics of eye movements and postural movements performed in each task and each group separately. In our previous studies of young adults (Bonnet, Davin, & Baudry, 2019;[START_REF] Bonnet | Relations between Eye Movement, Postural Sway and Cognitive Involvement in Unprecise and Precise Visual Tasks[END_REF]Bonnet, Szaffarczyk, et al., 2017), eye and head movements had the strongest functional synergies. We therefore considered correlations between eye movements and head movements as our primary outcome.

For the PD patients, we also determined whether significant correlations between eye movements and postural movements could be influenced by one or more clinical variables. We performed partial correlations by controlling for the influence of five clinical variables (the MoCA score, the MDS-UPDRS motor score, the mean time since disease onset, the mean daily total levodopa equivalent dose, and the axial rigidity score). We focused on nonsignificant partial correlations, which might indicate that a clinical variable might weaken (and therefore be a potential cause of) significant correlations between eye movements and postural movements.

All analyses were performed with Statistica software (version 10, Statsoft Inc., Tulsa, OK, USA). The threshold for statistical significance was set to p<0.01.

Selection of the data to be analyzed

The SensoMotoric Instruments eye tracker recorded "0-values" (i.e. missing values) when participants' eyes were closed (e.g. during blinking) and when the pupil was greatly dilated. The latter issue was mainly due to the fact that the room lighting was turned off so that the participants could see the images more clearly. Eye movement data files in which there were more than 20% of 0-values (missing values because of blinking or pupil dilation) were excluded, and only high-quality recordings were analyzed. The remaining visual files contained 93.40 ± 5.66% of the data, on average (PD patients: 94.04 ± 5.62%; age-matched participants: 94.27 ± 5.36%). For eye movement data, outliers accounted for respectively 2.04% and 0.75% of the data per spreadsheet for the PD patients and the age-matched participants, respectively. For postural movement data, outliers accounted for respectively 0.72% and 0.75% of the data per spreadsheet for the PD patients and the age-matched participants, respectively.

RESULTS

Coupling between eye movements and postural movements in PD patients and in agematched controls

In the search task, the PD patients had significant, positive Pearson correlations between eye and postural movements in the search task (Figure 2A, B, C, D, E andF). There were three significant positive correlations between the eyes and the upper back and three other significant positive correlations between the eyes and the lower back (Figure 2A, B, C, D, E andF). In contrast, the age-matched controls did not exhibit any significant correlations between eye movements and postural movements in the search task. In the free-viewing control task, no significant correlations were found in either group, ns. When controlling for the influence of the four clinical variables (see the Methods), the six above-mentioned correlations remained statistically significant. Hence, the clinical variables had no influence on the significant coupling shown in Figures 2A, B, C, D, E andF.

Additional analyses Description of eye and head angular yaw movements when performing the tasks

We analysed the angular rotations of the eyes and head (using the R and SD in the yaw (left-right) directions) in the free-viewing and search tasks. This descriptive analysis merely served to provide information on how much the participants rotated their eyes and head when exploring large images in visual tasks. Overall, the participants rotated their eye and body segments by between 90° and 110° (Table 1). Table 1. Angular movements (rotations) of eye and head movements in the left-right (eye) and yaw (head) directions.

In Panel B shows a significant correlation between Vup-down for eye movement and VAP for the lower back movement. Panel C shows a significant correlation between the mean velocity of eye movement in the left-right direction (Vleft-right) and of the upper back movement in the mediolateral axis (RML, in cm). Panel D shows a significant correlation between Vleft-right for eye movement and RML for the lower back movement. Panel E shows a significant correlation between the path length of eye movement and the path length of upper back movement. Panel F shows a significant correlation between the path length of eye movement and the path length of lower back movement. All correlations were significant at p < 0.01.

Performance in the search task

In the search task, the PD patients performed worse than the age-matched controls with regard to several variables (Table 2). In fact, the PD patients found fewer objects than the agematched controls (Table 2). The PD patients also made more mistakes than the age-matched controls about the objects found and exaggerated the number of objects found but these differences were not tested statistically. Despite the patients' worse performance, their confidence scores were similar to those of the age-matched controls (Table 2). Note. The proportion of inaccurate objects found was calculated as follows: total number of inaccurate objects found / total number of objects searched. The proportion of exaggeration was calculated as follows: number of objects declared to be found -number of objects really found)/total number of objects found. The eccentricity of the objects found corresponded to the proportion of the total number of objects in each of the three categories. The object was centered, middly centered or excentred when the closer limit of the object from the center of the panoramic display was < 20°, [20°-40°] and [40°-60°] (on the left or right of the panoramic display), respectively. The size of the objects found corresponded to the proportion of the total number of objects in each of the three categories. The object was smaller, mid-size or bigger when their size was < 5°, [5°-10°] and > 10°, respectively.

DISCUSSION

In the visual search task, the PD patients exhibited only positive (destabilizing) correlations between eye movements and upper/lower back the agematched controls did not exhibit any significant (i.e. stabilizing or destabilizing) correlations. Hence, the more the PD patients rotated their eyes to look at their environment, the more they swayed. The age-matched controls were better in controlling their posture because they did not exhibit any destabilizing relationships for eye movements vs. body movements. In general, the age-matched controls had a higher level of visual performance than the PD patients in the search task.

Positive correlations between eye movements and postural movements in the PD patients

In the visual search task, the PD patients showed positive correlations between eye movements and both upper body and lower body movements (see Figure 2). Hence, all the observed relationships between eye and postural movements were destabilizing. These results are not in line with those of other studies of PD-related impairments in the synergic control of the upright stance [START_REF] Falaki | Impaired synergic control of posture in Parkinson's patients without postural instability[END_REF](Falaki et al., , 2017b)). Falaki et al. studied multimuscle synergies controlling centre of pressure movements and evidenced reduced synergy for the maintenance of postural control [START_REF] Falaki | Impaired synergic control of posture in Parkinson's patients without postural instability[END_REF](Falaki et al., , 2017b) ) and for anticipatory synergy adjustments [START_REF] Falaki | Impaired synergic control of posture in Parkinson's patients without postural instability[END_REF]. These studies specifically focused on postural control. Our present results complement these literature reports by showing that PD impairs higher-order relationships between eye movements and postural movements and not only multimuscle synergies for postural control.

We did not observe any significant relationships between clinical variables and impaired synergy in PD patients (see the Results section). This result was expected, as all the patients had early-stage disease. Previous studies have already shown the absence of a relationship between clinical variables and postural variables in PD patients at Hoehn and Yahr stages I and II [START_REF] Bonnet | Parkinson's Disease-Related Impairments in Body Movement, Coordination and Postural Control Mechanisms When Performing 80° Lateral Gaze Shifts[END_REF][START_REF] Chastan | Discordance between measured postural instability and absence of clinical symptoms in Parkinson's disease patients in the early stages of the disease[END_REF]. This finding is consistent with Latash and Huang (2015)'s suggestion that impairments in synergy are as important as clinical impairments. The evaluation of impaired synergy (i.e. combined impairments in eye movements and postural movements) complements the evaluation of clinical impairments.

The existence of destabilizing relationships between eye movements and upper/lower body movements in PD patients might be due to impairments in the basal ganglia which notably. have an important role in the control of both eye and postural movements [START_REF] Hikosaka | Role of the Basal Ganglia in the Control of Purposive Saccadic Eye Movements[END_REF][START_REF] Visser | Role of the Basal Ganglia in Balance Control[END_REF]. When the basal ganglia are damaged, the affected individuals perform more irrelevant, reflexive saccades, which limits their ability to gather relevant, goal-directed information [START_REF] Hunt | Do people with Parkinson's disease look at task relevant stimuli when walking? An exploration of eye movements[END_REF]. Patients are limited in their acquisition and integration of visual information from the environment [START_REF] Hunt | Do people with Parkinson's disease look at task relevant stimuli when walking? An exploration of eye movements[END_REF]. It has been proposed that the basal ganglia have a role in the organization of muscle synergies in both humans and other animals [START_REF] Mileti | Muscle Synergies in Parkinson's Disease[END_REF]. Impairments in the basal ganglia, particularly in the caudate nucleus, substantia nigra pars reticulata and putamen [START_REF] Pasquini | Clinical implications of early caudate dysfunction in Parkinson's disease[END_REF][START_REF] Wang | Consistent decreased activity in the putamen in Parkinson's disease : A meta-analysis and an independent validation of resting-state fMRI[END_REF] might explain the observed PD-related limitations in adaptation of the magnitude of saccadic eye movement and postural control during a precise visual search task. These structures are also anatomically and functionally interconnected [START_REF] French | A Review of the Pedunculopontine Nucleus in Parkinson's Disease[END_REF]. The existence of destabilizing relationships between eye and upper/lower body movements in PD patients might be due to impairments in the pedunculopontine nucleus [START_REF] Ewenczyk | Antisaccades in Parkinson disease : A new marker of postural control[END_REF]; see also [START_REF] Gallea | Pedunculopontine network dysfunction in Parkinson's disease with postural control and sleep disorders[END_REF]. Future studies will need to establish which structures are correlated with PD-related impairments in synergic eye and postural movements.

Significant correlations between eye movements and postural movements are absent in older adults.

On one hand, our results showed that the age-matched controls did not show any destabilizing relationships at any level in the search task; hence, the controls were more capable than the PD patients to control their stance. On the other hand, the age-matched controls did not show any statistically significant stabilizing relationships between eye and postural movements in the search task (see Results section). The latter result contrasts with our previous findings in young adults (Bonnet, Davin, & Baudry, 2019;[START_REF] Bonnet | Relations between Eye Movement, Postural Sway and Cognitive Involvement in Unprecise and Precise Visual Tasks[END_REF]Bonnet, Szaffarczyk, et al., 2017). these three earlier studies, young adults showed significant stabilizing correlations between eye movements and postural movements in general and between eye and head movements in particular.

The literature data have evidenced an age-related reduction in functional synergies for postural control and/or motor action [START_REF] Asaka | Effects of Aging on Feedforward Postural Synergies[END_REF][START_REF] Olafsdottir | Elderly Show Decreased Adjustments of Motor Synergies in Preparation to Action[END_REF]. In our study, we did not find an age-related reduction in synergic eyeposture movements but we found an absence of such synergies. Thus, the absence of stabilizing (i.e. synergic) eye-postural movements in older adults was not expected. This result clearly shows that synergic relationships are fragile and can deteriorate with age. In a review, [START_REF] Seidler | Motor control and aging : Links to age-related brain structural, functional, and biochemical effects[END_REF] suggested that basal ganglia function is affected by age. Our present results in a population of age-matched older adults might testify to the presence of a neurological impairment that is only visible at the integrated level. Therefore, an analysis of behavioural synergy might be a powerful tool for detecting age-related impairments in eye and postural movements.

Results in the free-viewing task

In the free-viewing (control) visual task, the absence of significant relationships in the two groups was expected because this task did not require any synergic behavioural relationships between eye and postural movements [START_REF] Bonnet | A functional synergistic model to explain postural control during precise visual tasks[END_REF]. Our results show that precise, cognitively demanding search tasks can highlight subtle, PD-related impairments in the synergy between eye and postural movements.

Study limitations

Firstly, we did not control for (and could not determine) the respective contributions of goal-directed mechanisms and reflex mechanisms (such as the vestibulo-ocular reflex). The mechanisms' respective contributions are difficult to investigate because PD-related impairments can come from not only a combination of changes in oculomotor, postural and even vestibular systems (we believe) but also from a particular impairment (e.g. in the vestibular system). It is noteworthy that we observed PD-related impairments in the synergy between eye and upper/lower body movements but not in the synergy between eye and head movements. Hence, our findings might not be due to PD-related impairments in the vestibular system. Secondly, the medication taken by our patients might have influenced our results. This was not really an issue, since the relationships between eye movements and postural movements were already weaker in on-medication PD patients than in age-matched controls.

Hence, levodopa medication either does not change muscle synergies [START_REF] Mileti | Muscle Synergies in Parkinson's Disease[END_REF][START_REF] Rodrigues | Effects of saccadic eye movements on postural control stabilization[END_REF] or improves them (Falaki, Huang, Lewis, & Latash, 2017a;[START_REF] Latash | Neural control of movement stability : Lessons from studies of neurological patients[END_REF]. Therefore, we would have expected to see even stronger destabilizing relationships between eye movements and postural movements if the PD group had been offmedication. Future studies should focus on the contribution of medication. Thirdly, we did not find any significant stabilizing or destabilizing correlations between eye and postural movements in age-matched controls. We assume that this result was not biased because we found (i) significant destabilizing relationships in PD patients (see Figure 2) and (ii) significant stabilizing relationships in all previous studies with no more than 16 young adults (Bonnet, Davin, & Baudry, 2019;[START_REF] Bonnet | Relations between Eye Movement, Postural Sway and Cognitive Involvement in Unprecise and Precise Visual Tasks[END_REF]Bonnet, Szaffarczyk, et al., 2017).

Conclusion and future directions

We chose to perform ecological search and free-viewing tasks (rather than other visual tasks, such as gazing at specific targets displayed at a large visual angle and at high frequency) because people are more often used to looking freely at rich visual backgrounds in day-to-day situations. In this ecological context, our results are compelling insofar as they showed impairments in patients with mild PD (Hoehn and Yahr stage II, on average). The destabilizing relationships between eye movement and upper/lower body movements may contribute to the falls reported in PD patients, especially those with later-stage disease (Hoehn and Yahr stage > II). Here, we showed that the further the PD patients explored their environment, the more they swayed. We also know that the more PD patients sway, the more often they fall [START_REF] Contreras | Risk of Falls in Parkinson's Disease : A Cross-Sectional Study of 160 Patients[END_REF][START_REF] Duncan | Accuracy of Fall Prediction in Parkinson Disease : Six-Month and 12-Month Prospective Analyses[END_REF]. In conclusion, our first recommendation for PD patients is to avoid quick gaze shifts and especially high-amplitude gaze shifts involving rotation of the head and full body. This will certainly be difficult to comply with this recommendation in daily life. However, it is better to inform PD patients of the negative consequences of large, quick gaze shifts on their objective postural stability than to not inform them. A second recommendation (for physical therapists, this time) is to train PD patients to perform exercises that involve a combination of eye and postural movements, rather than exercises that involve eye movements alone or postural movements alone. These combined exercises might be more effective than separate exercises in enabling PD patients to recover goal-directed postural control.

ACKNOWLEDGMENT

We wish to thank Dr David Fraser (Biotech Communication, Damery, France) for his helpful advice on the English language in this paper. We thank Tanguy Davin (assistant engineer) for his help with programming MATLAB scripts to run the study, to automatize procedures both to run the study and to analyze the data. He also helped in writing the Method section. We thank France Parkinson for the grant that allowed us to perform this study (convention n° RD 1603). We also thank the 'Conseil Regional des Hauts-de-France' for their visionnAIRR grant (convention n°16000913). GRAPHICAL ABSTRACT. Our results showed that in the precise visual search task i) the PD patients showed significant positive linear correlations between eye movement and upper back/lower back movements (discussed as destabilizing relations); ii) the age-matched controls showed no significant relations in all our analyses

LIST OF ABBREVIATIONS

Figure 1 .

 1 Figure 1. A. The position of the participants with respect to the semi-circular panoramic display (radius: 2.04 m; height: 2 m) onto which the 12 images were projected. The participants stood 1.71 m behind the centre of the semi-circular panoramic display and therefore could see the images subtended by a visual angle of 100° at most. B. Two images shown during the study. The top image was shown in the precise search task because the participants had a list of objects to search for. This list was shown at the top of the image. The bottom image was shown in the free-viewing (control) task in which there were no objects to search for. The participants were told to look wherever they liked in the image. C. The experimental setup, with a participant standing on the force platform, wearing the Polhemus system's three markers and with the eye tracker fixed to the helmet.

Figure 2 .

 2 Figure 2. Significant correlations (according to Pearson's coefficient) between eye movements and body (lower back, upper back and/or head) movements in the precise search task in PD patients). Panel A shows a significant correlation between the mean velocity of eye movement in the up-down direction (Vup-down; in pixels×s -1 ) and the mean velocity of the upper back movement in the anteroposterior (AP) axis (VAP of the upper back, in cm.s -1 ).

  AP = anteroposterior; ML = medio-lateral; MoCA = Montreal Cognitive Assessment score; PD = Parkinson Disease; R = range; SD = standard deviation; UPDRS = United Parkinson Disease Rating Scale; V = mean velocity.

Table 2 .

 2 Visual performances in the precise search task.

		Patients with	Age-matched	One-way ANOVA
		Parkinson's disease	controls
	Mean number of objects	3.30±0.67	4.16±0.45	F(1,38)=22.27, p<0.01
	found
	Confident score	4.93±0.19	4.96±0.07	F(1,38)=0.71, p>0.01
	Proportion of inaccurate	3.13 %	0.78 %
	objects found
	Proportion of	11.25 %	4.51 %
	exaggeration
	Mean duration spent to	4.47±0.93 sec	5.04±0.70 sec	F(1,38)=4.8, p>0.01
	look at each object found
	Eccentricity of the objects	Centered: 62±22 %	Centered: 87±18 %
	found	Middly centered:	Middly centered:
		68±22 %	79±17 %
		Excentred: 68±20 %	Excentred: 84±19 %
	Size of the objects found	Smaller: 58±16 %	Smaller: 82±16 %
		Mid-size: 67±23 %	Mid-size: 80±20 %
		Bigger: 74±22 %	Bigger: 89±13 %

CONFLICT OF INTEREST

In the present manuscript, there is no conflict.

PUBLIC ACCESS REPOSITORY

Data of the results are available on a public access repository (Figshare) and can be found at: https://figshare.com/articles/dataset/Parkinson-posture/14269973

CONTRIBUTION OF THE AUTHORS

• Cédrick T. Bonnet: all parts of the work: conception, acquisition of all participants, analysis, interpretation, draft of the full manuscript, revision, collaboration, integrity of all parts of the manuscript, final approval. • Arnaud Delval: conception, recruitment and inclusion (medical assessment) of the PD patients, acquisition, analysis, interpretation, revision of the full manuscript for important intellectual content, collaboration, integrity of all parts of the manuscript, agreement to be accountable for all aspects of the manuscript, final approval. • Tarkeshwar Singh: analysis and interpretation of data for the manuscript, revision of the full manuscript for important intellectual content, collaboration, integrity of all parts of the manuscript, agreement to be accountable for all aspects of the manuscript, final approval. • Luc Defebvre: principal investigator of the study in the hospital, conception, recruitment and inclusion (medical assessment) of the PD patients, acquisition, analysis, interpretation, revision of the full manuscript for important intellectual content, collaboration, integrity of all parts of the manuscript, agreement to be accountable for all aspects of the manuscript, final approval.