Marius Hervé

Gaël Privault

Elzbieta Trzop

Shintaro Akagi

Yves Watier

Serhane Zerdane

Ievgeniia F Chaban

Ricardo Guillermo Torres Ramírez

Celine Mariette

Alix Volte

Marco Cammarata

Matteo Levantino

Hiroko Tokoro
email: tokoro@ims.tsukuba.ac.jp

Shin-Ichi Ohkoshi
email: ohkoshi@chem.s.u-tokyo.ac.jp

Eric Collet
email: eric.collet@univ-rennes.fr

Affiliations

Ultrafast and persistent photoinduced phase transition at room temperature monitored by streaming powder diffraction

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Introduction

The ability to control physical properties of materials on demand is an important challenge in science and technology. The use of light as external stimuli, spanning from UV to THz range, is very promising for contact-less, ultrafast and selective control. [START_REF] Basov | Towards properties on demand in quantum materials[END_REF][2][START_REF] Johnson | Memorial Lecture: From optical to THz control of materials[END_REF][START_REF] Koshihara | Challenges for developing photo-induced phase transition (PIPT) systems: From classical (incoherent) to quantum (coherent) control of PIPT dynamics[END_REF][START_REF] Kirschner | reversible phase transitions in all-inorganic perovskite nanocrystals[END_REF][START_REF] Mariette | Strain wave pathway to semiconductor-to-metal transition revealed by time-resolved X-ray powder diffraction[END_REF] Ultrashort laser pulses, containing a number of photons of the order of the number of active units within materials, open the way for ultrafast and cooperative switching of their physical properties through photoinduced polarons or lattice instabilities, [START_REF] Yue | Real-time observation of the buildup of polaron in alpha-FAPbI(3)[END_REF][START_REF] Smit | Ultrafast Pathways of the Photoinduced Insulator-Metal Transition in a Low-Dimensional Organic Conductor[END_REF][START_REF] Liu | Photoinduced multistage phase transitions in Ta(2)NiSe(5)[END_REF] with various applications for photonic devices or light-based technologies. For optically-driven photonic devices, memories, or actuators, the photo-response must combine important characteristics such as room temperature switching, wide thermal regime of bistability, photoinduced states that can persist long after stimuli, single laser shot with threshold switching and ultrafast dynamics. [START_REF] Bonhommeau | One shot laser pulse induced reversible spin transition in the spincrossover complex [Fe(C4H4N2){Pt(CN)4}] at room temperature[END_REF][START_REF] Teitelbaum | Real-Time Observation of a Coherent Lattice Transformation into a High-Symmetry Phase[END_REF] Molecular materials undergoing photoinduced phase transitions (PIPT) show various types of functionalities switchable under photoexcitation, such as conductivity, ferroelectricity, colour... [START_REF] Gao | Mapping molecular motions leading to charge delocalization with ultrabright electrons[END_REF][START_REF] Collet | Laser-induced ferroelectric structural order in an organic charge-transfer crystal[END_REF][START_REF] Ishikawa | Large and ultrafast photoinduced reflectivity change in the charge separated phase ofEt2Me2Sb[Pd(1,3-dithiol-2-thione-4,5-dithiolate)2[END_REF][START_REF] Halcrow | Spin-crossover materials : properties and applications[END_REF][START_REF] Collet | Aperiodic spin state ordering of bistable molecules and its photoinduced erasing[END_REF][START_REF] Ridier | Finite Size Effects on the Switching Dynamics of Spin-Crossover Thin Films Photoexcited by a Femtosecond Laser Pulse[END_REF][START_REF] Hoffmann | Persistent photoinduced effects in high-Te superconductors[END_REF][START_REF] Schaniel | Photoinduced Nitrosyl Linkage Isomers Uncover a Variety of Unconventional Photorefractive Media[END_REF][START_REF] Xue | Photon-Induced Reversible Phase Transition in CsPbBr3 Perovskite[END_REF] Among molecular materials, cyanide-bridged bimetallic assemblies are very attractive compounds exhibiting inter-metallic charge-transfer (CT), which allows for multifunctional properties that can be reversibly controlled by light, including magnetism, ionic conduction or optical properties such as rotation of light polarisation. [START_REF] Verdaguer | Molecules to build solids: high TC molecule-based magnets by design and recent revival of cyano complexes chemistry[END_REF][START_REF] Liu | Reversible valence tautomerism induced by a single-shot laser pulse in a cobalt-iron Prussian blue analog[END_REF][START_REF] Tokoro | Multifunctional Material: Bistable Metal-Cyanide Polymer of Rubidium Manganese Hexacyanoferrate[END_REF][START_REF] Zhang | Charge-Transfer Phase Transition of a Cyanide-Bridged Fe(II) /Fe(III) Coordination Polymer[END_REF][START_REF] Sato | Photoinduced Magnetization of a Cobalt-Iron Cyanide[END_REF][START_REF] Volatron | Photo-induced magnetic bistability in a controlled assembly of anisotropic coordination nanoparticles[END_REF][27][START_REF] Ohkoshi | 90-degree optical switching of output second-harmonic light in chiral photomagnet[END_REF][START_REF] Ohkoshi | Light-induced spincrossover magnet[END_REF][START_REF] Tokoro | Visible-Light-Induced Reversible Photomagnetism in Rubidium Manganese Hexacyanoferrate[END_REF][START_REF] Aguila | Switchable Fe/Co Prussian blue networks and molecular analogues[END_REF][START_REF] Cammarata | Charge transfer driven by ultrafast spin transition in a CoFe Prussian blue analogue[END_REF][START_REF] Azzolina | Out-of-equilibrium lattice response to photo-induced charge-transfer in a MnFe Prussian blue analogue[END_REF][START_REF] Azzolina | Exploring Ultrafast Photoswitching Pathways in RbMnFe Prussian Blue Analogue[END_REF][START_REF] Barlow | Ultrafast photoinduced dynamics in Prussian blue analogues[END_REF][START_REF] Johansson | Directly probing spin dynamics in a molecular magnet with femtosecond time-resolution[END_REF][START_REF] Ohkoshi | Photomagnetism in Cyano-Bridged Bimetal Assemblies[END_REF] Here we present a material of this class, Rb0.94Mn0.94Co0.06[Fe(CN)6]0.98•0.2H2O (RbMn0.94Co0.06Fe), which exhibits a wide thermal hysteresis centred at room temperature (Supplementary Methods 1) resulting from coupled CT and symmetry-breaking (SB). We used a streaming powder diffraction technique developed to study its ultrafast and permanent PIPT, induced by a single laser shot at room temperature. The time-resolved X-ray diffraction data show that at low fluence the system remains in a tetragonal and expanded lattice distorted by CT polarons, which are responsible for reverse Jahn-Teller relaxation. Above threshold excitation, the lattice expansion due to photoinduced polarons switches the material from its low temperature Mn III Fe II tetragonal ground state to the persistent Mn II Fe III cubic phase, on the 100 ps timescale. The multiscale nature of the process underlines the key role of elastic interactions in the threshold and cooperative phase transition.

Results

Room-temperature charge-transfer bistability

The synthesized RbMn0.94Co0.06Fe material exhibits a CT-based phase transition similar to the one of the pure RbMnFe(CN)6 system and derivatives, as characterized by magnetic, X-ray diffraction and infra-red measurements (Supplementary Methods 1). [START_REF] Azzolina | Landau theory for non-symmetry-breaking electronic instability coupled to symmetry-breaking order parameter applied to Prussian blue analog[END_REF] Fig. 1 shows the thermal hysteresis associated with the Mn II Fe III  Mn III Fe II bistability, monitored through the thermal dependence of the MT product (molar magnetic susceptibility M and temperature T). The MT value at low temperature (LT) is characteristic of the Mn III (S=2)Fe II (S=0) state. Upon warming it increases around 𝑇 ↑ = 328 K to reach a value characteristic of the Mn II (S=5/2)Fe III (S=1/2) high temperature (HT) state. Upon cooling the MT value suddenly drops back to LT value around 𝑇 ↓ = 253 K, resulting in a 75 K wide hysteresis. Compared to the pure RbMnFe(CN)6 material, for which the thermal hysteresis is centred at 267 K (𝑇 ↓ = 231 K and 𝑇 ↑ = 304 K), [START_REF] Azzolina | Landau theory for non-symmetry-breaking electronic instability coupled to symmetry-breaking order parameter applied to Prussian blue analog[END_REF] Co doping allows chemical tuning of the bistability regime, centred at room temperature (290.5 K) for RbMn0.94Co0.06Fe. The infrared spectra in the HT and LT phases are characteristic of the Mn III Fe II and Mn II Fe III states. Powder X-ray diffraction data show important changes of lattice parameters between the HT cubic space group F4 ̅ 3m (ac = 10.5495(6) Å) and the LT tetragonal space group F4 ̅ 2m (at = 10.0051(11) Å, ct = 10.4744(16) Å). This volume contraction between the cubic HT (𝑉 HT ) and tetragonal LT (𝑉 LT ) phases, corresponds to a large volume strain: 𝑣 s = 𝑉 LT -𝑉 HT 𝑉 HT ≃ -0.1. The use of the non-conventional F4 ̅ 2m LT space group allows for direct description of the symmetry breaking between the two phases, which is due to collective Jahn-Teller distortion along the z axis, stabilizing the Mn III LT state. As introduced for RbMnFe, [START_REF] Azzolina | Landau theory for non-symmetry-breaking electronic instability coupled to symmetry-breaking order parameter applied to Prussian blue analog[END_REF] the cubic→tetragonal symmetry-breaking ferroelastic distortion of the crystalline lattice, due to the loss of the three-fold rotational symmetry, is characterized by its amplitude:

(𝑇) = 2 √3 ( 𝒄 LT (𝑇)-𝒂 LT (𝑇) 𝒂 HT ) (1)
In the framework of Landau theory of phase transitions, [START_REF] Azzolina | Landau theory for non-symmetry-breaking electronic instability coupled to symmetry-breaking order parameter applied to Prussian blue analog[END_REF] we have shown that it is also necessary to

where 𝑁 Mn II Fe III and 𝑁 Mn III Fe II denote the number of sites in each CT states. We have shown for similar materials that, during thermal equilibrium phase transitions, the elastic couplings of both parameters 𝑞 and  to 𝑣 s are responsible for their simultaneous changes from the LT Mn III Fe II tetragonal (𝑞 = -1,  ≠ 0) to the HT Mn II Fe III cubic (𝑞 = 1,  = 0) phases and for widening the thermal hysteresis. [START_REF] Azzolina | Landau theory for non-symmetry-breaking electronic instability coupled to symmetry-breaking order parameter applied to Prussian blue analog[END_REF] As shown in Fig. 1a, the change of electronic state is associated with colour change between Mn III Fe II tetragonal and Mn II Fe III cubic states, both stable at room temperature. The LT-to-HT phase transition can also be induced by laser irradiation at room temperature within the thermal hysteresis. As shown in Fig. 1a, the characteristic colour change occurs at the laser spots, leading to a permanent photoinduced HT phase. In the following, we are interested in understanding the ultrafast out-of-equilibrium dynamics induced by a picosecond laser pulse and driving the Mn III Fe II tetragonal → Mn II Fe III cubic PIPT within the thermal hysteresis regime of bistability.

Time-resolved crystallography with powder streaming

Understanding the underlying physical mechanisms driving PIPT is mandatory for controlling by light the peculiar properties of these materials. Various ultrafast studies, making use of different techniques, were performed on cyanide-bridged bimetallic assemblies, such as Fe-Fe, Co-Fe, Mn-Fe or V-Cr, and focused on the ultrafast formation of local and transient CT states. [START_REF] Cammarata | Charge transfer driven by ultrafast spin transition in a CoFe Prussian blue analogue[END_REF][START_REF] Azzolina | Exploring Ultrafast Photoswitching Pathways in RbMnFe Prussian Blue Analogue[END_REF][START_REF] Barlow | Ultrafast photoinduced dynamics in Prussian blue analogues[END_REF][START_REF] Arnett | Excitation Dephasing, Product Formation, and Vibrational Coherence in an Intervalence Charge-Transfer Reaction[END_REF][START_REF] Weidinger | Transient absorption studies of vibrational relaxation and photophysics of Prussian blue and ruthenium purple nanoparticles[END_REF][START_REF] Moritomo | Photoinduced Phase Transition into a Hidden Phase in Cobalt Hexacyanoferrate as Investigated by Time-Resolved X-ray Absorption Fine Structure[END_REF][START_REF] Zerdane | Out-of-equilibrium dynamics driven by photoinduced charge transfer in CsCoFe Prussian blue analogue nanocrystals[END_REF][START_REF] Asahara | Ultrafast dynamics of reversible photoinduced phase transitions in rubidium manganese hexacyanoferrate investigated by midinfrared CN vibration spectroscopy[END_REF] In the present work we investigate by streaming powder diffraction (Fig. 1b) the out-of-equilibrium structural dynamics of a macroscopic PIPT in RbMn0.94Co0.06Fe, resulting from a single laser shot excitation within the thermal hysteresis. Conventional time-resolved diffraction techniques consist in exciting a system with a pump pulse and monitoring the change in the X-ray diffraction signal at a given delay t. [START_REF] Mariette | Strain wave pathway to semiconductor-to-metal transition revealed by time-resolved X-ray powder diffraction[END_REF][START_REF] Collet | Laser-induced ferroelectric structural order in an organic charge-transfer crystal[END_REF][START_REF] Johnson | Non-equilibrium phonon dynamics studied by grazing-incidence femtosecond X-ray crystallography[END_REF][START_REF] Beaud | A time-dependent order parameter for ultrafast photoinduced phase transitions[END_REF][START_REF] Chergui | Photoinduced Structural Dynamics of Molecular Systems Mapped by Time-Resolved X-ray Methods[END_REF][START_REF] Ichikawa | Transient photoinduced 'hidden' phase in a manganite[END_REF][START_REF] Elsaesser | Perspective: structural dynamics in condensed matter mapped by femtosecond x-ray diffraction[END_REF] Then, the system rapidly recovers the ground state to be excited anew, so that stroboscopic pump-probe experiments can be performed at high repetition rates (> 1Hz) on the same sample. However, this technique cannot be used to study non-reversible phenomena, such as photoswitching within thermal hysteresis. In parallel, serial crystallography, where time-resolved diffraction is performed on fresh single crystals for each laser shot, was largely applied in biology and recently developed for small-unit-cell systems. [START_REF] Yun | Early-stage dynamics of chloride ion-pumping rhodopsin revealed by a femtosecond X-ray laser[END_REF][START_REF] Coquelle | Chromophore twisting in the excited state of a photoswitchable fluorescent protein captured by time-resolved serial femtosecond crystallography[END_REF][START_REF] Pande | Femtosecond structural dynamics drives the trans/cis isomerization in photoactive yellow protein[END_REF][START_REF] Stockler | Towards pump-probe single-crystal XFEL refinements for small-unitcell systems[END_REF] Regrettably, it cannot be used in the present case because of i) the limited number of accessible Bragg peaks, given the small cell parameters and high symmetry of RbMn0.94Co0.06Fe and ii) the small size of the crystals giving rise to weak single crystal diffraction. Therefore, we developed a streaming powder diffraction technique (Fig. 1b & Supplementary Methods 2&3), which allows for studying out-of-equilibrium photoswitching dynamics inside thermal hysteresis by time-resolved X-ray diffraction (TR-XRD). We used the optical pump -X-ray probe configuration at the ERSF ID09 beamline [START_REF] Cammarata | Chopper system for time resolved experiments with synchrotron radiation[END_REF] with ~35 ps experimental time resolution. The sample was excited with 1 ps laser pulses. The photoinduced Mn III Fe II →Mn II Fe III CT occurs within 200 fs over a broad excitation spectral range including Fe to Mn CT band (around 500 nm) and Mn-centred band (around 650 nm). [START_REF] Tokoro | Visible-Light-Induced Reversible Photomagnetism in Rubidium Manganese Hexacyanoferrate[END_REF] We excited the sample at 650 nm, which is a good compromise between excitation efficiency and penetration depth (of the order of sample thickness).

The typical size of the crystals (0.90.3 µm), smaller than the laser pump penetration depth (1.9 µm), [START_REF] Ohkoshi | The dielectric constant in a thermal phase transition magnetic material composed of rubidium manganese hexacyanoferrate observed by spectroscopic ellipsometry[END_REF] ensures a relatively homogeneous excitation at the scale of individual crystals dispersed in ethanol (transparent at 650 nm). Streaming powder diffraction experiments consist in performing TR-XRD measurements on a collection of micro-crystals dispersed in solution and streamed through a liquid jet, where they interact with the optical pump and delayed X-ray probe beams, at 1 kHz repetition rate in the present case (Fig. 1b). The overall global fraction of LT crystals exposed to laser shot in the streamed suspension is small. In addition, this suspension circulates through a cooling loop set below 𝑇 ↓ , which allows to bring back to the ground LT phase the crystals eventually photoconverted. This ensures that each powder diffraction pattern is measured from a fresh batch of crystals initially in the ground LT state. This closed-loop circulation allows for long data acquisition time, often required in TR-XRD measurements. Both the HT and LT phases were measured at the room temperature of the jet. The lattice parameters of the HT (aHT = 10.544(2) Å) and the LT (aLT = 10.023(2) Å, cLT = 10.461(2) Å) phases, obtained from the Rietveld refinements in the streaming powder diffraction set-up, agree with conventional X-ray powder diffraction data (Supplementary Fig. 6), which shows the quality of the data obtained with streaming powder diffraction. Hereafter we present time-resolved streaming powder diffraction results, which made it possible to track the out-of-equilibrium structural dynamics towards the persistent PIPT, within the thermal hysteresis at room temperature.

Out-of-equilibrium structural dynamics

Fig. 2 shows the evolution of the diffraction pattern of RbMn0.94Co0.06Fe crystals, in the initial LT phase, and measured at room temperature as a function of pumpprobe delay t. We used a Rietveld refinement (Supplementary Methods 3) to track the time evolution of the lattice parameters, phase fractions, and bond lengths for the different phases coming into play. At negative delays the diffraction pattern corresponds to the initial Mn III Fe II LT phase (F4 ̅ 2m space group), where the characteristic Jahn-Teller ferroelastic distortion translates through the splitting of the (h00) and (00h)

Bragg peaks. At positive delays we had to consider two time-dependent phases: a tetragonal phase that describes the evolution of the original phase, called hereafter the photoexcited tetragonal (PT) phase, and a photoinduced cubic (PIC) phase, whose crystalline structure is similar to the HT phase.

Fig. 2a shows that for low pumping fluence (15 mJ.cm -2 ) the initial LT Bragg peaks shift with delay.

The structural Rietveld refinements of the diffraction patterns provide the evolution of the lattice parameters and of the ferroelastic distortion (𝑡)(𝑐(𝑡) -𝑎(𝑡))/𝑎 HT for each individual delay t.

The lattice parameters of the PT lattice evolve within 100 ps: cPT decreases while aPT = bPT increase, resulting in a global volume increase (1 Å 3 ) and decrease of . The lattice parameters relax back to initial equilibrium values within 10 µs. Similar lattice distortions were observed at low excitation density under photoexcitation outside thermal hysteresis (below 𝑇 ↓ ) of the Rb0.94Mn[Fe(CN)6]•2.5H2O sample. [START_REF] Azzolina | Out-of-equilibrium lattice response to photo-induced charge-transfer in a MnFe Prussian blue analogue[END_REF] Since RbMnFe and derivative materials exhibit no thermal expansion, [START_REF] Tokoro | Zero Thermal Expansion Fluid and Oriented Film Based on a Bistable Metal-Cyanide Polymer[END_REF] it was shown that such long-range lattice deformation results from the structural trapping of photoinduced Mn II Fe III CT state of higher volume, which forms within 200 fs and decays within 10 µs. [START_REF] Azzolina | Exploring Ultrafast Photoswitching Pathways in RbMnFe Prussian Blue Analogue[END_REF] At low fluence, the dynamical lattice response therefore includes two components: i) an impulsive lattice response driven by collective reverse Jahn-Teller reorganizations, elongating aPT and shrinking cPT, and ii) a volume expansion induced by the long-lived photo-induced CT Mn II Fe III small polarons, populating antibonding Mn 𝑑(𝑥 2 -𝑦 2 ) orbitals. The Rietveld analysis also reveals that a small fraction of photoinduced cubic phase (PIC) appears on the 100 ps timescale (xPIC ≃ 10 %).

The dynamical behaviour drastically changes when the laser fluence is increased to 115 mJ.cm -2 (Fig. 2c): the diffraction pattern of the initial LT phase almost disappears within 75 ps, while the Bragg peaks of the PIC phase form, characterized by the merging of (h00) and (00h) Bragg peaks. Here again, Rietveld refinements enable further understanding of the underlying dynamics at high fluence.

After photoexcitation, the lattice parameters aPT and cPT converge within 100 ps towards the cubic lattice parameters. This is due to a larger and global reverse Jahn-Teller distortion, characterized by the fast drop of 𝜂 for the tetragonal lattice. In addition, the fraction of photoexcited tetragonal phase xPT also drops within 75 ps, while the fraction xPIC of PIC phase growths rapidly and approaches 100 %. Overall, Rietveld analysis shows that high laser fluence drives within 100 ps a global photoinduced Mn III Fe II tetragonal → Mn II Fe III cubic phase transition inside the thermal hysteresis and of the photoexcited tetragonal lattice at different fluences (Fig. 3b), and through the fluence dependence of 𝑥 PIC (𝑡) (Fig. 3c). Again, these parameters change within 100 ps. This timescale matches the ratio between crystals' radius (450 nm) and sound velocity (c = 4300 m.s -1 56 ), which is characteristic of an elastically-driven process. [START_REF] Azzolina | Out-of-equilibrium lattice response to photo-induced charge-transfer in a MnFe Prussian blue analogue[END_REF][START_REF] Bertoni | Elastically driven cooperative response of a molecular material impacted by a laser pulse[END_REF] Indeed, since RbMnFe derivatives exhibit no thermal expansion, [START_REF] Tokoro | Zero Thermal Expansion Fluid and Oriented Film Based on a Bistable Metal-Cyanide Polymer[END_REF] the initial lattice expansion or ferroelastic distortion cannot be explained by a simple laser heating of the lattice. As can be seen on the evolution of 𝑥 PIC (𝑡), a second and weaker additional conversion step is observed for intermediate fluences around 10 ns, and is very likely due to slower heat diffusion as observed in spin-crossover materials. [START_REF] Bertoni | Elastically driven cooperative response of a molecular material impacted by a laser pulse[END_REF][START_REF] Bertoni | Temperature dependence of the cooperative out-of-equilibrium elastic switching in a spin-crossover material[END_REF][START_REF] Enachescu | Theoretical approach for elastically driven cooperative switching of spin-crossover compounds impacted by an ultrashort laser pulse[END_REF] Fig. 3c shows the fluence-dependent evolution of xPIC at delays of 100 ps, 1 ns, 10 ns and 500 ns. It clearly evidences a threshold fluence (≃ 10 mJ.cm -2 ) above which the Mn III Fe II tetragonal → Mn II Fe III cubic PIPT is fully established within less than 1 ns.

For fluences below threshold, Fig. 3b shows that the less bonding photoinduced CT Mn II Fe III state is responsible for a relative volume expansion ∆𝑣 of the photoexcited tetragonal lattice, therefore explaining the lattice expansion. In this polaronic regime, ∆𝑣 increases with excitation density, while the lattice of the photoexcited crystals retain tetragonal symmetry.

On the other hand, anisotropic lattice distortions in this regime are explained by the anisotropic reorganization around the Mn-N6 octahedron, as evidenced in Fig. 3b & S7: the structural refinements show elongation of the Mn-Nx,y bonds and a contraction of the Mn-Nz ones. This is the direct microscopic evidence of the structural reorganization around the Mn sites from D4h in Mn III Fe II configuration towards Oh for Mn II Fe III . This structural reorganisation is characteristic of the reverse Jahn-Teller distortion associated with the photoinduced CT polarons, [START_REF] Azzolina | Out-of-equilibrium lattice response to photo-induced charge-transfer in a MnFe Prussian blue analogue[END_REF][START_REF] Azzolina | Exploring Ultrafast Photoswitching Pathways in RbMnFe Prussian Blue Analogue[END_REF] which are localized at the level of a single Co-N-C-Fe unit as previously characterized by time-resolved IR spectroscopy. [START_REF] Asahara | Ultrafast dynamics of reversible photoinduced phase transitions in rubidium manganese hexacyanoferrate investigated by midinfrared CN vibration spectroscopy[END_REF] This collective local structural dynamic, driven by the photoinduced CT Mn II Fe III polaron, is the driving force of the macroscopic anisotropic lattice deformations: aPT elongates, cPT contracts and therefore 𝜂 decreases.

Above 10 mJ.cm -2 , the PIC phase forms within 100 ps and the conversion rate saturates with an almost complete conversion above 100 mJ.cm -2 . This corresponds to the PIPT regime, with a persistent fraction of PIC phase. Such a threshold response and fluence dependence is characteristic of cooperative PIPT. [START_REF] Ridier | Finite Size Effects on the Switching Dynamics of Spin-Crossover Thin Films Photoexcited by a Femtosecond Laser Pulse[END_REF][START_REF] Bertoni | Elastically driven cooperative response of a molecular material impacted by a laser pulse[END_REF][START_REF] Bertoni | Temperature dependence of the cooperative out-of-equilibrium elastic switching in a spin-crossover material[END_REF][START_REF] Enachescu | Theoretical approach for elastically driven cooperative switching of spin-crossover compounds impacted by an ultrashort laser pulse[END_REF] Together, the ultrafast nature of the phase transformation and its nonlinear fluence dependence clearly show that the PIC phase forms through a nucleation process, where a critical fraction of photoinduced CT Mn II Fe III polarons can drive macroscopic PIPT. Since is hard to estimate precisely the number of absorbed photons, we estimate percentage of photoexcited units by considering that quantum efficiency is of the order of unity and volume changes linearly with CT fraction. [START_REF] Azzolina | Out-of-equilibrium lattice response to photo-induced charge-transfer in a MnFe Prussian blue analogue[END_REF][START_REF] Azzolina | Exploring Ultrafast Photoswitching Pathways in RbMnFe Prussian Blue Analogue[END_REF] Compared to the relative volume expansion from LT to HT phases (∆𝑣 ~ 0.1) the relative photoexpansion of the PT phase at 62 mJ.cm -2 (∆𝑣 ~ 0.01, in Fig. 3b) allows to estimate that 10 % of the MnFe units are photoexcited around excitation threshold.

For each regime, the mechanism at play is deeply related to the evolution of the elastic deformation of the crystals, in particular the volume expansion and the release of the Jahn-Teller distortion. Below threshold excitation, the lattice pressure brings within 10 µs the higher volume photoinduced Mn II Fe III polarons back to their lower volume ground Mn III Fe II state, as characterized by the Mn-N bonds evolution (Fig. 3b). Above threshold, CT polarons drive larger volume expansion, which favours the cubic Mn II Fe III phase of higher volume. The overall threshold photoresponse results therefore from the competition between pushing forces of the lattice on the CT polarons, driving their relaxation towards lower volume ground state, and pulling forces of the expanded lattice on ground Mn III Fe II sites, favouring switching towards higher volume Mn II Fe III state.

The role of elastic cooperativity in the present dynamics can be understood within the framework of Landau theory of phase transition, which we have previously used to describe thermal equilibrium phase transitions in materials exhibiting coupled change of electronic state and symmetry, such as RbMnFe. [START_REF] Azzolina | Landau theory for non-symmetry-breaking electronic instability coupled to symmetry-breaking order parameter applied to Prussian blue analog[END_REF][START_REF] Collet | Coupled spin cross-over and ferroelasticity: revisiting the prototype [Fe(ptz)6](BF4)2 material[END_REF][START_REF] Collet | Coupling and decoupling of spin crossover and ferroelastic distortion: Unsymmetric hysteresis loop, phase diagram, and sequence of phases[END_REF] Indeed, the relevant Landau potential should include both the order parameter q monitoring CT and the symmetry breaking ordeohkor parameter  (Supplementary Discussion 1). These parameters couple to the volume strain 𝑣 s . Inside the thermal hysteresis, both the Mn II Fe III cubic phase (𝜂 HT = 0, 𝑞 = 1) and Mn III Fe II tetragonal phase (𝜂 LT ≠ 0, 𝑞 = -1) can be stable. Hereafter we focus our attention on the more usual Landau potential corresponding to the cubic → tetragonal symmetry breaking phase transition: [START_REF] Salje | Linear-quadratic order parameter coupling and multiferroic phase transitions[END_REF][START_REF] Carpenter | Spontaneous strain as a determinant of thermodynamic properties for phase transitions in minerals[END_REF][START_REF] Carpenter | Elastic anomalies in minerals due to structural phase transitions[END_REF]

𝐺(𝜂) = 1 2 𝑎′𝜂 2 + 1 3 𝑏𝜂 3 + 1 4 𝑐𝜂 4 (3)
where the renormalized coefficient 𝑎 ′ = 𝑎 + 2𝜆 𝜂 𝑣 s includes both the usual temperature-induced symmetry breaking term (𝑎 = 𝑎 0 (𝑇 -𝑇 SB )) and the elastic coupling of the ferroelastic distortion to 𝑣 s (𝑣 s 𝜂 2 < 0) where 𝑣 s < 0 stabilises the tetragonal phase. [START_REF] Salje | Phase Transitions in Ferroelastic and Co-elastic Crystals[END_REF][START_REF] Chernyshov | Landau theory for spin transition and ordering phenomena in Fe(II) compounds[END_REF] a changes sign at the symmetry-breaking temperature 𝑇 SB , which stabilizes 𝜂 = 0 above 𝑇 SB and 𝜂 ≠ 0 below, while the symmetry-allowed 𝜂 3 term limits the F4 ̅ 3m → F4 ̅ 2m ferroelastic transition to first-order. Inside the thermal hysteresis, both the high-volume cubic phase (𝜂 HT = 0, 𝑣 s = 0) and low-volume tetragonal phase (𝜂 LT ≠ 0, 𝑣 s < 0) can be stable, due to the elastic coupling to 𝑣 s of the ferroelastic distortion 𝜂. Fig. 4 shows the 𝐺(𝜂) curve obtained from equation (3) when cubic and tetragonal phases are equally stable (left panel). To describe the lattice response to photoexcitation, we should consider the results from the time-resolved measurements discussed above, which bring key information on the mechanism behind macroscopic and ultrafast PIPT within the thermal hysteresis. The photoinduced Mn II Fe III CT polarons, forming within 200 fs and leaving up to 10 µs, [START_REF] Azzolina | Exploring Ultrafast Photoswitching Pathways in RbMnFe Prussian Blue Analogue[END_REF] are sources of anisotropic elastic deformation waves, which propagate at crystal's scale and equilibrate with the lattice on the elastic timescales (100 ps). Compared to the initial LT volume strain 𝑣 s , the volume strain 𝑣 s * of the photoexcited lattice is modified by the small polarons resulting from photoinduced CT (𝛥𝑞) and responsible for volume expansion ∆𝑣:

𝑣 s * = 𝑣 s + Δ𝑣 (4)
The associated photoinduced volume expansion destabilizes the tetragonal lattice, through the elastic coupling cost term  Δ𝑣𝜂 2 > 0 (with Δ𝑣 > 0). Fig. 4a depicts the mechanism at play in the transient polaronic regime at low fluence: the weak volume expansion δ𝑣 reduces the ferroelastic distortion in the transiently equilibrated PT state (𝜂 PT < 𝜂 LT ), which remains tetragonal due to the energy barrier towards the cubic phase. As the photoinduced CT polarons decay within 10 µs, the system relaxes back to the ground LT state with initial volume strain 𝑣 s and ferroelastic distortion 𝜂 LT . In the persistent PIPT regime above threshold fluence (Fig. 4b), the high laser fluence induces larger volume expansion Δ𝑉 and therefore larger elastic coupling cost ( Δ𝑉𝜂 2 ), that fully destabilizes the tetragonal lattice. As a result, full conversion towards the high-volume cubic symmetry (𝜂 = 0) is observed within the elastic timescale. The system then relaxes towards the equilibrium Mn II Fe III cubic phase, corresponding to the HT phase. As the energy barrier with the tetragonal Mn III Fe II phase appears anew after global relaxation, this photoinduced HT phase is persistent at room temperature, and more generally over the wide thermal hysteretic regime of bistability.

The present results demonstrate the possibility to induce by a single laser shot an ultrafast PIPT at room temperature, in a functional material designed to exhibit a wide regime of thermal bistability. Compared to previous single laser shot photoswitching, observed in another RbMnFe derivative at low temperature by static X-ray diffraction studies, [START_REF] Azzolina | Single Laser Shot Photoinduced Phase Transition of Rubidium Manganese Hexacyanoferrate Investigated by X-ray Diffraction[END_REF] our time-resolved analysis also brings key information on the outof-equilibrium structural and symmetry dynamics. Indeed, these long-range structural reorganizations can only be directly monitored by ultrafast diffraction techniques. The cooperative tetragonal Mn III Fe II → cubic Mn II Fe III photoinduced phase transition, which occurs above threshold excitation fluence, is persistent and is driven by elastic cooperativity. The energy barrier in the thermodynamic potential between the tetragonal Mn III Fe II and cubic Mn II Fe III phases has two important roles. On the one hand, it is responsible for the threshold fluence necessary to induce the cubic phase, as a critical volume expansion is required to destabilize the tetragonal lattice. On the other hand, this volume expansion stabilizes the photoinduced cubic Mn II Fe III phase, which is persistent over a wide range of bistability.

The streaming powder diffraction set-up we developed made it possible to track structural dynamics at microscopic (Mn-N bonds) and macroscopic (lattice parameters) scales through the non-reversible symmetry and volume changes associated with the photoinduced tetragonal Mn III Fe II → cubic Mn II Fe III phase transition. Such ultrafast photoinduced phase transitions induced by a single laser pulse are very promising for optically-driven devices, memories, or actuators exhibiting room temperature switching within a wide thermal regime of bistability between states that can persist long after stimuli. As streaming powder technique can be easily implemented at X-ray Free-Electron Laser beamlines for femtosecond X-ray studies, we also foresee a broad range of applications to study out-of-equilibrium dynamics for non-reversible phenomena in various functional materials. [START_REF] Mariette | Strain wave pathway to semiconductor-to-metal transition revealed by time-resolved X-ray powder diffraction[END_REF][START_REF] Chergui | Photoinduced Structural Dynamics of Molecular Systems Mapped by Time-Resolved X-ray Methods[END_REF] Setup for Streaming Powder Diffraction Time-resolved X-ray diffraction (TR-XRD) measurements were performed on time-resolved beamline ID09 at the European Synchrotron Radiation Facility (ESRF). [START_REF] Cammarata | Chopper system for time resolved experiments with synchrotron radiation[END_REF] The experiment was performed while the ESRF storage ring was operated in 7/8+1 filling mode with a single bunch current of 4 mA, corresponding to an X-ray pulse duration of ~35 ps (HWHM). The X-ray central energy was 14.963 keV (λ = 0.824 Å, 2 % bandwidth), and the beam was focused to a spot size of 2525 µm 2 at the sample position. To perform laser pump-X-ray probe measurements, isolated Xray pulses are synchronized with a 1 ps laser, whose spot size was 210 µm x 250 µm. The larger laser spot size ensures a homogeneous excitation of the sample in the volume probed by the X-rays. The laser wavelength was set to 650 nm, exciting crystals on the red edge of their absorption spectrum, which results in a penetration depth of about 1.9 µm, 54 larger than the average size of the crystals (0.9  0.3 µm). Additionally, we studied how photo-response changes with laser fluence, which was varied between 0 and 150 mJ.cm -2 .

For the present streaming powder diffraction study (Supplementary Methods 2-4), crystals of RbMn0.94Co0.06Fe were dispersed in ethanol and streamed through a free-flowing liquid jet to interact with laser and X-ray pulses. The jet size was 300 µm, and the crystal-solvent weight ratio was 1:270, which corresponds to ~500 crystals per shot in the X-ray focus. The flow rate of the jet was set to 1.2 L.h -1 (jet velocity of 5 m.s -1 at the interaction region). As a consequence, a fresh ensemble of crystals, which was not excited by previous laser shots, is interacting with the laser pump -X-ray probe beams. The chosen flow rate limits the observation time window to 10 µs for pump-probe measurements. Jet was circulated in closed loop, and a cooling device was inserted in the liquid circulation loop in order to compensate for the non-reversible photo-conversion of the already-excited crystals (Supplementary Methods 2). X-ray diffraction patterns were recorded in transmission geometry using a Rayonix MX170-HS CCD detector with an integration time of 5 s for each image.

The delay between the laser pump pulse and the X-ray probe pulse was varied from -3 ns to +10 µs, and we typically collected 20 diffraction images for each delay. As crystals are randomly oriented in the liquid jet, diffraction images consisted of rings that were azimuthally integrated using pyFAI. [START_REF] Ashiotis | The fast azimuthal integration Python library: pyFAI[END_REF] We normalized each image over the Q-range of [5.5; 6.0] Å -1 , where diffraction signal remains constant along the dynamics.

XRD Analysis

The X-ray scattering patterns of RbMn0.94Co0.06Fe were obtained by subtracting the background scattering from the solvent. This was done by subtracting the pattern of pure ethanol, followed by polynomial fitting of residual background (Supplementary Methods 3&4). Rietveld refinement of the X-ray diffraction data was performed using TOPAS-Academic version 6 software, [START_REF] Coelho | TOPAS and TOPAS-Academic: an optimization program integrating computer algebra and crystallographic objects written in C++[END_REF] with a fixed wavelength of 0.827389 Å -1 (15 keV as calibrated at the ID09 ESRF beamline). For pure LT and HT phases, refinement was done over the scattering vector Q-range [0.9; 5.0] Å -1 using the structures found from powder diffraction, and including refinement of lattice parameters and isotropic Debye-Waller thermal factors. Given the symmetry change between tetragonal LT phase and cubic HT phase, refinement was done considering space group F4 ̅ 2m for LT phase and F4 ̅ 3m for HT phase in order to directly compare both phases, as explained in our recent work. [START_REF] Azzolina | Landau theory for non-symmetry-breaking electronic instability coupled to symmetry-breaking order parameter applied to Prussian blue analog[END_REF] Sometimes a weak contribution from HT phase was observed in the equilibrium X-ray diffraction data. Analysis of time-resolved and fluence-dependent diffraction patterns was performed following the same procedure as for pure phases. Background subtraction and Rietveld refinement were performed at each delay. More details are provided in Supplementary Methods 3 and 4. through a liquid jet. The X-ray beam probes the sample at a time delay t after laser pumping within the hysteresis, which may eventually convert crystals from LT (blue) to permanent HT phase (red).

The suspension in the reservoir circulates through a cooling device (230 K < 𝑇 ↓ ), bringing crystals back to the ground LT phase, before injecting them anew at room temperature in the pump-probe beams.

  Fig.3ashows the crystalline structure of the PIC phase, measured at 1 ns and 36 mJ.cm -2 . The

.

  Figure 1. Charge-transfer bistability at room temperature. a Pictures of 66 mm 2

Figure 2 .

 2 Figure 2. Time-resolved X-ray diffraction. Diffracted signal along the scattering vector Q (top)

Figure 3 .

 3 Figure 3. Structural reorganizations towards photoinduced cubic phase. a Room temperature

Figure 4 .

 4 Figure 4. From local charge-transfer polarons to macroscopic photoinduced phase transition

Acknowledgements

The authors gratefully acknowledge the Agence Nationale de la Recherche for financial support under grant ANR-19-CE30-0004 ELECTROPHONE, ANR-19-CE07-0027 SMAC, ANR-19-CE29-0018 MULTICROSS and the European Synchrotron Radiation Facility (ESRF) for provision of synchrotron radiation facilities under proposal number CH-6162. G.P. thanks Région Bretagne for partial PhD funding (ARED PHONONIC). I.C. acknowledges financial support from Région Bretagne under grant SAD. E.C. thanks the University Rennes and the Fondation Rennes 1 for funding. This work was carried out in the frame of a JST FOREST Program (JPMJFR213Q), a JSPS Grant-in-Aid for Scientific Research (B) (22H02046), Scientific Research (A) (20H00369), and Yasaki Memorial Fundation for Science and Technology. We are grateful to Yugo Nagane (the University of Tokyo) and Mayuko Tanaka (University of Tsukuba) for their technical support in synthesizing the sample.

Data Availability

Raw data generated in this study and codes used to process the data have been deposited on the Zenodo repository [START_REF] Herve | Raw Data and Codes for the Article "Ultrafast and persistent photoinduced phase transition at room temperature monitored by streaming powder diffraction[END_REF] under the DOI: https://doi.org/10.5281/zenodo.10227842. Any additional information is available from the corresponding authors upon reasonable request.

Methods

Synthesis and characterization of RbMn0.94Co0.06Fe

The Rb0.94Co0.06Mn0.94[Fe(CN)6]0.98•0.2H2O material was synthesized and characterized by various techniques (Supplementary Methods 1). The crystals are plate-shaped, with good crystallinity and average size of 0.9  0.3 µm. Magnetic measurements indicate that the compound exhibits a thermal phase transition between the HT and LT phases with transition temperatures 𝑇 ↓ = 253 K and 𝑇 ↑ = 328 K. The XRD powder patterns at room temperature and Rietveld analysis correspond for the HT phase to a cubic space group (F4 ̅ 3m) with a lattice constant of aHT = 10.5495(6) Å and for the LT phase to a tetragonal space group (F4 ̅ 2m) with aLT = bLT = 10.0051 (11) Å and cLT = 10.4744( 16) Å. CCDC #2259400 (LT) and #2259401 (HT) contain the supplementary crystallographic data for this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif.

Static photoinduced measurements on RbMn0.94Co0.06Fe films

RbMn0.94Co0.06Fe in pure ethanol was deposited on a glass substrate and dried to obtain films of the material (Fig 1a). Films were cooled below 180 K to reach the Mn III Fe II LT phase and warmed above 350 K to reach the Mn II Fe III HT phase. A 532 nm CW laser (FSDL-532-100T, Frankfurt Laser Company) with 100 mW output power was used to photoexcite during 3 s the LT phase at room temperature (0.50.45 mm² spot size (FWHM) at the film position, power density of 120 W.cm -2 ) to photoinduce the HT phase. For Fig. 1a, photos were taken with a camera, in the following order and for the same thin film: crystal deposition and cooling at 180 K, photo of the LT phase at room temperature, laser irradiation and photo of the irradiated film at room temperature, heating to 350 K and photo of the heated film.

Author contributions

Competing interests

The authors declare no competing interests.

Additional information

Supplementary information The online version contains supplementary information available at…

Correspondence and requests for materials should be addressed to E.C.

is then persistent at room temperature, due to the energy barrier with the initial tetragonal Mn III Fe II phase, matching the HT phase (red curve).


domain: info.info-ai info phys

text:
Karim Haddada
email: haddada.karim@gmail.com

Mohamed Ibn Khedher
email: mohamed.ibn-khedher@irt-systemx.fr

Olfa Jemai
email: olfa.jemai@isimg.tn

Comparative study of Deep Learning architectures for Early Alzheimer Detection

Keywords: Deep learning, Transfer learning, Early Alzheimer disease detection, MRI images

Alzheimer's disease (AD) is a chronic and irreversible brain disorder, making early detection crucial in managing its progression. This study aims to compare multiple deep learning models for the early detection of Alzheimer's disease. The research employs advanced techniques, including convolutional neural networks (CNNs) and transfer learning models. Specifically, six deep neural network techniques -2DCNN, VGG19, DenseNet121, Inception-V3, MobileNet and ResNet101v2 are utilized to classify and identify different stages of AD, such as NonDementia, Very Mild Dementia, Mild Dementia and Moderate Dementia. The comparative study is validated on the Kaggle dataset of 6400 MRI images. The results demonstrate that the CNN outperforms the fine-tuned architectures with an accuracy of 99.14% and AUC of 99.84%.

I. Introduction

Alzheimer's disease (AD) is a devastating neurodegenerative disorder that affects millions of people worldwide, primarily those over the age of 65. The disease causes a decline in cognitive function, including memory loss, language difficulties, and impaired judgment. Early detection is essential for effective treatment and management, as it can significantly improve patients' quality of life.

Currently, there is no definitive cure for Alzheimer's disease, but there are ways to temporarily slow down its symptoms and enhance the quality of life for patients. Therefore, extensive research is devoted to discovering effective treatments and preventing the disease's progression over time, also a critical need for innovative technology that can automatically support Alzheimer's patients to perform daily live activities [START_REF] Snoun | Deeplearning-based human activity recognition for alzheimer's patients' daily life activities assistance[END_REF], [START_REF] Ben Taleb | Reinforcement learning for assistance of alzheimer's disease patients[END_REF], [START_REF] Snoun | A reinforcement learning and transformers based intelligent system for the support of alzheimer's patients in daily life activities[END_REF]. Deep learning (DL) and machine learning (ML) algorithms, in particular, have demonstrated tremendous promise in several computer vision tasks [START_REF] Jmila | Adversarial machine learning for network intrusion detection: A comparative study[END_REF], [START_REF] Snoun | Towards a deep human activity recognition approach based on video to image transformation with skeleton data[END_REF], [START_REF] Snoun | A multimodal hypovigilance detection system based on fuzzy logic and transfer learning[END_REF] and are currently being investigated as a potential exploration tool for Alzheimer's disease.

Recently, deep learning techniques, such as Convolutional Neural Networks (CNNs) [START_REF] Esi Turkson | Classification of alzheimerâs disease using deep convolutional spiking neural network[END_REF], [START_REF] Mehmood | A deep siamese convolution neural network for multiclass classification of alzheimer disease[END_REF], [START_REF] Sathiyamoorthi | A deep convolutional neural network based computer aided diagnosis system for the prediction of alzheimer's disease in mri images[END_REF], [START_REF] Bae | Identification of alzheimer's disease using a convolutional neural network model based on t1-weighted magnetic resonance imaging[END_REF], [START_REF] Jiang | Classification of alzheimerâs disease via eight-layer convolutional neural network with batch normalization and dropout techniques[END_REF] and Transfer learning methods [START_REF] Mehmood | A transfer learning approach for early diagnosis of alzheimerâs disease on mri images[END_REF], [START_REF] Hadeer | Deep learning approach for early detection of alzheimerâs disease[END_REF], have shown great potential in detecting AD using brain imaging data. CNNs were the first deep learning models proposed for image processing. The principle of CNN is to exploit spatial correlations between pixels of the image through convolutional layers. CNNs are a type of deep neural network that can learn complex patterns and structures within the data using convolutional layers. CNNs trained from scratch may struggle to generalize well when the training data is limited or when the target task is significantly different from the training task. Transfer Learning, on the other hand, benefits from the learned features of the pre-trained model, which can improve generalization and performance, especially when the tasks are related or share similar features. According to the literature, CNNs have limited ability to detect long-distance dependencies. CNNs are designed to detect local patterns in the input image and are less effective at detecting long-range dependencies, which may be crucial for certain tasks. Indeed, a significant number of filters would be required to encode such data.

Due to the limited availability of related datasets and the imbalance of available AD datasets, AD detection is a difficult task. To address these issues, researchers have employed various techniques, including transfer learning and Stratified Cross-Validation (SCV), to enhance the accuracy and robustness of AD classification models. Transfer learning involves leveraging pre-trained models to accelerate training on new tasks, allowing the use of deep learning models in small datasets. Stratified crossvalidation is a widely used method for training and evaluating deep learning models that addresses the imbalance issue in AD datasets. SCV ensures that the distribution of data across different folds is proportional to the class distribution, improving the reliability and robustness of performance evaluation [START_REF] Ibn Khedher | Improving decision-making-process for robot navigation under uncertainty[END_REF], [START_REF] Hatem Ibn-Khedher | Mathematical programming approach for adversarial attack modelling[END_REF], [START_REF] Ibn | Analyzing adversarial attacks against deep learning for robot navigation[END_REF], [START_REF] Ibn Khedher | Dynamic and scalable deep neural network verification algorithm[END_REF], [START_REF] Mellouki | Abstract layer for leakyrelu for neural network verification based on abstract interpretation[END_REF].

Our contributions are as follows:

• Using our 2DCNN, excellent results have been achieved for the early detection of Alzheimer's disease, surpassing the results reported in the literature. 2DCNN has demonstrated superior performance in detecting early signs of Alzheimer's disease compared to other methods.

• We propose a comparative study of recent Deep

Learning based architectures including CNN and some transfer learning methods. VGG19 and DenseNet121 demonstrates a high capability for the early detection of Alzheimer's disease.

The rest of this paper is organized as follows. Section II presents an overview of the proposed approaches for early detect Alzheimer Disease. Section III details our proposed evaluation pipeline. In Section IV, we present our obtained results on the selected dataset. Finally, section V presents our conclusions and our potential suggestions for future directions.

II. State of the art

Early diagnosis of Alzheimer's disease (AD) using Machine Learning algorithms has gained significant attention in recent years.

Algorithms for early AD can be categorized into two main types: supervised models and unsupervised models.

A. Supervised Machine Learning models

Regarding supervised models, one of the most popular models is the Convolutional Neural Network (CNN), which has been used for the diagnosis of AD from brain MRI images [START_REF] Esi Turkson | Classification of alzheimerâs disease using deep convolutional spiking neural network[END_REF], [START_REF] Mehmood | A deep siamese convolution neural network for multiclass classification of alzheimer disease[END_REF], [START_REF] Sathiyamoorthi | A deep convolutional neural network based computer aided diagnosis system for the prediction of alzheimer's disease in mri images[END_REF], [START_REF] Bae | Identification of alzheimer's disease using a convolutional neural network model based on t1-weighted magnetic resonance imaging[END_REF], [START_REF] Jiang | Classification of alzheimerâs disease via eight-layer convolutional neural network with batch normalization and dropout techniques[END_REF]. The CNN architecture consists of convolution, pooling, batch normalization, dropout, fully connected, and softmax layers, as well as activation functions. Several pre-defined CNN architectures have been introduced and widely used, such as ResNet,VGGNet [START_REF] Mehmood | A transfer learning approach for early diagnosis of alzheimerâs disease on mri images[END_REF], [START_REF] Hadeer | Deep learning approach for early detection of alzheimerâs disease[END_REF], AlexNet, GoogleNet, Inception. Among these, ResNet and VGGNet [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF] were the most popular architectures used for early AD.

Deep Neural Networks (DNNs) are a type of artificial neural network that enables the learning of complex representations of data by utilizing multiple layers of neurons. DNNs have been successfully applied in various fields, including image and speech recognition, natural language processing, and most recently, in the diagnosis of Alzheimer's disease (AD). In the AD diagnosis domain, DNNs have been utilized to analyze medical data, such as MRI scans [START_REF] Amoroso | Deep learning reveals alzheimer's disease onset in mci subjects: Results from an international challenge[END_REF], [START_REF] Manzak | Automated classification of alzheimerâs disease using deep neural network (dnn) by random forest feature elimination[END_REF], EEG [START_REF] Kim | Detection of early stage alzheimerâs disease using eeg relative power with deep neural network[END_REF] and genetic measures [START_REF] Wang | Deep learning-based brain transcriptomic signatures associated with the neuropathological and clinical severity of alzheimer's disease[END_REF], as well as clinical data, such as speech transcripts [START_REF] Orimaye | Deep language space neural network for classifying mild cognitive impairment and alzheimer-type dementia[END_REF] for classification of AD . Despite the challenges associated with training DNNs, they have demonstrated promising outcomes in AD diagnosis and continue to be an active area of research. In addition to CNN, Recurrent Neural Networks (RNNs) can process sequential data like time series, speech, and text. RNNs [START_REF] Lee | Predicting alzheimerâs disease progression using multi-modal deep learning approach[END_REF] can be used to analyze and predict patterns in patient data over time, such as changes in cognitive function or brain imaging results, in the context of Alzheimer's disease. One example of a RNN model used in AD context is the Long Short-Term Memory (LSTM). LSTMs can selectively remember or forget information over time, making them well-suited for modeling long-term dependencies in sequential data. In [START_REF] Hong | Predicting alzheimerâs disease using lstm[END_REF], an LSTM network is constructed to capture the temporal relationship among features and the subsequent stage of Alzheimer's Disease.

In addition to CNNs and RNNs, other supervised models have also been investigated, such as the Multilayer Perceptron (MLP) [START_REF] Bhagyashree | Diagnosis of alzheimerâs disease using multi-layer perceptron network[END_REF], Support Vector Machines (SVMs) [START_REF] Sharma | Alzheimer's patients detection using support vector machine (svm) with quantitative analysis[END_REF], and Random Forest (RF) [START_REF] Velazquez | Random forest model for feature-based alzheimer's disease conversion prediction from early mild cognitive impairment subjects[END_REF]. These models have been applied to different types of data, such as MRI images, cerebrospinal fluid (CSF) biomarkers, and neuropsychological test results, to predict the onset of AD.

Transformers were initially proposed by Vaswani et al [START_REF] Vaswani | Attention is all you need[END_REF] for machine translation tasks and have since achieved state-of-the-art performance in various Natural Language Processing (NLP) tasks, such as language translation, sentiment analysis, and question-answering.In addition to NLP, transformers have also been applied to computer vision tasks such as image classification, object detection and image segmentation. In the field of Alzheimer's disease diagnosis, many types of transformers are used. It uses a self-attention mechanism to process sequences of input data, allowing it to capture long-term dependencies and relationships between different parts of the input sequence. In the field of Alzheimer's disease diagnosis, several studies utilizing transformers have emerged .

B. Unsupervised Machine Learning models

The objective of the initial phase of Machine Learning algorithm is to extract more complex features from the input data. To accomplish this, unsupervised deep learning techniques were developed.Auto-Encoders (AE) and GAN, which are further explained in this section, are two of the most commonly used models.

Unsupervised learning is distinct from supervised learning as it operates on unlabeled data, relying only on the input data itself. One approach in unsupervised learning is to use autoencoders, which are a type of neural network architecture that learns to reconstruct input data by first encoding it into a lower-dimensional representation and then decoding it back to the original data. In the context of diagnosing Alzheimer's disease (AD), autoencoders [START_REF] Vu | Non-white matter tissue extraction and deep convolutional neural network for alzheimerâs disease detection[END_REF], [START_REF] Kruthika | Cbir system using capsule networks and 3d cnn for alzheimer's disease diagnosis[END_REF] have been utilized to extract features and identify patterns in brain MRI images. The features learned in the middle layer of an autoencoder can then be extracted and utilized for unsupervised pre-training of feature extraction and dimension reduction. Subsequently, a classifier such as SVM or Softmax [START_REF] Liu | Early diagnosis of alzheimer's disease with deep learning[END_REF] can be employed for classification purposes. Another unsupervised model that has been used for AD diagnosis is the Generative Adversarial Network (GAN). GANs are neural networks that can generate new data samples that are similar to the input data. In the context of AD diagnosis, GANs [START_REF] Bai | A novel alzheimerâs disease detection approach using gan-based brain slice image enhancement[END_REF] have been used to generate synthetic brain MRI images that can be used to augment the training data for supervised models.

To summarize, various deep learning models have been employed for early diagnosis of Alzheimer's disease using both supervised and unsupervised approaches. These models have been applied to various types of data and have shown promising results in early AD.

III. Methods

In this section, we describe the evaluation method employed in this study. The flowchart of the method is illustrated in Fig. 1 . The input is an open-source dataset from the Kaggle platform composed of 6400 MRI associated with 4 stages of Alzheimer. The output is the patient's predicted Alzheimer disease stage.

Fig. 1 depicts the methodology employed to investigate an accurate and early diagnosis solution for AD. This approach targets the challenges outlined in the introduction. It primarily involves the utilization and comparison of our 2D CNN approach alongside fine-tuned deep learning models such as MobileNet, ResNet101, Inceptionv3, DenseNet121, and VGG19 for the purpose of diagnosing and classifying AD.

A. Our 2D CNN approach

In our 2D CNN, we didn't perform image segmentation as it's not necessary when using a CNN. The CNN extracts the best features present in an image according to the defined classes. The slices obtained are directly input to the CNN for attribute extraction. The CNN architecture used in this work is illustrated in Fig. 1. We define three convolutional layers that apply 3 × 3 filters for feature extraction. Each layer increases the number of filters by multiplying by 2. The number of filters used in the first, second, and third layers were 16, 32, and 64 respectively. During the training process, the filters are automatically adjusted for activation in the most relevant features using Rectified linear units (ReLU) as activation function. Pooling layers follow the convolutional layers, where a single value replaces the values belonging to a region in the features map. This reduces the number of attributes and prepares the input for the next convolutional layer. Maxpool function with a 2 × 2 window is used for the pooling layers to obtain the highest values of each region. After the last pooling layer, a dropout layer is defined to ignore a certain number of neurons in the network, which reduces overfitting on CNN during training. The rate of neurons to be ignored in the network is set to 0.25 for the dropout layer. A Flatten layer is used to transform the multidimensional input tensor into a one-dimensional output tensor, which is an essential component of a neural network. Then, four dense layers with 128, 64, 32, and 4 units, respectively, are defined. Finally, the softmax activation function is used for multi-class classification, where each element of the output vector represents the probability that the input belongs to a particular class.

B. Transfer learning

Transfer learning is a machine learning technique that involves repurposing a pre-trained model, which has already been trained on a large dataset for a specific task, for a new and similar task with a smaller dataset. By leveraging the learned features of the pre-trained model, transfer learning enhances the performance of the model and avoids the need to start from scratch. In this work, five transfer learning approaches, namely VGG19, DenseNet121, Inceptionv3, MobileNet, and ResNet101v2, are employed to identify early detection of Alzheimer's disease. These models utilize the principles of transfer learning to leverage their pre-trained knowledge and contribute to improved accuracy and efficiency in detecting Alzheimer's disease at an early stage.

When using these five transfer learning approaches for early detection of Alzheimer diseases, it is a common practice to freeze the base layers of a pre-trained model and add new layers on top for fine-tuning. This can be achieved by following the steps below:

• Instantiate a pre-trained base model and load its weights. The base model consists of pre-trained layers that have been trained on a large dataset. • Freeze the base layers by setting their trainable attribute to False. This ensures that the weights of these layers will not be updated during training. the necessary input and target data, along with the appropriate batch size and number of epochs. Hereafter is a description of the initial architectures of the five transfer learning approaches utilized in our work prior to fine-tuning:

1) VGG19: VGG19 is one of the variations of the VGG models. It is composed of 19 layers, including 16 convolution layers, 3 fully connected layers, 5 MaxPool layers and 1 SoftMax layer. Aside from VGG19, other variations of the VGG model include VGG11, VGG16, and others.

The network architecture involved feeding a fixed-size RGB image of (224 × 224) pixels, forming a matrix of shape (224, 224, 3), as input. Prior to this, the only preprocessing step performed was subtracting the mean RGB value from each pixel, computed across the training set. Convolutional kernels of (3 × 3) size were employed, with a stride size of 1 pixel, to capture the entire image notion. To maintain the spatial resolution of the image, spatial padding was utilized. Max pooling was conducted over 2 × 2 pixel windows with a stride of 2. Rectified linear units (ReLU) were applied to introduce non-linearity and enhance the model's classification performance and computational efficiency. The architecture implemented three fully connected layers, with the first two having a size of 4096, and finally a softmax function is used.

2) MobileNet: MobileNet is a convolutional neural network architecture that is comprised of several types of Fig. 1: Flowchart of used methods layers, including convolutional, depthwise separable convolutional, and fully connected layers. In our work, we used the Mobilenet architecture due to its lightweight nature. Mobilenet achieves its lightweight design by utilizing depthwise separable convolutions, which perform convolutions on each color channel separately instead of combining them, resulting in fewer computations compared to standard convolution. This approach filters input channels and generates new features via separate depthwise and pointwise convolution operations, which considerably reduces the computational complexity of the model. The Mobilenet architecture is a linear stack of multiple neural networks, starting with a flat layer that transforms the input from the previous layer. A hidden layer with 256 neurons and SoftMax activation follows, followed by a Dropout layer to prevent overfitting. The convolution in the Mobilenet network is a separable convolution, which filters features based on convolution kernels and feature combinations, generating new representations. The depthwise separable convolution further breaks down the filtering process, leading to a significant reduction in computational cost. The Mobilenet model also incorporates two hyperparameters, the width coefficient, and the resolution coefficient, to further optimize the computational cost. The width coefficient is the ratio of the total number of convolution kernels used by each module to the standard Mobilenet, thereby reducing the need for secondary parameters. To implement Transfer Learning, the base layers of the model are frozen, and new layers are added as per the requirements of the problem being solved.

3) ResNet101v2: ResNet101v2 is a CNN architecture that has been applied to various computer vision tasks, including medical image analysis. With 101 layers, this architecture was developed to tackle the problem of vanishing gradients in deep neural networks. The ResNet architecture uses residual connections, which enables the gradient to flow through the network with ease, making it more trainable. The architecture also includes average pooling layers, which reduce the spatial dimensions of the feature maps while retaining important information. This pooling operation helps decrease the computational complexity of the model and aids in capturing the most relevant features. The final layers of InceptionV3 are fully connected layers responsible for classification. They leverage the extracted features to make predictions and are typically followed by a softmax activation function, which produces class probabilities.

5) DenseNet121: DenseNet121 is a popular CNN architecture introduced by Facebook AI Research. It consists of 121 layers and belongs to the DenseNet family of models. DenseNet121 is known for its dense connectivity pattern and efficient parameter usage. It includes dense blocks, transition blocks, and a classification head. Dense blocks connect every layer to each other, promoting feature reuse and better gradient flow. Transition blocks downsample feature maps, maintaining important information. Global average pooling and fully connected layers are used for classification. DenseNet121 has achieved state-of-the-art performances in image classification, object detection, and image segmentation tasks.

IV. Experimental results

A. Dataset

The Alzheimer dataset used in this study was obtained from Kaggle 1 and consists of 6400 images associated with four Alzheimer's stages: Mild Demented, Moderate Demented, Non Demented and Very Mild Demented (896, 64, 3200 and 2240 images respectively). The dataset is divided into two parts, training and test. The training dataset is then divided into training and validation, with a respectively ratio, 80% and 20%.

B. Evaluation Protocol

The proposed models are evaluated using a stratified cross-validation approach. The purpose of stratified crossvalidation is to maintain the same class distribution in each fold as in the original dataset, ensuring a representative evaluation. The dataset is divided into 5 equalsized folds. Stratification ensures that each fold contains a proportional representation of samples from each class, preserving the overall class distribution. This helps prevent any bias that may arise from imbalanced class distributions.

In this study, various evaluation metrics are calculated for each fold: accuracy, AUC, precision, recall and F1 score. The average and standard deviation of these metrics across all folds are then reported, offering a comprehensive understanding of the models' performances. The five metrics used for the evaluation can be defined, as below, using standard terms: True Positives (TP), False Positives (FP), True Negatives (TN) and False Negatives (FN).

• Accuracy: The percentage of correctly classified samples.

Accuracy(ACC) =

T P + T N T P + T N + F P + F N • P recision: The proportion of true positive predictions over the total number of positive predictions .

1 https://www.kaggle.com/datasets P recision = T P T P + F P • Recall: (sensitivity): It is the correct positive result amount to all relevant sample amount (i.e., the number of correctly classified AD cases divided by the total number of actual AD cases).

Recall(sensitivity) = T P T P + F N • F 1score: The harmonic mean of precision and recall, which balances both metrics.

F 1score = 2T P 2T P + F P + F N • AU C: Area under the curve of the receiver operating characteristic (ROC) curve: A metric that measures the ability of the model to distinguish between AD and non-AD cases across different thresholds.

T P R(sensitivity) = T P T P + F N F P R(1 -specif ity) = F P F P + T N These metrics can provide valuable insights into the performance of Alzheimer's classification models and help researchers and clinicians make informed decisions.

C. Results

In order to perform multi-class Alzheimer diseases classification, we have introduced 2DCNN and several fine-tuned transfer learning models such as VGG19, DenseNet121, Inceptionv3, MobileNet and ResNet101v2. These models were evaluated based on their accuracy and other metrics and compared to other state-of-the-art models. In Table I, a Comparison of the performances of the proposed models with the state-of-the-art models is presented. Fig. 2 illustrates the performance metrics, including accuracy, AUC, precision, recall and F1 score, of our models and its graphical representation.

The confusion matrix, shown in Fig. 3, for multi-class Alzheimer's disease classification consists of a square matrix with dimensions corresponding to the number of classes or stages of Alzheimer's disease being classified. Each cell in the matrix represents the number of instances that were predicted to belong to a particular class (rows) but actually belonged to a different class (columns). In our example, the rows represent the true class labels, while the columns represent the predicted class labels. Each value in the matrix represents the number of instances from the true class that were classified into the predicted class. The confusion matrix shows that almost all the images are correctly classified. This proves that our 2DCNN model is excellent for early prediction of Alzheimer's disease.

D. Discussion

In this section, we provide a performance comparison between our proposed models and state-of-the-art models and discuss the findings. According to the results presented in Table I, it is evident that our proposed 2DCNN model surpasses the other models in terms of performance. Specifically, the proposed 2DCNN model achieves the highest accuracy and AUC score for prediction, with an accuracy of 99.14% and an AUC score of 99.84%. The fine-tuned VGG19 model achieves the second-highest accuracy of 97.67% and an AUC score of 99.80%, while the fine-tuned DenseNet121 model achieves the third-highest accuracy of 97.09% and an AUC score of 99.61%.

On the other hand, Sharma and Kalpna in [START_REF] Sharma | A deep learning based convolutional neural network model with vgg16 feature extractor for the detection of alzheimer disease using mri scans[END_REF], using The reasons why CNN surpasses transfer learning models can be summarized as follows:

• CNN is trained from scratch for the specific task of classifying and recognizing AD stages. This taskspecific training enables the CNN model to adapt and optimize its parameters exclusively for Alzheimer's disease detection. In contrast, transfer learning models were pretrained on unrelated datasets and may not be as tailored to the specific task at hand. • CNN can be fine-tuned and optimized using domainspecific data. By training the CNN model on a large dataset of brain MRI images, it becomes more specialized in understanding the unique features related to Alzheimer's disease. Transfer learning models, which were pretrained on different tasks, may not possess the same level of domain expertise.

It's important to note that the performance of different models can vary depending on the dataset, task, and specific implementation. The superiority of CNN over transfer learning models in this study does not imply that CNN is always superior in all tasks.

4 )

 4 InceptionV3: The InceptionV3 architecture is a powerful CNN model developed by Google for image classification tasks. It was designed to address the challenge of capturing both local and global image information effectively. InceptionV3 consists of multiple convolutional layers that perform various operations, including convolutions with different filter sizes (1 × 1, 3 × 3 and 5 × 5). These convolutions are applied in parallel, allowing the model to capture features at different scales and abstraction levels. The outputs of the parallel operations are then concatenated along the channel axis, combining the information learned from different filter sizes. To enhance training and regularization, InceptionV3 incorporates auxiliary classifiers at intermediate layers. These auxiliary classifiers introduce additional loss functions during training, encouraging the network to learn more discriminative features. This helps improve the gradient flow throughout the network and can prevent the vanishing gradient problem.

Fig. 2 :

 2 Fig. 2: Performance and comparative analysis of our different deep learning classifiers

Fig. 3 :

 3 Fig. 3: Confusion matrix of the 2DCNN model (C 1 = Mild_Demented, C 2 = Moderate_Demented, C 3 = Non_Demented, C 4 = Very_Mild_Demented)

  V. ConclusionsIn this paper, we evaluated several neural network architectures for early Alzheimer's disease detection based on MRI images. The initial architecture utilized in this study is a 2D CNN, while the remaining architectures are based on the Transfer Learning principle. More specifically, the MobileNet, ResNet101v2, Inception-v3, DenseNet121, and VGG19 models are fine-tuned and employed for the detection of Alzheimer's disease across multiple classes. The experimental results demonstrate the effectiveness of the proposed 2D CNN architecture as well as the finetuned VGG19, DenseNet121 and MobileNet models for Alzheimer's classification.Future work includes exploring and studying new architectures such as Transformers and Convmixer for multiclass AD stage classifications and evaluating their performance. Additionally, data augmentation methods and the utilization of other databases will be considered to validate and compare the obtained results with existing findings.

•

  Add new layers on top of the base model. In our work, we add one additional dense layer and a softmax layer for the Alzheimer diseases classification task. • Compile the model with the categorical cross-entropy loss function and Adam optimizer. • Train the model on the training data by providing

TABLE I :

 I Comparison of the proposed models with the state-of-the-art models

	Reference	Model	Accuracy(%) AUC(%) Sensitivity(%) Precision(%) F1-Score(%)
	[35]	Vgg16	90.40	96.90	90.40	90.50	90.4
	[36]	AlexNet	94.53	-	100	-	94.12
	[36]	ResNet50	58.07	-	81.46	-	75.14
	[37]	DenseNet-169	88.70	-	-	-	-
	[37]	ResNet-50	81.98	-	-	-	-
	[9]	CNN	96.00	-	-	-	-
	Proposed 2DCNN	2DCNN	99.14	99.84	99.14	99.14	99.08
	Fine-tuned MobileNet	MobileNet	96.87	99.06	96.82	96.91	97.28
	Fine-tuned Resnet101v2	Resnet101v2	85.78	97.38	83.93	87.23	87.74
	Fine-tuned Inception_V3 Inception_V3	94.65	98.80	94.57	94.73	94.85
	Fine-tuned DenseNet121	DenseNet121	97.09	99.61	97.06	97.13	97.51
	Fine-tuned VGG19	VGG19	97.67	99.80	97.45	97.80	98.25


Rallou Thomopoulos
email: rallou.thomopoulos@inra.fr

Serafim Bakalis
email: serafim.bakalis@nottingham.ac.uk

CONSUMER DEMAND FOR SUSTAINABLE VERSUS LOW-COST FOOD PRODUCTS: AN AGENT-BASED MODELLING APPROACH

Keywords: Sustainable food systems, Agent-based modelling, Simulation, Consumer choice, Product price

This paper introduces a first agent-based modelling attempt to food system simulation. It aims to capture the balance of supply and demand for sustainable versus low-cost food products. It contributes in modelling consumer choice and more specifically understanding the tensions between price and sustainability.

INTRODUCTION

With global population reaching 9bn by 2050 and significant environmental challenges, there is an increasing need to increase sustainability of the food chain. Although consumers demand more sustainable products, price is often driving consumer choice (Hoek et al., 2016;[START_REF] Asioli | Making sense of the "clean label" trends: A review of consumer food choice behavior and discussion of industry implications[END_REF]. Sustainable products are typically at a higher cost leaving them to a competitive disadvantage. There has been a lot of effort in understanding and modelling consumer choice and more specifically understanding the tensions between price and sustainability. A range of theoretical frameworks have been proposed [START_REF] Ho | Modeling the Psychology of Consumer and Firm Behavior with Behavioral Economics[END_REF][START_REF] Asioli | Making sense of the "clean label" trends: A review of consumer food choice behavior and discussion of industry implications[END_REF]. Higher prices of sustainable products do not only involve higher costs of raw materials but also costs of production as they tend to be specialized small scale production units. It would be of interest to explore a scenario where by adoption of sustainable products from consumers costs are reduced.

Agent-based models (ABMs) offer a way of taking into account actor heterogeneity, social interaction and interdependence, adaptation, and decisionmaking at different levels. In land use-related environmental science, they have affirmed as a way to capture complex characteristics of socioecological systems [START_REF] Epstein | Growing artificial societies: social science from the bottom up. Complex adaptive systems[END_REF][START_REF] Berger | Agent-based spatial models applied to agriculture: a simulation tool for technology diffusion, resource use changes and policy analysis[END_REF][START_REF] Parker | Multiagent systems for the simulation of land-use and land-cover change: a review[END_REF], Mathiews et al. 2007[START_REF] Rammel | Managing complex adaptive systems -A coevolutionary perspective on natural resource management[END_REF][START_REF] Bakalis | Modeling human decisions in coupled human and natural systems: review of agent-based models[END_REF][START_REF] Filatova | Spatial agent-based models for socio-ecological systems: Challenges and prospects[END_REF]. In this family of ABM works, ecology and geography predominate [START_REF] Grimm | Ten years of individualbased modelling in ecology: what have we learned and what could we learn in the future?[END_REF][START_REF] Parker | Multiagent systems for the simulation of land-use and land-cover change: a review[END_REF], Verburg and Veldkamp 2005[START_REF] Castella | Participatory simulation of landuse changes in the northern mountains of Vietnam: the combined use of an agent-based model, a role-playing game, and a geographic information system[END_REF][START_REF] Railsback | Agent-based simulation platforms: review and development recommendations[END_REF]. However social sciences are also present in such ABM approaches to assess and explain the complexity of human decision-making processes and behaviors (Schelling 1971[START_REF] Epstein | Growing artificial societies: social science from the bottom up. Complex adaptive systems[END_REF], Simon 1997[START_REF] Janssen | An adaptive agent model for analysing co-evolution of management and policies in a complex rangeland system[END_REF][START_REF] Janssen | Empirically based, agent-based models[END_REF][START_REF] Robinson | Comparison of Empirical Methods for Building[END_REF]). In food science, historically based on processoriented studies, ABMs are absent from the range of models used to approach food quality and, as a more recent concern, food sustainability. Recent studies related to multi-agent systems applied to the agri-food sector are those based on the argumentation theory [START_REF] Bourguet | An artificial intelligencebased approach to deal with argumentation applied to food quality and public health policy[END_REF], Thomopoulos et al. 2015, Thomopoulos and Paturel 2017), some of them coupled with system dynamics simulation (Thomopoulos et al. 2017). Social issues are deeply involved in food system sustainability. Consumer demand, environmental awareness, willingness to pay, acceptability of products, spreading of new food habits, are key compounds to analyze the system. Therefore, there is a pressing need and a real challenge to take into account foodrelated social behaviors and integrate them in decision making. In this paper, a first ABM attempt to food system simulation is proposed. It aims to model the balance of supply and demand for sustainable versus lowcost food products. More specifically we aim to understand the dynamics of adoption of a certain product (namely sustainable vs low-price) including the scenario of changing price with an increase of consumption.

METHODS

Agent-based modelling

Agent-based models are computational models used to simulate the actions and interactions of autonomous agents, individual or collective, in order to assess their effects on the system as a whole. They attempt to reproduce and predict the emergence of complex phenomena induced from the micro-scale of systems to a macro-scale. Their principle is that simple local behavioral rules generate complex global behavior. An overview of their early history can be found in Samuelson (2000) and more recent developments in Samuelson and Macal (2006).

The CORMAS platform

The CORMAS platform (http://cormas.cirad.fr/) used for the model construction was specially developed by CIRAD for simulating natural resource management and is oriented towards the representation of interactions between stakeholders about resource use. In CORMAS, entities are categorized into three types: spatial entities describing the space at different aggregation levels, passive entities, which are objects that can be manipulated by social agents, and social entities which can make decisions, move, and interact with other agents.

Our hypotheses

 A cognitive theory of food perceptions and needs, explained in the following scenario description.  3 triggers for sustainable/low-price product demand: (i) environmental crises; (ii) neighborhood effect;

(iii) low-price attractiveness.

Scenario description

The system we intend to model is a country made of a population of N citizens. Two products are sold in this country: low-cost products, which have a constant and low price, and sustainable products, whose price is initially higher but may vary in time. Each citizen is characterized by: (i) a constant level of need for sustainability regarding food (environment preservation, ethics, health, etc.); (ii) a variable level of perception of sustainable products. Depending on these two levels, citizens have a behavior regarding their food product demand, either sustainable if their perception exceeds their needs, or low-cost if their needs exceed their perception. Moreover, each citizen has some resistance to change. Environmental crises break out with a given frequency. Then all citizens get the information, all the more since they are close to the crisis event.

Their perception of sustainable products may be impacted. Each citizen also communicates with his/her direct neighbors and may change his/her diet depending on theirs. Finally, citizens are influenced by low-price attractiveness.

The objective of the simulation model is to provide a view of the evolution of both food behaviors and product prices in time. Initial parameters subject to stochasticity are the distribution of need and perception levels in the population, with respect to some initial proportions of sustainable and low-cost buyers.

RESULTS

Conceptual model

Modelling choices for price-related variables

In this scenario and for simplicity we selected a linear decrease of price with the volume of sales/consumers.  Price function for the sustainable product: linear function of the number of buyers, from the initial price of the sustainable product when the number of buyers is reduced (e.g. 5% as an initialization), to the low-cost product price if the whole population buys the sustainable product.

 Feedback influence of price on consumer perception: linear function from 'no effect' when the price of sustainable products is high (initial situation) to 'total effect' when the price of sustainable products decreases to the level of low-cost products.

When a consumer's perception surpasses this consumer's needs for sustainable products, this consumer becomes a sustainable buyer, which participates in reducing the price of sustainable products.

Simulation results

Figure 2 displays a running simulation. Each step has been calibrated to correspond to about one month, so that the whole simulation corresponds to 15 to 20 years approximately. The associated evolutions of the volumes of sales and of product prices are given in the graphs of Figure 3.

Figure 2: A running simulation. The left part shows consumer perceptions of sustainable products (the darker, the higher). The right side displays resulting consumer behaviors, green for sustainable buyers, purple for low-cost buyers. The yellow stars materialize the occurrence of an environmental crisis which tends to increase people's perceptions of sustainable products, all the more since the crisis localization is close.

Figure 3: In the top part, evolution of the number of low-cost buyers (in red) and of sustainable buyers (in pink). In the bottom, evolution of the prices of the sustainable product (in green) and of the lowcost product (in blue).

DISCUSSION

We can see in this simulation, after some initial period of oscillations, a decrease of the price of the sustainable product until it reaches the low product price, jointly with a decrease of low-price buyers and an increase of sustainable buyers.

We showed that this result is depending on the ratio between environmental crisis frequency and lowprice attractiveness. The latter is represented in our model as the ratio between the initial sustainable price and the initial low-product price (which remains constant). This is illustrated in Figure 4.

Figure 4: Sensitivity analysis shows that a stability in the simulation results obtained is reached for a given ratio between environmental crisis frequency and low-price attractiveness. This ratio seems to be close to the ratio between the initial product prices (sustainable price/low-cost price).

CONCLUSION

The overall work demonstrates the potential to develop an ABM to predict the dynamics of change of a food chain. Environmental crisis has led to adoption of sustainable products from consumers.

We also evaluated the effect of a reduction of product price with an increasing volume. After a certain period where oscillations in the number of adopters of sustainable products occurred and once a critical mass of consumers was adopted we observed that the number of adopters increased dramatically. The above results indicate that there is the potential of extending the reach of sustainable products by reaching a critical mass. An interesting issue to further investigate is how the public opinion is formed in case of environmental crisis, including situations where the assumptions that the whole population is well informed and the news reliable do not hold.

Figure 1

 1 Figure 1 displays the UML class diagram associated with the simulation model.

Figure

  Figure 1: Class diagram