Mariette

Maja Musse
email: maja.musse@cemagref.fr

Mireille Cambert

François Mariette

NMR study of water distribution inside tomato cells: Effects of water stress

Keywords: NMR, T 1, T 2, plant cells, cell compartments

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INTRODUCTION

NMR relaxation time measurements have been used in several studies to investigate vegetal cells [START_REF] Hills | The influence of chemical and diffusive exchange on water proton transverse relaxation in plant tissues[END_REF][START_REF] Snaar | Probing water compartements and membrane permeability in plant cells by H NMR relaxation measurements[END_REF]. The relaxation signals from vegetal cells have generally been described by a multi-exponential behaviour reflecting different water compartments. The longitudinal (T 1 ) and transverse relaxation (T 2 ) times are known to be related to the water status in the 1 compartments, i. e. water content, water mobility and interactions between water and macromolecules.

Although NMR has been shown to be able to provide useful information about plant cells, the interpretation of the results is not always straightforward. The number of NMR signal components undoubtedly depends on the plant tissue, but it seems also be related to the measurement protocol and the fitting method, even when the classical CRMG sequence is used. For example, a single component of T 2 relaxation decay was reported in the whole tomato fruit by Tu et al. [START_REF] Tu | Tomato quality evaluation by component force and NMR spin-spin relaxation time[END_REF], while four components were found in the tomato pericarp by Duval et al. [START_REF] Duval | NMR study of tomato pericarp tissue by spin-spin relaxation and water self-diffusion[END_REF], Marigheto [START_REF] Marigheto | Characterisation of Ripening and Pressure-Induced Changes in Tomato Pericarp Using NMR Relaxometry[END_REF] as well as in our previous studies [START_REF] Musse | An investigation of the structural aspects of the tomato fruit by means of quantitative nuclear magnetic resonance imaging[END_REF][START_REF] Musse | Monitoring the postharvest ripening of tomato fruit using quantitative MRI and NMR relaxometry[END_REF]. Furthermore, the precise assignment of the NMR signal components to the specific subcellular compartments is still a subject of debate. Only a few investigations have been focused on the attribution of the NMR signal components by probing water compartments. Three components of the NMR signal of apple cells have been assigned to the vacuole, cytoplasm and wall/extracellular space by following the uptake of Mn 2+ in the tissue [START_REF] Snaar | Probing water compartements and membrane permeability in plant cells by H NMR relaxation measurements[END_REF]. The same component attributions were also proposed by Hills and Remigereau [START_REF] Hills | NMR studies of changes in subcellular water compartmentation in parenchyma apple tissue during drying and freezing[END_REF]. More recently, four components in T 2 were found in apple cells and were assigned to the vacuole, cytoplasm, extracellular space and cell wall [START_REF] Marigheto | Two-dimensional NMR relaxation studies of apple quality[END_REF][START_REF] Sibgatullin | Analysis of diffusion and relaxation behavior of water in apple parenchymal cells[END_REF]. However, this attribution and distribution does not necessarily hold for all types of plant tissue. For example, Mariette et al. [START_REF] Mariette | Effet du procédé de friture sur la répartition et l'état de l'eau mesurée par RMN et IRM[END_REF] extracted four components in potatoes, and the component with the shortest T 2 was attributed to the non-exchangeable starch protons, while other components were attributed to the water protons from the different cell compartments (vacuole, cytoplasm and cell wall). Two components have been extracted in carrot parenchymal tissue and they were attributed to the vacuole and cell wall/extracellular space compartments [START_REF] Hills | NMR studies of changes in subcellular water compartmentation in parenchyma apple tissue during drying and freezing[END_REF]. In the latter study, a numerical cell model describing cell compartments was used to explain the NMR signal. Furfaro et al. [START_REF] Furfaro | Multidimensional NMR cross-correlation relaxation study of carrot phloem and xylem. Part I: Component assignment[END_REF] recently proposed another interpretation of the T 2 relaxation in carrot xylem and phloem tissues, assuming a contribution of non-exchangeable macromolecular protons. From a T 1 -T 2 relaxation experiment performed at 23.4 MHz they identified three components and attributed them to cell water and to non-exchangeable protons from pectin and cellulose. For the tomato [START_REF] Duval | NMR study of tomato pericarp tissue by spin-spin relaxation and water self-diffusion[END_REF][START_REF] Musse | An investigation of the structural aspects of the tomato fruit by means of quantitative nuclear magnetic resonance imaging[END_REF], the components were assigned to the vacuole, cytoplasm, cell wall and non-exchangeable solute protons by association with the potato, but no experiment was performed to validate this hypothesis. Most recently, Marigheto et al [START_REF] Marigheto | Characterisation of Ripening and Pressure-Induced Changes in Tomato Pericarp Using NMR Relaxometry[END_REF] met difficulties with the assignment of the four relaxation time components in green tomato pericarp. Indeed, two interpretations were proposed according to the experiments. The first interpretation associated the components with water in the vacuole, extra-cellular water, cytoplasm and the more rigid components of the cell wall and the second one associated the components with water in the vacuole, cytoplasm/extra-cellular compartments, cell wall and water inside starch granules. Further investigations are therefore necessary to make progress in the attribution of NMR signal components to the subcellular compartments for different plant types. This step is essential to provide accurate interpretation of the measurements and thus make it possible to use all the potential of NMR techniques in plant research.

Quantitative information about subcellular compartments via relaxation time measurements can be particularly relevant for investigation of different processes undergone by plants, such as ripening, osmotic dehydration and water stress. The macroscopic transport of water through tissues is mainly controlled by the microscopic distribution of water and air on a cellular and subcellular distance scale and by the extent of membrane permeability. It is thus important to investigate the subcellular water distribution that occurs during plant transformations in order to understand macroscopic water transfer. This is very important for use in environment-stressed plant research and can be applied for assisting agricultural practices. Such studies have been performed on different types of plant for different processes. For example, changes in subcellular water distribution during ripening have been studied in tomato [START_REF] Marigheto | Characterisation of Ripening and Pressure-Induced Changes in Tomato Pericarp Using NMR Relaxometry[END_REF][START_REF] Musse | Monitoring the postharvest ripening of tomato fruit using quantitative MRI and NMR relaxometry[END_REF] and banana [START_REF] Raffo | Ripening of banana fruit monitored by water relaxation and diffusion H-1-NMR measurements[END_REF] fruits. Iwaya-Inoue et al. [START_REF] Iwaya-Inoue | H-1-NMR method enables early identification of degeneration in the quality of sweet potato tubers[END_REF] monitored post-harvest sweet potato tubers by NMR and identified changes in water status in subcellular compartments produced by water stress. Capitani et al. [START_REF] Capitani | In Situ Investigation of Leaf Water Status by Portable Unilateral Nuclear Magnetic Resonance[END_REF] used a portable unilateral NMR instrument to detect in field conditions the water status of leaves of herbaceous crops, mesophyllous trees, and natural Mediterranean vegetation.

The aim of the study presented here was to provide greater understanding of NMR relaxation in tomato pericarp tissue in order to improve the knowledge about the attribution of NMR signal components to subcellular compartments and to investigate the response of specific cell compartments to water stress. Two kinds of experiment were performed. Firstly, two dimensional T 1 -T 2 relaxation correlation experiments were carried out on fresh pericarp samples, allowing the identification of the molecular species based on the T 1 /T 2 ratio [START_REF] Godefroy | 2D relaxation/diffusion correlations in porous media[END_REF].

This experiment made it possible to determine whether all components of the multiexponential NMR signals corresponded to subcellular water compartments, and particularly whether the component characterised by the shortest T 2 relaxation time corresponded to water or to non-exchangeable protons from non-aqueous molecules. Pericarp samples were then exposed to water stress and the NMR measurements were performed continuously during water rebalancing. Tomatoes were considered at two physiological stages by using green and red ripe tomato samples.

MATERIALS AND METHODS

Plant Material

Tomatoes (Lycopersicon esculentum Mill. var Admiro) provided by the CTIFL (Centre Technique Interprofessionel des Fruits et Légumes, France) were used in this study. Fruits for T 2 , T 1 and T 1 -T 2 measurements were picked at the late green stage (tomato colour code 3-4, CTIFL, France) and left to ripen in a constantly aerated ripening chamber under controlled conditions (18°C and 55%RH) for eight days. Five measurements were performed on samples taken from individual fruits. In the water stress experiment, a freshly picked green tomato (colour code 3-4) was compared to a freshly picked red tomato (colour code 7).

Three measurements were performed on samples taken from the green tomato, and three on samples from the red tomato.

For each experiment, samples were prepared from the outer pericarp as follows: a one centimeter thick slice was cut in the equatorial region perpendicular to the pedicle axis, and cylinders (0.8 cm in diameter) were cut into the tissue. Samples were wiped to remove water from the broken cells and then placed in NMR tubes and closed with a cap.

After NMR experiments, the water content of samples was estimated by measuring differences in weight after drying in an oven at 103°C for 24 hours.

A partial drying protocol used for the water stress experiment that significantly decreases the water content without cell deterioration was set up prior to measurements. In this protocol, cylindrical tissue samples were dried under different temperatures and pressures and cells were observed using a microscope (Nikon Eclipse 80i) equipped with a digital camera (DXM 1200c). The optimum protocol was found to be drying of samples in a vacuum oven at 30°C and 50 mbar for 45min. Samples were placed in caps during drying.

Methods

NMR Relaxometry measurements were performed on a 20 MHz (0.47 T) spectrometer (Minispec PC-120, Bruker, Karlsruhe, Germany) equipped with a thermostatted probe at 18°C.

Separate T 2 and T 1 measurements were performed in addition to the T 1 -T 2 measurements for comparison purposes. The Carr-Purcell-Meiboom-Gill (CPMG) sequence was used for T 2 measurement, 90°-180° pulse spacing was 0.1 ms and 5000 even echoes were recorded.

The duration of the 90 ° pulse was 18 μs. Data were averaged over eight acquisitions and the recovery delay was 12 s. T 1 was measured using a saturation recovery sequence (SR).

One hundred points were acquired from 30 ms to 12 s. The saturation-recovery times (t SR )

were spaced according to :

3 0 f 0 SR ) 1 n ( ) t t ( x ) 1 i ( t ) i ( t - - + + = (1)
with t 0 and t f the first and the last recovery time, respectively, and n the number of recovery time values used.

The relaxation curves were fitted by Scilab software according to the maximum entropy method (MEM) [START_REF] Mariette | Continuous relaxation time distribution decomposition by MEM[END_REF], which provides a continuous distribution of relaxation components without any assumption concerning their number, and the Levenberg-Marquardt algorithm which will fit the data to the multi term exponential model. In the latter method, the number of terms that best described the relaxation curve was determined by examining the residual plots and the values of the coefficient of determination. T 2 relaxation curves were fitted using the Levenberg-Marquardt algorithm according to the equation:

) T / t exp( I ) t ( I i , 2 i i 0 2 T - = ∑ (2)
where I 0i is the intensity of the i th exponential at the equilibrium state and T 2,i the characteristic transverse relaxation time for the i th exponential. T 1 relaxation curves were fitted using the equation:

)) T / t exp( 1 ( I ) t ( I i , 1 i i 0 1 T - α - = ∑ (3)
where I 0i is the intensity of the i th exponential at the equilibrium state and T 1,i the characteristic longitudinal relaxation time for the i th exponential. α  is a parameter that takes into account the 90° RF pulse imperfection, its value is being generally equal to one for a perfect 90° RF pulse .

A 90° RF pulse was added before the CPMG sequence to provide a sequence for 2D T 1 -T 2 correlation measurements (SR-CPMG). The experiment was performed by varying t SR and consequently the T 1 -weighting of the CPMG decay. We used 50 values between 30 ms and 12 s for t SR .

Assuming that each compartment is characterised by one T 2 and one T 1 value, the signal amplitude is given by the following equation:

I(t CPMG ,t SR )= (4) )) T / t exp( 1 )( T / t exp( I i 1 SR i 2 CPMG i i , 0 - α - -

∑

The SR-CPMG 2D signals were fitted by a two-step method. Firstly each CPMG curve was fitted by a multi-exponential model in order to extract the T 2 values and the associated intensities. The intensity of each T 2 component was then plotted against t SR and the SR decay curve was fitted with equation 3.

In the experiment to study water rebalancing inside cell compartments after subjection to water stress, T 2 relaxation decay was measured on the samples before applying the stress, at the end of the stress application and every three minutes for three hours. The measurements were performed on samples taken from the green and red tomatos.

In addition to the dynamic measurements, the amplitudes of the NMR signals from the outer pericarp before and after subjection to water stress were compared to the amplitude of the signal expected from distilled water. The signals expected from water were calculated for each sample as the product of its mass, its water content and the signal of the distilled water per unit mass. Three samples of the outer pericarp of the red tomatoes were used for this experiment. The aim was to investigate whether all components of the multi-exponential NMR signals could be assigned to water in cell compartments.

RESULTS AND DISCUSSION

T 2 , T 1 and T 1 -T 2 Correlation Measurements

Relaxation decays of the CPMG sequence were well described by four separate components, and decays of the SR sequence by two components, independently of whether the adjustment method was continuous (MEM) or discrete (Levenberg-Marquardt algorithm).

An example of transverse and longitudinal relaxation distribution is given in Fig. 1 and the results obtained by the discrete adjustment method are summarized in Table1 (1D). The relaxation times and relative signal amplitudes were in excellent agreement with our previous results performed on other tomato samples [START_REF] Musse | An investigation of the structural aspects of the tomato fruit by means of quantitative nuclear magnetic resonance imaging[END_REF].

Component 4 with the longest T 2 of 1528 ms and 68% relative signal intensity, was associated with the water in the vacuole, because of its high relative intensity and its long relaxation time. In fact, the vacuole contains about 50-80% of the cell water according to tissue [START_REF] Hopkins | Physiologie végétale[END_REF]. The attribution of the other three components is less obvious. However, it can be assumed that component 3, with T 2 =551 ms and a relative signal intensity of 20%, was associated with the cytoplasm, while component 2, with T 2 =134 ms and a relative signal intensity of 9%, was assigned to the cell wall. The interpretation of components 2 and 3 is not straightforward at this stage as the cell wall of some vegetable tissues may contain a greater water fraction than cytoplasm [START_REF] Hopkins | Physiologie végétale[END_REF]. The interpretations given in the literature [START_REF] Duval | NMR study of tomato pericarp tissue by spin-spin relaxation and water self-diffusion[END_REF][START_REF] Marigheto | Characterisation of Ripening and Pressure-Induced Changes in Tomato Pericarp Using NMR Relaxometry[END_REF][START_REF] Sibgatullin | Analysis of diffusion and relaxation behavior of water in apple parenchymal cells[END_REF] differ for component 1, characterised by the lowest T 2 and the lowest relative intensity, and no experimental study to our knowledge has attributed it to a specific cell compartment.

T 2 relaxation curves recorded in the SR-CPMG experiments with different t SR delays were fitted independently, according to the four-exponential transverse relaxation decay given by equation 2 for i=4, and T 2 values are plotted in Fig. 2. Variation in T 2 was relatively low: about 28% for the component 1, 19% for component 2, 12% for component 3 and 4% for component 4. For the components 1 and 2 variation in T 2 was calculated for t SR <3000 ms, and for the components 3 and 4 variation in T 2 was calculated for t SR <9000 ms. In the case of components 3 and 4, the first seven points were eliminated from the statistics as aberrant, due to the low signal to noise ratio of the NMR signal for the very short t SR values, and consequently to less accurate fitting. The T 2 variation for each t SR value was in agreement with the standard deviation of the T 2 parameter. It can be concluded that the T 2 value did not depend on t SR , indicating that each signal component corresponded to a distinct proton population.

After fitting the CPMG signal, the intensities of each component were plotted according to t SR to provide a T 1 relaxation decay for each T 2 component. T 1 was calculated for each component by fitting with the mono-exponential function given by equation 3 for i=1 (Fig. 3).

The T 2 and T 1 values obtained with the SR-CPMG sequence are summarized in Table 1 (2D) and compared to the values obtained with the independent T 2 and T 1 measurements. T 2 values obtained from the two methods matched well, whereas slight differences were observed in T 1 measurements. These differences can be explained by the relatively low signal to noise ratio of the SR-CPMG sequence compared to the 1D SR measurements, and consequently to the less accurate fitting procedure. The SR-CPMG experiment confirmed that the relaxation decay of the pericarp tissue can be described for both T 1 and T 2 by four relaxation components. Despite we used a two-step method for the fitting of the T 1 -T 2 data, a number of relaxation component is in agreement with the results from Marigheto [START_REF] Marigheto | Characterisation of Ripening and Pressure-Induced Changes in Tomato Pericarp Using NMR Relaxometry[END_REF] obtained with a 2D inverse-Laplace transformation. Fig. 4 shows the T 1 -T 2 distribution map. The T 1 /T 2 ratio was constant, which demonstrated that the relaxation mechanisms involved were the same in all subcellular compartments [START_REF] Hurlimann | Quantitative characterization of food products by two-dimensional D-T-2 and T-1-T-2 distribution functions in a static gradient[END_REF].

It was greater than 1 (shown as a dashed line in the T 1 -T 2 plot) as expected for biological tissues, because of the presence of chemical exchanges. Thus all components corresponded to water protons, and no other non-exchangeable protons were detected in this experiment. Regarding these results and the results from our previous measurements which showed that the component 1 was present throughout the maturation process [START_REF] Musse | Monitoring the postharvest ripening of tomato fruit using quantitative MRI and NMR relaxometry[END_REF], the component with the shortest transverse relaxation time could correspond to water in organelles, such as plastids.

Water stress experiment

Comparison of the pure water NMR signal and the NMR signals from the tomato pericarp before and after applying water stress.

The amplitudes of the signals from the fresh and dried tomatoes were compared with the expected signals from the pure water calculated for the same mass and water content as the tomato samples. The results are summarised in Table 2. The difference between the measured signal intensity (I M ) and the expected signal intensity (I E ) was insignificant for both the fresh and the dried tomatoes. Compared to what would have occurred if the nonexchangeable protons had contributed to the signal, the expected signals from the pure water were always slightly lower than the measured signals. The small difference between the signals could be explained by the slight contribution of the exchangeable protons to the water signal. These results demonstrated that only the water signal was detected and that all components of the NMR signal corresponded to water in different cell compartments. The hypothesis of non-exchangeable protons contributing to the NMR signal was thus no longer valid, the signal of other proton-containing compounds decayed too quickly for our measurements. The results of this experiment were in agreement with the results in the previous section.

Rebalancing of water in tomato tissue after applying water stress

Fig. 5 shows the variations in NMR signal intensity (CPMG) according to time before and after applying water stress for the green (a) and the red tomatoes (b). The intensity was normalised by the initial sample mass in order to make a direct comparison between the experiments. Four components were detected in the NMR signal in all measurements, indicating that the internal cell organization was not affected by water stress and that water was distributed between the same cell compartments. It should be noted that the observation under the microscope showed that the cell walls were not damaged by drying. respectively. After applying the water stress, the sum of signal intensities was constant during the rebalancing period (i.e. 10.79±0.01 and 10.27±0.06 for the green and red tomatoes, respectively), demonstrating that no dehydration occurred during the three hours.

The curves shown in

However, the water distribution between cell compartments of the fresh samples (first point of the curve, Fig. 5) differed between different ripening stages. The relative signal intensities of each cell compartment at different points are given in Table 3. For the fresh green tomato, the water in the vacuole corresponded to about 70±5% of the total amount of water, compared to 64±5% for the fresh red tomato while the relative intensity of component 2 of the fresh samples was lower for the green than for the red tomato. These results are consistent with the results of our previous study [START_REF] Musse | Monitoring the postharvest ripening of tomato fruit using quantitative MRI and NMR relaxometry[END_REF].

After applying water stress for both the green and red tomatoes, there was a decrease in the signal intensity of components 4 and 3, corresponding to the vacuole and the cytoplasm, respectively. The intensity of component 2 increased slightly for the green tomato and decreased for the red tomato, whereas the intensity of component 1 increased for both tomatoes. Hills et al. [START_REF] Hills | NMR studies of changes in subcellular water compartmentation in parenchyma apple tissue during drying and freezing[END_REF] demonstrated that air-drying of apple parenchyma tissues resulted in loss of water from the vacuole, with very little change in the water content of the other compartments. However, the protocol used in their experiments included a 24-hour storage period between the drying step and the NMR measurements. This storage period was used to allow water equilibration between cell compartments. Our experiments demonstrated that in the absence of an equilibrium period both compartments (vacuole and cytoplasm) were affected by water stress. The difference in behaviour of component 2 between the green and the red tomatoes could indicate the attribution to the cell wall as the ability of the cell wall to retain water is influenced by changes in cell wall polysaccharides during fruit ripening.

Further studies are however required to verify this hypothesis.

In the second part of the curves (describing the rebalancing period), the intensity of component 4 decreased significantly and the intensity of component 3 increased for both the green and red tomatoes. The intensity of component 2 increased significantly for the green tomato, while it remained almost constant for the red tomato. Component 1 was stable for both tomatoes. It should be noted here that component 1 was present in this experiment for both green and red tomatoes according to our previous study [START_REF] Musse | Monitoring the postharvest ripening of tomato fruit using quantitative MRI and NMR relaxometry[END_REF].

The NMR experiment reflected here one of the roles of the vacuole, i.e. to supply water to other cell compartments when a cell is submitted to stress. The vacuole therefore continued to transfer water to the cytoplasm and the cell wall, thus ensuring cell endurance. It can be computed form the data presented in Fig. 5 that in the case of the green tomato the relative signal intensity loss (reflecting relative amount of water loss) produced by water stress was lower for component 4 than for component 3 (48% and 72%, respectively) while it was almost equal for these two components for the red tomato (53% and 57%, respectively), demonstrating that the response of specific cell compartments to water stress depends on the stage of maturation of the fruit.

On Fig 6 a and b, T 2 is plotted according to time before and after applying water stress for green and red tomatoes, respectively. T 2 remained unchanged for the green tomato between the first two points (corresponding to time before and after applying water stress) for components 4 and 3, while it decreased slightly for component 2, and increased slightly for component 1. In the red tomato, T 2 decreased between the first two points for all components, except for component 4.

The T 2 of water is known to be related to water content and membrane permeability [START_REF] Van Der Weerd | Nuclear magnetic[END_REF].

Thus for constant membrane permeability a decrease in the water content should induce a decrease in T 2 . Moreover, the T 2 of water is sensitive to the structure and the mobility of the surrounding molecules through chemical exchange and dipole-dipole interactions. Therefore a change in the non-aqueous molecular mobility should induce a change in the T 2 of water.

According to these two mechanisms, we expected a decrease in T 2 for components 4 and 3, which exhibited the greatest signal loss. However, the T 2 of these components remained constant after applying water stress for the green tomato, whereas it decreased only slightly for component 3 of the red tomato. It should be noted here that lack of sensitivity of the T 2 to the water loss was also reported by Hills et al. [START_REF] Hills | NMR studies of changes in subcellular water compartmentation in parenchyma apple tissue during drying and freezing[END_REF]. In fact, in apple cells the T 2 of the vacuole remained constant for below 45% of the mass loss, which corresponds to the mass loss which occurred in our experiment. This could therefore mean that T 2 is not sensitive to relatively low water loss (in the case of the tomato pericarp, 50% of the mass loss corresponds to 5% of water content loss). The slight variation observed can be explained by changes in the structure of molecules dispersed in the water phase induced by the treatment.

The latter phenomenon could explain the differences observed between the green and red tomatoes. Another hypothesis is that lack of significant changes in T 2 is due to a decrease in the permeability induced by the treatment.

During the rebalancing period the correlation between the T 2 and the NMR signal amplitude observed was as expected. For the green tomato, both the T 2 and the signal intensity of the vacuole decreased. For the red tomato, the T 2 of the vacuole decreased for about 20 minutes and after that it remained mostly constant, corresponding to the decrease in signal intensity which was marked at the beginning of the post-stress period. The T 2 was constant for the other components.

CONCLUSION

The aim of this study was to improve understanding of the multi-exponential NMR relaxation in the tomato fruit pericarp and of the link between signal components and the cell compartments. Moreover, we investigated by NMR the water rebalancing between cell compartments after water stress.

The two-dimensional T 1 -T 2 correlation measurements, and the comparison of the amplitude of the NMR signals from the distilled water and the outer pericarp before and after water stress, showed clearly that all components corresponded to the water in different cell compartments, and that no signal from non-exchangeable molecules was observed.

The NMR study of the response of specific cell compartments to water stress showed that the vacuole represented the water pool for other cell compartments and that it exchanged water principally with components probably corresponding to the cytoplasm and cell wall. This experiment demonstrated that the NMR relaxation technique is suitable for the continuous monitoring of water rebalancing between subcellular compartments of plant tissues. The method is promising for use in environment-stressed plant research, although further studies are required to evaluate it for physiological changes of water content.

The results of this study represent an important step towards further studies on NMR signal analysis, although more work remains to be done to understand the origin of the signal component with the lowest relaxation time and its accurate assignment to a cell Table 3 : Relative signal intensifies for four cell compartments before applying water stress (i), immediately after (ii) and three hours later (iii) for the green and red tomatoes.

  Fig 5 can be divided into two parts: the first part (the first (i) and second (ii) points) related to the effects of water stress on the cells and second part (from point (ii) to the end (point (iii)) describing the water transfer between the subcellular compartments during the three-hour rebalancing period. The sum of the signal intensities of the four components did not significantly differ between the green and the red tomatoes. For the first point (fresh samples) it was 20.55 and 20.49 for the green and the red tomatoes,

Fig. 1 :

 1 Fig. 1 :Transverse (a) and longitudinal (b) relaxation time distributions (MEM) for a tomato

Fig. 2 :

 2 Fig. 2 : Four transverse relaxation time components for a tomato pericarp sample according

Fig. 3 :

 3 Fig. 3 : SR-CPMG signal decomposed according to T 2 distribution (•-component 1, ▲component 2, ■ -component 3 and ♦ -component 4) according to t SR . T 1 was calculated for each component by fitting with the mono-exponential model.

Fig. 4 :

 4 Fig. 4 : T 1 -T 2 correlation spectrum for the tomato pericarp. The solid line corresponds to the linear fit of the experimental data and the dashed line corresponds to T 1 =T 2 .

Fig. 5 :

 5 Fig. 5 : Variations in the NMR signal intensity (CMPG) for four cell compartments according

Fig. 6 :

 6 Fig. 6 : Variation in transverse relaxation times (CMPG) for four cell compartments (log scale)

Table 1 :

 1 Results from the T 2 (CPMG), T 1 (SR) and T 1 -T 2 correlation measurements (SR-CPMG) on the tomato pericarp sample obtained by the discrete adjustment method. Mean values and standard deviations result from five individual measurements. Table 2 : Amplitudes of the signals from the fresh and dried tomatoes were compared with the expected signals from the pure water at the same mass and water content as the tomato samples. The signals were normalised by the mean sample masses.

  Fig.1

  Fig.4

  %] I 22 [%] I 23 [%] I 24 [%] I 21 [%] I 22 [%] I 23 [%] I 24 [%]

	Tab. 3	Tab. 2	1 D		2 D
	I 21 [Point i 3±0	T 2 (CPMG) I [%] T 2 [ms] I [%] T 1 [ms] T 1 (SR) Measured signal T 2 -T 1 (SR-CPMG) I [%] T 2 [ms] T 1 [ms] Green tomato Red tomato Expected signal Water content [%] intensity (I E ) [a.u.] intensity (I T ) [a.u.] 4±1 22±5 70±5 3±0 9±1 24±4 64±5
	Component 1 4±1 Point ii 9±0 10±2 12±2 69±4 18±4 18±4 455±66 8±0	7±4 12±1 21±3 59±4 18±4 233±50
	Component 2 9±1 Fresh tomato Point iii 9±0 20±2 24±2 47±4 10±0 16±2 28±4 46±3 134±10 82±4 1644±44 13±5 121±20 362±23 95.2±0.0 3.94±0.01 3.91±0.01
	Component 3 20±3 551±49 Dried tomato 90.6± 0.1	24±4 2.08±0.01	552±66	908±88 2.03±0.01
	Component 4 68±5 1528±62	56±11 1573±67 1705±119

Acknowledgment: The authors wish to thank CTIFL for their collaboration, especially Mr.

Letard. We also thank Miss Drouet for her contribution to the experimental work.

REFERENCES :


Olivier Bouré

Nazim Fatès
email: nazim.fates@loria.fr

Vincent Chevrier
email: vincent.chevrier@loria.fr

Boure

Robustness of Cellular Automata in the Light of Asynchronous Information Transmission

Keywords: asynchronous cellular automata, discrete dynamical systems, robustness, phase transitions, directed percolation

Cellular automata are classically synchronous: all cells are simultaneously updated. However, it has been proved that perturbations in the updating scheme may induce qualitative changes of behaviours. This paper presents a new type of asynchronism, the β-synchronism, where cells still update at each time step but where the transmission of information between cells is disrupted randomly. We experimentally study the behaviour of β-synchronous models. We observe that, although many effects are similar to the perturbation of the update, novel phenomena occur. We particularly study phase transitions as an illustration of a qualitative variation of behaviour triggered by continuous change of the disruption probability β.

Introduction

Cellular automata are a parallel, spatially-extended, model of computation, which has been studied as an alternative to the sequential computing models, for instance Turing machines. By their very structure, they are well-suited for modelling natural phenomena and for the design of massively parallel algorithms. These mathematical objects have been mostly considered in the synchronous case, that is, when all their components are simultaneously updated. However, this hypothesis of perfect synchrony is somehow inadequate when modelling systems that are subject to noise or non-ideal information transmission, as often met in various natural systems or in asynchronous parallel computing devices. This is why authors tackled the question of whether a cellular automaton is robust to non-ideal updating, either without changing its local transition rule (e.g. [START_REF] Fatès | An experimental study of robustness to asynchronism for elementary cellular automata[END_REF]), or by adding adapted constructs (e.g. [START_REF] Peper | Self-timed cellular automata and their computational ability[END_REF]).

The aim of this paper is to study the robustness of cellular automata by considering the possibility of disruptions in the transmission of information between cells. To this end, we describe the updating process of the system in the frame of a cellular cycle. This cycle consists of two steps: (a) the local computation and (b) the transmission of the updated state of a cell to its neighbours. This dichotomy induces two related types of asynchronism: the update-wise asynchronism, or α-synchronism, which disrupts step (a) and a novel type of asynchronism, the influence-wise asynchronism or β-synchronism, which disrupts step (b) and perturbs the interaction between a cell and its neighbours.

The α-synchronous updating is now a relatively well-studied perturbation, whose effect can either be "inoffensive" or trigger drastic qualitative changes, such as phase transitions [START_REF] Fatès | Asynchronism induces second order phase transitions in elementary cellular automata[END_REF][START_REF] Fatès | Does Life resist asynchrony?, Game of Life Cellular Automata[END_REF][START_REF] Grilo | Effects of asynchronism on evolutionary games[END_REF]. Our goal is now to examine the effects of β-synchronism on simple cellular automata, in particular, in comparison with those of α-synchronism. We aim at extending the range of perturbations, in order to gain insight on how complex collective behaviour emerges from numerous simple local interactions.

On the formal side, this type of asynchronism leads us to extend the cell state space in order to distinguish the eigenstate, the actual state of the cell, and the observable state, the state of the cell which is observed by its neighbours. The corresponding definitions are introduced in Sec. 2 while experimental observations are presented in Sec. 3. We then analyze more particularly the occurrence of phase transitions in Sec. 4 and proceed to bring discussion in Sec. 5.

Asynchronous Cellular Automata

Cellular Automata

A synchronous cellular automaton is a discrete dynamical system defined by A = {L, Q, N , f } where :

• L ⊂ Z d the array of the cellular space, where an element of L represents a cell.

• Q is a nonempty finite set of states.

• N ⊂ L is a finite set of vectors called the neighbourhood, which associates to a cell the set of its neighbouring cells. N and L are such that for all c ∈ L and for all n ∈ N , the neighbour c + n is in L.

• f is the local transition rule, which defines the next state of a cell according to the state of this cell and the ones of its neighbours.

A configuration x t represents the state of the automaton at time t; it is defined as a function x t : L → Q which maps each cell to a state. Classically, cellular automata are synchronously updated, meaning that at each time the local transition rule is applied simultaneously of all cells. The global transition function is therefore defined as x t+1 = F (x t ), so that, for N = {n 1 , ..., n k }: A triplet represents a cell with its state (y) and the observable states of its neighbours (x for left cell and z for right). The prime sign indicates states that are updated. Solid arrows show the updates of states, otherwise states are conserved from the previous step. In α-synchronism, the local transition update applies with probability α and in β-synchronism, the information transmission is applied with probability β. The lower arrow is the local transition update, which performs y = f (x, y, z) when applied. The forking upper arrow is the information transmission, which performs z L = y and x R = y when applied.

∀c ∈ L, x t+1 (c) = f x t (c), x t (c + n 1 ), ..., x t (c + n k ) . α β y' z x y z x z' x' y' t x' R z' L y L y R y' L y' R
Without loss of generality, we assume that the neighbourhood N does not contain the cell itself. This hypothesis is necessary to explicitly represent the flow of information between a cell and its neighbours. Note that this does not restrict the expressiveness of f since the current state of a cell x t (c) is always a parameter of f , possibly not taken into account in the transition calculus.

Asynchronism as a disruption of cell activity

Cell cycle. The update of a cell can be represented by a cell cycle, which we decompose into two steps (see Fig. 1):

• the state update step, where a cell changes its state according to the local transition function.

• the information transmission step, where the cell transmits the updated state to its neighbours.

We give an example of the cell cycle for the different updating schemes on Fig. 2. The update-wise asynchronous updating in cellular automata, or αsynchronism [START_REF] Fatès | Does Life resist asynchrony?, Game of Life Cellular Automata[END_REF], is defined as follows: at each time step, each cell is updated with a fixed probability α, or else left unchanged. We introduce a new type of asynchronism, the β-synchronism, where each cell is always updated but the transmission of the new state to the neighbourhood is realized with a fixed probability β. As a result, both perturbations consist in applying one of the two steps of the cell cycle with a probability defined as the synchrony rate.

Please note that for the sake of simplicity, α and β denote both the type of asynchronism and the associated synchrony rates. Formally, we introduce a selection function ∆ α : N → P(L) which returns for time t the subset of cells to be updated, where each cell has a probability α to be selected. Note that when α = 1 the updating is fully synchronous and the system is deterministic. The global transition function becomes ∀t ∈ N, ∀c ∈ L, N = {n 1 , ..., n k }:

f ∆α (x t (c)) = f (x t (c), x t (c + n 1 ), ..., x t (c + n k )) if c ∈ ∆ α (t) x t (c) otherwise.
β-synchronism. To define this new asynchronism, we need to extend the classical definition of cellular automata by taking into account the difference between the eigenstate of a cell, and the observable state.

Let us consider a cellular automaton

A = {L, Q, N , f }. From A we derive a new cellular automaton A = {L, Q , N , f } where : • Q = Q 2 is the new set of states.
For a given configuration x t , a cell state is denoted by

x t = x t e x t o with
x t e : L → Q the eigenstate of the cell, and x t o : L → Q the state of the cell observable by its neighbourhood.

• the local transition function f is splitted into two parts to decompose its action: state update and information transmission.

We thus write f = f t • f u , so that:

• f u : Q k+1 → Q is the update function, which computes the new state of the cell based on its eigenstate and the observable state of the neighbours:

f u e o , . o 1 , ..., . o k = f (e, o 1 , ..., o k ) o • f t : Q → Q is the transmission function,
which replaces the observable state by the eigenstate:

f t e o = e e .
We use a similar selection function ∆ β as introduced for α-synchronism, where cells are selected with probability β, and define the global transition function as, ∀t ∈ N, ∀c ∈ L, N = {n 1 , ..., n k }:

f ∆ β (x t (c)) = f t • f u (x t (c), x t (c + n 1 ), ..., x t (c + n k )) if c ∈ ∆ β (t) f u (x t (c), x t (c + n 1 ), ..., x t (c + n k )) otherwise.

Models studied

Now that we defined our two updating schemes for cellular automata, we choose to study their effects on well-studied models.

The Game of Life. This 2-dimensional cellular automaton is expressed in our formalism as A GL = {L, N , {0, 1} , f } where:

• L = {Z/LZ} × {Z/LZ} is a square grid of size L with periodic boundary conditions.

• N = {c ∈ L, ||c|| = 1} represents the 8-cell (Moore) neighbourhood.

• the local transition function f : Q 9 → Q is outer-totalistic, that is, it can be written f (x t (c), x t (c + n 1 ), ..., x t (c + n 8 )) = δ(x t (c), s) where s = c ∈N x t (c ).

If x t (c) = 0, then δ(x t (c), s) = 1 if s = 3 0 otherwise. (Birth rule) If x t (c) = 1, then δ(x t (c), s) = 1 if s ∈ {2, 3} 0 otherwise. (Survival rule)
Elementary Cellular Automata (ECA). An ECA is a 1-D binary cellular automaton with nearest-cell neighbourhood, whose transition function is determined according to Wolfram's notation. In our formalism, ECAs are denoted by A E = {L, N , {0, 1} , f } where:

• L = {Z/LZ} is a 1-dimensional ring.
• N = {-1, +1}, i.e. the 2-cell neighbourhood.

• the local transition function δ is determined by its code:

W = f (0, 0, 0).2 0 + f (0, 0, 1).2 1 + ... + f (1, 1, 1).2 7 .

Qualitative observations

We now observe qualitatively the effect of β-synchronism. We are in particular interested in knowing whether its effects will differ from those of α-synchronism.

In the rest of this paper, initial conditions are constructed by assigning to each cell an equal value to the observable state and the eigenstate, and choosing this value randomly and uniformly in {0, 1}.

The Game of Life

Figure 3 shows sample configurations appearing for random initial conditions after a transient time of 1000, which was observed sufficient for reaching a steady state. We observe that the behaviour separates into two distinct phases for both types of asynchronism: the system converges to a fixed point for values of the synchrony rate α or β higher than 0.9, whereas a labyrinth-like pattern appears for values lower than 0.9. To quantify this observation, we use the following macroscopic parameters:

• the density is the ratio of cells with state (or eigenstate) 1.

• the activity is the ratio of unstable cells. A cell is said unstable at time t if a synchronous update modifies its state (or eigenstate).

Note that the latter definition cannot be readily transposed for the β-synchronism. Indeed what does a synchronous update mean for a cell whose eigenstate and observable state are different? As neighbourhood knowledge is always absolute in synchronous cellular automata, we reckon that it should also be the cases for the estimation of activity. Therefore, using the notations defined in Fig. 1, we define a cell as instable if y = f (y L , y, y R ). Figure 4 compares the two types of asynchronism through the activity parameter after a steady state has been reached. At first sight, the β-synchronous Game of Life seems to react in a similar way to its α-synchronous counterpart :

1. A singularity occurs for the activity parameter in the synchronous case (α, β = 1). This phenomenon has been explained for α-synchronism as the loss of stable periodic patterns when a noise is introduced no matter how small (α < 1) [START_REF] Blok | Synchronous versus asynchronous updating in the "Game of Life[END_REF]. The same explanation stands for β-synchronism. This means that the traditional construct to make the Game of Life Turinguniversal, that is, by using periodical patterns such as gliders, no longer holds. Nevertheless, the question of the Turing-universality of the asynchronous Game of Life still stands.

2. The macroscopic behaviour confirms the phase separation for a critical value of the synchrony rate α c (resp. β c ). For values α > α c (resp. β > β c ), the system converges to a stable fixed point of low density, which constitutes a passive phase. However, for values α < α c (resp. β < β c ), the system enters an active phase, characterized by the labyrinth pattern (though less regular in β-than in α-synchronism).

The comparison of both types of asynchronism reveals a similar reaction of the Game of Life, but this calls for a closer observation of the characteristics of the phase transition (see Sec. 4.1).

Elementary Cellular Automata

Through the search of reflexive and complementary symmetries, it is possible to narrow down the number of ECA to study from the 256 possible models to 88 inequivalent ECA. To quantify these observations on the 88 ECA, we use the following macroscopic parameters:

• the density d, as defined above.

• the 01 block density (00, respectively) is the ratio of successive cells with states 0 -1 (resp., 0 -0). When comparing visually the plot profiles of the 88 minimal ECA under α-and β-synchronism1 , it can be observed that most rules display similar reactions to both types of asynchronism. However, some surprising divergences of plot profile appeared for lower synchrony rates, for example for ECA 6, 22, 72 and 200. For the sake of conciseness, we choose to focus on a single rule, namely ECA 50 and leave for future work the exhaustive study of the entire range of phenomena.

The case of ECA 50. A notable phenomenon appears for the β-synchronous ECA 50. The study of the density parameter (see Fig. 5a) shows a similar plot profile for both α-and β-synchronism, including a phase transition, which suggests that ECA 50 reacts in a similar way to these two perturbations. However, a difference of behaviour is observed with the visual inspection of the evolution of the automaton over a few time steps (see Fig. 6). For instance, for the same synchrony rate 0.75, the patterns in β-synchronism appear much more regular than α-synchronism. In particular, in the long run clusters of 0-states appear less frequently, and 111 patterns seem inexistent.

This can be explained for values of β close to 1 with the following observations:

• the checkerboard regions (i.e. alternated 0s and 1s) are robust to the βsynchronism, that is, the few anomalies (regions of 0s) that disturb the • the pairs (i.e. 00 and 11 pairs) follow a non-biased random walk, and annihilate when they meet.

As a consequence, if L is even, in the long run, the system tends to be covered by checkerboard patterns, which is a new property exclusive to β-synchronism. This example illustrates the qualitative differences that may occur between α-and β-synchronous cellular automata. More specifically, we note that ECA 50 is more regular for β-synchronism than for α-synchronism. This regularity can be "seen" as the proximity between rule ECA 50 and the inversion rule ECA 51, which differ only by one bit in their transition table. As the rule ECA 51 is insensitive to β-synchronism (the inversion rule is independent from the neighbours state), there also seems to exist a proximity in terms of global behaviour between ECA 50 and 51 for the β-synchronism. One may thus wonder what is the origin of such a radical difference between the two updating schemes. In the next paragraph, we endeavour to capture this difference in the framework of stochastic cellular automata.

Stochastic cellular automata.

It is straightforward to describe α-synchronous ECA in terms of elementary stochastic cellular automata, that is, to define them with a function f α : {0, 1} 3 → [0, 1] which associates to each neighbourhood state (x, y, z) the probability to update to 1. Indeed, the local rule f α is simply obtained as the barycentre of the transition function f and the identity rule Id with weights α and 1 -α, respectively.

By contrast, β-synchronism requires us to extend the state space in order to differentiate the eigenstate and the observable state. As a result, the number of inputs of the transition table is doubled to take into account whether the two states are synchronized, that is, if the eigenstate and the observable state are identical.

As said before, ECA 51 is insensitive to β-synchronism. However, this particular property of ECA 51 does not appear readily on its transition table (see Table 1). This makes the stochastic table all the more "cryptic", as the divergence from synchronism cannot be deduced from the reading of the table, unlike α-synchronism. This shows that the difference between the two types of asynchronism is non-trivial, and justifies the experimental approach adopted to study these systems.

Study of phase transitions

The occurrence of phase transitions is probably one of the most remarkable phenomena that arises in asynchronous cellular automata: there exists a non-trivial value of the synchrony rate, the critical threshold, which separates two distinct qualitative behaviours of the system. We are now interested in measuring quantitatively this phenomenon.

The Game of Life

As pointed out in Sec. 3.1, the activity parameter reveals the existence of two distinct phases in the α-and β-synchronous Game of Life. For the α-synchronism, the phase transition has been proved to be second-order [START_REF] Blok | Synchronous versus asynchronous updating in the "Game of Life[END_REF], that is, if the macroscopic measures that describe the behaviour are continuous, their derivative curve is discontinuous for a critical value of the synchrony rate.

How to determine the critical synchrony rate α c or β c ? A simple method to measure it consists in estimating the singularity point where the slope jumps from null to infinity, but for practical purposes, this method is rather imprecise as it introduces systematic biases. In order to reduce them, we follow a different protocol (see e.g. [START_REF] Grassberger | Synchronization of coupled systems with spatiotemporal chaos[END_REF]):

1. We fix β, start from a random initial condition and let it evolve for a given number of steps.

2. We monitor the evolution of the order parameter for a long simulation time until we observe a sub-critical or super-critical behavior. In a loglog plot, a concave curve occurs for the passive phase (activity converges to zero) and a convex curve for the active phase (activity converges to a non-zero value). As we expect the order parameter to follow a power law K.t -δ near criticality, the curve at critical value should appear as a straight line of slope -δ.

3. We repeat the experiment with a value closer to the critical point until a satisfactory precision is reached (here 10 -3 ).

For the α-synchronous Game of Life, it has been measured that α c = 0.9083 and that the evolution of the order parameter of this value followed a power law in the form K.t -δ with δ 2D = 0.451 [START_REF] Fatès | Does Life resist asynchrony?, Game of Life Cellular Automata[END_REF], which is the expected critical exponent for the directed percolation universality-class in 2 dimensions [START_REF] Hinrichsen | Nonequilibrium critical phenomena and phase transitions into absorbing states[END_REF]. Figure 7 shows that the measures for β-synchronous updating. They also confirm the directed percolation hypothesis, with β c = 0.945.

Elementary Cellular Automata

Among the 88 minimal ECAs, it has been observed that several ECA display second-order phase transition in their α-synchronous version. These rules were proved to belong to the directed percolation class [START_REF] Fatès | Asynchronism induces second order phase transitions in elementary cellular automata[END_REF], and have been divided into 3 distinct subclasses:

• rules 18, 26, 50, 58, 106 and 146 are the DP hi class, for which the active phase of density (respectively the passive phase) occurs for α > α c (resp. α < α c ).

• rules 6, 38 and 134 are the DP low class, where active and passive phases are inverted.

• rule 178 is the sole element of the DP 2 class, where the density is stable in average but for which a phase transition appears for the 01-block-density.

Our observations show that these subclasses react differently to β-synchronism:

1. The rules of the DP hi class have shown little behavioural change between the two types of asynchronism: the phase transition appearing for the density parameter in Fig. 5 is conserved.

For α-synchronism, the same protocol was applied to ECA 50 [START_REF] Fatès | Asynchronism induces second order phase transitions in elementary cellular automata[END_REF], and displayed good evidence of a directed percolation phenomenon for α c = 0.6282.

For β-synchronism (see Fig. 7), it appears the behaviour at critical synchrony rate β c = 0.601 is in good agreement with a power law of critical exponent δ 1D = 0.1595, the expected value for the directed percolation.

2. Surprisingly enough, no phase transition was observed for the three rules of the DP low class in β-synchronism (see ECA 6 in Fig. 5), leaving a constant null-density convergent phase for any value of β.

3. Finally, the ECA 178 (DP 2 class) reproduced a similar plot profile for the 01-block-density (see ECA 178 in Fig. 5).

These first results show how rich the study of β-synchronism can be, and that although α-and β-synchronism are intuitively similar in their mechanisms, their effects may differ radically.

Discussion

This paper presented a formalism for a new type of asynchronous updating in cellular automata, the β-synchronism, based on the disruption of information transmission between cells. We compared this perturbation to α-synchronism and observed from a macroscopic point of view that α-and β-synchronism had similar effects. It was observed that β-synchronous updating also produces phase transitions, but for a smaller set of rules than α-synchronism. In particular, we remarked that there was no phase transition for the three ECA of the DP low class but we have no explanation for this phenomenon so far. By studying ECA 50 more closely, we could exhibit an example for which a macroscopic parameter behaviour was similar for both types of asynchronism, but for which novel lower-scale properties were observed.

This extension is a first step towards a unified view of asynchronism for cellular automata, based on the idea of cell cycles. There exist plenty of other ways the cellular activity can be perturbed. This raises the question of how the space of perturbations can be described.

Although this paper focused on cellular automata, the idea behind the βsynchronism can be transposed to other synchronous collective systems, such as lattice-gas cellular automata, neural networks or multi-agent systems. By extending the range of variations applied to the definition of the discrete dynamical systems, can we perfect our understanding of the robustness of natural systems? Can we gain some insight from their mechanisms to improve the design of spatially-extended computing models?

Figure 1 :

 1 Figure1: Representation of a cell cycle, for a cell with two neighbours, denoted by indices L and R. A triplet represents a cell with its state (y) and the observable states of its neighbours (x for left cell and z for right). The prime sign indicates states that are updated. Solid arrows show the updates of states, otherwise states are conserved from the previous step. In α-synchronism, the local transition update applies with probability α and in β-synchronism, the information transmission is applied with probability β. The lower arrow is the local transition update, which performs y = f (x, y, z) when applied. The forking upper arrow is the information transmission, which performs z L = y and x R = y when applied.

α = 1 α 1 Figure 3 :

 113 Figure 3: Game of Life configurations for different values of α and β (0 is denoted by a white space, 1 by blue). These are obtained with the software FiatLux for identical initial states, for a transient time t ≈ 1000. The first row shows the different behaviors obtained with different values of α-synchronism. The second row displays them for the same values of β-synchronism.

Figure 4 :

 4 Figure 4: Steady-state values of the activity parameter for the Game of Life with synchrony rates α and β, averaged over a sample of 50 initial conditions of size L=100, t ∈ [10000, 11000]. Inset: a closeup for values α, β ∈]0.9, 1].

Figure 5 :

 5 Figure 5: Steady-state values of various observation parameters for different ECA. The grid size is F = 1000, transient time is T t = max(10000, 1000/α) (β respectively) and averages are obtained for a sampling time T s = max(1000, 100/α) (resp. β).

Figure 6 :

 6 Figure 6: Configurations for ECA 50 under different updating schemes (0 is denoted by a white space, 1 by blue).

  probabilities for updated states to become 1 after an βsynchronous transition of ECA 51 (inversion rule). Each column represents a possible state for a cell and its neighbourhood (upper box) and the associated probability to become a 1 for the output of the transition function (lower box). pattern are quickly restored to the original pattern.

Figure 7 :

 7 Figure 7: Phase transition analysis for the Game of Life (top) and ECA 50 (bottom). The straight lines follow a power law f (t) ∼ t -δ . For the Game of Life, measures are averaged over 25 samples of size 800 × 800. For the ECA 50, measures are averaged over 25 samples of size 20000. The straights line give the expected critical exponent for directed percolation: δ 2D = 0.451 (top) and δ 1D = 0.1595 (bottom).

Table 1 :

 1 Table of

for the complete results: http://www.loria.fr/ ~boure/results/ecaparambeta/