V Deniau

J Rioult

Radiated Power Measurements of Electronic Equipments in Three-Dimensional TEM Cells

Three-dimensional TEM cells are new test facilities which permit us to measure the total radiated power of equipment under test, in the lower frequencies, without having to place it in different positions. Firstly, this paper briefly presents the threedimensional concept and the optimisation works carried out to improve the impedance matching of the plates. Then, after a summary on the experimental approaches used for radiation measurements in TEM and 3-D TEM cells, radiation measurements results of an off-the-shelf electronic equipment performed in both test facilities are compared in order to study the reproductibility and the repeatability of the results. The final section of this paper focuses on the useful test volume of the 3-D TEM prototype.

INTRODUCTION

Among the test facilities used today to characterise electronic equipments and systems, TEM cells are the only known test facilities that cover the low frequency band. However, in all variants of such TEM cells, the full three-dimensional characterization of equipment under test (EUT) can only be obtained by rotating it. Three-Dimensional TEM (3-D TEM) cells [START_REF] Klingler | On the Use of Three-Dimensional TEM Cells for total Radiated Power Measurements[END_REF] were therefore developed in order to regroup the electromagnetic properties of three conventional TEM cells into one to avoid moving the EUT. Moreover, by decreasing the size of the cell, this tool may be extended to radiation measurements of electronic components working on large frequency range. However, this paper only focuses on electronic equipments tests.

THE THREE-DIMENSIONAL CONCEPT

Knowing that, in the lower frequencies an EUT must be characterized according to three orthogonal coupling planes, measurements performed in conventional TEM cells require placing the EUT in three different orthogonal positions regarding to the unique strip line of the cavity. Originally [START_REF] Klingler | On the Use of Three-Dimensional TEM Cells for total Radiated Power Measurements[END_REF], Three-Dimensional TEM cells were therefore created to include three plates placed in three different orthogonal positions in order to generate three orthogonal components of electric and magnetic fields. However, by including three additional plates opposite the first three plates, we can use the prototype in a balanced mode that can be very beneficial in term of repeatability and useful test volume. On the one hand, for radiation measurements, we reduce the probability that the total radiated power obtained depends on the position of the EUT by collecting the voltages appearing at twelve ports instead of six ports. On the other hand, for immunity testing, we can reduce the gradient of the components of the electric field in the inner volume of the cell by feeding in phase the twoby-two opposite plates by half the input power. This could allow us to increase the useful test volume.

The prototype developed, represented in figures 1 and 2, is a 1-meter cubic cell. It permits us to have symmetrical situations according to the three x, y and z-axes.

IMPEDANCE MATCHING OF THE PLATES

The first resonance frequency of the considered cavity appears at 215 MHz. Consequently, above 200 MHz, the TEM functioning is not maintained due to the resonance's appearance. We therefore tried to improve the impedance matching of the plates up to 215 MHz in order to increase the transmission of power from the power source to the interior of the cavity.

Contrary to conventional TEM cells, in 3-D TEM cells the plates can not be symmetrically placed inside the cavity, and the rules used to determine the appropriate dimensions of the strip line giving a characteristic impedance of 50 Ω in a conventional TEM cell, cannot be exploited. We then developed a generic analytical optimisation method [START_REF] Deniau | Détermination des dimensions optimales des septa dans une cellule TEM-3D[END_REF] for an asymmetrical structure, allowing us to determine the adequate dimensions of the plates according to the cavity's geometry.

Firstly, inspired by the approach to conventional TEM cells [START_REF] T I P P E T A N D | Radiation Characteristics of Electrically Small Devices in a TEM Transmission Cell[END_REF], we obtained a characteristic impedance relation in function with the dimensions of the prototype cross section (Figure 3).

             + +         + + + ≅ 2 2 coth 1 ln 2 2 1 2 coth 1 ln 2 2 2 2 1 2 62 . 376 b g b g b W b W Z c π π π π (1)
Then, to define the best couple of values, we calculated the appropriate slope for the transition section of the plates to conserve a constant ratio between the dimensions of the upper and the inferior parts of the cross section to the current flowing along the plate [START_REF] Deniau | Research of the Optimal Dimensions and Position of the Plates in a 3D-TEM Cell[END_REF]. Due to the length of the transition section being imposed at 10 cm, the slope of the transition imposed a unique b 2 value and the W associated value to get a characteristic impedance of 50 Ω.

To illustrate the results, figure 4 presents reflection Sparameters obtained at the extremity of one of the plates between 10 MHz and 210 MHz. We only considered the frequencies inferior to 210 MHz due to the inevitable increase of the reflection coefficient resulting from the appearance of the first resonance. Knowing that this TEM cell can be used up to 375 MHz, we represented the reflection S-parameters obtained for the frequency band 10 MHz -370 MHz.

Regarding the levels of reflection in figures 4 and 5, we can see that the analytical method developed allows us to obtain, on an asymmetrical structure, an impedance matching as good as that of a conventional TEM cell.

RADIATED POWER MEASUREMENTS

In order to measure total radiated power in TEM cells, the EUT needs to be placed according to three orthogonal orientations as represented in figure 6 [START_REF] Deniau | Détermination des dimensions optimales des septa dans une cellule TEM-3D[END_REF].

Fig. 6: Experimental approach in TEM Cell

For each position in which the EUT is placed, voltages Va and Vb are measured with a spectrum analyser at both ports of the TEM cell. The total power radiated is then obtained by using the relation ( 2):

∑ = + ⋅ ⋅ = 3 1 2 0 2 0 ) ( 3 1 2 i bi ai c V V e Z Z Pt λ π (2)
Where V ai and V bi are the voltages measured at both ports for the three orientations in Volts, e 0 is the normalized electric field for an input power of 1 Watt in the TEM cell and Z c is the characteristic impedance of the cell.

In the case of the measurement performed in the 3-D TEM cell, it is not necessary to change the position of the EUT and the total radiated power is given by (3):

∑ = + ⋅ ⋅ = 6 1 2 0 2 0 ) ( 6 1 2 i bi ai c V V e Z Z Pt λ π (3)
Where V ai and V bi are the voltages measured at both ports of the six plates in Volts.

We carried out a comparison of total radiated power measured in both test facilities. For each test facility, three series of tests were performed in order to observe the measurements repeatability. In the TEM cell the three tests were identical and in the 3-D TEM cell, we modified the orientation of the EUT between the three tests in order to verify that the measurements results did not depend on the position. Moreover, the EUT mainly radiating between 50 MHz and 100 MHz, we only considered the results obtained in this frequency band.

Firstly, we calculated the average results of the three series of tests performed in each test facility. The averages total radiated power collected in both cells are presented in figure 7. The blue curve represents the average of the three results collected in the 3-D TEM cell and the red curve illustrates the average of the three results obtained in the TEM cell. x y

Vb

Figure 8 shows a relatively good agreement between the two average results of the tests performed in both cells, except for some differences appearing at certain frequencies (see arrows).

REPEATABILITY RESULTS

In order to observe the repeatability of the measurements carried out in both cells, we have presented the results of the three series of tests for the TEM and the 3-D TEM cells in figures 8 and 9. Through these results, we can see some problems of repeatability regarding the differences appearing between the three results of tests performed in each cell. Effectively, we see some significant differences at certain frequencies between the three results for each test facility. However, we also see that the greater differences, which are highlighted by arrows in the figures, appear at the same frequencies in the results obtained with the conventional TEM cell and the 3-D TEM prototype.

USEFULL TEST VOLUME

In the case of the 3-D TEM cell, the space between a plate and the opposite wall is greater than in a conventional TEM cell due to the strip line not being centred. Consequently, the gradient of the electromagnetic field in the centre of the cavity is relatively strong if only one plate is fed [START_REF] Klingler | Measuring Radiation of Small Electronic Equipment in Three-Dimensional TEM Cells[END_REF]. Figure 10 represents an example of the simulated total electric field distribution created by the plate X in the yz-plane for an input power of 1 Watt. Effectively, considering the centred 40cm x 40 cm surface, we observe a strong gradient of the zcomponent of the electric field from approximately 5 V/m to 13 V/m. However, the use of the opposite plates in balanced mode permits us to improve the field uniformity and the useful test volume. Effectively, by symmetrically feeding the plates X and X' by ½ Watt, the gradient of the z-component of electric field along the z-axis can be significantly reduced as highlighted by the simulated field distribution in figure 11. We can see that in the 40 cm x 40 cm centred surface, we conserve a z-component of electric field which contained between 8 and 12 V/m.

CONCLUSION

Concerning the radiation testing, we obtained a satisfying reproductibility compared to the results of radiation testing performed in a conventional TEM cell. We also saw that the repeatability between three series of tests, is as good as that of a conventional TEM cell whereas we placed the EUT in different positions between the three series of tests carried out in the 3-D TEM prototype. Moreover, simulations results showed that the uniformity of the electric field could be improved in the central volume of the cell by symmetrically feeding the opposite plates. In the final paper, this part will be completed by field measurements. Through these observations, Three-Dimensional TEM cells could prove an attractive solution for EMC immunity or radiation testing which avoid moving the EUT. Furthermore, knowing that the upper frequency limit is inversely proportional to the dimensions of the structure, small Three-Dimensional TEM cell can be envisaged for small electronic components testing up to relatively high frequencies.

z(cm)

22 -24 20 -22 40 18 -20 16 -18 y (cm) 14 -16 -40 40 12 -14 10 -12 8 -10

Fig. 1 :Fig. 2 :

 12 Fig. 1: 6-plate 3-D TEM cells

Fig. 3 :

 3 Fig. 3: 6-plate 3D-TEM cell's cross section The relation (1) permitted us to determine couples of values (W, b 2 ) for the width of the plate and the gap between the plates and the walls giving a characteristic impedance of 50 Ω.



Fig. 4 :Fig. 5 :

 45 Fig. 4: Reflection S-parameters at one of the connectors However, this figure shows that the reflection coefficient stays inferior to -20 dB up to 193 MHz. Keeping in mind that the first resonance frequency is 215 MHz, this result is perfectly satisfying. As a reference, we measured the reflection Sparameters at each port of a standard AR TC3020A TEM cell which works up 375 MHz. These results are presented in figure 5. Transition section 1 m

Fig. 7 :

 7 Fig. 7: Average of three results collected in both cells

Fig. 8 :Fig. 9 :

 89 Fig. 8: Total radiated power collected in TEM cell

Fig. 10 :

 10 Fig. 10: Total electric field in the yz-plane cell

Fig. 11 :

 11 Fig. 11: Total electric field in the yz-plane cell


Tasnime N Akbaraly

Archana Singh-Manoux

Adam G Tabak

Markus Jokela

Marianna Virtanen

Jane E Ferrie

Michael G Marmot

Martin J Shipley

Mika Kivimaki

Overall diet history and reversibility of the metabolic syndrome over 5 years: the Whitehall II

Keywords: Diet, Female, Humans, Logistic Models, Male, Metabolic Syndrome X, metabolism, pathology, Middle Aged, Prospective Studies Metabolic syndrome, Prospective study, the Alternative Healthy Eating Index

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

%

% [START_REF] Wild | The Global burden of the metabolic syndrome and its consequences for diabetes and cardiovascular disease . Metabolic syndrome[END_REF] an increased risk of type 2 diabetes (T2D) and cardiovascular diseases (CVD). Lifestyle modification, such as increased physical exercise[ and diet therapies , , may have a beneficial impact on MetS. We examined a set of dietary guidelines targeting major chronic 2 ] [3 4 ] diseases known as the Alternative Healthy Eating Index (AHEI) . Our goal was to examine whether adherence to AHEI was associated [START_REF] Mccullough | Diet quality and major chronic disease risk in men and women: moving toward improved dietary guidance[END_REF] with reversion of the MetS over a 5-year period in a middle-aged population.

RESEARCH DESIGN AND METHODS

Data came from phases 3 (1991 1993), 5 (1997 1999) and 7 (2002 2004) of the Whitehall II study of London based office workers

- - - [6 ]
. From the 3698 participants with complete data on MetS at phases 3 and 5; diet and covariates at phase 5, and the outcome (MetS status at phase 7), the 339 MetS cases at phase 5 formed the study population for the main analysis.

At each phase, MetS was defined using the National Cholesterol Education Program Adult treatment Panel III criteria . The [START_REF]Executive Summary of The Third Report of The National Cholesterol Education Program (NCEP) Expert Panel on Detection, Evaluation, And Treatment of High Blood Cholesterol In Adults (Adult Treatment Panel III)[END_REF] definition of each MetS component plus details of other measurements are given in the footnote of and have been described Table 1 previously . The 5-year reversion of MetS was defined as not having MetS at phase 7. [START_REF] Akbaraly | Association between metabolic syndrome and depressive symptoms in middle-aged adults: results from the Whitehall II study[END_REF] Dietary intake data were collected via a validated 127-item food frequency questionnaire (FFQ) both at phases 3 and 5. AHEI [9 -10 ] score was created by summing its 9 component scores (1/fruit, 2/vegetable, 3/ratio of white to red meat, 4/ fat, 5/ratio of [START_REF] Mccullough | Diet quality and major chronic disease risk in men and women: moving toward improved dietary guidance[END_REF] trans polyunsaturated to saturated fat, 6/total fiber, 7/nuts and soy, 8/alcohol consumption and 9/long-term multivitamin use); a higher score corresponded to greater adherence (Appendix-Table A1). We used the AHEI measures in two ways: the Phase 5 score for the main analysis and the average AHEI score across phases 3 and 5, to take into account longer-term adherence to AHEI, in subsidiary analysis on the 337 participants with complete dietary data from phases 3 and 5. Levels of AHEI adherence at phase 5 were comparable to those reported in two large American cohorts (Appendix-Table A2).

[5 ]

Logistic regression models examined the association between tertiles of the AHEI scores at phase 5 and reversion of MetS and of its components at phase 7, sequentially adjusted for age, sex, ethnicity (White/non-White) and total energy intake (kcal/day) (Model 1) and for educational attainment (no academic qualification/lower secondary/higher secondary/university degree/higher university degree), marital status (married or cohabiting/living alone), smoking (current/former/non smoker), persistence of MetS at phase 5 (having MetS both at phases 3 and 5), depressive symptoms, and intensity of physical activity (high/medium/low) (Model 2). No significant [START_REF] Akbaraly | Association between metabolic syndrome and depressive symptoms in middle-aged adults: results from the Whitehall II study[END_REF] interaction between AHEI scores and covariates (including sex) was observed. Analysis was conducted using the SAS software, version 9.1 (SAS Institute).

RESULTS

Among the 339 participants with MetS at phase 5, 158 (46.6 ) recovered by phase 7. Characteristics of the participants (as a function % of MetS reversion and AHEI category) are shown in Appendix-Tables A3 andA4. After controlling for potential confounders, adherence to AHEI was associated with increased odds of MetS reversion over the 5-year follow-up ( ). This association was stronger among Table 1 participants with MetS at both phases 3 and 5 (n of reversions/total n 56/155, OR 3.74, 95 CI:1.37 10.2). The AHEI-MetS reversion

= = % -
association was particularly evident among participants with central obesity and among those with high triglycerides ( ). Table 1 Furthermore, adherence to AHEI was associated with a 5-year reversion of the high triglyceride component (n 276/767, OR 1.61, 95 = = % CI:1.12 2.33) and with an indication of reversion of central obesity (n 75/481, OR 1.42, 95 CI:0.75 2.68). Analyses with average

- = = % -
AHEI score across phases 3 and 5 as the exposure largely replicated these findings (Appendix-Table A5).

CONCLUSION

In the present report, we show that adherence to dietary guidance for healthy eating, the AHEI, is associated with reversion of the MetS in a middle-aged population. While several studies have investigated the diet MetS prevalence and incidence relationships, the impact of diet on MetS reversion has only been studied in two clinical trials assessing adherence to Mediterranean-Diet in two Mediterranean countries. One trial of 180 Italian subjects found the Mediterranean -Diet intervention to led to reversion of MetS , the [START_REF] Esposito | Effect of a mediterranean-style diet on endothelial dysfunction and markers of vascular inflammation in the metabolic syndrome: a randomized trial[END_REF] other larger trial (n 1224 Spanish) suggested that the observed effect was due to the effect of nut supplements rather than the Med-Diet as = a whole . Even though the clinical utility of MetS as an independent predictor of CVD has been challenged , our findings, from a

[4 ] [START_REF] Despres | Abdominal obesity and metabolic syndrome[END_REF] non-Mediterranean country, are novel and strengthen evidence of the potential impact of diet in countering increasing levels of risk factors associated with CVD and T2D.

The observed stronger impact of AHEI on MetS reversion in participants with central obesity and high triglycerides is plausible as the reduction of visceral fat leads to a decrease in the flux of free fatty acids which is hypothesized to be responsible for insulin-resistance [START_REF] Bergman | Why visceral fat is bad: mechanisms of the metabolic syndrome[END_REF] a key feature in MetS pathophysiology . This interpretation assumes that central obesity mainly reflects visceral fat as the majority of

- [13 ]
the subjects were males and there was a higher reversion of MetS in men. Further research is needed to examine these and other plausible mechanisms, such as counteracting oxidative stress (and related insulin resistance ) via antioxidants from fruits, vegetables and [START_REF] Ando | Metabolic syndrome and oxidative stress[END_REF] long-term multivitamin use; and the lowering of high triglyceride levels (which are linked to reduction of inflammation processes involved in MetS ) as a result of increased consumption of polyunsaturated fat, nuts, soy and a reduced consumption of trans fat.

[15 ]

Limitations of this study include: the small sample size that is not fully representative of the British population and that does not [START_REF] Marmot | Cohort Profile: the Whitehall II study[END_REF] allow ethnic group sub-stratification other than White/non-White, limiting the generalizability of our findings; the lack of objective measure of physical activity; and, the use of the FFQ, recognized to be less precise than diary questionnaires, to assess diet.

Despite these limitations, our findings emphasize the potential benefits of adherence to the dietary recommendations of the AHEI in middle-aged individuals with MetS, especially those with central obesity or high triglyceride levels.

make the study possible. Continuing data collection on this study is funded by the Medical Research Council, National Institute on Aging (R01AG013196, R01AG034454), the National Heart, Lung, and Blood Institute (R01HL036310), and the British Heart Foundation.

Footnotes:

Contributors: TNA, MK, AS-M, and MGM designed the study; TNA conducted the statistical analyses and co-wrote the initial and final drafts, and is guarantor. MK, MJS, AGT, MJ, MV, MGM, JEF and AS-M co-wrote the final draft.

TNA had full access to all of the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis Model 1: Adjusted for sex, age, ethnicity and energy intake. Model 2: Model 1 additionally adjusted for education, marital status, smoking habits, physical activity, persistence of the metabolic syndrome, and depressive symptoms.

+

Ackowledgements:

We thank all participating men and women in the Whitehall II Study; all participating Civil Service departments and their welfare, personnel, and establishment officers; the Occupational Health and Safety Agency; and the Council of Civil Service Unions. The Whitehall II Study team comprises research scientists, statisticians, study coordinators, nurses, data managers, administrative assistants and data entry staff, who Overall Diet and Reversion of Metabolic Syndrome Diabetes Care . Author manuscript

Table 1

Association between adherence to the alternate healthy eating index (AHEI) and 5-yeart reversion of the metabolic syndrome (MetS). Metabolic syndrome defined using the NCEP definition [START_REF]Executive Summary of The Third Report of The National Cholesterol Education Program (NCEP) Expert Panel on Detection, Evaluation, And Treatment of High Blood Cholesterol In Adults (Adult Treatment Panel III)[END_REF] based on the presence of 3 or more of the following: waist circumference: men >102 cm, women >88 cm; serum triglycerides: 1.7 mmol/L; ≥ HDL cholesterol: men <1.04 mmol/L, women < 1.29 mmol/L; blood pressure: 130 mmHg/ 85 mmHg systolic over diastolic pressure; fasting glucose: 6.1 mmol/L or presence of T2D. Waist

Adherence to AHEI , participants (n reversion cases)


Hedi Ben Amor
email: hedi.ben-amor@imag.fr

Jacques Demongeot
email: jacques.demongeot@imag.fr

Nicolas Glade
email: nicolas.glade@imag.fr

Mnesic Evocation: An isochron-based analysis

Mnesic evocation occurs under the action of a stimulus. A successful evocation is observed as the overrun of a certain threshold of the neuronal activity followed by a medical imaging instrument like a PET-scanner. Within the neural system, this successful evocation corresponds to an effective activity that induces other activations in other parts of the brain and conscious actions. Populations of coupled neuronal oscillators can dynamically store information in the form of a periodic attractor of large dimension. In this context, the overrun of such an activity threshold is due to a maximization of the global activity of the population of oscillators. It is allowed by a synchronized activity of the -neuronal -oscillators, which can be provided by the action of an external stimulation. One can hold this process as an elementary process of mnesic evocation. We use an isochron-based analysis to understand the relevant aspects of this synchronization phenomenon. The temporal gap resulting from the perturbation of a large population of uncoupled oscillators (initially distributed in an equal manner on the latent phase) gives us a direct characterization of the phase space. We obtain a method to classify the phase space into fast an slow synchronization regions, thus allowing a qualitative understanding of the behaviour adopted by oscillators in response to perturbations.

I. INTRODUCTION

By studying the isochrons profil of a dynamical system having a limit cycle attractor, one can classify the phase space into different regions : fast and slow synchronization regions. A population of oscillators having the same dynamics can be synchronized via a common perturbation undergone in a certain direction and having a sufficient amplitude to bring them all to fast or slow synchronization areas. In this article, we focus on 3 different dynamical systems, the van der Pol oscillator a well-known self-driven dynamical system [START_REF] Der Pol | Frequency demultiplication[END_REF], the phospho-fructo kinase (PFK) that can show in certain conditions periodic dynamics of the fructose-6phosphate (the product) and the available chemical energy (ATP/ADP) [START_REF] Ricci | Influence of phosphoenolpyruvate on the dynamic behavior of phosphofructokinase of escherichia coli[END_REF], and more particularly on the Wilson-Cowan neuronal oscillator. The Wilson-Cowan model [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF] consists in a couple of inhibitory and excitatory neurons. Its dynamical behaviour exhibits a limit cycle attractor and an unstable equilibrium at the origin. We highlight the synchronization phenomena and their analysis, starting with a description of the isochron-based analysis and the maximum phase shift analysis. We determin the isochrons of the limit cycles of these systems in their whole phase spaces. Then we perform several simulations (perturbation and relaxation) which show correlations between phase synchronization and the amplitude of perturbation through a measure of the maximum phase shift. Then we discuss about their importance in the understanding of the mnesic evocation process.

A. Definition of the isochrons

Let us consider the space E ⊂ ℜ n (where ℜ is the set of the real numbers) of all states of a dynamical system and the times set T ⊂ ℜ (containing time 0), and denote by x(t) the state of the system at time t. A trajectory (or flow) ϕ is an application from the Cartesian product E × T onto E defined by: ∀(x, t) ∈ E × T , then: (i) ϕ(x, t) = x(t), (ii) x(0) = x, (iii) ϕ(ϕ(x, t), s) = ϕ(x, t + s). The orbit of a state y iterated by the flow ϕ is the set of points O(y) = {ϕ j(y) } j∈T . We define the Birkhoff limit set L(x) of any state x of E taken as initial condition of the trajectory ϕ(x, .), as the set of temporal accumulation points of this trajectory. If B(y, ε) denotes the set of states at distance of y less than ε (B(y, ε) is reduced to {y} in the discrete case): L(x) = {y ∈ E; ∀ε > 0, ∀t ∈ T, ∃s ∈ T / s > t and O(ϕ(x, s)) ∩ B(y, ε) = ∅} The attractor basin B(A) of a subset A of E is the set of all initial conditions x not in A, but such as L(x) ⊂ A. Let us denote L(A) = ∪ x∈A L(x). A is the set A completed by all possible shadow trajectories [START_REF] Bowen | Limit sets for axiom a diffeomorphisms[END_REF]. An attractor A verifies [START_REF] Cosnard | On the definitions of attractors[END_REF], [START_REF]Attracteurs: une approche déterministe[END_REF]: i) A is a fixed set for the composed set operator LoB : A = L(B(A)), where ii) there is no set C such as A ⊂ C ⊂ A, C = A, verifying i), iii) there is no set D ⊂ A, D = A, verifying i) and ii). An attractor A is invariant in the dual operations consisting firstly in considering all the trajectories of its basin from all initial conditions not in A, but finishing their life in A, and secondly to restrict them to their ends of life (Figure 1). Fig. 1. Definition of an attractor (the character 'star') and its basin (the gray bubble) in the case of a Boolean network, whose the states space E is the hypercube {0, 1} 11 (left). Initial conditions are indicated in the gray disks and attracting conditions for the majority rule and synchronous iterations (the state of a node equals 1 if its activating neighbours in state 1 are equally or more numerous than the inhibiting ones) are given outside blue disks (right).

Let us suppose now that the attractor A is a limit cycle, i.e., if we denote by p the period of the limit cycle, A = {a 0 , a 1 = ϕ(a 0 , 1), ..., , a p-1 = ϕ(a 0 , p -1)} and there is a natural isomorphism ψ between A and the set S = {0, ..., τ -1}. By denoting T s = {t ∈ T /t = s + kτ } k∈N,s∈S we have: T = ∪ s∈S T s . The isochron I s of phase s is the attractor basin of {ψ(s) = a s } for the flow ϕ s (equal to ϕ on E × T s ) and B(A) = ∪ s∈S I s . If T = ℜ+ (the positive real numbers set), I s is transversal to A, i.e., the tangent vector to I s at the state a s is not tangent to A [START_REF] Guckenheimer | Isochrons and phaseless sets[END_REF], [START_REF] Freire | Limit cycles and lie symmetries[END_REF]. Examples of isochrons of some dynamical systems are shown below in Figure 2. The fibration of equally phase-distributed isochrons is a good tool for analysing the phase space. The convergence (resp. the divergence) of isochrons indicates a slow synchronization region (resp. a fast synchronization region). This furnishes a quantitative information about the direction and the strength of the perturbation to perform in order to synchronize a homogeneous population of oscillators. (Up) A nonsymmetrical system: the PFK enzymatic balance. Isochrons and the limit cycle attractor of the PFK system obtained for a very high precision (> 10 6 ). Note that the region in the bottom is very sensitive to numerical imprecision. In this region, the isochrons are overlapped. This indicates a sensitivity to numerical precision and corresponds to a slow synchronization region. (Down) Isochrons and the limit cycle attractor of the van der Pol oscillator. Fast (resp. slow) synchronization regions, or isochron divergence (resp. isochron convergence) regions of equally phase-distributed isochrons, are located overhead and underneath (resp. on the right and on the left) of the limit cycle in the phase space.

B. Numerical resolution of isochrons

All points delayed by a period T (period of the limit cycle) are on the same isochron because they have the same latent phase. The determination of these points can be done by using numerical methods. Whatever the numerical method used is, checking if points of the state space belong to the isochron of a chosen phase can be obtained by verifying, when they converge until the attractor via the equations of the dynamical system, if they admit that phase of the attractor once considered a certain limit. Several numerical methods are possible: a systematic exploration of the state space by dividing it in small areas, a random exploration in all the state space, or a local random exploration with a guidance of the research area along a direction given by the points already founded (a kind of intelligent paintbrush). We pull out a point of the state space randomly, then we compute a discrete version of the differential equations until the trajectory reaches for the first time a point belonging to a neighbourhood of fixed thickness 10 -k of the attractor, where k is a thickness parameter. The phase of this point on the attractor is memorized. This phase minus the phase of the nearest isochron of interest is compared to a tolerance value in order to determine whether this point can be considered as belonging on the isochron or not. This tolerance value is equal to T 2np (Figure 3) where T is the period of the attractor considered as a limit cycle, n is the number of isochrons considered and p ∈ [1, + inf[ is a precision parameter.

C. Maximum phase shift computation

Once considered a population of uncoupled oscillators of same nature, it may be of interest to have a measure of the synchronization that occurs after a perturbation tranlating the state of all the oscillators to an other region at the same time, i.e. conserving that way the shape of the limit cycle (see Figures 8 and9). The isochronal fibration gives a qualitative idea of the phase shift or on the contrary of the resynchronization which may result from a perturbation affecting this population. The divergence (resp. convergence) of equally distributed isochrons in a particular region of the phase space means indeed that this region is a fast (resp. slow) synchronization one. A quantitative measurement can be calculated from this fibration. We computed the phase shift between the two isochrons containing the set of n points (P i ) i∈ [1,n] obtained after a translation of the limit cycle as follows:

• The phase φ i of each P i is calculated as described in the previous section. • The vector [P i ] i is sorted by order of the increasing values of φ i .

• The vector ((∆Φ i,i+1 ) i∈[1,n-1] , ∆Φ n,0 ) is determined. ∆Φ i,i+1 = φ i+1 -φ i for i ∈ [1, n -1] and ∆Φ n,0 = 2π -(φ n -φ 0 ). • The maximum phase shift is ∆Φ M ax = 2π - M ax((∆Φ i,i+1 ) i∈[1,n-1] , ∆Φ n,0 ).
Those steps are repeated for several values of the amplitude between 0 and R max (maximum perturbation). When the system is symmetrical, we compute the maximum phase shift along an arbitrary direction of perturbation 10. This is not valid when the dynamical system is asymmetrical like the phospho-fructo kinase (PFK) system (see Figure 4). White regions correspond to regions where the phase is not well-defined (very strong sensitivity to numerical imprecision). This synchronization map is very irregular and clearly shows that it is hard to sycnhronize the PFK (Up) due to a (strong non linearity and asymmetry of the system). Synchronizing the whole system needs to choose very precisely the direction and amplitude of the perturbations. This is very different for the van der Pol (Down) system where regions of synchronization (underneath and overhead) and desynchronization (on the right and on the left) are well defined and localized in the phase space.

II. MNESIC EVOCATION

Let us consider a network (neural or genetic) made of several subsystems, like modules, identical or different, and weakly or strongly connected: in Figure 5, M1 is an arbitrary subnetwork and M2 is made of a simplified Hippocampus one-layered network, with one Cyto-Architectural 1 (CA1) and one Cyto-Architectural 3 (CA3) pyramidal neuron, one Entorhino-Cortical neuron (EC) and one Inter-Neuron (IN), all interconnected. If we simplify this module in a subnetwork of size 2, we obtain a structure called negative regulon, with one negative circuit and 2 positive self-loops. We will consider in the following that this subnetwork is sequentially repeated in a chain of modules, each of them being weakly linked to the following at the level of the CA3 neurons (the X i 's in Figure 6): CA3 neurons send axons with excitatory synapses to CA1 and CA3 neurons, these latter sending axons with excitatory synapses to EC neurons through the Subiculum (SB), and EC forming global inhibitory connections with CA3 neurons through Inter-Neurones (IN) of Dentate Gyrus (DG) [START_REF] Buzsaki | Feed-forward inhibition in the hippocampal formation[END_REF], [START_REF] Gluck | Computational models of hippocampal function in memory[END_REF], [START_REF] Hefft | Asynchronous GABA release generates long-lasting inhibition at a hippocampal interneuron-principal neuron synapse[END_REF], [START_REF] Bartesaghi | Input-output relations in the entorhinal cortex-dentate-hippocampal system: evidence for a non-linear transfer of signal[END_REF], [START_REF] Mori | Recruitment of an inhibitory hippocampal network after bursting in a single granule cell[END_REF]. The Wilson-Cowan system WC [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF] used to simulate the dynamics of a subnetwork represented by a negative regulon (Figure 6) is given by 2n differential equations: ∀i = 1, ..., n, dX i /dt = -X i /a + tanh(bX i )tanh(bY i ) + kX i-1 , dY i /dt = -Y i /a + tanh(bX i ) + tanh(bY i ). This system is closed to a Hopfield neural network, when b is positive and large, and a is negative and large [START_REF] Tonnelier | Synchronization and desynchronization of neural oscillators: comparison of two models[END_REF]. Trajectories and isochrons of WC are represented in Figure 7, showing fast (resp. slow) regions, i.e., zones where the flow runs fast (resp. slow). When the system WC is stimulated through a perturbation S translating all the initial conditions on the limit cycle C into a fast region (Figure 8 and9), then all the return trajectories go to the same phase x1, provoking a synchronization of all the activities X i 's. In other terms, if the X i 's before the stimulation are dispatched uniformly on the limit cycle C, then their sum denoted A(t) = i=1,...,n X i (t) is about zero, because the positive values of the X i 's are compensated by the negative ones. If the X i 's run in phase, A(t) behaves like nX i (t) and can be detected for example by a Pet-scanner of a functional MRI device (Figure 11). Some dynamical systems like the PFK system are not symmetrical and have numerous regions of slow synchronization as revealed by their isochron fibration (fig. 2) and by their synchronization map (fig. 4). In those cases, translating the system to these regions causes a loss of the synchrony of the X i 's. If the coupling between the WC subnetworks is made at the level of the CA3 neurons (whose activities are the X i 's) is weak (which correspond to a permanent translation to slow synchronization areas 10), then a desynchronization occurs (Figure 11-1), allowing the exit out the perseveration behaviour (Figure 11-2). If the neuronal activity is noised, the desynchronization obtained with a weak coupling is not perfect (Figure 11-3), as well as the perseveration (Figure 11-4), both behaviours exhibiting a small residual synchronized activity. In case of synchronization, the global activity A(t) of the CA3 neurons "evokes" the common limit cycle attractor [START_REF] Csicsvari | Fast network oscillations in the hippocampal ca1 region of the behaving rat[END_REF] of the X i 's and can be considered as the phenomenologic result of the recall of a mnesic temporal pattern stored in the network [START_REF] Demongeot | Positive feedback circuits and memory[END_REF], [START_REF] Samsonovich | A simple neural network model of the hippocampus suggesting its pathfinding role in episodic memory retrieval[END_REF]. If the succession of states to be transiently evoked is only a part of this attractor, then a fast desynchronization is needed, which is ensured by a weak coupling (Figure 12). The richness of the stored souvenirs comes from the number and the complexity of the common attractors of the X i 's. If their number increases (e.g., from 1 to 2) as well their complexity (e.g., passing from the circular limit cycle coming from a fixed configuration of focus type through a Hopf bifurcation, to a chaotic behaviour through doubling period bifurcations), then we can locally store and evoke complicated temporal patterns coding for complex cognitive entities or serving for moving objects detection in a complex scene [START_REF] Hayashi | Moving object detection algorithm inspired by the sequence detection in the hippocampus and its digital LSI implementation[END_REF]. Such a bifurcations landscape can be simply obtained from the simulation of a fully connected Hopfield network of size 16, involved in a learning process, as shown on Figure 13 [START_REF] Nérot | Mémorisation par forc ¸age neuronal des dynamiques chaotiques dans les modèles connexionnistes récurrents[END_REF]. The places were the isochrons diverge (resp. converge) are fast (resp. slow) synchronization regions. This allows us to classify the phase space so as to determine the minimal or the suitable strenght and direction of perturbation to have an acceptable phase synchronization. Synchronization corresponds to an increase of the global activity of a population of coupled oscillators. The resulting signal in a neural cortex becomes measurable by fMRI. Such a synchronized signal reveals the occurence of a conscious event like mnesic evovation. The desynchronization of neurons is then ensured by a weak coupling between oscillators, which in other term corresponds to a permanent perturbation towards slow synchronization regions. Then, the understanding of the phase synchronization by analysing the phase space constitutes a powerful tool for designing new storage systems based on populations of coupled oscillators, as proposed in [START_REF] Ben-Amor | The isochronal fibration: Characterization and implication in biology[END_REF], and an advance in the researches on the memory and mnesic evocation.

Fig. 2 .

 2 Fig.2. Examples of Isochrons of 2 dynamical systems. (Up) A nonsymmetrical system: the PFK enzymatic balance. Isochrons and the limit cycle attractor of the PFK system obtained for a very high precision (> 10 6 ). Note that the region in the bottom is very sensitive to numerical imprecision. In this region, the isochrons are overlapped. This indicates a sensitivity to numerical precision and corresponds to a slow synchronization region. (Down) Isochrons and the limit cycle attractor of the van der Pol oscillator. Fast (resp. slow) synchronization regions, or isochron divergence (resp. isochron convergence) regions of equally phase-distributed isochrons, are located overhead and underneath (resp. on the right and on the left) of the limit cycle in the phase space.

Fig. 3 .

 3 Fig. 3. Graphical description of the measure of precision Let us consider 8 isochrons equally spaced. The largest acceptable tolerance is T /16, ie. the phase interval between 2 isochrons is equally divided. The precision parameter is used to express the tolerance parameter as follow: Tolerance = T /(P recision * 2 * N isochrons )

Fig. 4 .

 4 Fig. 4. Maximum phase shift of the PFK (Up) and the van der Pol (Down) oscillators in response to perturbations (Up) The maximum phase shift of the PFK. (Down) The maximum phase shift of the van Der Pol. Dark (resp. bright) colors correspond to fast (resp. slow) synchronization regions.White regions correspond to regions where the phase is not well-defined (very strong sensitivity to numerical imprecision). This synchronization map is very irregular and clearly shows that it is hard to sycnhronize the PFK (Up) due to a (strong non linearity and asymmetry of the system). Synchronizing the whole system needs to choose very precisely the direction and amplitude of the perturbations. This is very different for the van der Pol (Down) system where regions of synchronization (underneath and overhead) and desynchronization (on the right and on the left) are well defined and localized in the phase space.

Fig. 5 .

 5 Fig. 5. Modular structure, in which the module M1 is arbitrary and M2 represents Hippocampus subnetwork.

Fig. 6 .

 6 Fig. 6. Scheme of a sequential modular structure made of a chain of negative regulons, where X1 represents the activity of CA3 and X2 the activity of CA1 (see anatomy of CA3, CA1, DG and SB in the cartouche)

Fig. 7 .

 7 Fig. 7. (Up) Isochronal and trajectory landscape for the Wilson-Cowan dynamics of a negative regulon, showing fast (resp. slow) regions, where the velocity is high (resp. low). (Down) Fast (resp. Slow) synchronization regions in dark colors (resp. bright colors) of the Wilson-Cowan oscillator obtained by computing the maximum phase shift in response to perturbations.

Fig. 8 .Fig. 9 .Fig. 10 .

 8910 Fig. 8. Synchronizing stimulation S translating the limit cycle C in a fast region of the states space E

Fig. 11 .

 11 Fig. 11. Post-stimulation global activity A(t) of CA3 neurons (A(t) = i=1,...,n X i (t)), with a weak deterministic intra-CA3 coupling showing a fast desynchronization (1), without coupling showing a perseveration (2), with a weak noised coupling showing a fast desynchronization followed by a residual synchronization (3) and with noise without coupling showing a slow desynchronization (4)

Fig. 13 .

 13 Fig.[START_REF] Mori | Recruitment of an inhibitory hippocampal network after bursting in a single granule cell[END_REF]. Bifurcations of the attractors of the dynamics (X 1 (t-1), X 1 (t)) for the first of 16 neurons fully connected in a classical Hopfield neural network, when a function W of its synaptic weights w ij vary during a learning process, showing alternance of chaotic and periodic behaviours

ACKNOWLEDGMENT

This work was supported by the Virtual Physiological Human Network of Excellence of the European Community (VPH-NoE).