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Section: Scientific Foundations

Multiscale system theory: analysis of transfers of energy and information among scales

We consider networks or ensembles of cells of the same type modeled by (2 ) with a single base model and different parameters $ \theta$i . In this case the solution of (2 ) may never live in the synchronization manifold and it is of theoretical and practical interest to study the deviations from this manifold.

We are specially interested by large networks with a particular structure, like e.g. possibly infinite binary trees as it is the case for hemodynamic networks (e.g. the coronary tree). When using thermodynamically consistent reduced order models for the cells (e.g. cardiac cells and coronary vessels for the heart or fuel cell systems) to model the multiscale systems we want to study, a natural question arises: what is the relation between the multiscale structure of the $ \theta$i and the structure of energy and information u , y among scales? The inverse problem is the principal motivation: gaining information on the $ \theta$i from multiscale analysis of y .

Large deviations and singularity spectra ; scaling invariant models

Two possible approaches for describing the transfer of energy among scales are the following: Looking at the way a given positive measure $ \mu$ is distributed at the successive scales of regular nested grids (denoted Gn at resolution n ), or looking at the manner the wavelet coefficients of a square integrable function g decay to 0 along the scales. This can be done by using ideas initially used by physicists in order to describe the geometry of turbulence and then formalized by mathematicians in the so-called multifractal formalism ( [101] , [102] , [91] , [130] , [108] ).

On the one hand one uses tools coming from statistical physics and large deviations theory in order to describe asymptotically for each singularity value $ \alpha$ the logarithmic proportion of cubes C in Gn (the dyadic grid of level n ) such that the mass distributed in C is approximately equal to the power $ \alpha$ of the diameter of C , i.e. Im35 ${\#956 {(C)}\#8776 2^{-n\#945 }}$ . This yields a sequence of functions fn of $ \alpha$ called large deviation spectrum, which describes statistically the heterogeneity of the distribution of the measure at small scales. Another tool associated with this spectrum consists in the partitions functions

Im36 ${\#964 _n{(q)}=\mfrac 1nlog_2\munder \#8721 {C\#8712 G_n}\#956 {(C)}^q.}$

They are Laplace transforms closely related to the functions fn .

The same quantities can be associated with the L2 function g by replacing the masses $ \mu$(C) by the wavelet coefficients |dC(g)| .

In practice, the functions fn and $ \tau$n can be computed and are used to exhibit a scaling invariance structure in a given signal as soon as they remain quasi constant when n ranges in some non trivial interval. This approach proves to be efficient in detecting scaling invariance in energy dissipation and velocity variability in fully developed turbulence [101] as well as in the heart-beat variability [125] , [107] and in financial time series [120] . Scaling invariance in heart-beat variability is one of our research directions (see Section 3.2). It should reflect the heterogeneous spatiotemporal distribution of the energy in the cardiac cells and should be related to models of this phenomenon.

On the other hand one uses tools from geometric measure theory such as Hausdorff measures and dimensions in order to have a geometrical description of the (fractal) sets of singularities Im37 $S_\#945 $ obtained as the sets of those points x at which the sequences $ \mu$(Cn(x)) or |dCn(x)(g)| behaves asymptotically like Im38 $2^{-n\#945 }$ , where Im39 $\mfenced o=( c=) C_n{(x)}$ is the sequence of nested cubes in the grids Gn that contain the point x . The singularity spectrum obtained by computing the Hausdorff dimension of the sets Im37 $S_\#945 $ yields a finer description of the heterogeneity in the energy distribution than the statistical one provided by large deviations spectra. But this object is purely theoretical since it necessitates the resolution to go until $ \infty$ .

Since the tools described above are efficient in physical and social phenomena, it is important to investigate models of measures and functions having such properties and develop associated statistical tools of identification. Such models do exist and have been studied for a long time ( [119] , [111] , [132] , [104] , [109] , [78] , [100] , [81] , [83] , [82] ) but few satisfactory associated statistical tools have been developed. We shall study new models of scaling invariant measures, signed multiplicative cascades, and wavelet series. In particular we will be inspired by the model proposed in [113] of cascading mechanisms for the evolution of wavelet coefficients of the solution of the Euler equation. It could be used to construct a model for the multiscale control of cardiac cellular energetics and, as we already said above, a model for the heart-beat variability.

These works will contribute to one of the theoretical aspects developed in the team, which consists in studying and classifying statistically self-affine and multifractal mathematical objects.

Multiscale signals analysis & dynamical systems. Example of the cardiovascular system

Analysis of Heart Rate Variability (HRV), the beat-to-beat fluctuations in heart rate, has many clinical applications. The observation of the 1/f shape of the HRV spectrum has been strengthened recently by using techniques of multifractal signals processing. These techniques quantify a signal temporal irregularity for instance by constructing an histogram of the “coarse-grained” Hölder exponents computed on finer and finer nested grids. This leads to the so-called large deviations spectrum, which describes the frequency at which each Hölder exponent occurs. This is a way to estimate variability. One can say that some scale invariance holds when the large deviation spectrum weakly depends on the scale in the nested grid. Such a scale invariance has been observed on RR signals, and one concluded that the largest the range of the exponents, the better the patient's health. In particular the multifractal large deviation spectrum is shown to be a useful tool to study the long-term fluctuations for the diagnosis of some pathologies like congestive heart failure.

HRV analysis can be completed considering Blood Pressure Variability (BPV). For example joint analysis of short-term HRV and BPV leads to the baroreflex sensitivity (BRS), the gain of the parasympathetic feedback loop, a useful index of parasympathetic activity that has a prognostic value in several situations (myocardial infarction, heart failure of diabetic patients): low BRS is correlated with mortality in patients with heart failure. In the case of BPV, 1/f shaped spectra have also been observed and it has been found that sympathetic nerve traffic and BPV follow comparable self-similar scaling relationships. In both case, HRV or BPV, the physiological origins of these long-term fluctuations remain mysterious. The goal of this study is to provide methods and tools to improve variability analysis for a better understanding of these fluctuations.

Our method will be to associate multiscale signal analysis and mathematical models whenever it will be possible. The ANR project DMASC has started in 2009 on these questions.


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