Christophe LETELLIER

C. Lainscsek, P. Rowat, L. Schettino, D. Lee, D. Song, C. Letellier & H. Poizner
Finger tapping movements of Parkinson’s disease patients automatically rated using nonlinear delay differential equations
Chaos, 22, 013119, 2012. Online

- Abstract
Parkinson’s disease is a degenerative condition whose severity is assessed by clinical observations of motor behaviors. These are performed by a neurological specialist through subjective ratings of a variety of movements including 10-s bouts of repetitive finger-tapping movements. We present here an algorithmic rating of these movements which may be beneficial for uniformly assessing the progression of the disease. Finger-tapping movements were digitally recorded from Parkinson’s patients and controls, obtaining one time series for every 10 s bout. A nonlinear delay differential equation, whose structure was selected using a genetic algorithm, was fitted to each time series and its coefficients were used as a six-dimensional numerical descriptor. The algorithm was applied to time-series from two different groups of Parkinson’s patients and controls. The algorithmic scores compared favorably with the unified Parkinson’s disease rating scale scores, at least when the latter adequately matched with ratings from the Hoehn and Yahr scale. Moreover, when the two sets of mean scores for all patients are compared, there is a strong (r=0.785) and significant (p < 0.0015) correlation between them.

L. A. Aguirre & C. Letellier
Nonstationarity signatures in the dynamics of global nonlinear models
Chaos, 22, 033136, 2012. Online

- Abstract
The aim of this paper is to learn how to recognize a posteriori signatures that nonstationarity leaves on global models obtained from data. To this end the effects of nonstationarity on the dynamics of such models are reported for two benchmarks. Parameters of the Rössler and Lorenz models are varied to produce nonstationary data. It is shown that not only the rate of change of the varying parameter but also which recorded variable is used to estimate global models may have visible effects on the results, which are system-dependent and therefore difficult to generalize. Although the effects of nonstationarity are not necessarily obvious from the phase portraits, the first-return map to a Poincar-é section is a much more adequate tool to recognize such effects. Three examples of models previously obtained from experimental data are analyzed in the light of the concepts discussed in this paper.

C. Letellier & L. A. Aguirre,
Required criteria for recognizing new types of chaos : Application to the “cord” attractor
Physical Review E, 85, 036204, 2012. Online

- Abstract
After suggesting criteria to recognize a new system and a new attractor—and to make a distinction between them—the paper details the topological analysis of the “cord” attractor. This attractor, which resembles a cord between two leaves, is produced by a three-dimensional system that is obtained after a modification of the Lorenz-84 model for the global atmospheric circulation [1]. The nontrivial topology of the attractor is described in terms of a template that corresponds to a reverse horseshoe, that is, to a spiral Rössler attractor with negative and positive global π twists. Due to its particular structure and to the fact that such a system has two variables from which the dynamics is poorly observable, this attractor qualifies as a challenging benchmark in nonlinear dynamics.

R. Naeck, D. Bounoiare, U. S. Freitas, H. Rabarimanantsoa, A. Portmann, F. Portier, A. Cuvelier, J.-F. Muir & C. Letellier,
Dynamics underlying patient-ventilator interactions during nocturnal noninvasive ventilation,
International Journal of Bifurcation & Chaos, 22 (2), 1250030, 2012. Online

- Abstract
Noninvasive ventilation is a common procedure for managing patients having chronic respiratory failure. The success of this ventilatory assistance is often linked with patient’s tolerance that is known to be related to the quality of the synchronization between patient’s spontaneous breathing cycles and ventilatory cycles delivered by the ventilator. Thirty-four sleep sessions (more than 5000 ventilatory cycles each) were automatically investigated using a specific algorithm processing airflow and pressure time series. Four groups of patients were defined according to the interplay between asynchrony events and leaks. Different mechanisms that depend on sleep stages were thus evidenced. A Shannon entropy was also proposed as a new sleep fragmentation quantification methodology.

F. Denis & C. Letellier,
Chaos theory : A fascinating concept for oncologists,
Cancer - Radiothérapie, 16 (3), 230-236, 2012. Online

- Abstract
The oncologist is confronted daily by questions related to the fact that any patient presents a specific evolution for his cancer : he is challenged by very different, unexpected and often unpredictable outcomes, in some of his patients. The mathematical approach used today to describe this evolution has recourse to statistics and probability laws : such an approach does not ultimately apply to one particular patient, but to a given more or less heterogeneous population. This approach therefore poorly characterizes the dynamics of this disease and does not allow to state whether a patient is cured, to predict if he will relapse and when this could occur, and in what form, nor to predict the response to treatment and, in particular, to radiation therapy. Chaos theory, not well known by oncologists, could allow a better understanding of these issues. Developed to investigate complex systems producing behaviours that cannot be predicted due to a great sensitivity to initial conditions, chaos theory is rich of suitable concepts for a new approach of cancer dynamics. This article is three-fold : to provide a brief introduction to chaos theory, to clarify the main connecting points between chaos and carcinogenesis and to point out few promising research perspectives, especially in radiotherapy.

J.-M. Ginoux & C. Letellier
Van der Pol and the history of relaxation oscillations : Toward the emergence of a concept,
Chaos, 22, 023120, 2012. On line

- Abstract
Relaxation oscillations are commonly associated with the name of Balthazar van der Pol via his paper (Philosophical Magazine, 1926) in which he apparently introduced this terminology to describe the nonlinear oscillations produced by self-sustained oscillating systems such as a triode circuit. Our aim is to investigate how relaxation oscillations were actually discovered. Browsing the literature from the late 19th century, we identified four self-oscillating systems in which relaxation oscillations have been observed : (i) the series dynamo machine conducted by Gérard-Lescuyer (1880), (ii) the musical arc discovered by Duddell (1901) and investigated by Blondel (1905), (iii) the triode invented by de Forest (1907), and (iv) the multivibrator elaborated by Abraham and Bloch (1917). The differential equation describing such a self-oscillating system was proposed by Poincaré for the musical arc (1908), by Janet for the series dynamo machine (1919), and by Blondel for the triode (1919). Once Janet (1919) established that these three self-oscillating systems can be described by the same equation, van der Pol proposed (1926) a generic dimensionless equation which captures the relevant dynamical properties shared by these systems. Van der Pol’s contributions during the period of 1926-1930 were investigated to show how, with Le Corbeiller’s help, he popularized the “relaxation oscillations” using the previous experiments as examples and, turned them into a concept.

M. Frunzete, J.-P. Barbot & C. Letellier
Influence of the singular manifold of nonobservable states in reconstructing chaotic attractors,
Physical Review E, 86, 026205, 2012. Online

- Abstract :
It is known that the reconstructed phase portrait of a given system strongly depends on the choice of the observable. In particular, the ability to obtain a global model from a time series strongly depends on the observability provided by the measured variable. Such a dependency results from (i) the existence of a singular observability manifold, Msobs, for which the coordinate transformation between Rm and the reconstructed space is not defined and (ii) how often the trajectory visits the neighborhood UMsobs of Msobs. In order to clarify how these aspects contribute to the observability coefficients, we introduce the probability of visits of Msobs and the relative time spent in UMsobs to construct a new coefficient. Combined with the symbolic observability coefficients previously introduced [2] (only taking into account the existence of Msobs), this new coefficient helps to determine the specific role played by the location of Msobs with respect to the attractor, in phase portrait reconstruction and in any analysis technique.

F. Denis & C. Letellier
Radiotherapy and chaos theory : The tit and the butterfly…
Cancer/Radiothérapie, 16 (5–6), 404–409, 2012 on line (in French)

- Abstract
Although the same simple laws govern cancer outcome (cell division repeated again and again), each tumour has a different outcome before as well as after irradiation therapy. The linear-quadratic radiosensitivity model allows an assessment of tumor sensitivity to radiotherapy. This model presents some limitations in clinical practice because it does not take into account the interactions between tumour cells and non-tumoral bystander cells (such as endothelial cells, fibroblasts, immune cells…) that modulate radiosensitivity and tumor growth dynamics. These interactions can lead to non-linear and complex tumor growth which appears to be random but that is not since there is not so many tumors spontaneously regressing. In this paper we propose to develop a deterministic approach for tumour growth dynamics using chaos theory. Various characteristics of cancer dynamics and tumor radiosensitivity can be explained using mathematical models of competing cell species.

V. Messager, R. Gilmore & C. Letellier
Henri Poincaré and the principle of relativity
Contemporary Physics, 53 (5), 397-415, 2012. Online

- Abstract
Often considered as the last ‘encyclopedist’, Henri Poincare´ died one hundred years ago. If he was a prominent man in 1900 French Society, his heritage is not so clearly recognised, particularly in France. Among his too often misunderstood works is his contribution to the theory of relativity, mainly because it is almost never presented within Poincaré ’s general approach to science, including his philosophical writings. Our aim is therefore to provide an historical account of the main steps (experimental as well as theoretical) which led Poincaré to contribute to the theory of relativity. Starting from the optical experiments which led to the inconsistency of the classical (Galilean) composition law for velocities to explain light propagation, we introduce the FitzGerald and Lorentz contraction which was viewed as the ‘sole hypothesis’ to explain the Michelson and Morley experiment. We then show that Poincaré ’s contribution starts with a discussion of the principles governing the mechanics and was built step by step up to express in all its generality the principle of relativity. Poincaré thus showed the invariance of the Maxwell equations under the Lorentz transformation. In doing so, he also discovered the right composition law for velocities. Poincaré ’s approach to philosophy is detailed to help the reader to understand what a theory meant to him.

G. G. Rodrigues, U. S. Freitas, D. Bounoiare, L. A. Aguirre & C. Letellier
Leakage estimation using Kalman filtering in noninvasive mechanical ventilation,
IEEE Transactions on Biomedical Engineering, 60 (5), 1234-1240, 2012 . On line

- Abstract
Noninvasive mechanical ventilation is today often used to assist patient with chronic respiratory failure. One of the main reasons evoked to explain asynchrony events, discomfort, unwillingness to be treated, etc. is the occurrence of nonintentional leaks in the ventilation circuit which are difficult to account for because they are not measured. This paper describes a solution to the problem of variable leakage estimation based on a Kalman filter driven by airflow and the pressure signals, both of which are available in the ventilation circuit. The filter was validated by showing that based on the attained leakage estimates, practically all the untriggered cycles can be explained.

C. Quérel, O. Grondin & C. Letellier
State of the art and analysis of control oriented NOx models
SAE Technical Paper, 2012-01-0723, 2012, online

- Abstract
Future pollutant emissions legislations are expected to be increasingly stringent. To reduce Nitrogen Oxides (NOx) emissions produced by Diesel engines, advanced combustion technologies - like Low Temperature Combustion (LTC) -, vehicle hybridization and NOx after-treatment systems - such as Selective Catalytic Reduction (SCR) systems - can be considered, leading to a growing demand for NOx models.

[1] L. A. Aguirre & C. Letellier, Investigating observability properties from data in nonlinear dynamics, Physical Review E, 83 066209, 2011.

[2] C. Letellier & L. A. Aguirre, Symbolic observability coefficients for univariate and multivariate analysis, Physical Review E, 79, 066210, 2009.

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