From nonlinear dynamics to biological data
University of Normandie
CORIA, Avenue de l’Université, F-76800 Saint-Etienne du Rouvray (FRANCE)
Summary:
I. SOME BACKGROUND IN DYNAMICAL SYSTEMS
A. State space
The concepts of dynamical and its associated state space will be introduced. The role of singular points and their stability will be briefly introduced, and their different types (in two-dimensional space) discussed. The concept of bifurcation will be illustrated (yet in two-dimensional space). When the attractor becomes recurrent (periodic, quasi-periodic, chaotic, ...), it is useful to investigate its properties through the concept of surface of section (commonly designed as Poincaré section). For instance, the limit cycle becomes a periodic point, and a stability analysis similar to those developed for singular points can be performed, although with a different stability condition: for this latter reason, it is highly recommended to distinguish these two types of points (singular points and periodic points) which are too often designed in the literature as “fixed point”.
B. Route to chaos and properties of chaos
From a Poincaré section, a bifurcation diagram can be constructed. Some examples will be provided and some guidelines for reading them. A few main route to chaos will be discussed: period-doubling cascade, intermittency (with its equivalent in fluid mechanics), Ruelle-Takens scenario, Curry-Yorke scenario…
When the dynamics is strongly dissipative and low-dimensional, it is useful to compute a first-return to a Poincaré section which often can offer a natural partition of the dynamics and thus allow the construction of a generating dynamics symbolic. Basic elements of symbolic dynamics will be introduced for labelling periodic orbits. It will be explained how a Shannon entropy can be easily computed when a symbolic dynamics is introduced. A link between each domain and the strips of paper-sheet model (template) will be briefly illustrated.
C. A taxonomy of chaos
Various examples of chaotic dynamics will be introduced using a taxonomy of chaos. It is based on the bounding tori, first-return map, the number of positive and negative Lyapunov exponents and, ultimately, the concept of template. Chaos, hyperchaos, toroidal chaos, conservative chaos, and their variant will be discussed. The challenging points of some of them will be exhibited. Some examples from the real world will be presented when known.
II. BIOMEDICAL APPLICATIONS
A. Atrial fibrillation as an intermittency
Atrial fibrillation is a chronic disease that affects the heart's electrical activity and, consequently, its beating. Once this pathology introduced, it will be explained how it can be viewed as an intermittency. Indeed, the route to chaos (or turbulence) through intermittency is an interesting conceptual framework for understanding the slow evolution of the chronic disease from compensation (stable) to decompensation (acute). The heart variability will thus be investigated from tachograms, that is, from RR-intervals (as data from a Poincaré section). A first-return map on ∆RRn+1 = RRn+1 − RRn — introduced in 2009 [5, 6] and now in some commercial Holter — are used, and some guidelines to read their underlying structure will be introduced. They will be investigated by using a symbolic dynamics, an entropy (computed from the symbolic dynamics) [8]. The load on atrial fibrillation is also computed. These results are briefly compared with “common” markers as SDNN, SDANN, etc. [11].
B. A clinical follow-up for patients treated for a lung cancer
From a simple three-dimensional cancer model describing the interactions between host, immune and tumor cells [1], an observability analysis exhibit that “measuring” host cells provides the best observability of the state space. Taking advantage of this, we developed a follow-up weekly self-evaluated symptoms sent with a web application to automatically trigger an alert when a relapse is detected. It will be shown that such a follow-up allows to improve the survival [2].
C. A data-driven model for prostate cancer
Prostate cancer is hormone-dependent whose progression is quite slow: typically patients can live more than 15 years with a prostate cancer. This type of cancer can be tracked by measuring the PSA level with a common blood analysis. A four-dimensional mode, considering the global status of the patient, the PSA level, the hormono-dependent tumor cells, and the hormono-independent tumor cells is designed and its parameter values are estimated from the PSA level (and the drug prescription) using a genetic algorithm [3]. It is thus shown that the strength of the patient and the stiffness of the tumor growth can be estimated from these parameter values. It is then possible to use that model for optimizing the protocol for drug delivery.
III. ADVANCED STUDIES
A. Observability and controlability
Commonly, once a time series is recorded in a system (numerical or experimental), no attention is paid to the variable measured. This is mainly due to the Takens theorem which states that an object of dimension d is diffeomorphically reconstructed in a space of dimension 2d + 1 using delay or derivative coordinates [12]. Nevertheless, one of the condition of the Takens theorem is that the measurement function ξ = h(x) must be generic, meaning that applying a perturbation to it must not change the result. Unfortunately, often the measured variable are the result of a measurement function which is not generic and the Takens theorem does no longer holds.
In control theory, there is a useful concept to know whether a state of the dynamical system under study can be retrieved from the measured data: this is called observability. For short, this is related to an observability matrix Oξ d which is the matrix of the d successive Lie derivatives of the measured variable (here we consider that the first Lie derivative is the measured variable itself). It will be shown how a first (approximated) assessment of observability can be performed using a fluence graph. Using a variable providing a global observability of the state space is relevant for determining the state of the system, modeling, parameter idenfitication, computing the embedding dimension, etc.).
The dual of observability is controlability, to assess how it is possible to control the system under study. According to Kalman [7] and Lin [10], observability is the dual of controlability (established for linear system). Yet using the fluence graph, it is will be shown how the dual - a derivative of the system - of a measured variable can be identified.
B. Flat control law
Global observability and global controllability are used to optimally place sensors and actuators [9]. Then a feedback linearization can be used for designing the actuating signal applied to the selected derivative through the actuator. To do that, an additional condition must be fulfilled: the actuating signal must be written in terms of the measured variable and its time derivatives [4]. It is thus shown how systems can be controlled to any drive dynamics. The case of two y-diffusively coupled Rössler systems is explicitly treated as well as a random network of 28 coupled Rössler systems.
Practical session
The participants are invited to send (up to Saturday July 6th) the lecturer Christophe.Letellier@univ-rouen.fr a few slides explaining the context of their studies for which they would need to address some questions with the help of techniques borrowed from the nonlinear dynamical systems theory. In turn, the lecturer will try to provide some suggestions.
Bibliography
[1] L. G. De Pillis and A. Radunskaya. The dynamics of an optimally controlled tumor model: A case study. Mathematical and Computer Modelling, 37(11):1221–1244, 2003.
[2] F. Denis, E. Basch, A.-L. Septans, J. Bennouna, T. Urban, A. C. Dueck, and C. Letellier. Two-year survival comparing web-based symptom monitoring vs routine surveillance following treatment for lung cancer. JAMA, 321:306–307, 2019.
[3] C. Draghi, F. Denis, A. Tolédano, and C. Letellier. Parameter identification of a model for prostate cancer treated by intermittent therapy. Joural of Theoretical Biology, 461:117–132, 2019.
[4] M. Fliess, J. L. Lévine, P. Martin, and P. Rouchon. Flatness and defect of non-linear systems: introductory theory and examples. International Journal of Control, 61(6):1327–1361, 1995.
[5] U. Freitas, E. Roulin, J.-F. Muir, and C. Letellier. Identifying chaos from heart rate: The right task? Chaos, 19(2):028505, 2009.
[6] E. Fresnel, E. Yacoub, U. Freitas, A. Kerfourn, V. Messager, E. Mallet, J.-F. Muir, and C. Letellier. An easy-to-use technique to characterize cardiodynamics from first-return maps on ∆RR-intervals. Chaos, 25(8):083111, 2015.
[7] R. Kalman. On the general theory of control systems. IFAC Proceedings Volumes, 1(1):491–502, 1960. 1st International IFAC Congress on Automatic and Remote Control, Moscow, USSR, 1960.
[8] C. Letellier. Estimating the Shannon entropy: Recurrence plots versus symbolic dynamics. Physical Review Letters, 96:254102, 2006.
[9] C. Letellier and J.-P. Barbot. Flatness for an optimal control of chaotic systems using a minimal numbers of sensors and actuators. Chaos, 31:103114, 2021.
[10] C.-T. Lin. Structural controllability. IEEE Transactions on Automatic Control, 19(3):201–208, 1974.
[11] F. Shaffer and J. P. Ginsberg. An overview of heart rate variability metrics and norms. Front Public Health, 5:258, 2017.
[12] F. Takens. Detecting strange attractors in turbulence. Lectures Notes in Mathematics, 898:366–381, 1981.
