**VI. Scientific Program**

Course 1

Title: Introduction to Symbolic Dynamics

Abstract :

Symbolic Dynamics is a very important domain for Dynamical Systems. The general results hold for subshifts and a lot of proofs consists in constructing a conjugacy between a given dynamical system and a symbolic one (coding).

The course will present main basic notions and tools such as words, languages, recurrence, complexity function, topological entropy.

Examples of subshifts of finite types, markov shifts, sofic systems will be studied.

A brief introduction to ergodicity will be done (operators, Von Neumann theorem. Birkhoff theorem. Frequency).

A special attention will be put on substitutions as the Fibonacci and the Thue-Morse substitutions which are example of quasi-crystal (low complexity but not periodic).

If time permits it, the last lecture will focus on coding for linear automorphisms of the torus.

Bibliography:

Lind & Marcus: Symbolic dynamics and coding.

B. Kitchens: Symbolic dynamics.

A. Katok & B. Hasselblatt: Introduction to dynamical systems.

Course 2

Title: Searching for deterministic chaos in biological data

Chaotic dynamics have always been strongly appealing to biologists. From a modeling viewpoint for instance, the first studies of chaotic dynamics in difference equations (e.g. the logistic map) were actually motivated by population biology [1]. The question of whether chaotic dynamics are at work in biology is another very active field of research since the late 1970's. Given some measurement of a biological system (e.g. an univariate time series), can one provide reasonable evidence that the dynamics of the underlying system exhibits low-dimensional deterministic chaos? Over the years, a number of algorithms and quantitive indices have been proposed to solve this problem, with applications mostly in population dynamics, epidemiology, cardiac rhythms, neuroscience or intra-cellular chemical reactions [2]. These indices rely on various rationales, including characterization of the putative strange attractor after embedding of the measured time series; quantification of recurrence in the data or quantification of the symbolic sequence derived from the data [3]. In most case, shuffling strategies (surrogates) can additionally be applied for statistical testing [4]. In spite of this profusion, convincing proof of chaotic dynamics in biological systems is usually difficult to make because the chaotic indices are mostly efficient on long, clean and stationary time series whereas experimental measurements are usually short, noisy and possibly non-stationary. However, in some application domains, chaotic indices can be used for diagnosis e.g. to classify the time series in two classes, such as healthy vs pathological.

In this course, I will introduce the major analysis methods developed to try and evidence chaos from experimental measurements in biology. Illustrations will mainly come from neuroscience, from the analysis of microscopic dynamics (spike trains from single neurons) to that of large, macroscopic neuronal networks (as measured by electroencephalograms) [5]. Although the question of the presence of chaotic dynamics in the brain is still open [6], the derived indices offer interesting opportunities to quantify electroencephalograms traces and e.g. predict the emergence of epilepsy crises [7]. Moreover, they offer an interesting framework for theorization of the operation of neuronal network [8,9].

Bibliography

1. May, R.M. (1976) Simple mathematical models with very complicated dynamics, Nature 261, 459 - 467.

2. Lesne, A. (2006) Chaos in biology, Biology Forum 99, 413-428.

3. Kantz, H. and Schreiber, T. (2004) Nonlinear Time Series Analysis, 2nd edition, Cambridge University Press, Cambridge.

4. Schreiber, T. and Schmitz A. (2000) Surrogate time series, Physica D 142, 346-382

5. El Boustani, S. and Destexhe, A. (2010) Brain dynamics at multiple scales: can one reconcile the apparent low-dimensional chaos of macroscopic variables with the seemingly stochastic behavior of single neurons?, Int. J. Bifurcation Chaos 20, 1687–1702.

6. Korn, H. and Faure, P. (2003) Is there chaos in the brain? II. Experimental evidence and related models, C.R. Biologies 326, 787-840.

7. Subramaniyam, N.P., Donges, J.F. and Hyttinen, J. (2015) Signatures of chaotic and stochastic dynamics uncovered with ε-recurrence networks, Proc. R. Soc. A 471, 20150349.

8. Siri, B., Berry, H., Cessac, B., Delord, B. and Quoy, M. (2008) A mathematical analysis of the effects of Hebbian learning rules on the dynamics and structure of discrete-time random recurrent neural networks. Neural Comput. 20, 2937-2966.

9. Naudé, J., Cessac, B., Berry, H. & Delord, B. (2013) Effects of Cellular Homeostatic Intrinsic Plasticity on Dynamical and Computational Properties of Biological Recurrent Neural Networks, J Neurosci. 33, 15032-15043.

Course 3

Title: Ordinal symbolic dynamics with physiological data applications

Abstract:

Ordinal symbolic dynamics is a promising new approach to the investigation of time series and the systems behind them with the important properties that it is conceptually simple and relatively robust with respect to noise. The idea behind it is to consider the order relation between the values of a time series instead of the values

themselves. Roughly speaking, a given time series is transformed into a series of order patterns describing the up and down in the original series. Then the distribution of ordinal patterns obtained is the base of the analysis.

Since Bandt and Pompe introduced permutation entropy in their celebrated paper in 2002, the idea has been intensively discussed. For example, there are interesting results and open questions concerning the relation of permutation entropy to conventional ergodic concepts in dynamical systems like Kolmogorov-Sinai entropy. On the other hand, the tools of ordinal symbolic dynamics are being used in real time series analysis with a remarkable success, e.g. for identifying and discriminating different brain states in epilepsy research and in anesthesiology, for heart rate analysis and for testing independence.

The course starts with a short introduction into dynamical systems and symbolic dynamics with a special focus on nonlinear data analysis. In particular some well known entropies are discussed, on the one hand the theoretical concept of Kolmogorov-Sinai entropy, on the other hand more empirical measures like the

Sample entropy and the Approximate entropy. Then we focus on ordinal symbolic dynamics and permutation entropy. The course includes work both on real world physiological data and synthetic date coming from well-known dynamics like e.g. from the logistic family.

Course 4-5

Title: Introduction to (geometric) Dynamical Systems

Abstract of the course:

This course will give basic notion on general and geometric dynamical systems. Main examples will by non uniformly hyperbolic diffeomorphisms as Axiom-A or Anosov. The first part will focus

on different notions of recurrence as periodic points, recurrent points, chain recurrent points, non-wandering points. alpha and omega limit sets. Invariant set. Minimal sets.

Then, one shall define and study Anosov and Axiom-A diffeos. The main results are the definition and construction of the stable and unstable manifolds, the shadowing lemma, the construction of the Markov partition which allows conjugacy to a subshift of finite type. If time permits the theorem of structural stability will be stated.

In the last part one will trackel ergodic properties as invariant measures and the Birkhoff theorem. Then one will define/study the Lyapunov exponents (for Uniform. Hyper. Diffeos). The case of attractors (as the so famous Lorenz attractor and Smale Solenoid) will be presented.

Bibliography

Rufus Bowen Lecture Notes 470 Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms .

A. Katok & B. Hasselblatt: Introduction to dynamical systems.

Course 6

Title: Applications of ODEs and Dynamical Systems in Biology

Abstract :

This course will illustrate the use of ordinary differential equations and dynamical systems in the modeling of biological phenomena. The first part of the course illustrates how ODEs can be used to model biolog- ically relevant phenomena, with a stronger focus on population dynamics. We start from the full solution of simple linear and non-linear tractable models and finally introduce the principles of qualitative analysis.

In the second part of the course we discuss systems involving two interacting populations. After a brief introduction on linear systems making use of basic linear algebra, we focus on the qualitative analysis of non-linear models with examples taken from population biology and epidemiology.

In a third part, we will introduce Lyapunov functions to assess asymptotic stability, and discuss the existence of limit cycles observed in some biological phenomena. We will then generalize the qualitative analysis of dynamical systems to more than two dimensions.

Bibliography

Allman, E. S. and J. A. Rhodes. 2004, Mathematical Models in Biology: An Introduction. Cambridge University Press.

Auger, P., C. Lett, and J.-C. Poggiale. 2010, Mod ́elisation math ́ematique en ́ecologie. Cours et exercices corrig ́es. Dunod.

Baca ̈er, N. 2010, A Short History of Mathematical Population Dynamics. Springer, 1st edn.

Gillespie, J. H. 2004, Population genetics. A concise guide. Johns Hopkins Uni- versity Press, 2nd edn.

Gotelli, N. G. 1998, A primer of ecology. Sinauer, 2nd edn. Pav ́e, A. 1984, Mod ́elisation en biologie et en ́ecologie. Al ́eas.