# Sesje gości zagranicznych

**dr Krzysztof Bartoszek**

Departament of Mathematics, Uppsala University, Sweden

*Phylogenetic comparative methods: applying modern probabilistic methods to evolutionary biology***dr Paweł Pilarczyk**

Institute of Science and Technology, Austria

*Automatic classification of global dynamics in multi-parameter systems***dr Jonathan Touboul**

College de France & Inria

*The complex interplay between structure and function in the brain: from experiments to theory*

### Automatic classification of global dynamics in multi-parameter systems

A dynamical system is a mathematical concept for describing an object varying in time, using a fixed rule that depends on the current state of the object (and not on its past). Dynamical systems can be used to describe a variety of phenomena, such as the growth of a population or spread of an infectious disease. In the first part of the talk, I am going to introduce a framework for automatic classification of global dynamics in a dynamical system depending on a few parameters (such as fertility rates or disease transmission rates). A set-oriented topological approach will be used, based on Conley’s idea of a Morse decomposition (see [3]), combined with rigorous numerics, graph algorithms, and computational algebraic topology. This approach allows to effectively compute outer estimates of all the recurrent dynamical structures encountered in the system (such as equilibria or periodic solutions), as perceived at a prescribed resolution. It thus provides an automatic computational method for concise and comprehensive classification of all the dynamical phenomena found across the given parameter ranges (see [1], [2]). The method is mathematically rigorous (a.k.a. computer-assisted proof), and has a potential for wide applicability thanks to the mild assumptions on the system.

In the second part of the talk, I am going to discuss a few interesting applications. First, I am going to show a population model which contradicts the commonly held belief that the initial state of a population doesn’t matter, because it will eventually stabilize at the equilibrium (see [1]). Next, I am going to show a model which shows that increasing the use of pesticides may sometimes actually increase the amount of pests (see [4]). Finally, I am going to show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior, not normally found in the simple epidemic models (see [5]).

**Bibliography:**

[1] Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, P. Pilarczyk, *A database schema for the analysis of global dynamics of multiparameter systems*, SIAM J. Appl. Dyn. Syst., Vol. 8, No. 3 (2009), 757-789.

[2] J. Bush, M. Gameiro, S. Harker, H. Kokubu, K. Mischaikow, I. Obayashi, P. Pilarczyk, *Combinatorial-topological framework for the analysis of global dynamics*, Chaos, Vol. 22, No. 4 (2012), 047508.

[3] C. Conley, *Isolated invariant sets and the Morse index*, CBMS Regional Conference Series in Math., no. 38, Amer. Math. Soc., Providence, RI, 1978.

[4] E. Liz, P. Pilarczyk, *Global dynamics in a stage-structured discrete-time population model with harvesting*, J. Theoret. Biol., Vol. 297 (2012), 148-165.

[5] D.H. Knipl, P. Pilarczyk, G. Rost, *Rich bifurcation structure in a two-patch vaccination model*, submitted.

### The complex interplay between structure and function in the brain: from experiments to theory

**a.** Topology of visual cortex: some theoretical thoughts and novel data, towards a theory of topological optimization

**b.** The role of noise, disorder and heterogeneity in the dynamics of large cortical networks

In these lectures. I will give an overview of two instances in which mathematical analysis lead to a better understanding of cortical function. Both will revolve around collective dynamics in the mammalian brain. In this domain, an ever-increasing amount of data challenges our understanding of cortex and function. At the center of this problem is the inter-relationship between structure and function of cortical networks, on which we will try to shed new light in the course of our lectures.

**a.** I will start by presenting some works dealing with the structure and function of the visual cortex. In the early visual cortex of higher mammals, information is processed within functional maps whose layout is thought to underlie visual perception. Here, I will preserit a few theoretical thoughts, simulations and experimental data on the possible principles at the basis of their architecture. as well as their possible role in perception. I will present new data on spatial frequency preference representation in cat: we evidence the presence of a continuous map with singularities around which the map organizes as an electric dipole potential. Mathematically, I will show that both architectures are the most parsimonious topologies ensuring local exhaustive representation. Eventually, I will show using computer simulations how these topologies may improve coding capabilities.

**b.** Another question that is largely open and that may be addressed using mathematics is the role of noise and disorder in the large-scale dynamics of neuronal networks. In order to investigate these questions, I will introduce the main mathematical tools. from the domain of probability theory. that are used in the modeling of large-scale neuronal networks involved at functional scales in the brain. Limits of large networks with complex topologies will be derived, and I will show how levels of noise shape the collective response, in particular I will present a particularly interesting transition to global synchronization in the network as noise is increased.