Ergodic theory for complex systems: a rigorous study of dynamics on heterogeneous networks

Project Information

ErgThComplexSys

Grant agreement ID: 843880

DOI

10.3030/843880

Project closed

EC signature date 10 December 2019

Start date 1 September 2020

End date 31 August 2023

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 257 619,84

EU contribution

€ 257 619,84

Coordinated by

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

Periodic Reporting for period 2 - ErgThComplexSys (Ergodic theory for complex systems: a rigorous study of dynamics on heterogeneous networks)

Reporting period: 2022-09-01 to 2023-08-31

Summary of the context and overall objectives of the project

The mathematical study of coupled dynamical systems, i.e. systems with multiple components that interact with
one another, started with the formulation of Newton’s universal law of gravitation when astronomers and mathematicians
addressed the question on the stability of the Solar System. Since then, it has become clear that even weak
interactions between different agents can have drastic consequences on their evolution and make their dynamics very
hard to predict. Nowadays, models of networks of coupled dynamical systems appear in many areas of science:
physics, chemistry, biology and also engineering and human sciences. Examples range from solid state physics to
astrophysics, reaction-diffusion equations, transportation systems, the Internet, social networks, opinion models, etc.
This vast spectrum of applications has stimulated a broad interdisciplinary endeavour to predict and control the behaviour
of so-called ”complex systems”, i.e. large ensembles of units coupled through an intricate interaction web.
The main goal of this project is to study networks of coupled dynamical systems, and how the interplay between
structure of the interactions and local dynamics shapes their evolution. The systems under study are inspired by
paradigmatic examples in biology, and relate to questions of primary importance in this field. For example: how does
the dynamics of coupled neurones shape the functionality of the brain? How does it give rise to stable spiking patterns
such as gamma-band oscillations, and, at the same time, undergo transitions from healthy to pathological states?
A rigorous mathematical framework to satisfactorily explain these mechanisms is currently
not available.

So far, most studies on coupled systems have been limited to numerical methods due to the considerable difficulties
encountered in formulating general theoretical descriptions. Instead, this project aims to develop rigorous
mathematical approaches, accompanied with computational methods when necessary, to describe the emergence of
global behaviour in complex heterogeneous systems, underlining its dependence on the microscopic features. At the
interface between pure and applied mathematics, the strategy is to rely on results from abstract ergodic theory, especially
those recently obtained for non-uniformly hyperbolic systems, to address concrete archetypes exhibiting
real-world features.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

We started by investigating dynamical systems obtained by coupling two maps: one of the maps is chaotic exhibiting erratic behavior, and the other exemplify the degrade towards a resting state. We successfully recovered the statistical properties of these maps, in particular focusing on the presence of chaos and of stationary measures describing the asymptotic statistical behavior.

We then moved to a rigorous analysis of a class of coupled dynamical systems was presented. Here two distinct types of components, one excitatory and the other inhibitory, interact with each other. This borught our rigorous mathematical analysis closer to real-world systems inspired by biology where excitation, inhibition, and threshold crossing are important features. Further investigation of driver-driven systems considered the case where the driver is generated by a dynamical system with asymptotically large expansion.

Another approach we took is in the study of coupled maps in the infinite limit, i.e. when the number of components tends to infinity; this is a mathematical abstraction -- any real world system is made by only finitely many components -- but a crucial one that allows to give simpler descriptions of the evolution of real world phenomena. The main actor in this case is an operator called the "Self-consistent Transfer Operator" that describes the evolution of the infinite limit. Our investigation has been crucial in determining the properties of this operator and describe the corresponding infinite limit.

In studying these systems in the infinite limit, we focused on several aspects that were previously out of the reach of existing techniques. In particular we developed new techniques to study synchronisation in this context, i.e. the tendency tendency of the coupled maps to evolve in a coordinated way. We provided sufficient conditions for synchronization to be a stable phenomenon and therefore observable in a specific system. Our investigation crucially addressed the case where interactions are heterogeneous, and we incorporated graphons in our setup of coupled maps; these are combinatoric objects thatcapture the structure of the interactions.

Finally, we looked at models of interacting systems reproducing the evolution of some genetic networks. In this context, we focused on the effect of delay in the interactions and how this determines the presence of synchronization.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

1) We gave rigorous mathematical proofs of existence of ergodic probability measures in coupled systems whose components have behaviours inspired by real-world systems (excitation/inhibition, threshold crossing,...). This work shed new light on

2) We provided new techniques to study chaotic coupled maps and, in particular, their ergodic theoretical properties. The new spaces of measures that we have introduced promise to shed new light in the study of coupled systems in discrete time. Crucially, our techniques can deal with the notoriously hard case of heterogeneous interactions.

3) The techniques above made a decisive step forward in studying the relation between the dynamics of high-dimensional, but finite coupled systems, and their thermodynamic limit, and provided a new framework for the study of coupled systems. In this limit the onset of synchronization and chimera states was investigated.

4) We advanced the study of infinite limits of coupled maps: by adapting previous techniques to the study of synchronisation in coupled maps, by providing new tools to study equilibrium states in the infinite limit, by analysing the effect of an heterogeneous interaction structure in the infinite limit.

5) We studied driver-driven coupled systems assuming high chaoticity in the driving system and stated our results in terms of approximate ergodic properties, which provided a new angle under which to study such systems. Beyond shedding light to the study of coupled systems, this laid another bridge between deterministic chaotic systems and completely random systems.

6) We gave a rigorous mathematical description of how synchronization is favoured by a time delay in the interaction between degrade and fire oscillators modelling networks of genes.

Network structure of inhibitory components interacting with an excitatory chaotic environment