Leading experts in the field will introduce the different conference themes interactively in tutorials on the first day of the conference week.
Date and Venue
Monday, March 19, 2018
- Arkady Pikovsky – Universität Potsdam,
- Michael Rosenblum – Universität Potsdam,
- Ralph G. Andrzejak – UPF Barcelona,
- Aneta Stefanovska – University of Lancaster.
- Hierarchy of synchronization models, Arkady Pikovsky:
In this tutorial I discuss how synchronization phenomena can be described with models of different complexity. First I outline the phase reduction method and elementary properties of interaction of two coupled oscillators. Then properties of phase lattices are presented. Finally, I will describe analytical methods due to Watabe-Strogatz and Ott-Antonsen suitable for description of synchronization in mean-field models, and will discuss more complex synchronized states beyond applicability of these approaches.
Synchronization approach to data analysis, Michael Rosenblum:The tutorial presents data analysis techniques based on the theory of coupled oscillators. The approach assumes that the multi-variate time series under study are outputs of weakly-coupled self-sustained oscillators and that the signals are appropriate for phase estimation. The main idea is to infer a model for phase dynamics of the observed network. Analysis of this model yields the strength of directed links and, thus, represents an approach to the connectivity problem, relevant for physiology, neuroscience, and other fields. We demonstrate that our technique provides effective phase connectivity which is close, though not identical to the structural one. However, for weak coupling we achieve a good separation between existing and non-existing connections. We also discuss how the frequencies and phase response curves of interacting units can be estimated. Next, we extend the approach to cover the case of pulse-coupled neuron-like units, where only times of spikes can be registered, so that the data represent point processes.
A brief introduction to nonlinear time series analysis, Ralph G. Andrzejak:Suppose that you have measured a signal from a dynamical system and you want to know if the system is deterministic or stochastic. In this tutorial we will illustrate how nonlinear time series analysis can help you with this problem. We at first show how delay coordinates allow us to reconstruct an estimate of the system’s state space representation. The nonlinear prediction error is then used to quantify the degree to which similar momentary states evolve in similar ways. Based on several examples we show that this predictability is a sensitive but not specific criterion for a deterministic system. Finally, by using the concept of surrogate signals we arrive at a more specific test for determinism. During the tutorial we will use Matlab simulations to demonstrate the different steps of analysis. The underlying scripts will be provided to the attendees.
Understanding real-world systems from the perspective of time-varying dynamics, Aneta Stefanovska:Chaotic dynamics has long been used to model complex behaviour. Another approach is through stochastic dynamics, where complexity is considered attributable to random noise, e.g. by seeking parameters quantifying the noise without attempting to understand the roles of individual components contributing to the complexity. We will introduce a new class of systems where complexity may arise deterministically without being chaotic. Their main characteristics are their oscillatory nature and time-dependent interactions, modulating their amplitudes, phases, couplings, or a combination of these. We will illustrate that systems of this class are very robust and that perturbing them can often, counterintuitively, increase their stability, rather than destabilising them. Their stability is not evaluated asymptotically, but via finite-time Lyapunov exponents, or via synchronization analysis.
The participants should have some background in nonlinear dynamics and time series analysis. They must bring laptops installed with MATLAB/Octave if they want to take part in the practical activities.