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Résumé |
In this talk I will try to give an overview of recent achievements in the field of complexity. The science of complexity is a new and extremely trans-disciplinary field of research. Many definitions have been given so far and yet the concept has resisted categorization, but, nevertheless becomes more and more attractive for many researchers in different fields. One of the points of agreement among the scientists is that complex systems are those composed of a large number of interacting elements, so that the collective behaviour of those elements goes far beyond the simple sum of the individual behaviours [1]. In the last twenty years, one of the most studied collective behaviours has been spontaneous synchronization in networks of elements with simple dynamics, e.g., oscillators [2]. Synchronization occurs in physical/biological systems over a wide range of spatial/temporal scales, e.g., in electrochemical oscillators, laser arrays, animal flocking, pedestrians on footbridges, etc [3]. Besides the synchronous firings of cardiac cells required to keep the heart beating, synchrony is required for technological applications, e.g., in electrical power-grids, the generators must lock to the grid frequency [4]. Synchrony also has undesirable effects, e.g., in brain circuits, it can be related to epilepsy. Interactions among identical units can even lead to more complex collective behaviours, e.g., in fully coupled systems, to quasi-periodic and chaotic evolutions [5], while the presence of non-local coupling can induce the emergence of new states with broken symmetry, termed ”Chimera States”, where synchronized and desynchronized populations coexist [6]. Chimera states have been reported in experiments in opto-electronic devices, mechanical oscillator networks and electronic delayed oscillators [7]. An extremely powerful exact method recently developed is the Ott-Antonsen Ansatz [8], which allows to rewrite the dynamics of fully-coupled networks of phase oscillators in terms of few collective variables. The success of the approach has led to hundreds of recent publications in applied mathematics and physics. A recent application [9] to spiking neural networks has opened up the perspective to apply this method to computational neuroscience and to derive exact models able to describe at a macroscopic level the dynamics of a neural population. |
