The subject of this thesis is the role of noise in interacting systems, with an eye to applications in Biology. This study is based on the Kuramoto model, a model of one-dimensional interacting and noisy rotators which admits a synchronization type transition, and on some of its generalizations. The first part of the thesis (made in collaboration with G. Giacomin, K. Pakdaman and X. Pellegrin) is about the "Active Rotators" model, a generalization of the Kuramoto model in which each rotators has its own inner dynamics, that can be chosen excitable. We prove rigorously that the global system may have a very different behavior from the one of an isolated rotator. In particular noise and interaction may induce periodic behaviors. In the second part (made in collaboration with G. Giacomin and E. Luçon), we study the disordered Kuramoto model, in the limit of vanishing disorder. We show in particular that when the disorder is not symmetric the model admits a periodic solution, and we give an asymptotic of its speed. The third part (made in collaboration with G. Giacomin and L. Bertini) is devoted to the long time behavior of the Kuramoto model : for times proportional to the size of the system the rotators keep a synchronized profile, which performs a Brownian motion at the limit of an infinite population. Finally, in the last part, I study the validity of a phase reduction in the noise induced escape problem, for models close to reversibility.
