We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e. a model on the complete graph with a suitably renormalized interaction parameter. The purpose of this talk is twofold: (1) to explain that this proximity can be established in great generality on finite time horizon; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with N large but finite, notably the values of N can be reached in simulations or that correspond to the typical number of interacting units in a biological system. `
I will review some properties of models of one-dimensional chains driven off equilibrium by temperature or force gradients. In particular I will deal with two paradigmatic systems, the Hamiltonian XY model and the discrete nonlinear Schr"odinger equation.
S. Iubini, S. Lepri, R. Livi, A. Politi Boundary-induced instabilities in coupled oscillators arXiv:1401.2846 Phys. Rev. Lett. 112, 134101 (2014).
S. Iubini, S. Lepri, R. Livi, A. Politi Coupled transport in rotor models New J. Phys. 18(2016)083023.
Active fluids contain a large number of active agents which consume energy in order to move or exert mechanical forces. Examples include intracellular flows driven by the activity of molecular motors in networks of biopolymers inside cells and bacterial swarms and suspensions of microswimmers. The talk surveys two representative examples for spatiotemporal self-organisation in active fluids: (i) mechano-chemical waves in small droplets (microplasmodia) of the slime mold Physarum polycephalum and (ii) the so called -mesoscale turbulence- in dense suspensions of swimming bacteria. We show that a variety of mechanochemical patterns inside the droplets can be reproduced in a model, that considers Physarum cells as an active poroelastic two-phase medium coupled to an intracellular calcium dynamics. The experimentally observed net motion of the droplets is obtained if one assumes a stick-slip motion of the droplets on the substrate. Mesoscale turbulence in bacterial suspensions is found to be caused by the interplay of active motion, local alignment of swimmers and long-range hydrodynamic interactions with the help of a continuum model derived from the dynamics of individual swimmers.
The phenomenology of the Kuramoto model (continuum limit) strongly relies on the nonlinear stability of its stationary states. To understand and to rigorously assert stability in this infinite-dimensional setting have been long-standing challenges, and show similar features of the Landau damping in the Vlasov equation. In this talk, I will review results on stability conditions and asymptotic stability of various stationary states, that mathematically confirm the intuited phenomenology and its dependence on parameters.
We revisit the dynamics of a prototypical model of balanced activity in networks of spiking neutrons. A detailed investigation of the thermodynamic limit for fixed density of connections (massive coupling) shows that, when inhibition prevails, the asymptotic regime is not asynchronous but rather characterized by a self-sustained irregular, macroscopic (collective) dynamics. So long as the connectivity is massive, this regime is found in many different setups: leaky as well as quadratic integrate-and-fire neurons; large and small coupling strength; weak and strong external currents.
Ekkehard Ullner, Antonio Politi, Alessandro Torcini, Ubiquity of collective irregular dynamics in balanced networks of spiking neurons arXiv:1711.01096
