**Spontaneous Synchronization**

K=0.2K=0.6 | Question: How can a collection of individual oscillators, each with their own characteristic properties, act collectively and come into sync? The search for answers to this basic question has in in the last few decades spawned an entire research area. The short answer is that synchronization depends sensitively on how and how strongly the individual units are connected and how diverse their properties are. The Kuramoto model is probably the simplest mathematical model that sheds some light on the phenomenon of mass synchronization of biological oscillators, such as fireflies, crickets, neurons, etc. Loosely speaking, each oscillator tries to modulate the frequency of all the other oscillator so as to bring them more into phase with itself. For low coupling the system is not able to synchronize at all. At a certain critical value (0.16), sync starts to happen, and as the coupling is increased the degree of sync also increases. In one modification, we have included network dynamics into these synchronization studies. In other words, the network of oscillator coupling is now allowed to evolve in time based on specific "learning rules." In such systems, the dynamical richness is much increased. One can observe in numerical simulations the self-assembly of such networks, based on the mutual reinforcement of synchronization and learning. We have also found a class of experimental systems that are effectively governed by Kuramoto-like models. The picture below (PRE 92, 052912) shows the time evolution of the dynamics frequencies of coupled Wien-bridge oscillators. When the coupling strength is sufficiently strong, all frequencies quickly converge (top panel). We are also exploring the experimental realization of chimera-like states in lattices of Wien-bridge oscillators. These are symmetry-broken dynamical states that are characterized by a coexistence of synchronized and decoherent oscillators. | 1) Kuramoto model simulations 2) Hebbian learning and synchronization 3) Wien-bridge oscillator network as a experimental toy model 4) Chimera-like states |