Synchrony, Unstable Attractors, and Beyond - Does the Structure of a Neural Network Control its Dynamics?

Pulse-coupled oscillators constitute a paradigmatic class of dynamical systems interacting on networks because they model a variety of biological systems including flashing fireflies and chirping crickets as well as pacemaker cells of the heart and neural networks. Synchronization is one of the most simple and most prevailing kinds of collective dynamics on such networks. Here we demonstrate, how breaking different symmetries of the network dynamics affects collective synchronization, often leading to the breaking of synchrony.

Globally coupled, symmetric networks without interaction delays attract every random initial condition towards the completely synchronous state. However, we show that the presence of delays or structured network connectivity lead to completely different phenomena: exponentially many periodic attractors, attracting yet unstable periodic orbits, long chaotic transients, and the coexistence of irregular, asynchronous with regular, synchronous dynamics. Furthermore, we investigate the speed of synchronization in structured networks using random matrix theory. Although, as might be expected, the speed of synchronization increases with increasing coupling strengths, it stays finite even for infinitely strong interactions. The source of this speed limit is determined by the connectivity structure of the network.