Symmetry breaking and avalanche shapes in modular neural networks.

Modularity is as a key characteristic of structural and functional brain networks across species and spatial scales. We investigate the stochastic Wilson-Cowan model on a modular network in which synaptic strengths differ between intra-module and inter-module connections. The system exhibits a rich phase diagram comprising symmetric (with low and high activity) and "broken symmetry" phases. Symmetric phases are characterized by the same low or high activity in all the modules, while the broken symmetry phases are characterized by a high activity in a subset of the modules and low activity in the remaining ones. There are two lines of critical points, the first between the low activity symmetric phase and the high activity symmetric phase, and the second between the low activity symmetric phase and a broken symmetry phase with one active module. At those lines the system shows a critical behavior, with power law distributions in the avalanches. Avalanche shapes differ systematically along the two lines: they are symmetric or right-skewed at the transition with the symmetric phase, but become left-skewed over intermediate durations along critical line with the broken symmetry phase. These results provide a theoretical framework that accounts for both symmetric and left-skewed neural avalanche shapes observed experimentally, linking modular organization to critical brain dynamics.