How synaptic heavy tails and modularity shape chaotic activity in randomly connected neural networks

Understanding how network connectivity shapes neural dynamics is central to both theoretical neuroscience and artificial intelligence. In this talk, I will discuss how deviations from classical homogeneous random connectivity alter the emergence and nature of chaotic activity in recurrent neural networks.

In the first part, motivated by growing experimental evidence that the distribution of cortical synaptic weights exhibits heavy tails, I will focus on networks with power-law distributions of weights. I will show that heavy-tailed connectivity fundamentally reshapes the transition to chaos. In networks of binary neurons, only networks with heavy-tailed weights display a continuous transition to chaos accompanied by scale-free neuronal avalanches. In networks with heavy-tailed weights and smooth activation functions, finite-size effects play a crucial role: while infinite-size mean-field theory predicts ubiquitous chaos, finite networks undergo a slow transition between quiescent and chaotic regimes. As a result, these networks exhibit critical-like behavior over a wider range of the control parameter compared to their Gaussian counterparts. At the same time, heavier tails are associated with a lower dimensionality of chaotic activity.

In the second part, I will turn to modular and hierarchical network connectivity structures. Using mean-field theory and simulations, I will show that modularity introduces a rich dynamical phase diagram with distinct low- and high-dimensional chaotic regimes, separated by a crossover region characterized by low values of the maximal Lyapunov exponent and participation ratio dimension, but with high values of the Lyapunov dimension. Surprisingly, chaos can be attenuated either by adding noise to strongly modular networks or by introducing modular structure into otherwise random connectivity. Extending the model to include a multilevel, hierarchical connectivity reveals that a loose balance of activity across levels naturally drives the system toward the edge of chaos.