Broad onset of chaos with toroidal dynamics in finite-size random neural networks

Randomly connected neural networks undergo a transition from a stable fixed point to chaos as the coupling strength increases. In the thermodynamic limit, this transition has been shown theoretically to occur abruptly at a critical point. In finite-size systems, however, a variety of bifurcation cascades appear between the stable fixed point and chaos. In this study, we systematically characterize routes to chaos in finite-size random neural networks. By analyzing individual realizations, we identify multiple scenarios, including the Ruelle–Takens–Newhouse route, torus doubling, and fractalization, as well as chaotic dynamics with persistent toroidal geometry. We also study these behaviors at the ensemble level by quantifying the fraction of chaotic trajectories as a function of coupling strength and system size. The resulting finite-size crossover sharpens with increasing system size and exhibits empirical scaling trends, providing a statistical characterization of the broad onset of chaos in finite random networks.