Abstract
In the modeling of a variety of models of large-scale systems of interacting
agents, mathematical models consider that the coupling terms representing the
interactions between agents are scaled according to the network size to prevent
divergence of the coupling terms. In a previous paper of ours and follow-ups
from colleagues, we have explored the behavior of a large-scale system of
interacting agents with diffusive coupling. It was shown that the system
converges to a Dirac mass, that is free to evolve according to the intrinsic
dynamics. We generalize these results to more general coupling functions, and
show that, under specific conditions, the divergence of the coupling term does
not preclude the well-posedness of limit equations. Instead, the system carries
solutions associated to a balanced, so that the net input where leading order,
vanishes. We also explore convergence to the balanced state. Such balanced
regimes were widely observed in neuroscience and associated with complex
regulatory mechanisms. This results offer an alternative, minimalistic
perspective on the balance of excitation/inhibition in the brain.