Neural circuits generate cognition, sensation, and behavior through the coordinated activity of many interconnected units. Understanding how these functions emerge dynamically and what connectivity structures support this emergence is a central challenge in neuroscience. This challenge is compounded by neural circuits' essential features: large numbers of components (neurons), nonlinear dynamics, complex recurrent interactions, and plastic connectivity. This thesis develops theoretical approaches to tackle this complexity, using tools from physics, particularly dynamical mean-field theory (DMFT), to analyze how connectivity structure shapes collective neuronal dynamics and computational functions in nonlinear recurrent neural networks.