How high dimensional neural dynamics are confined in phase space

High dimensional dynamics play a vital role in brain function, ecological systems, and neuro-inspired machine learning. Where and how these dynamics are confined in the phase space remains challenging to solve. Here, we provide an analytic argument that the confinement region is an M-shape when the neural dynamics show a diversity, with two sharp boundaries and a flat low-density region in between. Despite increasing synaptic strengths in a neural circuit, the shape remains qualitatively the same, while the left boundary is continuously pushed away. However, in deep chaotic regions, an arch-shaped confinement gradually emerges. Our theory is supported by numerical simulations on finite-sized networks. This analytic theory opens up a geometric route towards addressing fundamental questions about high dimensional non-equilibrium dynamics.