How high dimensional neural dynamics are confined in phase space

Abstract

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.

Figures (5)

• Figure 1: The free energy of the reference has a transition at $g_c=1$.
• Figure 2: The geometry of the phase pace confining high-dimensional recurrent dynamics. Local entropy (b) and its derivative (a) as a function of the distance for different values of $g$. If $g<1$, the local entropy profile is a spike near the origin. If $g>1$, the profile shows an M-shaped structure and then arched structure, and the corresponding position becomes further away from the reference with increasing $g$.
• Figure 3: The confinement is repelled from the reference when increasing the gain parameter.
• Figure 4: Distribution of low-speed solutions ($\|\dot{\mathbf{x}}\|<10^{-3}$) around a fixed point ($\|\dot{\mathbf{x}}\|<10^{-6}$) for network dynamics with $N=100$ and $g=3.0$. Each histogram represents the distribution around a specific reference.
• Figure 5: Low-dimensional representation of fixed points for network dynamics with $N=100$ and different values of $g$. Visualization is carried out via the nonlinear dimensionality reduction method Isomap Sci-2000 (see details in the SM). Fixed points are found using the Levenberg-Marquart method NO-2006 (see SM for details) with the speed threshold $\varepsilon=10^{-6}$ .