New math cracks non-normal neural network dynamics
How random connections shape neural variability—solved with free probability.
Jacob Zavatone-Veth derived a closed functional equation for the stationary covariance spectrum of discrete-time recurrent dynamics with random non-normal Gaussian weights. Using a free-probability approach, the work characterizes how principal component variance distributes in noise-driven dynamics. It analyzes tail eigenvalue behavior in the critical regime and notes why continuous-time analogs lead to an infinite hierarchy of equations rather than a closed scalar equation, with concluding comments on the relevance to comparing models of non-normal dynamics to neural data.
- Uses free probability to derive a closed functional equation for the stationary covariance spectrum of discrete-time RNNs with random non-normal weights
- Reveals distinct tail eigenvalue behavior in the critical regime for discrete-time vs continuous-time dynamics
- Highlights that continuous-time dynamics lead to an intractable infinite hierarchy of equations, limiting direct analysis
Why It Matters
Provides a rigorous analytic framework for interpreting PCA of neural data from random recurrent networks.