Discontinuous transition to chaos in a canonical random neural network

TL;DR

This paper investigates a classic random neural network model and discovers a novel discontinuous transition to chaos driven by the shape of the nonlinear transfer function, expanding understanding of neural dynamics.

Contribution

It generalizes the SCS model to include odd saturating nonlinearities and reveals a new type of abrupt chaos transition caused by the transfer function's curvature.

Findings

Discontinuous chaos transition occurs when the transfer function's slope at zero is a local minimum.

Chaos emerges suddenly via an attractor-repeller fold, with the Lyapunov exponent non-zero at onset.

The transition is characterized by a sudden jump in chaotic behavior, not gradual.

Abstract

We study a paradigmatic random recurrent neural network introduced by Sompolinsky, Crisanti, and Sommers (SCS). In the infinite size limit, this system exhibits a direct transition from a homogeneous rest state to chaotic behavior, with the Lyapunov exponent gradually increasing from zero. We generalize the SCS model considering odd saturating nonlinear transfer functions, beyond the usual choice $\phi(x)=\tanh x$. A discontinuous transition to chaos occurs whenever the slope of $\phi$ at 0 is a local minimum (i.e., for $\phi'''(0)>0$). Chaos appears out of the blue, by an attractor-repeller fold. Accordingly, the Lyapunov exponent stays away from zero at the birth of chaos.