Vanishing transition width in the thermodynamic limit

Establish that, in the thermodynamic limit N → ∞ of the massively coupled excitatory–inhibitory rate network defined by Eq. (1) with short‑term synaptic depression acting on excitatory‑to‑excitatory synapses and synaptic strengths scaling as 1/√K, the intermediate transition region between the homogeneous fixed point and the rate‑chaos regime shrinks to zero width, yielding an abrupt transition at a single critical synaptic strength J0 = Jc.

Background

The paper analyzes a massively coupled excitatory–inhibitory rate network with short‑term synaptic depression (STD) on excitatory‑to‑excitatory synapses. For finite networks, the authors observe three regimes as the global synaptic strength J0 increases: a homogeneous fixed point at small J0, a complex transition region featuring heterogeneous stationary or oscillatory states, and a rate‑chaos regime at large J0.

Based on numerical evidence and theoretical tools (random matrix stability analysis and dynamical mean‑field methods), the authors report that the transition region narrows as the system size grows. They explicitly conjecture that, in the thermodynamic limit, the width of this transition region vanishes, implying an abrupt onset of chaos at a critical coupling, analogous to sharp transitions previously reported in related rate and spiking models.