**Left column:** Global stability analysis in the single-population network. **A** Illustration of network architecture. **C** Upper panel: Input-output relationship for external Poisson drive shown in gray. In addition for for the noiseless case (blue) and the noisy case (red). The inset shows the gray curve over a larger input range. Lower panel: for different rates of the external Poisson drive from black to light gray. Intersections with the identity line (dashed) mark fixed points of the system, which are shown in **E** as a function of *ν*_{ext}. **F** Flux in the bistable case for in black, in blue, and modified system in red. Intersections with zero (dashed line) mark fixed points. The inset shows an enlargement close to the LA fixed point. Horizontal bars at top of figure denote the size of the basin of attraction for each of the three settings. **Right column:** Global stability analysis in the network of two mutually coupled excitatory populations. **B** Illustration of network architecture. **D** Flow field and nullclines (dashed curves) for and separatrices (solid lines), LA fixed point (rectangle), HA fixed point (cross) and unstable fixed points (circles) for in black, in blue, and in red. The red separatrix and the red unstable fixed point coincide with the black ones. **G** Enlargement of D close to the LA fixed points. Flow field of original system shown in black, of modified system in red.