Dynamical regimes for the network with adaptation or synaptic filtering with additive external noise.

Numerical integration of the dynamics with units receiving additive external white noise, as a proxy for spiking noise. A: Network with adaptive neurons. B: Network with synaptic filtering. Colored lines correspond to the firing rate of individual neurons, the black line indicates the population average activity. Insets: complex eigenspectrum λw/s of the dynamic matrix at the fixed point. Dots: eigenvalues of the connectivity matrix used in the network simulation. Solid line: theoretical prediction for the envelope of the eigenspectrum. i. Both the network with adaptation and synaptic transmission are stable, the external noise generates stationary fluctuations around the fixed point. ii. The network with synaptic filtering undergoes a zero-frequency bifurcation. Noise adds fast temporal variability in the firing rates. The network with adaptation remains stable, and the fluctuations are larger in amplitude. iii. The network with adaptation undergoes a Hopf bifurcation. The firing rate activity combines the fast fluctuations produced by white noise and the chaotic activity with an oscillatory component. iv. The network with adaptation shows highly irregular activity, and strong effects due to the activation and saturation bounds of the transfer function. Parameters as in Fig 4, external noise ση = 0.06.