Comparison of the phase portraits of excitatory-inhibitory networks.

A. Architecture of the large network of spiking neurons. B. Architecture of the network of two complex-valued firing-rate units. C–E. A sparse, randomly-connected network of QIF, EIF, or LIF neurons, respectively. For each connection strength, 50 ms of the firing rate of the excitatory population is shown in black. Stability diagram of the corresponding two-unit complex-valued rate-model network is superimposed on each panel, where orange indicates a stable limit-cycle, and white a stable fixed-point. A constant external input was also included with mean and variance set to produce a baseline firing rate of 50 Hz and a CV of 0.1 when and were zero. F. Sample excitatory (top, red) and inhibitory (bottom, blue) dynamics from both the spiking (dark) and rate (light) EIF networks with and (green square in D). An exemplary spike raster of 50 neurons from each population (excitatory/inhibitory, respectively) is overlaid on the firing rate curves of both networks. Horizontal scale bar = 10 ms. Vertical scale bar = 10 Hz. G. Power spectra of excitatory (top) and inhibitory (bottom) units from both networks, with spiking network in darker shades and rate network in lighter shades, as in F. Both networks have a dominant frequency near 50 Hz. Curves represent mean power spectra from all parameters in D for which both networks are oscillatory (standard error comparable to line width). H. Cross-correlation between excitatory and inhibitory units for EIF spiking (dark purple) and rate (light purple) networks. Both networks exhibit maximal correlation at a small positive phase shift, indicating that inhibitory oscillations follow closely behind excitatory oscillations. As in G, curves represent means over all parameters producing oscillations in D, with standard error smaller than line width.