Figure 5. Stability Analysis of a Two-Dimensional Firing Rate Model.

(A) Bifurcation diagram of a firing rate model as a function of the inhibitory decay timescale τ_i and inhibitory projection width σ_i. The excitatory projection width and time constant are fixed at σ_e = 0.1 and τ_e = 5 ms respectively. Color represents the wavenumber with the largest real part of eigenvalue and the gray region is stable. Top-down modulation of inhibitory neurons modeling attention expands the stable region (black dashed).

(B) Left: the real part of eigenvalues as a function of wavenumber for increasing τ_i, when σ_i = σ_e. Right: three consecutive spike raster snapshots of a spiking neuron network with σ_i = σ_e and slow inhibition (same network as in Figure 4 in the unattended state).

(C) Same as (B) for σ_i larger than σ_e. Right: spike raster snapshots of a spiking neuron network with broad inhibitory projections, where the excitatory and the inhibitory projection widths of layer 3 were ${\alpha }_{\text{e}}^{\left(3\right)}=0.1$ and ${\alpha }_{\text{i}}^{\left(3\right)}=0.2$, respectively. Other parameters were the same as in (B).