Fig. 1

Bifurcation diagrams of (1) w.r.t. the slope parameter σ, when \(I_{\mathrm{thalamus}}=0\) and \(J(\mathbf {x},\mathbf {y})\) is a difference of Gaussians on a square (above) or hexagonal (bottom) domain with periodic boundary conditions. The vertical axis is the norm of the sigmoid of the stationary solution. Black lines correspond to unstable solutions and brown lines to stable ones. Each bifurcated solution yields a torus of identical steady states up to translations. Red dots are additional bifurcation points where the bifurcated branches have not been computed. The connectivity is tuned such that the first bifurcation point for the nonlinear gain σ is a supercritical \(\mathbf{D}_{4}\)-pitchfork (respectively, \(\mathbf{D}_{6}\)-pitchfork) bifurcation with wave-vector \(\|\mathbf{k}\|=1\) for which the spots are stable, while the stripes are not. The size of the square cortex is \(8\times 2\pi\) and the numerical mesh size is 1024² (top) and \(3\times512^{2}\) (bottom), the threshold is \(T = 0.1\) and \(\sigma_{\mathrm{loc}} = \pi\cdot0.395\). We only used the 15 eigenvalues with the largest real parts to find the bifurcation points. See Appendix B for details concerning the numerical methods