Extended Data Fig. 3: Stability of population profile and fixed-point analysis.
a, The stability of the shape of the population profile depends on JE and JI (shown for N = 6). ‘Unstable’ regime: the population activity diverges over time. ‘Homogeneous’ regime: the network generates a stable activity profile that is uniform across the entire network. ‘Inhomogeneous’ regime: the network generates a stable bump of activity that persists at a discrete set of orientations in the absence of input. Dashed lines indicate optimal values of JE for which the network generates a set of marginally stable solutions that can persist at any orientation in the absence of input. b-c, Fixed point conditions from the equations for bump orientation (panel b; fodd = 0) and relative amplitude (panel c; feven = 1/JE). See Supplementary Note for details. b, Heatmap of fodd (w, ψ) for densely sampled bump widths \(w\in \left.\right[2\uppi /N,2(N-1)\pi /N\left.\right)\) and orientations ψ ∈ [0, 2π). Red and blue regions correspond to fodd > 0 and fodd < 0 (which drive the bump orientation to the right and left, respectively). White regions indicate fodd = 0, which correspond to potential fixed points \({{\mathcal{X}}}^{* }=({a}^{* },{w}^{* },{\psi }^{* })\) at which the bump can stably persist. Note that at ψ = (θc + θd)/2, d = c, c + 1, fodd (w, ψ) = 0 regardless of the value of w. c, Contours of constant feven(w, ψ), shown for 10 evenly spaced values of 1/JE between and including 1/12 and 1/2.4. These contours indicate a necessary (but not sufficient) relationship between w and ψ for stationary bump solutions. d-i, Eigenvalues of linearized system about fixed points with orientation ψ* = θj (panels d-f) or ψ* = (θj + θj+1)/2 (panels g-i), j = 1…N. See Supplementary Note for details. d, g, Eigenvalue λψ depends on JE. This eigenvalue corresponds to changes in orientation near the fixed points and is the sole determinant of stability of the fixed points. Note that when the set of fixed points corresponding to ψ* = θj, j = 1…N, is stable, the other set of fixed points corresponding to ψ* = (θj + θj+1)/2, j = 1…N, is unstable, and vice-versa. The remaining two eigenvalues λ+ (panels e, h) and λ− (panels f, i) depend on both JE and JI but are always negative in the parameter regime that generates bump-like profiles (region above black line; compare to ‘inhomogeneous; stable’ in panel a). Panels b,c,e,f,h,i were generated using redblueu.m (https://www.mathworks.com/matlabcentral/fileexchange/74791-redblue-colormap-generator-with-zero-as-white-or-black) and magma.m (https://www.mathworks.com/matlabcentral/fileexchange/51986-perceptually-uniform-colormaps).
