Fig. 4: Smaller networks require more fine-tuning and are less robust to noise.

a, Top: log of net drift speed (color map) as a function of J_E and N. Red circular markers indicate optimal values of \({J}_{\textrm{E}}^{* }\); darker blue colors indicate slower (that is, better) drift rates. Suboptimal networks achieve better performance as N increases. Bottom: to estimate tolerance around an optimal value of \({J}_{\textrm{E}}^{* }\), we compute the local change in net drift speed with respect to J_E (turquoise lines) that will achieve performance below some threshold (horizontal dashed black line, illustrated for a threshold of 0.1 rad s⁻¹). b, For a given N (different colors), larger values of local excitation require less fine-tuning to achieve the same performance. Solid lines mark the analytic tolerance given in equation (7); filled circles indicate the numerically estimated tolerance about each optimal value of \({J}_{\textrm{E}}^{* }\). Results were computed for a threshold value of 0.001 rad s⁻¹, and are shown for all evenly sized networks between N = 6 and N = 20. c, Given a fixed value of \({J}_{\textrm{E}}^{* }\), the tolerance increases linearly with N. Results are shown for \({J}_{\textrm{E}}^{* }=4\), the only optimal value of local excitation that remains unchanged with even N. d, Top: error variance between the current and initial bump positions in a small, optimally tuned network with additive Gaussian noise. Numerical results are shown for three different optimal values of \({J}_{\textrm{E}}^{* }\), and with a noise variance σ² = (A/6)², where A = 0.2 is the bump amplitude. Bottom: beyond 10 s, the error variance grows linearly over time, following a diffusion equation with slope 2D (where D is the diffusion coefficient). We use 1/2D as a measure of noise robustness, with lower diffusion signifying higher robustness. e, Consistent with d, larger optimal values of \({J}_{\textrm{E}}^{* }\) lead to higher noise robustness for a fixed N. f, Given a fixed value of \({J}_{\textrm{E}}^{* }\) (shown for \({J}_{\textrm{E}}^{* }=4\)), noise robustness increases linearly with N, and is inversely proportional to noise variance σ² (shown for σ² = (A/6)² × [1, 4, 9, 16, 25]). Dashed lines indicate best linear fits; see Extended Data Fig. 9 for fit coefficients.