Fig. 3: Mean field stationary statistics.
From: Myelin-induced gain control in nonlinear neural networks
a Stationary mean of the average over network nodes \(\bar{u}\) computed numerically (triangles) plotted alongside the stationary mean μS computed analytically (dotted lines, see Methods section Eqs. (18), (22)) for Gamma distributed delays, and across various conduction velocities considered (c = 5.0 m/s, blue c = 2.5 m/s, black; c = 1.0 m/s, red) and various orders of noise variances D. The gray shaded area is bounded by the limit cases where there are no delays (top, c → ∞) or alternatively if delays become infinite (bottom, c → 0). Both of these limits were computed analytically. Significant deviations between numerical and analytical results emerge for strong noise variances D (marked by asterisks). b Equivalent plot for uniformly distributed delays. c Stationary variance (i.e., σ2) of the network ensemble average \(\bar{u}\) for Gamma distributed delays, computed both numerically (triangles) and analytically (dotted lines), see Methods section for details. Here, deviations appear for large noise variances D (marked as asterisks). d Same as (c) for uniformly distributed delays. In (a, c), k = 2 and θ = 4 mm/c. In (b, d), \({l}_{\max }=10\) mm. The network is fully connected (i.e., ρ = 1) and synaptic weights are both positive and identical (i.e., \({J}_{ij}=\bar{J}\)). Simulations were performed using the Euler-Maruyama scheme, with a population of N = 100 neurons over trials of duration T = 500 secs with steps of 1 ms. Other parameters are \({J}_{ij}=\bar{J}=0.9\), ρ = 1, I = 0, and the activation function is nonlinear with parameters β = 2, h = 0.5.
