Fig. 4: Delay-induced firing and gain control as a function of noise and input strength.

a Mean network firing rate as a function of noise variance D for various axonal conduction velocities (c = 5.0 m/s, blue; c = 2.5 m/s, black; c = 1.0 m/s, red). Firing rates are computed by multiplying the network firing probability F by a factor of 100. These rates expectedly scale with D, but are also amplified by higher conduction velocity. 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. The theory (dashed line) agrees well with the simulation of the full spiking network. b Network gain G or susceptibility as a function of D for different conduction velocities. These theoretical curves represent the slope of the curves in (A), and reveal an optimal noise strength that decreases with conduction velocity. Gray lines represent the same cases as in (a). c Mean firing rate as a function of constant input strength I for different values of c. Theory (dashed line) is accurate up to ~I = 0.045, after which discrepancies grow due to the nonlinear behavior of the network and the fact that the theory does not account for increased spiking noise variance in the network simulations. d Susceptibility as a function of input strength I for different c values. The theory is accurate again over the same range as in (c). Network simulations for higher values of I reveal a susceptibility peak due to the inflection point present in (c). The network is assumed to fully connected (i.e., ρ = 1) and synaptic weights are both positive and identical (i.e., \({J}_{ij}=\bar{J}\)). The model assumes a distribution of axonal lengths and delays are Gamma-distributed with k = 2 and θ = (4 mm/c). Other parameters are \(\bar{J}=0.9\), ρ = 1, and the activation function is nonlinear with parameters β = 2, h = 0.5.