Fig 2. Soft-threshold readout error.

(a) The soft-threshold and the delay create variations in the membrane potential dynamics V_i(t), which in turn create variations in the readout $\widehat{x}\left(t\right)$. When the membrane potentials (top, purple) surpass threshold, the neurons spike probabilistically, and the first-spike time is an exponential random variable with standard deviation $\frac{1}{N\rho }$. After a first spike, the number of spurious spikes that occur during the delay is a Poisson random variable, with standard deviation $\sqrt{\lambda }$, and each spike inhibits the membrane potentials V_i(t) by 1 through recurrent connectivity (see the first paragraph of Results for recurrent connectivity). These variations in spike-timing and spurious spikes carry through to the readout $\widehat{x}\left(t\right)$ (bottom, blue). Note that since the network input is constant, the readout encoding this input should produce a constant output as closely as possible; however, these variations instead increase the deviation (light blue shaded) from the mean readout ${\langle \widehat{x}\left(t\right)\rangle }_t$ (black horizontal line). (b) Readout error as a function of the mean number of spurious spikes λ and the delay δ. Top: for three different values of delay (blue, red, purple), λ is varied in computer simulations (N = 32, dots) and Eq 10 (solid curves), revealing both the U-shaped dependence of the readout error σ_readout and an excellent match between theory and experiment. Bottom: the optimal readout error ${\sigma }_{readout}^{*}$ (black) and the associated optimal λ* increase as a function of delay δ according to Eqs 12 and 11.