Fig. 6

Percentage-relative error of the correlation calculated between the first-order perturbative expansion and the numerical simulation of the neural network (left) and the probability \(\mathscr{P}\) defined by (4.13) (right), for \(\sigma=10^{-3}-1\). The error is small (\({<}3.5\%\)) even for relatively large values of the perturbative parameter (\(\sigma \thicksim1\)), which proves the goodness of the perturbative approach. ε% increases considerably for \(\sigma\gg1\), but this result has not been shown, since such values correspond to biologically unrealistic levels of randomness for a neural network. On the other hand, the figure shows that \(\mathscr{P}\approx1\), which further confirms the legitimacy of the Taylor expansion (3.2) and therefore the validity of our results. Clearly \(\mathscr{P}\) decreases with σ because a larger variance brings the membrane potential closer to the borders defined by the radius of convergence