Correction to: Scientific Reports https://doi.org/10.1038/s41598-021-96745-2, published online 02 September 2021

The original version of this article contained an error in the legends of Figure 3 and Figure 4, where it was erroneously stated that the Lyapunov exponent plotted in Figure 3 and Figure 4b is the largest Lyupanov exponent.

In Figure 3,

“Enhanced responsiveness corresponds to regions closer to instability. (a) The largest stability Lyapunov exponent \(\lambda\) (real part) of asynchronous dynamics as a function of the heterogeneity of inhibitory neurons (\({\sigma }_{I}\)). Different colors indicate different parameters of the baseline external drive and the strength of excitatory-excitatory quantal conductance (\({\nu }_{0},{Q}_{EE}\)), i.e. black (1.5 Hz, 1.5 nS), red (3 Hz, 1.5 nS), blue (2 Hz, 1.5 nS) and orange (3 Hz, 1.65 nS). Symbols are located at the value of \({\sigma }_{I}\) for which the responsiveness R is maximum (same color code as continuous line). Different symbols indicate different amplitudes A of the input, diamond (\(A=0.1\) Hz), star (\(A=0.5\) Hz) and dot (\(A=1\) Hz). (b) The larger stability Lyapunov exponent \(\lambda\) of asynchronous dynamics as a function of the heterogeneity of inhibitory neurons (\({\sigma }_{I}\)) and the strength of excitatory-excitatory quantal conductance \({Q}_{EE}\) for a baseline external drive \({\nu }_{0}=1.5\) Hz. The dotted (diamond) line is the responsiveness R for an input amplitude \(A=0.1\) Hz (\(A=1 Hz\)) and \({Q}_{EE}=1.5\) nS (as in Figs. 1 and 2). Such responsiveness has been properly rescaled on the y-axes (i.e. multiplied by an ad-hoc factor) in order to fit in the image.”

now reads:

“Enhanced responsiveness corresponds to regions closer to instability. (a) The second largest stability Lyapunov exponent \(\lambda\) (real part) of asynchronous dynamics as a function of the heterogeneity of inhibitory neurons (\({\sigma }_{I}\)). Different colors indicate different parameters of the baseline external drive and the strength of excitatory-excitatory quantal conductance (\({\nu }_{0},{Q}_{EE}\)), i.e. black (1.5 Hz, 1.5 nS), red (3 Hz, 1.5 nS), blue (2 Hz, 1.5 nS) and orange (3 Hz, 1.65 nS). Symbols are located at the value of \({\sigma }_{I}\) for which the responsiveness R is maximum (same color code as continuous line). Different symbols indicate different amplitudes A of the input, diamond (\(A=0.1\) Hz), star (\(A=0.5\) Hz) and dot (\(A=1\) Hz). (b) The second largest stability Lyapunov exponent \(\lambda\) of asynchronous dynamics as a function of the heterogeneity of inhibitory neurons (\({\sigma }_{I}\)) and the strength of excitatory-excitatory quantal conductance \({Q}_{EE}\) for a baseline external drive \({\nu }_{0}=1.5\) Hz. The dotted (diamond) line is the responsiveness R for an input amplitude \(A=0.1\) Hz (\(A=1 Hz\)) and \({Q}_{EE}=1.5\) nS (as in Figs. 1 and 2). Such responsiveness has been properly rescaled on the y-axes (i.e. multiplied by an ad-hoc factor) in order to fit in the image.”

In Figure 4,

“Heterogeneous networks admit more dynamical regimes compared to homogeneous networks. (a) Inhibitory neurons population stationary firing rate \({r}_{I}^{*}\) in function of \({\sigma }_{I}\) for \(\overline{{E_{L}^{I} }} = - 70\) mV. Black line indicates stable (\(\lambda <0\)) asynchronous state, dashed blue line indicates unstable (\(\lambda >0\)) asynchronous state where a limit cycle appears (red lines indicates maximum and minimum value of \({r}_{I}\) in time). (b) The largest stability Lyapunov exponent \(\lambda\) (real part) of the asynchronous state as a function of the average resting potential of inhibitory neurons \(\overline{{E_{L}^{I} }}\) and their standard deviation \({\sigma }_{I}\). Whenever \(\lambda >0\) the asynchronous state is unstable and a stable limit cycle appears. In direct network simulations we observe sparsely synchronous oscillations (see raster plots in panel (c) where we use \(\overline{{E_{L}^{I} }} = - 70\) mV and \({\sigma }_{I}=\mathrm{0,0.12}, 0.2\) from top to bottom). In these simulations \({Q}_{EE}=1.53\) nS.”

now reads:

“Heterogeneous networks admit more dynamical regimes compared to homogeneous networks. (a) Inhibitory neurons population stationary firing rate \({r}_{I}^{*}\) in function of \({\sigma }_{I}\) for \(\overline{{E_{L}^{I} }} = - 70\) mV. Black line indicates stable (\(\lambda <0\)) asynchronous state, dashed blue line indicates unstable (\(\lambda >0\)) asynchronous state where a limit cycle appears (red lines indicates maximum and minimum value of \({r}_{I}\) in time). (b) The second largest stability Lyapunov exponent \(\lambda\) (real part) of the asynchronous state as a function of the average resting potential of inhibitory neurons \(\overline{{E_{L}^{I} }}\) and their standard deviation \({\sigma }_{I}\). Whenever \(\lambda >0\) the asynchronous state is unstable and a stable limit cycle appears. In direct network simulations we observe sparsely synchronous oscillations (see raster plots in panel (c) where we use \(\overline{{E_{L}^{I} }} = - 70\) mV and \({\sigma }_{I}=\mathrm{0,0.12,0.2}\) from top to bottom). In these simulations \({Q}_{EE}=1.53\) nS.”

As a result, in the Results section, under the subheading ‘Optimal responsiveness comes from pushing the network at the edge of a dynamical transition’,

“The maximum exponent \(\lambda\) is reported in Fig. 3a for different parameter setups.”

now reads:

“The real part of the second largest exponent  \(\lambda\) is reported in Fig. 3a for different parameter setups.”

The original article has been corrected.