# Neuro-Inspired Computing with Spin-VCSELs
## 3 Results and Discussion
### The Role of Delay Time \(\tau_{D}\)
The authors found that VCSELs connected to longer delay lines quickly move towards chaotic regimes with increasing feedback rates, whereas VCSELs connected to shorter delay lines would have various plateaus of steady state behavior, interspersed between chaotic regions, as the feedback rate is increased. To further investigate the effect of \(\tau_{D}\) and \(\eta\) on the dynamical regime, we prepared two RC systems with the optimal parameters corresponding to the red crosses in Figure 2a,b. We studied the modal output power as the system was injected with a constant value instead of masked data. The resulting orbit diagrams of the modal output power, as the feedback rate was varied, are shown in Figure 3a,b. In Figure 3a, we see the orbit diagram for a long delay time \(\tau_{D}=150\) ps. The laser has a steady state behavior for the lowest feedback rate, but it quickly moves towards a chaotic regime via a period-doubling route. In contrast, for a shorter delay line \(\tau_{D}=15\) ps (orbit diagram shown in Figure 3b), we see plateaus of steady state behavior, interspersed by periodic regimes, similar to findings in [32]. In Figure 3c,d, we show the Santa Fe timeseries prediction performance as a function of the feedback rate for \(\tau_{D}=150\) ps and \(\tau_{D}=15\) ps, respectively. Furthermore, we show the performance for each possible output layer. From both plots, it is clear that the best performances were achieved when the output layer consists of virtual nodes from both \(P^{+}\) and \(P^{-}\), whereas training on virtual nodes from a single mode gives the worst outcome. Training on the output polarization \(\left(POL_{out}=\frac{P^{+}-P^{-}}{P^{+}+P^{-}}\right)\) is not recommended either, since its performance is almost always worse. We see a brief interval for \(\eta\in[50,80]\) in Figure 3c,d  
Figure 3: Orbit diagram of the modal output power as the feedback rate \(\eta\) is varied for (**a**) \(\tau_{D}=150\) ps and (**b**) \(\tau_{D}=15\) ps. The \(NMSE\) for the Santa Fe timeseries prediction task as the feedback is varied for (**c**) \(\tau_{D}=150\) ps and (**d**) \(\tau_{D}=15\) ps. where the output layer formed by the output polarization has a better performance than the rest. However, this performance is nowhere near the minimum \(NMSE\). From the literature, we know that the best performing reservoirs are typically tuned to be at the edge of a periodic or chaotic dynamical regime before data are injected into the system [3; 33; 34]. It is at \(\eta=1\) ns\({}^{-1}\) for \(\tau_{D}=150\) ps and \(\eta=100\) ns\({}^{-1}\) for \(\tau_{D}=15\) ps, where we see a switch from a steady state to a periodic regime. These are also the feedback rates with which we achieve the best \(NMSE\) values. Another local minimum \(NMSE\) was found around \(\eta=15.5\) ns\({}^{-1}\) for \(\tau_{D}=15\) ps in Figure 3d, which corresponds to a switch from a steady state regime to a periodic regime in Figure 3b. The findings from Figure 3 confirm our assumption that the delay time \(\tau_{D}\) and feedback rate were intrinsically linked to the RC performance as opposed to previous RC setups, where the delay time is several orders of magnitude larger, and where the delay can be kept fixed when the feedback rate is changed [14; 17; 18; 27].