# Neuro-Inspired Computing with Spin-VCSELs
## 3 Results and Discussion
### The Role of Delay Time \(\tau_{D}\)  
In previous studies, the delay time \(\tau_{D}\) is often matched to the mask length \(\tau_{M}\)[14; 17; 18; 19]. Initially, we did the same, such that \((\tau_{D}=\tau_{M})\). This allowed us to observe the effects of feedback-induced dynamics (related to \(\tau_{D}\)) on the performance and at the same time we could find a combination of \(N\) and \(\theta\) that might work for our benchmark task (\(\tau_{M}=N\theta\)). In Figure 2a,b, we saw the results of these scans for a low feedback rate \(\eta=1\) ns\({}^{-1}\) and for a high feedback rate \(\eta=100\) ns\({}^{-1}\), respectively. For the system with a low feedback rate, we saw a rather large area with good performance (lighter colors) in contrast with the results for the system with a high feedback rate. We would expect the best performing regions to be aggregated at higher values of \(N\) and around a particular value of \(\theta\), because the node state diversity becomes poor at low \(N\) and the node spacing has to be around the period of the polarization oscillations. Here, we see two contrasting plots for the different feedback rates. For \(\eta=1\) ns\({}^{-1}\), the absolute best performance was achieved at \(N=50\) and \(\theta=3\) ps with \(NMSE=0.044\). For \(\eta=100\) ns\({}^{-1}\), the absolute best performance was achieved at \(N=30\) and \(\theta=0.5\) ps with \(NMSE=0.071\). These best points were shown as red crosses in Figure 2a,b. On both plots, we see the regions with good performances (lighter colors) that coalesce around hyperbolic curves, corresponding to constant mask length \(\tau_{M}\) and the delay time \(\tau_{D}\), as \(\tau_{D}=\tau_{M}\). For the low feedback system, we found the best performance at \(\tau_{D}=\tau_{M}=150\) ps and a larger region of good performance at \(\tau_{D}=\tau_{M}=228\) ps. These are shown as white dashed lines in the left plot. For the strong feedback system, we found the best performance at \(\tau_{D}=\tau_{M}=15\) ps, shown as a white dashed line in the right plot. We observe that longer delay lines are required to obtain good performance for a low feedback rate and vice versa. Furthermore, for low feedback rates, the best performances are found near the middle of the hyperbolic line, showing a trade-off between the number of nodes \(N\) and the node spacing \(\theta\). For high feedback rates, a similar trend is seen, where the performance worsens as the number of nodes \(N\) is decreased. This makes sense as the  
Figure 2: Santa Fe timeseries prediction performance, indicated by the \(NMSE\), color-coded unto the parameter space spanned by \(N\) and \(\theta\) for (**a**) \(\eta=1\) ns\({}^{-1}\) and (**b**) \(\eta=100\) ns\({}^{-1}\). The delay time was fixed to match the mask length, \(\tau_{D}=\tau_{M}\). The red crosses denote the minimum \(NMSE\) achieved over the scanned space and the white dashed lines are a guide to the eye, denoting a constant delay time \(\tau_{D}\) and hence constant mask length \(\tau_{M}\). diversity of states of the virtual nodes is reduced if \(N\) is too small and hence the nonlinear memory capacity of the system will deteriorate [30]. It seems that the delay time, in combination with the feedback rate, has a very profound effect on the performance of our setup via feedback-induced dynamics. This is in stark contrast with previous delay-based RC setups using edge-emitting semiconductor lasers, where the delay time \(\tau_{D}\) had no such significant role in the RC performance [20; 31]. The results indicate that \(\tau_{D}\) and \(\eta\) affect the dynamical regime of the VCSEL considerably. This observation was supported by the findings in [32], where the nonlinear dynamics of the spin-VCSEL subjected to feedback was studied.