# Neuro-Inspired Computing with Spin-VCSELs
## 3 Results and Discussion
### Decoupling \(\tau_{D}\) and \(\tau_{M}\)  
Now that we established that the delay time \(\tau_{D}\) is intrinsically linked to the RC performance, we decouple the mask length \(\tau_{M}\) from the delay time \(\tau_{D}\), such that we can find the optimal number of nodes \(N\) and optimal node spacing \(\theta\). There have already been studies where the delay time was longer than the mask length (\(\tau_{D}>\tau_{M}\)), with state-of-the-art performances [16; 20; 31]. This mismatch between mask length and delay time is favorable for RC performances, because the interconnection between nodes over multiple round trips becomes more complex. In our case, however, the delay time \(\tau_{D}\) is already fixed at rather low values. For the reservoir with weak feedback, the delay time is fixed at \(\tau_{D}=150\) ps and for the reservoir with strong feedback at \(\tau_{D}=15\) ps. Here, we expect that the case where \(\tau_{D}<\tau_{M}\) will be favored, since \(\tau_{M}=N\theta\) and we need a sufficient number of nodes \(N\) to be able to perform the tasks.  
We did the same parameter scan as in Figure 2, but this time with constant delay times (\(\tau_{D}=150\) ps for \(\eta=1\) ns\({}^{-1}\) and \(\tau_{D}=15\) ps for \(\eta=100\) ns\({}^{-1}\), corresponding to the white lines in Figure 1) and the results are shown in Figure 4. We again see the best performing regions coalesce along hyperbolic curves only corresponding to a constant mask length \(\tau_{M}\) this time. For the reservoir with low feedback, we find the best performance \(NMSE=0.025\) at \(\theta=5.5\) ps and \(N=95\), corresponding to \(\tau_{M}=522.5\) ps. This point was depicted with a red cross in Figure 4. The mask length at this point corresponds to a processing speed of approximately 2 GSa/s.  
For the reservoir with strong feedback, shown in Figure 4b, we have a considerably larger part of parameter space which gives excellent performance. The best performance  
Figure 4: Santa Fe timeseries prediction performance, indicated by the \(NMSE\), color-coded unto the parameter space spanned by \(N\) and \(\theta\) for (**a**) \(\tau_{D}=150\) ps and \(\eta=1\) ns\({}^{-1}\), and (**b**) \(\tau_{D}=15\) ps and \(\eta=100\) ns\({}^{-1}\). 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 mask length \(\tau_{M}\
eq\tau_{D}\).  
for this reservoir is \(NMSE=0.012\), found at \(\theta=7.5\) ps and \(N=55\), corresponding to \(\tau_{M}=412.5\) ps. This point is depicted with a red cross. However, we observe another region with similar performances (\(NMSE\) around 0.015) around the hyperbolic curve corresponding to \(\tau_{M}=188\) ps (shown as a white dashed line). The points on this white curve have a shorter mask length and hence a higher processing speed as compared to the red cross. The processing speed for the points on this white line is 5.3 GSa/s, almost three times faster than that of the RC with low feedback. Both based on the \(NMSE\) as well as the processing speed, we found that our RC has superior qualities when operated with strong feedback rates.  
It is worth noting for the discussion later on, that at the best processing speed, a single masked datapoint traverses the delay line \(\frac{\tau_{M}}{\tau_{D}}=3.5\) times for the low feedback setting and 12.5 times for the high feedback setting.  
We also note that the lower left corner of Figure 4 shows high \(NMSE\) values, i.e., worse performances, which corresponds to the area where \(\tau_{D}<\tau_{M}\). In this region, either the number of nodes are not sufficient to deliver the nonlinear memory capacity needed to perform the task or the node spacing becomes too small for the laser to be able to follow the input data. These findings are similar to those in [20]. Similarly, in Figure 4b, we see the lower left corner turn darker, indicating worsening performances. This area is smaller, given that the delay time \(\tau_{D}=15\) ps is smaller than \(\tau_{D}=150\) ps in Figure 4a.