# Neuro-Inspired Computing with Spin-VCSELs
## 3 Results and Discussion
### The Role of Pumping Parameters  
Two other important parameters of the RC system are the electrical pump \(J_{0}\) and the optical pump amplitude \(\delta J\). We scanned these two parameters, for a reservoir with weak feedback and a reservoir with strong feedback corresponding to the red crosses in Figure 4a,b. The result of this scan is shown in Figure 5a,b, respectively. We observe that the best performance for the reservoir with weak feedback is found at higher electrical bias as well as higher spin pump amplitude. For the reservoir with strong feedback, the best performances are found at lower electrical biases and a fairly wide interval of optical pump amplitudes (between \(2J_{th}\) and \(8J_{th}\)). The points of best performances are indicated by the red crosses. For the low feedback reservoir, this point, corresponding to an \(NMSE\) of 0.015, was found at an electrical pump of \(7.5J_{th}\) and an optical pump amplitude of \(9.5J_{th}\). For the high feedback reservoir, the lowest \(NMSE=0.013\) was found at an electrical pump of \(3J_{th}\) and an optical pump amplitude of \(8J_{th}\). The pump parameters are optimized to rather high values for the reservoir with weak feedback and to low values for the reservoir with strong feedback. We presumed that these optimized parameter sets are such that there is a balance between the linear and nonlinear  
Figure 5: Santa Fe timeseries prediction performance, indicated by the \(NMSE\), color-coded unto the parameter space spanned by the electrical pump \(J_{0}\) and the optical pump amplitude \(\delta J\) for (**a**) \(\eta=1\,\mathrm{ns}^{-1}\), \(\tau_{D}=150\) ps, \(N=95\) and \(\theta=5.5\) ps and (**b**) \(\eta=100\,\mathrm{ns}^{-1}\), \(\tau_{D}=15\) ps, \(N=55\) and \(\theta=7.5\) ps. The red crosses denote the minimum \(NMSE\) achieved over the scanned space. memory capacity needed for the Santa Fe task. In Ref. [31], in the case of a standard semiconductor laser, [PERSON] et al. showed that the linear memory capacity drastically decreases as the mask becomes longer than the delay time. In our reservoirs, both with weak and strong feedback, the mask is longer than the delay time, namely \(\tau_{M}=3.5\tau_{D}\) for weak feedback and \(\tau_{M}=12.5\tau_{D}\) for strong feedback. The nonlinear memory capacity in these systems increases, since nodes related to one data sample (i.e., within one mask length) strongly mix with each other due to the mask looping around the delay line multiple times. This increase in nonlinear memory capacity comes at the expense of linear memory capacity, which is typically due to the interaction between nodes of subsequent data samples (i.e., over subsequent mask lengths). However, contrary to the system studied in [31], which have two characteristic timescales (namely the nodal response timescale and the feedback dynamics timescale), our system has three characteristic timescales that can couple nodes with each other. The first of these is the timescale introduced by the polarization dynamics, which governs the nodal response. The second is the timescale introduced by the feedback dynamics, which couples nodes separated by the delay time \(\tau_{D}\) and lastly, we have the timescale introduced by relaxation oscillations (ROs). The RO frequency and damping are strongly influenced by the pumping parameters. We prepared two reservoirs to look at the effect of ROs on the interaction between nodes. One reservoir was set up with a weak feedback rate ( \(\eta=1\,\mathrm{ns}^{-1}\) and \(\tau_{D}=150\) ps) and another with a strong feedback rate (\(\eta=100\) ns\({}^{-1}\) and \(\tau_{D}=15\) ps). The other parameters of these reservoirs were optimized for best performance in the Santa Fe task. Both systems were injected with a constant stream of zeros until they reached a steady state regime. Then, we applied a perturbation of length \(\theta\) and observed how this perturbation rippled through the system into the sum of the modal intensities.