## 2 Neural Network Architecture
### VCSEL-Neuron Modeling and Dynamic Regimes  
The model used to describe the two-section VCSEL-neuron is described by the following rate equations [38]:  
\[\frac{dn_{g}}{dt}\ =\ -\Gamma_{g}g_{g}\left(S-\frac{k_{e}\tau_{ph}\lambda}{hcV_{g} }P_{IN}-\sum_{i=1}^{N}\frac{\omega_{i}\tau_{ph}\lambda}{hcV_{g}}P_{io}\right) \left(n_{g}-n_{0g}\right)-\frac{n_{g}}{\tau_{g}}+\frac{I_{g}}{eV_{g}} \tag{1}\]  
\[\frac{dn_{a}}{dt}\ =\ \ -\Gamma_{a}g_{a}(n_{a}-n_{0a})S-\frac{n_{a}}{\tau_{g}} \tag{2}\]  
\[\frac{dS}{dt}\ =\ \Gamma_{g}g_{g}\left(n_{g}-n_{0g}\right)S+\Gamma_{a}g_{a}(n_{a }-n_{0a})S-\frac{S}{\tau_{ph}}+\beta B_{r}n_{g}^{2} \tag{3}\]  
The subscript \(g\) and \(a\) refer to gain and absorber area, respectively. \(S\) represents the photon density in the cavity and \(n_{g/a}\) is the electron density in the corresponding area. The term \(\frac{k_{e}\tau_{ph}P_{IN}\lambda}{hcV_{g}}\) in (1) simulates the electrically injected input, where \(k_{e}\) is the coupling strength of the external signal, \(\tau_{ph}\) the photon lifetime, \(P_{IN}\) is the power of the input electric signal, \(h\) is the Plank's constant, \(c\) is the speed of light, \(\lambda\) is the VCSEL's wavelength and \(V_{g}\) is the cavity volume. The term \(\sum_{i=1}^{N}\frac{\omega_{i}\tau_{ph}P_{io}\lambda}{hcV_{g}}\) represents the weighted sum of electrical inputs from presynaptic neurons where \(\omega_{i}\) is the weight of the \(i^{th}\) synapses and \(P_{io}\ =\ \frac{\eta_{i}\Gamma_{g}S_{i}V_{g}hc}{\tau_{ph}\lambda}\) is the power originating form the \(i^{th}\) presynaptic neuron, where \(\eta_{c}\) is the power coupling coefficient, \(\Gamma_{g}\) is the confinement factor and \(S_{i}\) is the photon density at the \(i^{th}\) presynaptic neuron. Other parameters used in this work are the electron's charge \(e\), pumpcurrent \(I_{g}\), the spontaneous emission coefficient \(\beta\), the bimolecular recombination coefficient \(B_{r}\), the differential gain at every section \(g_{g/a}\) and the transparency carrier density \(n_{0g/0a}\). There parameters used in this work are typical quantum-well VCSEL parameters and are provided in Table 1.  
The utilized VCSELs are biased in two distinctive dynamic regimes: the excitable regime and the spiking regime. In general, VCSELs in the excitable regime produce a spike only if the injected electrical stimuli exceed a certain threshold (integrate-and-fire) [9]. For \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Gain Section** & **Absorber Section** \\ \hline Cavity Volume V\({}_{g,a}\) & \(2.4\cdot 10^{-18}\) m\({}^{3}\) & \(2.4\cdot 10^{-18}\) m\({}^{3}\) \\ Confinement factor \(\Gamma_{g,a}\) & \(0.06\) & \(0.05\) \\ Carrier Lifetime \(\tau_{g,a}\) & \(1\) ns & \(100\) ps \\ Differential gain/loss \(g_{g,a}\) & \(2.9\cdot 10^{-12}\) m\({}^{3}\) s\({}^{-1}\) & \(14.5\cdot 10^{-12}\) m\({}^{3}\) s\({}^{-1}\) \\ Carriers at transparency n\({}_{0g,a}\) & \(1.1\cdot 10^{24}\) m\({}^{-3}\) & \(0.89\cdot 10^{24}\) m\({}^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Typical vertical cavity surface emitting lasers (VCSEL) parameters used in the simulation. example, for an electrical input power (bias current) of \(P_{IN}\) = 0.1 mW or lower no spike is detected (subthreshold input) while for higher \(P_{IN,\text{ the}}\) threshold condition is satisfied and the VCSEL produces a single spike (Figure 1a). Moreover, as the power level increases the latency of the spike decreases, encoding the input's strength in the timing of the spikes (temporal encoding). On the other hand, a VCSEL in the spiking regime constantly produces spikes of period \(T_{sp}\), where \(T_{sp}\) is the interspike interval [9]. Figure 1b demonstrates that for the investigated VCSELs, the variation of the \(T_{sp}\) is inversely proportional to the input power.