## 3 Brain Inspired Processing
### LSNN Architecture  
The proposed LSNN network is shown in Figure 5, and it is composed of 20 ALIF and two output neurons which discriminate between the two classes of the bridge. We isolated the most important configuration parameters of the training algorithms (hyper-parameters) and studied their impact in terms of LSNN computational complexity and internal activity with an exhaustive search.  
The hyperparameters that we explored can be divided into two classes, the network parameters and the training parameters (summarised in Table 1). The first class comprises all the constants that strictly concern the network, while the second class is composed of all the values used to reach better performances but that don't feature the network. The network parameters that we explored are:  
* The input number (I) [50; 150] is the number of input neurons in the first layer. A high number create a huge \(W^{I}\) matrix that cannot be trained by a small network, while a low number risks not extracting all the features of the input signal [31].
* \(\tau_{in}\)[20; 30] represents the decay of the membrane potential in the recurrent layer. With high values, the neuron will need more time to return to the resting potential, representing the membrane potential of the neuron if any inputs have not perturbed it.
* \(\tau_{o}\)[3; 10; 30] is the decay of the membrane potential of the output neurons.
* \(v_{thc}\)[0.01, 0.03]. The spike threshold (\(v_{th}\)) of the neurons is computing as: \(v_{th}=v_{thc}/(1-e^{-h/\tau_{in}})\). A high value of the \(v_{thc}\) is translated into a higher value of the threshold.
* \(\beta_{c}\)[1.7, 1.8] is used to compute the increase of the adaptive threshold (\(\xi\)) in the ALIF neurons. Higher values of this parameter are equivalent to a greater increase in the spike threshold.
* \(\tau_{ac}\)[0.5, 1] is used to compute the decay of the adaptive threshold (\(\tau_{a}\)). \(\tau_{a}\) should have a value comparable to the time length of the problem [19].  
The training parameters that we explored are the inference time (\(t_{inf}\)) expressed in ticks [5; 10; 20]. A tick in an SNN represents a cycle to complete the dynamics integration of all network neurons. The actual time depends on the implementation. The inference time is then the number of ticks that the network uses to process the input sample. For the regularisation coefficient we use (\(reg_{c}\)) [1; 100; 300] and for the regularisation rate we  
\begin{table}
\begin{tabular}{l l r} \hline \hline
**Parameter** & **Description** & **Explored Values** \\ \hline I & Input neurons & 50, 150 \\ \(\tau_{in}\) & Recurrent layer membrane potential decay & 20, 30 \\ \(\tau_{o}\) & Output layer membrane potential decay & 3, 10, 30 \\ \(v_{thc}\) & Spike threshold coefficient & 0.01, 0.03 \\ \(\beta_{c}\) & Adaptive threshold coefficient & 1.7, 1.8 \\ \(\tau_{ac}\) & Adaptive threshold decay & 0.5, 1.0 \\ \(t_{inf}\) & Inference ticks & 5, 10, 20 \\ \(reg_{c}\) & Loss regularization coefficient & 1, 100, 300 \\ \(reg_{r}\) & Firing rate regularization coefficient & 0.01, 0.001 \\ \hline \hline \end{tabular}
\end{table}
Table 1: LSNN parameters explored.  
Figure 5: SNN architecture.  
use (\(reg_{r}\)) [0.01, 0.001]. The two regularisation values are used to computing the loss of the network:  
\[L_{fr}=\frac{1}{2}\sum_{j}\left(\frac{1}{nT}\sum_{t=1}^{nT}z_{j}^{t}-reg_{r} \right)^{2} \tag{5}\]  
\[L=L_{p}+L_{fr}*reg_{c} \tag{6}\]  
Loss is composed of two parts (Equation (6)), the loss of the classification (\(L_{p}\)) that is computing with the cross-entropy and the weighted firing-rate loss (\(L_{fr}\)). The \(L_{fr}\) is computed as the difference between the firing rate activity of the network and the \(reg_{r}\) that is the desired firing rate [19].