The output of neuron \( i \) in layer 2 is thus,
$$ \begin{align} y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) \tag{10}\\ &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right] \tag{11} \end{align} $$where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads
$$ \begin{align} y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) \tag{12}\\ &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right) + b_1^3\right] \tag{13} \end{align} $$