Formula: Artificial Neuron (Q156)

$ \begin{aligned} \text{Let } \mathbf{x} = (x_1,\dots,x_n)^\top \in \mathbb{R}^n \text{ be the input vector} \\ \text{Let } \mathbf{w} = (w_1,\dots,w_n)^\top \in \mathbb{R}^n \text{ be the weight vector} \\ \text{Let } b \in \mathbb{R} \text{ be the scalar bias} \\ \text{The affine pre‑activation is } z = \mathbf{w}^\top \mathbf{x} + b = \sum_{i=1}^n w_i x_i + b \\ \text{Choose a non‑linear activation } \sigma: \mathbb{R} \to \mathbb{R} \\ \text{The artificial neuron output is } y = \sigma(z) \in \mathbb{R} \\ \text{Typical activations: } \sigma(z) = \mathbf{1}_{z \ge 0} \text{ (step)}, \quad \sigma(z) = (1+e^{-z})^{-1} \text{ (sigmoid)}, \quad \sigma(z) = \max(0,z) \text{ (ReLU)} \end{aligned} $