Let $f : \mathbb{R} \to \mathbb{R}$ be a D4364: Real function.
The neuron function with respect to $f$, $w \in \mathbb{R}^{N \times 1}$, and $b \in \mathbb{R}$ is the D4364: Real function \begin{equation} \mathbb{R}^{N \times 1} \to \mathbb{R}, \quad (x_1, \ldots, x_N) \mapsto f \left( b + \sum_{n = 1}^N w_n x_n \right) \end{equation}

Let $f : \mathbb{R} \to \mathbb{R}$ be a D4364: Real function.
The neuron function with respect to $f$, $w \in \mathbb{R}^{N \times 1}$, and $b \in \mathbb{R}$ is the D4364: Real function \begin{equation} \mathbb{R}^{N \times 1} \to \mathbb{R}, \quad x \mapsto f \left( w^T x + b \right) \end{equation}