McCulloch-Pitts neuron

Definition

$$\xymatrix{ I_1 \ar[drr]^{w_1} & & \\ \vdots \ar@{.>}[rr] & & *++[o][F-]{\sum} \ar@{<.}[ll]!<0.5em,1.75em> \ar@{<.}[ll]!<0.5em,-1.75em> \ar@{<-}[d]!<0em,1em>_(.9){\theta} \ar[r]^-{\text{net}} & *+[o][F-]{\mathcal{A}(\cdot)} \ar[r] & \text{Output}\\ I_n \ar[urr]_{w_n} & & & }$$
Diagram of a McCulloch-Pitts neuron

The McCulloch-Pitts neuron has:

• Inputs: $(I_1,\ldots,I_n)\in\mathbb{R}^n$
  • Usually each $I_i$ is confined to $[0,1]\subset\mathbb{R}$ or $[-1,1]\subset\mathbb{R}$
• A set of weights, one for each input: $(w_1,\ldots,w_n)\in\mathbb{R}^n$
• A bias: $\theta\in\mathbb{R}$
• An activation function, $\mathcal{A}:\mathbb{R}\rightarrow\mathbb{R}$
  • It is more common to see $\mathcal{A}:\mathbb{R}\rightarrow[-1,1]\subset\mathbb{R}$ or sometimes $\mathcal{A}:\mathbb{R}\rightarrow[0,1]\subset\mathbb{R}$ than the entire of $\mathbb{R}$

The output of the neuron is given by:

$$\text{Output}:=\mathcal{A}\left(\sum_{i=1}^n(I_iw_i)+\theta\right)$$