McCulloch-Pitts neuron/Definition

[math]\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} & & & } [/math]
Diagram of a McCulloch-Pitts neuron

The McCulloch-Pitts neuron has:

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

The output of the neuron is given by:

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

Neural Networks and Statistical Learning - Ke-Lin Du and M. N. S. Swamy