Consider the $1-$D Self$-$Organizing Map neural network with nodes numbered from $0$ to $255($not all are shown$)$ where the nodes on the two ends are also neighbors. The network has three inputs, each input value varying from $0$ to $255,$ which represent color values for red, blue and green. The nodes$$ weights are each initialized with uniformly distributed random values between $0$ and $255.$ The black node $($see the pdf file$)$ with index m $=2,$ has weights $(95,2,255)$ that are $$closest$$ to the input vector x$($k$)=[100,5,250]$ presented to the network at time index k$.$ The gray node with index n $=255,$ has weights $[50,100,10].$

Define the neighborhood function for the SOM where m and n refer to the nodes$$ indices thusly:

N$\backslash$left$($m$,$n$\backslash$right$)=\backslash$begin$\{$cases$\}0$&if$\backslash$:m$=$n$\backslash \backslash$ min$\backslash$left$(\backslash$left$|$m$-$n$\backslash$right$|,\backslash$left$|$m$-$n$\backslash$right$|-256\backslash$right$)$&if$\backslash$:m

e n$\backslash$end$\{$cases$\}$

which basically returns the $$distance$$ between two nodes $---$ i$.$e$.,$ the minimum number of nodes from one node m to another n$.$

Define the functions

f$\backslash$left$($z$\backslash$right$)=\backslash$begin$\{$cases$\}\backslash$frac$\{1\}\{$z$\}$&for$\backslash$:z$>0\backslash \backslash 0$&for$\backslash$:z$\backslash$le $0\backslash$end$\{$cases$\}$ and learning parameter $\backslash$eta $\backslash$left$($k$\backslash$right$)=\backslash$frac$\{2\}\{$ln$\backslash$left$(10+$k$\backslash$right$)\}$ where k corresponds to a time index $\{1<=$ k $<\backslash$infty $\}$ ; and the following update formula for the weights of a node n in which a node m corresponds to the node with weights that most closely matches the inputs $($i$.$e$.,$ node m is the node that MAXNET indicates $$wins$$ the competition among all other nodes$)$:

w$\_$n$\backslash$left$($k$+1\backslash$right$)=$w$\_$n$\backslash$left$($k$\backslash$right$)+\backslash$eta $\backslash$:$\backslash$left$($k$\backslash$right$)$f$\backslash$left$($N$\backslash$left$($m$,$n$\backslash$right$)\backslash$right$)\backslash$:$\backslash$left$[$x$\backslash$left$($k$\backslash$right$)-$w$\_$n$\backslash$left$($k$\backslash$right$)\backslash$right$]$

Using the update formula, calculate the first vector element of the updated weight$$vector wn to the nearest integer value for node n depicted in the graphic at time index k $=1.$ If a vector element is negative, just handle that in the mathematically appropriate manner.$$ In other words, use the negative value to complete any relevant computations.