As Mark notes, he posted about the Happycow:
  
  
    
  Look at this rotating brush, called the Happycow. From the manufacturer's description: Through automatic controls, the cows start up the machine by themselves by a slight lift of the brush.…
  
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In the comments @Neovison.vison  mentioned the self-groomer cat toy. I bought one and apparently so did Mark. The original link now goes to an unavailable item, but the item in Mark’s post above is the same.
  
  
    Closest thing I’ve ever found for cats was this little plastic nubbly brush that mounted on the corner of a wall  down at cat-height. My cat loved that thing. She could stimulate herself all day long if she wanted! Now that it’s discontinued, I wish I had taken that thing with me when I moved. 
ETA: Looks like other companies make similar doohickeys. Hey kitty, guess what you’re getting for Christmas!
   
 
Thanks, @Neovison.vison !
ETA : Here’s another available in grey or blue (I prefer the grey):
https://www.amazon.com/Flightbird-Groomer-Massage-Grooming-Massager/dp/B076889QC9/ref=pd_sbs_199_1/133-9749215-0356712?_encoding=UTF8&pd_rd_i=B076889QC9&pd_rd_r=96cf6e64-2bf1-11e9-8a86-1d7479451c7c&pd_rd_w=2wBGh&pd_rd_wg=dCGAe&pf_rd_p=588939de-d3f8-42f1-a3d8-d556eae5797d&pf_rd_r=7NSY1E78QH6T61ATVJ4Q&psc=1&refRID=7NSY1E78QH6T61ATVJ4Q