# paper.pdf ## Metadata - PDFFormatVersion=1.6 - Language=en-US - IsLinearized=true - IsAcroFormPresent=false - IsXFAPresent=false - IsCollectionPresent=false - IsSignaturesPresent=false - CreationDate=D:20260618113728+05'30' - Creator=Adobe InDesign 17.4 (Windows) - ModDate=D:20260618115052+05'30' - Producer=Adobe PDF Library 16.0.7 - Trapped=False - xmpmm:instanceid=uuid:ed574703-50c1-4040-a020-7bec77498c63 - xmpmm:originaldocumentid=xmp.did:A33916180F2068118083ACD1B377E157 - xmpmm:documentid=xmp.id:80007688-4976-814a-9d6d-c9eebe0a1032 - xmpmm:renditionclass=proof:pdf - xmpmm:derivedfrom=xmp.iid:fa7d92ae-a869-c044-85fe-fe35acc0a2b9xmp.did:fa7d92ae-a869-c044-85fe-fe35acc0a2b9xmp.did:A33916180F2068118083ACD1B377E157default - xmpmm:history=convertedfrom application/x-indesign to application/pdfAdobe InDesign 17.4 (Windows)/2026-06-18T11:37:28+05:30 - xmp:createdate=2026-06-18T11:37:28+05:30 - xmp:modifydate=2026-06-18T11:50:52+05:30 - xmp:metadatadate=2026-06-18T11:50:52+05:30 - xmp:creatortool=Adobe InDesign 17.4 (Windows) - xmprights:marked=False - photoshop:modifydate=2019-01-31T16:16:25 - photoshop:createdate=2019-01-24T11:43:46 - dc:format=application/pdf - pdf:producer=Adobe PDF Library 16.0.7 - pdf:trapped=False - extensisfontsense:slug=3023672065GraphikOpenType - PS3023672065--0Graphik-Bold1.0011904271029GraphikOpenType - PS1904271029--0Graphik-Regular1.0012428528031News Gothic MT OTOpenType - PS2428528031Monotype Typography0NewsGothicMTOT-Regular1.000543760319News Gothic MT OTOpenType - PS543760319Monotype Typography0NewsGothicMTOT-Bold1.0002785939051KnockoutOpenType - PS2785939051--0Knockout-68FullFeatherwt1.200423856041Graphik X CondOpenType - PS423856041--0GraphikXCond-Super1.0001526035435Graphik X CondOpenType - PS1526035435--0GraphikXCond-Bold1.0003093728306Graphik CondOpenType - PS3093728306--0GraphikCond-Black1.0011341118539Graphik CondOpenType - PS1341118539--0GraphikCond-Regular1.0011385893531GraphikOpenType - PS1385893531--0Graphik-Semibold1.0012986479693Graphik CompactOpenType - PS2986479693--0GraphikCompact-Regular1.0024223138361Graphik CompactOpenType - PS4223138361--0GraphikCompact-Semibold1.0023673047546Harding SevenOpenType - PS3673047546--0HardingSeven-Extrabold0.000103712566Harding SevenOpenType - PS103712566--0HardingSeven-Roman0.0002232452979Harding SevenOpenType - PS2232452979--0HardingSeven-Medium0.000631916388Harding SevenOpenType - PS631916388--0HardingSeven-Bold0.000157629484Publico BannerOpenType - PS157629484--0PublicoBanner-Black1.005 ## Contents ### Page 1 nature neurosciencehttps://doi.org/10.1038/s41593-026-02333-wArticleLearning shapes neural geometry in the primate prefrontal cortexIn the format provided by the authors and uneditedSupplementary information ### Page 2 Learning shapes neural geometry in the primate prefrontal cortexSupplementary information ### Page 3 Section 1.1: How to set selectivity parameters for optimal XOR decodingWe assume that neural activities 𝒙 of 𝑵 neurons are given by the following regression model𝒙 = 𝜷! + 𝛿#,#! 𝜷% + 𝛿&,&! 𝜷' + 𝛿#,#! 𝛿&,&! 𝜷( + 𝜼(1) Where 𝜷% , 𝜷' , 𝜷( are the regression ‘coefficients’ for colour, shape, and the interaction term, respectively, 𝛿#,#! = 1 if the colour 𝑐 is colour 1 and -1 otherwise (same for 𝛿&,&!for shapes 𝑠), and 𝜼~ 𝒩(𝟎, 𝚺)). Therefore,𝒙|𝑟) ~ 𝒩(𝝁*! , 𝚺))(2)𝒙|𝑟+ ~ 𝒩(𝝁*# , 𝚺+). (3) Where 𝑟) is one XOR condition (i.e., 𝑐 = 1, 𝑠 = 1 or 𝑐 = 2, 𝑠 = 2) and 𝑟+ is the other XOR condition (i.e., 𝑐 = 1, 𝑠 = 2 or 𝑐 = 2, 𝑠 = 1) and 𝚺+ is some noise covariance matrix. Note that with the inclusion of the interaction term, it is sufficient to separate the two XOR conditions. We now calculate 𝝁*! and 𝝁*# :𝝁*# = 𝜷! + 𝜷% + 𝜷' + 𝜷(2 + 𝜷! − 𝜷% − 𝜷' + 𝜷(2= 𝜷! + 𝜷((4) and,𝝁*# = 𝜷! − 𝜷% + 𝜷' − 𝜷(2 + 𝜷! + 𝜷% − 𝜷' − 𝜷(2= 𝜷! − 𝜷((5) Therefore,𝒙|𝑟) ~ 𝒩(𝜷! + 𝜷(, 𝚺+)(6)𝒙|𝑟+ ~ 𝒩(𝜷! − 𝜷(, 𝚺+). (7) Therefore, we need 𝜷( > 0 to be able to separate the two XOR conditions.Section 1.2: An energy cost on unnecessary neural activity ### Page 4 If we also consider minimising the squared norm of neural activity for each condition (i.e., an energy cost), we have𝔼 ;<𝒙#,&<+= = 𝔼>𝒙#,&?, 𝔼>𝒙#,&? + Tr(𝚺))(8) where the subscripts 𝑐 and 𝑠 correspond to colour and shape indices, respectively. Therefore,𝔼 ;<𝒙),+<+= = (𝜷! − 𝜷% + 𝜷' − 𝜷(), (𝜷! − 𝜷% + 𝜷' − 𝜷() + Tr(𝚺))(9)𝔼 ;<𝒙+,)<+= = (𝜷! + 𝜷% − 𝜷' − 𝜷(), (𝜷! + 𝜷% − 𝜷' − 𝜷() + Tr(𝚺))(10)𝔼 ;<𝒙),)<+= = (𝜷! − 𝜷% − 𝜷' + 𝜷(), (𝜷! − 𝜷% − 𝜷' + 𝜷() + Tr(𝚺))(11)𝔼 ;<𝒙+,+<+= = (𝜷! + 𝜷% + 𝜷' + 𝜷(), (𝜷! + 𝜷% + 𝜷' + 𝜷() + Tr(𝚺))(12) Therefore, the total mean energy cost 𝑚 is given by𝑚 = 1 4 D D 𝔼 ;<𝒙#,&<+=&-),+#-),+(13)= 𝜷!,𝜷! + 𝜷% ,𝜷% + 𝜷',𝜷' + 𝜷(,𝜷( + Tr(𝚺))(14)= ‖𝜷!‖+ + ‖𝜷%‖+ + ‖𝜷'‖+ + ‖𝜷(‖+ + Tr(𝚺))(15) To minimise 𝑚 while keeping 𝜷( > 0, which we need for performance, we can set𝜷! = 𝜷% = 𝜷' = 𝟎 which gives𝑚 = ‖𝜷(‖+ + Tr(𝚺))(16)