{"id":"https://openalex.org/W7207516487","doi":"https://doi.org/10.48550/arxiv.2609.02183","title":"Neural Logic, Invariance, and the Retina---McCulloch and Pitts","display_name":"Neural Logic, Invariance, and the Retina---McCulloch and Pitts","publication_year":2026,"publication_date":"2026-09-02","ids":{"openalex":"https://openalex.org/W7207516487","doi":"https://doi.org/10.48550/arxiv.2609.02183"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.02183","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02183","pdf_url":"https://arxiv.org/pdf/2609.02183","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.02183","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5003949897","display_name":"Nima Dehghani","orcid":"https://orcid.org/0000-0003-2032-8903","observed_orcids":["https://orcid.org/0000-0003-2032-8903"]},"institutions":[{"id":"https://openalex.org/I144112489","display_name":"McGovern Institute for Brain Research","ror":"https://ror.org/05ymca674","country_code":"US","type":"facility","lineage":["https://openalex.org/I144112489","https://openalex.org/I63966007"]},{"id":"https://openalex.org/I4210160315","display_name":"The NSF AI Institute for Artificial Intelligence and Fundamental Interactions","ror":"https://ror.org/04pvzz946","country_code":"US","type":"facility","lineage":["https://openalex.org/I121934306","https://openalex.org/I12912129","https://openalex.org/I136199984","https://openalex.org/I4210160315","https://openalex.org/I63966007"]},{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Dehghani, Nima","raw_affiliation_strings":[") McGovern Institute for Brain Research , Massachusetts Institute of Technology , Cambridge , Massachusetts 02139 , USA",") The NSF AI Institute for Artificial Intelligence and Fundamental Interactions (IAIFI) , Massachusetts Institute of Technology , Cambridge , Massachusetts 02139 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":") McGovern Institute for Brain Research , Massachusetts Institute of Technology , Cambridge , Massachusetts 02139 , USA","institution_ids":["https://openalex.org/I144112489","https://openalex.org/I63966007"]},{"raw_affiliation_string":") The NSF AI Institute for Artificial Intelligence and Fundamental Interactions (IAIFI) , Massachusetts Institute of Technology , Cambridge , Massachusetts 02139 , USA","institution_ids":["https://openalex.org/I4210160315","https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5003949897"],"corresponding_institution_ids":["https://openalex.org/I144112489","https://openalex.org/I4210160315","https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.4724000096321106,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.4724000096321106,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13308","display_name":"Cybernetics and Technology in Society","score":0.27320000529289246,"subfield":{"id":"https://openalex.org/subfields/1207","display_name":"History and Philosophy of Science"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10581","display_name":"Neural dynamics and brain function","score":0.07970000058412552,"subfield":{"id":"https://openalex.org/subfields/2805","display_name":"Cognitive Neuroscience"},"field":{"id":"https://openalex.org/fields/28","display_name":"Neuroscience"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/threshold-logic","display_name":"threshold logic","score":0.7559999823570251},{"id":"https://openalex.org/keywords/neural-computation","display_name":"neural computation","score":0.6660000085830688},{"id":"https://openalex.org/keywords/boolean-functions","display_name":"Boolean functions","score":0.5210000276565552},{"id":"https://openalex.org/keywords/linear-separability","display_name":"linear separability","score":0.8379999995231628},{"id":"https://openalex.org/keywords/periodic-orbits","display_name":"periodic orbits","score":0.8339999914169312},{"id":"https://openalex.org/keywords/heterarchy","display_name":"heterarchy","score":0.6269999742507935},{"id":"https://openalex.org/keywords/group-averaging","display_name":"group averaging","score":0.7319999933242798},{"id":"https://openalex.org/keywords/frog-retina","display_name":"frog retina","score":0.7319999933242798},{"id":"https://openalex.org/keywords/second-order-statistics","display_name":"second-order statistics","score":0.7390000224113464},{"id":"https://openalex.org/keywords/recurrent-neural-networks","display_name":"recurrent neural networks","score":0.37400001287460327},{"id":"https://openalex.org/keywords/history-of-neural-networks","display_name":"history of neural networks","score":0.3230000138282776}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.47189998626708984},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.4691999852657318},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.45579999685287476},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.428600013256073},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.42750000953674316},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.42399999499320984},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.41780000925064087},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.38040000200271606},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.37209999561309814},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.37130001187324524},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.3441999852657318},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33660000562667847},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.33480000495910645},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3172999918460846},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3005000054836273},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.2980000078678131},{"id":"https://openalex.org/C54271186","wikidata":"https://www.wikidata.org/wiki/Q1148456","display_name":"Computable function","level":2,"score":0.2976999878883362},{"id":"https://openalex.org/C29248071","wikidata":"https://www.wikidata.org/wiki/Q163310","display_name":"Turing machine","level":3,"score":0.29319998621940613},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.289900004863739},{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.2896000146865845},{"id":"https://openalex.org/C194989596","wikidata":"https://www.wikidata.org/wiki/Q211790","display_name":"Logical connective","level":2,"score":0.28929999470710754},{"id":"https://openalex.org/C2781089630","wikidata":"https://www.wikidata.org/wiki/Q21856745","display_name":"Realization (probability)","level":2,"score":0.273499995470047},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.2728999853134155},{"id":"https://openalex.org/C27508121","wikidata":"https://www.wikidata.org/wiki/Q210841","display_name":"Modal logic","level":3,"score":0.2703999876976013},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.266400009393692},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.2662999927997589},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2612999975681305},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.2590000033378601},{"id":"https://openalex.org/C38858127","wikidata":"https://www.wikidata.org/wiki/Q5441228","display_name":"Feed forward","level":2,"score":0.2565999925136566},{"id":"https://openalex.org/C116672817","wikidata":"https://www.wikidata.org/wiki/Q1454986","display_name":"Physical system","level":2,"score":0.2547000050544739},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.25450000166893005},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.25380000472068787},{"id":"https://openalex.org/C186661526","wikidata":"https://www.wikidata.org/wiki/Q13647261","display_name":"Sine","level":2,"score":0.2524000108242035},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.25189998745918274}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.02183","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02183","pdf_url":"https://arxiv.org/pdf/2609.02183","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.02183","doi":"https://doi.org/10.48550/arxiv.2609.02183","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.02183","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.02183","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02183","pdf_url":"https://arxiv.org/pdf/2609.02183","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7207516487.pdf","grobid_xml":"https://content.openalex.org/works/W7207516487.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W1956916615","https://openalex.org/W1932742286","https://openalex.org/W137689097","https://openalex.org/W4388253141","https://openalex.org/W2726146638","https://openalex.org/W2051429545","https://openalex.org/W4249335423","https://openalex.org/W7098113483","https://openalex.org/W2261950910","https://openalex.org/W4388318335"],"abstract_inverted_index":{"This":[0],"chapter":[1],"reconstructs":[2],"the":[3,14,34,53,64,78,197,206,247,253],"McCulloch-Pitts":[4],"program":[5],"as":[6,77,234],"a":[7,18,31,46,98,107,139,164,174,213,219],"physics":[8],"of":[9,17,81,92,132,252],"neural":[10],"computation":[11],"rather":[12],"than":[13],"familiar":[15],"cartoon":[16],"binary":[19,95],"neuron.":[20],"The":[21,151,167,192],"1943":[22],"logical":[23,43,82],"calculus":[24],"is":[25,85],"developed":[26,184],"in":[27],"both":[28],"directions:":[29],"given":[30,40],"net,":[32],"characterize":[33],"propositions":[35],"realized":[36],"by":[37,163,181],"its":[38,227],"activity;":[39],"an":[41,88,159],"admissible":[42],"expression,":[44],"construct":[45],"net":[47],"that":[48],"realizes":[49,122],"it.":[50],"We":[51,142],"recover":[52,226],"original":[54],"distinction":[55],"between":[56],"thresholded":[57],"excitatory":[58,72],"summation":[59],"and":[60,146,189,250],"absolute":[61],"inhibitory":[62],"veto-one":[63],"weighted-threshold":[65],"form":[66],"cannot":[67],"preserve":[68],"for":[69],"arbitrarily":[70],"large":[71],"inputs-and":[73],"read":[74],"unit-time":[75],"delay":[76],"physical":[79,175],"realization":[80],"depth.":[83],"Recurrence":[84],"treated":[86],"exactly:":[87],"autonomous,":[89],"deterministic":[90],"network":[91,176],"finitely":[93],"many":[94],"units":[96],"has":[97],"finite":[99,129,140],"state":[100],"space,":[101],"so":[102],"every":[103],"trajectory":[104],"eventually":[105],"enters":[106],"periodic":[108],"orbit-a":[109],"fact":[110],"about":[111],"finite-state":[112],"dynamics,":[113],"not":[114],"unbounded":[115],"Turing":[116],"computation.":[117],"A":[118],"single":[119],"threshold":[120,149],"element":[121],"only":[123],"linearly":[124],"separable":[125],"Boolean":[126,136],"functions,":[127],"whereas":[128],"feedforward":[130],"networks":[131],"them":[133],"synthesize":[134],"any":[135],"function":[137],"on":[138,170],"domain.":[141],"then":[143],"follows":[144],"McCulloch":[145],"Pitts":[147],"beyond":[148],"logic.":[150],"1945":[152],"heterarchy":[153],"paper":[154],"turns":[155],"cyclic":[156],"preference":[157],"into":[158],"obstruction":[160],"to":[161],"representation":[162],"scalar":[165],"utility.":[166],"1947":[168],"work":[169],"universals":[171],"asks":[172],"how":[173,212],"can":[177,217,232],"identify":[178],"inputs":[179],"related":[180],"nuisance":[182],"transformations,":[183],"here":[185],"via":[186],"group":[187],"averaging":[188],"feedback":[190],"canonicalization.":[191],"1959":[193],"frog-retina":[194],"study":[195],"makes":[196],"adequate-stimulus":[198],"question":[199],"experimental,":[200],"revealing":[201],"parallel":[202],"invariant":[203],"operations":[204],"before":[205],"brain":[207],"proper.":[208],"Spike-triggered":[209],"analysis":[210],"shows":[211],"nonlinearly":[214],"driven":[215],"neuron":[216],"have":[218],"vanishing":[220],"first-order":[221],"average":[222],"while":[223],"second-order":[224],"statistics":[225],"hidden":[228],"selectivity:":[229],"methodological":[230],"failure":[231],"masquerade":[233],"physiological":[235],"absence.":[236],"Modern":[237],"mathematical":[238],"tools":[239],"are":[240,255],"used":[241],"without":[242],"projecting":[243],"their":[244],"notation":[245],"onto":[246],"historical":[248],"papers,":[249],"limitations":[251],"idealization":[254],"stated":[256],"explicitly.":[257]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-04T00:00:00"}
