{"id":"https://openalex.org/W2157312889","doi":"https://doi.org/10.1109/tnn.2002.1031956","title":"Some extensions of a new method to analyze complete stability of neural networks","display_name":"Some extensions of a new method to analyze complete stability of neural networks","publication_year":2002,"publication_date":"2002-09-01","ids":{"openalex":"https://openalex.org/W2157312889","doi":"https://doi.org/10.1109/tnn.2002.1031956","mag":"2157312889","pmid":"https://pubmed.ncbi.nlm.nih.gov/18244521"},"language":"en","primary_location":{"id":"doi:10.1109/tnn.2002.1031956","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnn.2002.1031956","pdf_url":null,"source":{"id":"https://openalex.org/S42080949","display_name":"IEEE Transactions on Neural Networks","issn_l":"1045-9227","issn":["1045-9227","1941-0093"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069809502","display_name":"Mauro Forti","orcid":"https://orcid.org/0000-0002-3970-7201"},"institutions":[{"id":"https://openalex.org/I102064193","display_name":"University of Siena","ror":"https://ror.org/01tevnk56","country_code":"IT","type":"education","lineage":["https://openalex.org/I102064193"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"M. Forti","raw_affiliation_strings":["Dipt. di Ingegneria dell'Informazione, Siena Univ., Italy","Dipartimento di Ingegneria dellE28099Informazione, Universit\u00e0 di Siena, Siena, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipt. di Ingegneria dell'Informazione, Siena Univ., Italy","institution_ids":["https://openalex.org/I102064193"]},{"raw_affiliation_string":"Dipartimento di Ingegneria dellE28099Informazione, Universit\u00e0 di Siena, Siena, Italy","institution_ids":["https://openalex.org/I102064193"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5069809502"],"corresponding_institution_ids":["https://openalex.org/I102064193"],"apc_list":null,"apc_paid":null,"fwci":1.8619,"has_fulltext":false,"cited_by_count":48,"citation_normalized_percentile":{"value":0.90346247,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"13","issue":"5","first_page":"1230","last_page":"1238"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11347","display_name":"Neural Networks Stability and Synchronization","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T11347","display_name":"Neural Networks Stability and Synchronization","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T10320","display_name":"Neural Networks and Applications","score":0.9950000047683716,"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/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9894000291824341,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.690237820148468},{"id":"https://openalex.org/keywords/equilibrium-point","display_name":"Equilibrium point","score":0.6092000007629395},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.60546875},{"id":"https://openalex.org/keywords/sigmoid-function","display_name":"Sigmoid function","score":0.5720458626747131},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5109673738479614},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.4998900890350342},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4759245216846466},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.468715101480484},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.44722428917884827},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.4315072298049927},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.2268204391002655},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.12881487607955933},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11517602205276489},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.08572283387184143}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.690237820148468},{"id":"https://openalex.org/C94766913","wikidata":"https://www.wikidata.org/wiki/Q1530271","display_name":"Equilibrium point","level":3,"score":0.6092000007629395},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.60546875},{"id":"https://openalex.org/C81388566","wikidata":"https://www.wikidata.org/wiki/Q526668","display_name":"Sigmoid function","level":3,"score":0.5720458626747131},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5109673738479614},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4998900890350342},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4759245216846466},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.468715101480484},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.44722428917884827},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.4315072298049927},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2268204391002655},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.12881487607955933},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11517602205276489},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.08572283387184143},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1109/tnn.2002.1031956","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnn.2002.1031956","pdf_url":null,"source":{"id":"https://openalex.org/S42080949","display_name":"IEEE Transactions on Neural Networks","issn_l":"1045-9227","issn":["1045-9227","1941-0093"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks","raw_type":"journal-article"},{"id":"pmid:18244521","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/18244521","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE transactions on neural networks","raw_type":null},{"id":"pmh:oai:usiena-air.unisi.it:11365/9085","is_oa":false,"landing_page_url":"http://hdl.handle.net/11365/9085","pdf_url":null,"source":{"id":"https://openalex.org/S4377196319","display_name":"Use Siena air (University of Siena)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I102064193","host_organization_name":"University of Siena","host_organization_lineage":["https://openalex.org/I102064193"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W114737826","https://openalex.org/W1514578360","https://openalex.org/W1606202192","https://openalex.org/W1970381782","https://openalex.org/W1972828808","https://openalex.org/W1981479913","https://openalex.org/W2036458635","https://openalex.org/W2060592270","https://openalex.org/W2062194437","https://openalex.org/W2062210787","https://openalex.org/W2065134213","https://openalex.org/W2066263844","https://openalex.org/W2091616801","https://openalex.org/W2096862483","https://openalex.org/W2097113878","https://openalex.org/W2097138247","https://openalex.org/W2100885571","https://openalex.org/W2109653785","https://openalex.org/W2112246162","https://openalex.org/W2122029947","https://openalex.org/W2149943625","https://openalex.org/W2157965136","https://openalex.org/W2160121923","https://openalex.org/W2169095995","https://openalex.org/W2169122717","https://openalex.org/W2310420328","https://openalex.org/W3141151088","https://openalex.org/W3145540823"],"related_works":["https://openalex.org/W4385957115","https://openalex.org/W2061372042","https://openalex.org/W3047779762","https://openalex.org/W1520030019","https://openalex.org/W2071654592","https://openalex.org/W3104477175","https://openalex.org/W2088157920","https://openalex.org/W2147858351","https://openalex.org/W2410283908","https://openalex.org/W3109989318"],"abstract_inverted_index":{"In":[0],"a":[1,4,33,130,226],"recent":[2],"work,":[3],"new":[5],"method":[6,65,242],"has":[7,163],"been":[8],"introduced":[9],"to":[10,62,68,197,217,249,252],"analyze":[11],"complete":[12,39,70,107,152,254],"stability":[13,40,71,108,141,153,255],"of":[14,47,58,72,77,98,113,119,132,139,142,188,212,220,240,243,256],"the":[15,48,73,87,114,117,136,160,218,238,241],"standard":[16],"symmetric":[17,78,120],"cellular":[18],"neural":[19,49,79,122,133,161,194,204,258],"networks":[20,80,123,134,195,205],"(CNNs),":[21],"which":[22],"are":[23,90,177],"characterized":[24],"by":[25,32],"local":[26],"interconnections":[27],"and":[28,245],"neuron":[29,88,126,208],"activations":[30,89],"modeled":[31],"three-segment":[34],"piecewise-linear":[35],"(PWL)":[36],"function.":[37],"By":[38],"it":[41,192],"is":[42,61,105,147,154,187,211],"meant":[43],"that":[44,64,106,151],"each":[45],"trajectory":[46],"network":[50,162],"converges":[51],"toward":[52],"an":[53,82,95],"equilibrium":[54,167,180],"point.":[55],"The":[56,101,182,232],"goal":[57],"this":[59,185,235],"paper":[60,186,236],"extend":[63],"in":[66,184,234],"order":[67],"address":[69,253],"much":[74],"wider":[75],"class":[76,118,131],"with":[81,94,124,206,247],"additive":[83,121],"interconnecting":[84],"structure":[85],"where":[86,159,175],"general":[91],"PWL":[92,125,230,257],"functions":[93],"arbitrary":[96],"number":[97],"straight":[99],"segments.":[100],"main":[102],"result":[103],"obtained":[104],"holds":[109],"for":[110,156,172],"any":[111,222],"choice":[112],"parameters":[115],"within":[116],"activations,":[127],"i.e.,":[128],"such":[129],"enjoys":[135],"important":[137],"property":[138],"absolute":[140],"global":[143],"pattern":[144],"formation.":[145],"It":[146,210],"worth":[148],"pointing":[149],"out":[150],"proved":[155],"generic":[157],"situations":[158,174],"finitely":[164],"many":[165],"(isolated)":[166],"points,":[168],"as":[169,171],"well":[170],"degenerate":[173],"there":[176],"infinite":[178],"(nonisolated)":[179],"points.":[181],"extension":[183],"practical":[189],"importance":[190],"since":[191],"includes":[193],"useful":[196],"solve":[198],"significant":[199],"signal":[200],"processing":[201],"tasks":[202],"(e.g.,":[203,225],"multilevel":[207],"activations).":[209],"theoretical":[213],"interest":[214],"too,":[215],"due":[216],"possibility":[219],"approximating":[221],"continuous":[223],"function":[224],"sigmoidal":[227],"function),":[228],"using":[229],"functions.":[231],"results":[233],"confirm":[237],"advantages":[239],"Forti":[244],"Tesi,":[246],"respect":[248],"LaSalle":[250],"approach,":[251],"networks.":[259]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":3}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}
