{"id":"https://openalex.org/W7225317574","doi":"https://doi.org/10.1007/s11063-026-11893-8","title":"Lag-Synchronization and Global $$\\psi $$-Mittag-Leffler Stability of Generalized Fractional Neural Networks","display_name":"Lag-Synchronization and Global $$\\psi $$-Mittag-Leffler Stability of Generalized Fractional Neural Networks","publication_year":2026,"publication_date":"2026-11-05","ids":{"openalex":"https://openalex.org/W7225317574","doi":"https://doi.org/10.1007/s11063-026-11893-8"},"language":"en","primary_location":{"id":"doi:10.1007/s11063-026-11893-8","doi":"https://doi.org/10.1007/s11063-026-11893-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11063-026-11893-8","pdf_url":null,"source":{"id":"https://openalex.org/S140962798","display_name":"Neural Processing Letters","issn_l":"1370-4621","issn":["1370-4621","1573-773X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","ccf-c","cwts-core","dongbi-c","jufo-1","norway-1","poland-70","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Processing Letters","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1007/s11063-026-11893-8","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5149281897","display_name":"Afaq Ahmad","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I36001604","display_name":"University of Peshawar","ror":"https://ror.org/02t2qwf81","country_code":"PK","type":"education","lineage":["https://openalex.org/I36001604"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Afaq Ahmad","raw_affiliation_strings":["Department of Mathematics, University of Peshawar, Peshawar, KP, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Peshawar, Peshawar, KP, Pakistan","institution_ids":["https://openalex.org/I36001604"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5084862329","display_name":"Akbar Zada","orcid":"https://orcid.org/0000-0002-2556-2806","observed_orcids":["https://orcid.org/0000-0002-2556-2806"]},"institutions":[{"id":"https://openalex.org/I36001604","display_name":"University of Peshawar","ror":"https://ror.org/02t2qwf81","country_code":"PK","type":"education","lineage":["https://openalex.org/I36001604"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Akbar Zada","raw_affiliation_strings":["Department of Mathematics, University of Peshawar, Peshawar, KP, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-2556-2806","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Peshawar, Peshawar, KP, Pakistan","institution_ids":["https://openalex.org/I36001604"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103095115","display_name":"Sultan Hussain","orcid":"https://orcid.org/0000-0002-4727-2658","observed_orcids":["https://orcid.org/0000-0002-4727-2658"]},"institutions":[{"id":"https://openalex.org/I240666556","display_name":"Imam Mohammad ibn Saud Islamic University","ror":"https://ror.org/05gxjyb39","country_code":"SA","type":"education","lineage":["https://openalex.org/I240666556"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Sultan Hussain","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Sciences, Imam Mohammad Ibn Saud Islamic University (IMSIU), 11623, Riyadh, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Sciences, Imam Mohammad Ibn Saud Islamic University (IMSIU), 11623, Riyadh, Saudi Arabia","institution_ids":["https://openalex.org/I240666556"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5050873507","display_name":"Mohammad Esmael Samei","orcid":"https://orcid.org/0000-0002-5450-3127","observed_orcids":["https://orcid.org/0000-0002-5450-3127"]},"institutions":[{"id":"https://openalex.org/I167694869","display_name":"Bu-Ali Sina University","ror":"https://ror.org/04ka8rx28","country_code":"IR","type":"education","lineage":["https://openalex.org/I167694869"]}],"countries":["IR"],"is_corresponding":true,"raw_author_name":"Mohammad Esmael Samei","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran"],"raw_orcid":"https://orcid.org/0000-0002-5450-3127","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran","institution_ids":["https://openalex.org/I167694869"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5050873507"],"corresponding_institution_ids":["https://openalex.org/I167694869"],"apc_list":{"value":3090,"currency":"USD","value_usd":3090},"apc_paid":{"value":3090,"currency":"USD","value_usd":3090},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"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/T11347","display_name":"Neural Networks Stability and Synchronization","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"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/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/lag-synchronization","display_name":"lag synchronization","score":0.9269999861717224},{"id":"https://openalex.org/keywords/fractional-order-neural-networks","display_name":"fractional-order neural networks","score":0.8679999709129333},{"id":"https://openalex.org/keywords/impulsive-effects","display_name":"impulsive effects","score":0.8140000104904175},{"id":"https://openalex.org/keywords/time-varying-delays","display_name":"time-varying delays","score":0.9470000267028809},{"id":"https://openalex.org/keywords/impulsive-control","display_name":"impulsive control","score":0.8679999709129333},{"id":"https://openalex.org/keywords/drive-response-systems","display_name":"drive-response systems","score":0.7160000205039978},{"id":"https://openalex.org/keywords/existence-and-uniqueness-of-solutions","display_name":"existence and uniqueness of solutions","score":0.8870000243186951},{"id":"https://openalex.org/keywords/lyapunov-direct-method","display_name":"Lyapunov direct method","score":0.7409999966621399},{"id":"https://openalex.org/keywords/activation-functions","display_name":"activation functions","score":0.45399999618530273}],"concepts":[{"id":"https://openalex.org/C2778562939","wikidata":"https://www.wikidata.org/wiki/Q1298791","display_name":"Synchronization (alternating current)","level":3,"score":0.7728999853134155},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.6765000224113464},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.6442999839782715},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.609499990940094},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5940999984741211},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5541999936103821},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.5297999978065491},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.5224000215530396},{"id":"https://openalex.org/C139502532","wikidata":"https://www.wikidata.org/wiki/Q1122090","display_name":"Computational intelligence","level":2,"score":0.5041000247001648},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.47519999742507935},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4603999853134155},{"id":"https://openalex.org/C2779843651","wikidata":"https://www.wikidata.org/wiki/Q7390335","display_name":"SIGNAL (programming language)","level":2,"score":0.4343999922275543},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.38989999890327454},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.3472999930381775},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.34540000557899475},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.3345000147819519},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.3303999900817871},{"id":"https://openalex.org/C47822265","wikidata":"https://www.wikidata.org/wiki/Q854457","display_name":"Complex system","level":2,"score":0.3221000134944916},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.3147999942302704},{"id":"https://openalex.org/C143170015","wikidata":"https://www.wikidata.org/wiki/Q17007850","display_name":"Stability conditions","level":3,"score":0.31470000743865967},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.30880001187324524},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.30720001459121704},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3034000098705292},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.2989000082015991},{"id":"https://openalex.org/C50897621","wikidata":"https://www.wikidata.org/wiki/Q2665508","display_name":"Hybrid system","level":2,"score":0.28189998865127563},{"id":"https://openalex.org/C41949839","wikidata":"https://www.wikidata.org/wiki/Q1756677","display_name":"Stability theory","level":3,"score":0.2630000114440918},{"id":"https://openalex.org/C148043351","wikidata":"https://www.wikidata.org/wiki/Q4456944","display_name":"Current (fluid)","level":2,"score":0.2574000060558319}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s11063-026-11893-8","doi":"https://doi.org/10.1007/s11063-026-11893-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11063-026-11893-8","pdf_url":null,"source":{"id":"https://openalex.org/S140962798","display_name":"Neural Processing Letters","issn_l":"1370-4621","issn":["1370-4621","1573-773X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","ccf-c","cwts-core","dongbi-c","jufo-1","norway-1","poland-70","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Processing Letters","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s11063-026-11893-8","doi":"https://doi.org/10.1007/s11063-026-11893-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11063-026-11893-8","pdf_url":null,"source":{"id":"https://openalex.org/S140962798","display_name":"Neural Processing Letters","issn_l":"1370-4621","issn":["1370-4621","1573-773X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","ccf-c","cwts-core","dongbi-c","jufo-1","norway-1","poland-70","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Processing Letters","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2801475367","https://openalex.org/W2897894423","https://openalex.org/W2067002012","https://openalex.org/W2768082753","https://openalex.org/W2751488646","https://openalex.org/W4403365479","https://openalex.org/W2765924212","https://openalex.org/W4313201535","https://openalex.org/W4309474333","https://openalex.org/W2796327229"],"abstract_inverted_index":{"Fractional-order":[0],"neural":[1,13,55],"networks":[2,56],"with":[3,84],"impulsive":[4,50,127],"effects":[5],"and":[6,39,60,74,78,113,123,129,146,155,176,181,209,230,236],"delays":[7,158],"provide":[8],"realistic":[9,237],"models":[10,189],"of":[11,19,49,76,109,239],"complex":[12],"dynamics,":[14],"yet":[15],"the":[16,65,72,85,103,110,114,121,134,160,202,206,210],"synchronization":[17,182],"behavior":[18],"such":[20,35],"systems,":[21],"particularly":[22],"under":[23,126,159],"generalized":[24],"delay":[25,149,208,213,221],"structures,":[26],"remains":[27],"insufficiently":[28],"understood,":[29],"limiting":[30],"their":[31],"applicability":[32],"to":[33,141,187,214,223],"areas":[34],"as":[36,171],"secure":[37],"communication":[38],"signal":[40,241],"processing.":[41],"To":[42],"address":[43],"this,":[44],"we":[45,70,88],"model":[46,204],"a":[47,130,147,197,233],"class":[48],"\\(\\psi":[51,91,105,116,162],"\\)":[52,92,106,117,163],"-Hilfer":[53,164],"fractional-order":[54],"incorporating":[57],"both":[58,102],"simple":[59,154,207],"continuous":[61,156,211],"time-varying":[62,157,212],"delays.":[63],"Using":[64],"Banach":[66],"fixed":[67],"point":[68],"theorem,":[69],"establish":[71],"existence":[73],"uniqueness":[75],"solutions,":[77],"using":[79],"Lyapunov\u2019s":[80],"direct":[81],"method":[82],"combined":[83],"Razumikhin":[86],"approach,":[87],"derive":[89],"global":[90,104,115],"-Mittag-Leffler":[93],"stability":[94,108,180],"criteria.":[95],"Sufficient":[96],"algebraic":[97],"conditions":[98],"are":[99,194],"obtained":[100],"ensuring":[101],"-ML":[107,118],"drive":[111,122],"system":[112],"lag-synchronization":[119],"between":[120],"response":[124],"systems":[125],"control,":[128],"numerical":[131],"example":[132],"confirms":[133],"theoretical":[135],"predictions.":[136],"Unlike":[137],"prior":[138,188],"studies":[139],"restricted":[140],"Caputo":[142],"or":[143],"Riemann-Liouville":[144],"operators":[145,170],"single":[148],"type,":[150],"this":[151],"work":[152],"unifies":[153],"general":[161],"framework,":[165],"which":[166,191],"recovers":[167],"existing":[168],"fractional":[169],"special":[172],"cases,":[173],"thereby":[174,231],"extending":[175],"generalizing":[177],"previously":[178],"reported":[179],"results.":[183],"Moreover,":[184],"in":[185,190],"contrast":[186],"all":[192],"states":[193],"processed":[195],"through":[196,226],"single,":[198],"common":[199],"activation":[200,217],"function,":[201],"proposed":[203],"assigns":[205],"two":[215],"distinct":[216],"functions,":[218],"allowing":[219],"each":[220],"type":[222],"be":[224],"captured":[225],"its":[227],"own":[228],"nonlinearity":[229],"providing":[232],"more":[234],"flexible":[235],"representation":[238],"neuronal":[240],"transmission.":[242]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-11-06T00:00:00"}