{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T10:57:02Z","timestamp":1761649022440,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,18]],"date-time":"2022-05-18T00:00:00Z","timestamp":1652832000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Anhui Province of China","award":["No. 1908085MA01","KJ2019A0557","KJ2020A0503"],"award-info":[{"award-number":["No. 1908085MA01","KJ2019A0557","KJ2020A0503"]}]},{"name":"Natural Science Foundation of the Higher Education Institutions of Anhui Province of China","award":["No. 1908085MA01","KJ2019A0557","KJ2020A0503"],"award-info":[{"award-number":["No. 1908085MA01","KJ2019A0557","KJ2020A0503"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Owing to the symmetry between drive\u2013response systems, the discussions of synchronization performance are greatly significant while exploring the dynamics of neural network systems. This paper investigates the quasi-synchronization (QS) and quasi-uniform synchronization (QUS) issues between the drive\u2013response systems on fractional-order variable-parameter neural networks (VPNNs) including probabilistic time-varying delays. The effects of system parameters, probability distributions and the order on QS and QUS are considered. By applying the Lyapunov\u2013Krasovskii functional approach, H\u00f6lder\u2019s inequality and Jensen\u2019s inequality, the synchronization criteria of fractional-order VPNNs under controller designs with constant gain coefficients and time-varying gain coefficients are derived. The obtained criteria are related to the probability distributions and the order of the Caputo derivative, which can greatly avoid the situation in which the upper bound of an interval with time delay is too large yet the probability of occurrence is very small, and information such as the size of time delay and probability of occurrence is fully considered. Finally, two examples are presented to further confirm the effectiveness of the algebraic criteria under different probability distributions.<\/jats:p>","DOI":"10.3390\/sym14051035","type":"journal-article","created":{"date-parts":[[2022,5,18]],"date-time":"2022-05-18T23:14:26Z","timestamp":1652915666000},"page":"1035","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Quasi-Synchronization and Quasi-Uniform Synchronization of Caputo Fractional Variable-Parameter Neural Networks with Probabilistic Time-Varying Delays"],"prefix":"10.3390","volume":"14","author":[{"given":"Renyu","family":"Ye","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, Anqing Normal University, Anqing 246133, China"}]},{"given":"Chen","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Anqing Normal University, Anqing 246133, China"}]},{"given":"Axiu","family":"Shu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Anqing Normal University, Anqing 246133, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7183-9570","authenticated-orcid":false,"given":"Hai","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Anqing Normal University, Anqing 246133, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"102232","DOI":"10.1016\/j.bspc.2020.102232","article-title":"Influence of EMG-signal processing and experimental set-up on prediction of gait events by neural network","volume":"63","author":"Nardo","year":"2021","journal-title":"Biomed. 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