{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:35:03Z","timestamp":1760243703898,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,29]],"date-time":"2022-10-29T00:00:00Z","timestamp":1667001600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"2247-A National Leading Researchers Program of TUBITAK","award":["120C138","AP08856170"],"award-info":[{"award-number":["120C138","AP08856170"]}]},{"name":"Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan","award":["120C138","AP08856170"],"award-info":[{"award-number":["120C138","AP08856170"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we rigorously prove that unpredictable oscillations take place in the dynamics of Hopfield-type neural networks (HNNs) when synaptic connections, rates and external inputs are modulo periodic unpredictable. The synaptic connections, rates and inputs are synchronized to obtain the convergence of outputs on the compact subsets of the real axis. The existence, uniqueness, and exponential stability of such motions are discussed. The method of included intervals and the contraction mapping principle are applied to attain the theoretical results. In addition to the analysis, we have provided strong simulation arguments, considering that all the assumed conditions are satisfied. It is shown how a new parameter, degree of periodicity, affects the dynamics of the neural network.<\/jats:p>","DOI":"10.3390\/e24111555","type":"journal-article","created":{"date-parts":[[2022,10,30]],"date-time":"2022-10-30T04:57:34Z","timestamp":1667105854000},"page":"1555","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dynamics of Hopfield-Type Neural Networks with Modulo Periodic Unpredictable Synaptic Connections, Rates and Inputs"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2985-286X","authenticated-orcid":false,"given":"Marat","family":"Akhmet","sequence":"first","affiliation":[{"name":"Department of Mathematics, Middle East Technical University, Ankara 06531, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5572-2305","authenticated-orcid":false,"given":"Madina","family":"Tleubergenova","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aktobe Regional University, Aktobe 030000, Kazakhstan"},{"name":"Institute of Information and Computational Technologies CS MES RK, Almaty 050010, Kazakhstan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4878-4927","authenticated-orcid":false,"given":"Akylbek","family":"Zhamanshin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Middle East Technical University, Ankara 06531, Turkey"},{"name":"Department of Mathematics, Aktobe Regional University, Aktobe 030000, Kazakhstan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1073\/pnas.79.8.2554","article-title":"Neural networks and physical systems with emergent collective computational abilities","volume":"79","author":"Hopfield","year":"1982","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3088","DOI":"10.1073\/pnas.81.10.3088","article-title":"Neurons with graded response have collective computational properties like those of two-stage neurons","volume":"81","author":"Hopfield","year":"1984","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1250","DOI":"10.1109\/TNN.2006.875978","article-title":"A Hopfield Neural Network for Image Change Detection","volume":"17","author":"Pajares","year":"2006","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"640","DOI":"10.1109\/42.790463","article-title":"Abdominal organ segmentation using texture transforms and a Hopfield neural network","volume":"18","author":"Koss","year":"1999","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1109\/42.511759","article-title":"The Application of Competitive Hopfield Neural Network to Medical Image Segmentation","volume":"15","author":"Cheng","year":"1996","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_6","first-page":"3101","article-title":"Application of Hopfield neural network for facial image recognition","volume":"8","author":"Soni","year":"2019","journal-title":"IJRTE"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1016\/S0031-3203(00)00041-8","article-title":"Segmentation of FLIR images by Hopfield neural network with edge constraint","volume":"34","author":"Sang","year":"2001","journal-title":"Pattern Recognit."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1109\/42.141645","article-title":"Optimization neural networks for the segmentation of magnetic resonance images","volume":"11","author":"Amartur","year":"1992","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1016\/j.chaos.2006.06.035","article-title":"Exponential stability in Hopfield-type neural networks with impulses","volume":"32","author":"Mohammad","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1109\/72.896806","article-title":"Stability of asymmetric Hopfield networks","volume":"12","author":"Chen","year":"2001","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_11","first-page":"623","article-title":"Existence and exponential stability of anti-periodic solutions of Hopfield neural networks with impulses","volume":"216","author":"Shi","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1366","DOI":"10.1109\/72.809081","article-title":"Stability analysis of Hopfield type neural networks","volume":"10","author":"Juang","year":"1999","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1109\/72.317724","article-title":"Exponential stability and oscillation of Hopfield graded response neural network","volume":"5","author":"Yang","year":"1994","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.matcom.2006.05.027","article-title":"Almost periodic solutions for Hopfield neural networks with continuously distributed delays","volume":"73","author":"Liu","year":"2007","journal-title":"Math. Comput. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1007\/s00521-011-0655-x","article-title":"Almost periodic solution of impulsive Hopfield neural networks with finite distributed delays","volume":"21","author":"Liu","year":"2012","journal-title":"Neural Comput. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/j.nonrwa.2004.11.004","article-title":"Periodic oscillation for a class of neural networks with variable coefficients","volume":"6","author":"Guo","year":"2005","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1016\/j.neucom.2005.05.002","article-title":"Existence and exponential stability of almost periodic solutions for Hopfield neural networks with delays","volume":"68","author":"Liu","year":"2005","journal-title":"Neurocomputing"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1760","DOI":"10.1016\/j.neucom.2005.12.117","article-title":"On the almost periodic solution of generalized Hopfield neural networks with time-varying delays","volume":"69","author":"Liu","year":"2006","journal-title":"Neurocomputing"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.physleta.2005.06.008","article-title":"Existence and stability of periodic solution in impulsive Hopfield neural networks with finite distributed delays","volume":"343","author":"Yang","year":"2005","journal-title":"Phys. Lett. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1076","DOI":"10.1016\/j.chaos.2006.09.085","article-title":"Existence and exponential stability of almost periodic solution for Hopfield type neural networks with impulse","volume":"37","author":"Zhang","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5850","DOI":"10.1016\/j.na.2009.05.008","article-title":"Existence and stability of almost periodic solutions of Hopfield neural networks with continuously distributed delays","volume":"71","author":"Bai","year":"2009","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_22","unstructured":"Poincare, H. (1957). New Methods of Celestial Mechanics, Dover Publications."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Birkhoff, G. (1927). Dynamical Systems, American Mathematical Society.","DOI":"10.1090\/coll\/009"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cnsns.2016.04.007","article-title":"Unpredictable points and chaos","volume":"40","author":"Akhmet","year":"2016","journal-title":"Commun. Nonlinear Sci. Nummer. Simulat."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.cnsns.2016.12.015","article-title":"Poincare chaos and unpredictable functions","volume":"48","author":"Akhmet","year":"2017","journal-title":"Commun. Nonlinear Sci. Nummer. Simulat."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"33","DOI":"10.18514\/MMN.2019.2826","article-title":"Poincare chaos for a hyperbolic quasilinear system","volume":"20","author":"Akhmet","year":"2019","journal-title":"Miskolc Math. Notes"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"105287","DOI":"10.1016\/j.cnsns.2020.105287","article-title":"Shunting inhibitory cellular neural networks with strongly unpredictable oscillations","volume":"89","author":"Akhmet","year":"2020","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Akhmet, M., Tleubergenova, M., and Akylbek, Z. (2020). Inertial neural networks with unpredictable oscillations. Mathematics, 8.","DOI":"10.3390\/math8101797"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Akhmet, M. (2021). Domain Structured Dynamics: Unpredictability, Chaos, Randomness, Fractals, Differential Equations and Neural Networks, IOP Publishing.","DOI":"10.1088\/978-0-7503-3507-2ch2"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Akhmet, M., \u00c7in\u00c7in, D.A., Tleubergenova, M., and Nugayeva, Z. (2020). Unpredictable oscillations for Hopfield\u2013type neural networks with delayed and advanced arguments. Mathematics, 9.","DOI":"10.3390\/math9050571"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1007\/978-3-030-79357-9_59","article-title":"Unpredictable Oscillations of Impulsive Neural Networks with Hopfield Structure","volume":"76","author":"Akhmet","year":"2021","journal-title":"Lect. Notes Data Eng. Commun. Technol."},{"key":"ref_32","unstructured":"Sell, G. (1971). Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold Company."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Akhmet, M., Tleubergenova, M., and Zhamanshin, A. (2021). Modulo periodic Poisson stable solutions of quasilinear differential equations. Entropy, 23.","DOI":"10.3390\/e23111535"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Hartman, P. (2002). Ordinary Differential Equations, Birkhauser.","DOI":"10.1137\/1.9780898719222"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1555\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:05:32Z","timestamp":1760144732000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1555"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,29]]},"references-count":34,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["e24111555"],"URL":"https:\/\/doi.org\/10.3390\/e24111555","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,10,29]]}}}