{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:50:34Z","timestamp":1760147434626,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,2]],"date-time":"2023-02-02T00:00:00Z","timestamp":1675296000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11872327","20KJA190001"],"award-info":[{"award-number":["11872327","20KJA190001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Research Project of Jiangsu Colleges and Universities","award":["11872327","20KJA190001"],"award-info":[{"award-number":["11872327","20KJA190001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In view of the diversity of stimulated current that neurons may experience, an extended Hindmarsh\u2013Rose neuron model is proposed and the corresponding fractional-order neuron model, with no equilibrium point, is depicted. Additionally, various hidden attractors of the addressed neuron model are analyzed by changing system parameters and the order of fractional-order neuron system. Furthermore, hybrid projective synchronizations of the proposed neurons are investigated and schemes are obtained by designing suitable controllers according to fractional stability theory. Besides, the validity of the theoretical results is verified through numerical simulations. In short, the research results have potential application in revealing the dynamical behaviors of neuron system and controlling the behaviors of neuron into certain status.<\/jats:p>","DOI":"10.3390\/axioms12020157","type":"journal-article","created":{"date-parts":[[2023,2,2]],"date-time":"2023-02-02T06:11:53Z","timestamp":1675318313000},"page":"157","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Hybrid Projective Synchronization of Fractional-Order Extended Hindmarsh\u2013Rose Neurons with Hidden Attractors"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7727-3678","authenticated-orcid":false,"given":"Xuerong","family":"Shi","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zuolei","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,2]]},"reference":[{"key":"ref_1","first-page":"177","article-title":"Propagation of electrical signals along giant nerve fibres","volume":"140","author":"Hodgkin","year":"1952","journal-title":"Proc. R. Soc. Lond."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1038\/296162a0","article-title":"A model of the nerve impulse using two first-order differential equations","volume":"296","author":"Hindmarsh","year":"1982","journal-title":"Nature"},{"key":"ref_3","first-page":"257","article-title":"Mathematical models of threshold phenomena in the nerve membrane","volume":"17","author":"FitzHugh","year":"1955","journal-title":"Bull. Math. Biol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/0167-2789(85)90060-0","article-title":"Chaos in a three-variable model of an excitable cell","volume":"16","author":"Chay","year":"1985","journal-title":"Physica D"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/S0006-3495(81)84782-0","article-title":"Voltage oscillations in the barnacle giant muscle fiber","volume":"35","author":"Morris","year":"1981","journal-title":"Biophys. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1569","DOI":"10.1109\/TNN.2003.820440","article-title":"Simple model of spiking neurons","volume":"14","author":"Izhikevich","year":"2003","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s11071-019-05047-w","article-title":"Chaotic resonance in Hodgkin-Huxley neuron","volume":"97","author":"Baysal","year":"2019","journal-title":"Nonlinear Dynam."},{"key":"ref_8","first-page":"167","article-title":"Analysis of chaos route in hybridized FitzHugh- Nagumo neuron model","volume":"30","author":"Nobukawa","year":"2017","journal-title":"Trans. Inst. Sys. Ctrl. Info. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s11571-020-09570-0","article-title":"Various firing activities and finite-time synchronization of an improved Hindmarsh-Rose neuron model under electric field effect","volume":"14","author":"Wouapi","year":"2020","journal-title":"Cogn. Neurodyn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1007\/s11071-016-3073-x","article-title":"Hopf bifurcation control of the M-L neuron model with type I","volume":"87","author":"Huang","year":"2017","journal-title":"Nonlinear Dynam."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2050131","DOI":"10.1142\/S021812742050131X","article-title":"Stair-like frequency response of single neuron to external electromagnetic radiation and onset of chaotic behaviors","volume":"30","author":"Feng","year":"2020","journal-title":"Int. J. Bifurcat. Chaos"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1450037","DOI":"10.1142\/S0129065714500373","article-title":"A new work mechanism on neuronal activity","volume":"25","author":"Wang","year":"2015","journal-title":"Int. J. Neural Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1007\/s11071-018-4103-7","article-title":"The essence of neuronal activity from the consistency of two different neuron models","volume":"92","author":"Wang","year":"2018","journal-title":"Nonlinear Dynam."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1407","DOI":"10.1631\/FITEE.2100563","article-title":"Phase synchronization and energy balance between neurons","volume":"23","author":"Xie","year":"2022","journal-title":"Front. Inform. Technol. Electron. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1049\/el:19920477","article-title":"Neuron model with potential applications in pulse-stream neural nets","volume":"28","author":"Gan","year":"1992","journal-title":"Electron. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3529","DOI":"10.1007\/s11071-021-06647-1","article-title":"An extensive FPGA-based realization study about the Izhikevich neurons and their bio-inspired applications","volume":"105","author":"Karaca","year":"2021","journal-title":"Nonlinear Dynam."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1007\/s11071-019-05249-2","article-title":"Oscillation dynamics in an extended model of thalamic-basal ganglia","volume":"98","author":"Yu","year":"2019","journal-title":"Nonlinear Dynam."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1016\/j.neunet.2019.11.001","article-title":"Model-based optimized phase-deviation deep brain stimulation for Parkinson\u2019s disease","volume":"122","author":"Yu","year":"2020","journal-title":"Neural Networks"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1080\/17513758.2016.1162856","article-title":"Study on dynamic characteristics\u2019 change of hippocampal neuron reduced models caused by the Alzheimer\u2019s disease","volume":"10","author":"Peng","year":"2016","journal-title":"J. Biol. Dynam."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1007\/s00401-022-02462-z","article-title":"Decreased FAK activity and focal adhesion dynamics impair proper neurite formation of medium spiny neurons in Huntington\u2019s disease","volume":"144","author":"Lee","year":"2022","journal-title":"Acta Neuropathol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1293\/tox.2021-0056","article-title":"Histopathological changes of the spinal cord and motor neuron dynamics in SODI Tg mice","volume":"35","author":"Tanaka","year":"2022","journal-title":"J. Toxicol. Pathol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1007\/s00521-016-2681-1","article-title":"Study on electrical synapse coupling synchronization of Hindmarsh-Rose neurons under Gaussian white noise","volume":"30","author":"Wu","year":"2018","journal-title":"Neural Comput. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.amc.2019.02.046","article-title":"Bogdanov-Takens singularity in the Hindmarsh-Rose neuron with time delay","volume":"354","author":"Li","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"929","DOI":"10.1140\/epjst\/e2020-900247-7","article-title":"Route to hyperchaos and chimera states in a network of modified Hindmarsh-Rose neuron model with electromagnetic flux and external excitation","volume":"229","author":"Wang","year":"2020","journal-title":"Eur. Phys. J-Spec. Top."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.chaos.2018.08.027","article-title":"Desynchronization effects of a current-driven noisy Hindmarsh-Rose neural network","volume":"115","author":"Djeundam","year":"2018","journal-title":"Chaos Soliton. Fract."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s11571-022-09812-3","article-title":"Topology identification and dynamical pattern recognition for Hindmarsh-Rose neuron model via deterministic learning","volume":"17","author":"Chen","year":"2022","journal-title":"Cogn. Neurodynamics"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2627","DOI":"10.1007\/s11071-022-07331-8","article-title":"Stability analysis of the Hindmarsh-Rose neuron under electromagnetic induction","volume":"108","author":"Goulefack","year":"2022","journal-title":"Nonlinear Dynam."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/j.amc.2019.02.041","article-title":"Birth and death of spiral waves in a network of Hindmarsh-Rose neurons with exponential magnetic flux and excitable media","volume":"354","author":"Parastesh","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1479","DOI":"10.1007\/s11071-016-2773-6","article-title":"Model of electrical activity in a neuron under magnetic flow effect","volume":"85","author":"Lv","year":"2016","journal-title":"Nonlinear Dynam."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2101","DOI":"10.1140\/epjst\/e2019-900011-1","article-title":"Phase noise- induced coherence resonance in three dimension memristive Hindmarsh-Rose neuron model","volume":"228","author":"Lu","year":"2019","journal-title":"Eur. Phys. J.-Spec. Top."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"112071","DOI":"10.1016\/j.chaos.2022.112071","article-title":"Energy analysis of bursting Hindmarsh-Rose neurons with time- delayed coupling","volume":"158","author":"Moujahid","year":"2022","journal-title":"Chaos Soliton. Fract."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1007\/s11071-020-05495-9","article-title":"Hidden dynamics in a fractional-order memristive Hindmarsh-Rose model","volume":"100","author":"Yu","year":"2020","journal-title":"Nonlinear Dynam."},{"key":"ref_33","first-page":"2213","article-title":"Hopf bifurcation of the fractional-order hindmarsh- rose neuron model with time-delay","volume":"50","author":"Shi","year":"2021","journal-title":"Rocky Mt. J. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2050044","DOI":"10.1142\/S0218127420500443","article-title":"Adaptive synchronization of fractional-order coupled neurons under electromagnetic radiation","volume":"30","author":"Meng","year":"2020","journal-title":"Int. J. Bifurcat. Chaos"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"190689","DOI":"10.1109\/ACCESS.2020.3026842","article-title":"Adaptive fuzzy control for the generalized projective synchronization of fractional-order extended Hindmarsh-Rose neurons","volume":"8","author":"Liu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_36","first-page":"1887","article-title":"Synchronization of fractional order neurons in presence of noise","volume":"19","author":"Malik","year":"2022","journal-title":"IEEE ACM T. Comput. Bi."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Ramakrishnan, B., Parastesh, F., Jafari, S., Rajagopal, K., Stamov, G., and Stamova, I. (2022). Synchronization in a multiplex network of nonidentical fractional-order neurons. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6030169"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2521","DOI":"10.1007\/s11071-019-05145-9","article-title":"Advances in Lyapunov theory of Caputo fractional-order systems","volume":"97","author":"Ren","year":"2019","journal-title":"Nonlinear Dynam."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4402","DOI":"10.7498\/aps.58.4402","article-title":"A stability theorem about fractional systems and synchronizing fractional unified chaotic systems based on the theorem","volume":"58","author":"Hu","year":"2009","journal-title":"Acta Phys. Sin."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2235","DOI":"10.7498\/aps.58.2235","article-title":"A novel stability theorem for fractional systems and its applying in synchronizing fractional chaotic system based on back-stepping approach","volume":"58","author":"Hu","year":"2009","journal-title":"Acta Phys. Sin."},{"key":"ref_41","first-page":"87","article-title":"A model of neuronal bursting using three coupled first order differential equations","volume":"221","author":"Hindmarsh","year":"1984","journal-title":"Proc. R. Soc. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s11571-013-9273-x","article-title":"Dynamic behavior analysis of fractional-order Hindmarsh-Rose neuronal model","volume":"8","author":"Dong","year":"2014","journal-title":"Cognit. Neurodyn."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/s11071-019-05205-0","article-title":"Dynamical behavior and network analysis of an extended Hindmarsh-Rose neuron model","volume":"98","author":"Rajagopal","year":"2019","journal-title":"Nonlinear Dynam."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/j.chaos.2018.11.028","article-title":"Generalized synchronization of the extended Hindmarsh -Rose neuronal model with fractional order derivative","volume":"118","author":"Giresse","year":"2019","journal-title":"Chaos Soliton. Fract."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/157\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:22:17Z","timestamp":1760120537000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/157"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,2]]},"references-count":44,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["axioms12020157"],"URL":"https:\/\/doi.org\/10.3390\/axioms12020157","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,2,2]]}}}