{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T23:02:32Z","timestamp":1775602952214,"version":"3.50.1"},"reference-count":55,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,18]],"date-time":"2021-11-18T00:00:00Z","timestamp":1637193600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Previous studies on the synchronization behaviors of neuronal networks were constructed by integer-order neuronal models. In contrast, this paper proposes that the above topics of symmetrical neuronal networks are constructed by fractional-order Hindmarsh\u2013Rose (HR) models under electromagnetic radiation. They are then investigated numerically. From the research results, several novel phenomena and conclusions can be drawn. First, for the two symmetrical coupled neuronal models, the synchronization degree is influenced by the fractional-order q and the feedback gain parameter k1. In addition, the fractional-order or the parameter k1 can induce the synchronization transitions of bursting synchronization, perfect synchronization and phase synchronization. For perfect synchronization, the synchronization transitions of chaotic synchronization and periodic synchronization induced by q or parameter k1 are also observed. In particular, when the fractional-order is small, such as 0.6, the synchronization transitions are more complex. Then, for a symmetrical ring neuronal network under electromagnetic radiation, with the change in the memory-conductance parameter \u03b2 of the electromagnetic radiation, k1 and q, compared with the fractional-order HR model\u2019s ring neuronal network without electromagnetic radiation, the synchronization behaviors are more complex. According to the simulation results, the influence of k1 and q can be summarized into three cases: \u03b2&gt;0.02, \u22120.06&lt;\u03b2&lt;0.02 and \u03b2&lt;\u22120.06. The influence rules and some interesting phenomena are investigated.<\/jats:p>","DOI":"10.3390\/sym13112204","type":"journal-article","created":{"date-parts":[[2021,11,19]],"date-time":"2021-11-19T02:43:31Z","timestamp":1637289811000},"page":"2204","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["The Synchronization Behaviors of Coupled Fractional-Order Neuronal Networks under Electromagnetic Radiation"],"prefix":"10.3390","volume":"13","author":[{"given":"Xin","family":"Yang","sequence":"first","affiliation":[{"name":"Aeronautics Engineering College, Air Force Engineering University, Xi\u2019an 710038, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guangjun","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Air Force Engineering University, Xi\u2019an 710058, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xueren","family":"Li","sequence":"additional","affiliation":[{"name":"Aeronautics Engineering College, Air Force Engineering University, Xi\u2019an 710038, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dong","family":"Wang","sequence":"additional","affiliation":[{"name":"Aeronautics Engineering College, Air Force Engineering University, Xi\u2019an 710038, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1113\/jphysiol.1952.sp004764","article-title":"A quantitative description of membrane current and its application to conduction and excitation in nerve","volume":"117","author":"Hodgkin","year":"1952","journal-title":"J. Physiol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/S0006-3495(61)86902-6","article-title":"Impulses and physiological states in theoretical models of nerve membrane","volume":"1","author":"FitzHugh","year":"1961","journal-title":"Biophys. J."},{"key":"ref_3","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_4","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":"Soc. Lond. B Biol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7766","DOI":"10.1523\/JNEUROSCI.20-20-07766.2000","article-title":"High-frequency synchronization of neuronal activity in the subthalamic nucleus of Parkinsonian Patients with Limb Tremor","volume":"20","author":"Levy","year":"2000","journal-title":"J. Neurosci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0006-8993(73)90647-1","article-title":"Genesis of epileptic interictal spikes. New knowledge of cortical feedback systems suggests a neurophysiological explanation of brief paroxysms","volume":"52","author":"Ayala","year":"1973","journal-title":"Brain Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2189","DOI":"10.1088\/1009-1963\/14\/11\/008","article-title":"Transition to complete synchronization via near-synchronization in two coupled chaotic neurons","volume":"14","author":"Wang","year":"2005","journal-title":"Chin. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4404","DOI":"10.1209\/0295-5075\/83\/50008","article-title":"Synchronization transitions on small-world neuronal networks: Effects of information transmission delay and rewiring probability","volume":"83","author":"Wang","year":"2008","journal-title":"Europhys. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1395","DOI":"10.1007\/s11071-018-4134-0","article-title":"Synchronous dynamics in neural system coupled with memristive synapse","volume":"92","author":"Xu","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_10","first-page":"321","article-title":"Phase synchronization between two neurons induced by coupling of electromagnetic field","volume":"307","author":"Ma","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_11","first-page":"217","article-title":"Synchronizability of two neurons with switching in the coupling","volume":"350","author":"Parastesh","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_12","first-page":"194","article-title":"Impact of network structure on synchronization of Hindmarsh\u2013Rose neurons coupled in structured network","volume":"333","author":"Bandyopadhyay","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2465","DOI":"10.1140\/epjst\/e2019-800203-3","article-title":"Synchronization patterns in a blinking multilayer neuronal network","volume":"228","author":"Parastesh","year":"2019","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"123132","DOI":"10.1063\/1.5128495","article-title":"Synchronous patterns and intermittency in a network induced by the rewiring of connections and coupling","volume":"29","author":"Budzinski","year":"2019","journal-title":"Chaos"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Perez, T., Garcia, G.C., Eguiluz, V.M., Vicente, R., Pipa, G., and Mirasso, C. (2011). Effect of the Topology and Delayed Interactions in Neuronal Networks Synchronization. PLoS ONE, 6.","DOI":"10.1371\/journal.pone.0019900"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"053113","DOI":"10.1063\/1.4983838","article-title":"Effects of partial time delays on phase synchronization in Watts-Strogatz small-world neuronal networks","volume":"27","author":"Sun","year":"2017","journal-title":"Chaos"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2509","DOI":"10.1007\/s11071-017-3600-4","article-title":"Synchronization transitions induced by partial time delay in an excitatory-inhibitory coupled neuronal network","volume":"89","author":"Sun","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2339","DOI":"10.1007\/s11071-019-05395-7","article-title":"Initial-induced coexisting and synchronous firing activities in memristor synapse-coupled Morris-Lecar bi-neuron network","volume":"99","author":"Bao","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"He, S., Natiq, H., Banerjee, S., and Sun, K. (2021). Complexity and Chimera States in a Network of Fractional-Order Laser Systems. Symmetry, 13.","DOI":"10.3390\/sym13020341"},{"key":"ref_20","first-page":"136","article-title":"Wave propagation and synchronization induced by chemical autapse in chain Hindmarsh-Rose neural network","volume":"352","author":"Ge","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1007\/s12648-018-1181-9","article-title":"Dependence of synchronization transitions on mean field approach in two-way coupled neural system","volume":"92","author":"Shi","year":"2018","journal-title":"Indian J. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.neucom.2018.09.037","article-title":"Propagation of firing rate by synchronization in a feed-forward multilayer Hindmarsh-Rose neural network","volume":"320","author":"Ge","year":"2018","journal-title":"Neurocomputing"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"052914","DOI":"10.1103\/PhysRevE.89.052914","article-title":"Observation and characterization of chimera states in coupled dynamical systems with nonlocal coupling","volume":"89","author":"Gopal","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s11571-018-9505-1","article-title":"Burst synchronization in a scale-free neuronal network with inhibitory spike-timing-dependent plasticity","volume":"13","author":"Kim","year":"2019","journal-title":"Cogn. Neurodyn."},{"key":"ref_25","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":"Cogn. Neurodyn."},{"key":"ref_26","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."},{"key":"ref_27","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_28","doi-asserted-by":"crossref","first-page":"1615","DOI":"10.1007\/s11071-018-4648-5","article-title":"Dynamical behavior and synchronization in time-delay fractional-order coupled neurons under electromagnetic radiation","volume":"95","author":"Meng","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1335","DOI":"10.1038\/nn.2212","article-title":"Fractional differentiation by neocortical pyramidal neurons","volume":"1","author":"Lundstrom","year":"2008","journal-title":"Nat. Neurosci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1531","DOI":"10.1016\/j.camwa.2011.04.057","article-title":"Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay","volume":"62","author":"Wang","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"24","DOI":"10.3389\/fncom.2020.00024","article-title":"Stochastic Resonance Based Visual Perception Using Spiking Neural Networks","volume":"14","author":"Fu","year":"2020","journal-title":"Front. Comput. Neurosci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/BF00206239","article-title":"The fractional-order dynamics of brain-stem vestibulooculomotor neurons","volume":"72","author":"Anastasio","year":"1994","journal-title":"Biol. Cybern."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1007\/s11071-020-05757-6","article-title":"A fractional-order model for the novel coronavirus (COVID-19) outbreak","volume":"101","author":"Rajagopal","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_34","unstructured":"Li, R. (2014). Chaotic Synchronization of a Class of Fractional Order System, Air Force Engineering University."},{"key":"ref_35","first-page":"30289","article-title":"Multiple modes of electrical activities in a new neuron model under electromagnetic radiation","volume":"205","author":"Mi","year":"2016","journal-title":"Neurocomputing"},{"key":"ref_36","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":"Mi","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1016\/j.cnsns.2019.01.004","article-title":"Mohamadou. Firing and synchronization modes in neural network under magnetic stimulation","volume":"72","author":"Tabi","year":"2019","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1007\/s11071-020-05529-2","article-title":"Memristor synapse-coupled memristive neuron network: Synchronization transition and occurrence of chimera","volume":"100","author":"Bao","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/s00422-014-0642-2","article-title":"Spike-frequency adaptation of a two-compartment neuron modulated by extracellular electric fields","volume":"109","author":"Yi","year":"2015","journal-title":"Biol. Cybern."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"3673","DOI":"10.1016\/j.apm.2011.11.012","article-title":"Spiking patterns of a minimal neuron to ELF sinusoidal electric field","volume":"36","author":"Yi","year":"2012","journal-title":"Appl. Math. Model."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2107","DOI":"10.1007\/s11071-015-2129-7","article-title":"A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model","volume":"81","author":"Gu","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"080504","DOI":"10.7498\/aps.61.080504","article-title":"Identification of a stochastic neural firing rhythm lying in period-adding bifurcation and resembling chaos","volume":"61","author":"Gu","year":"2012","journal-title":"Acta Phys. Sin."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2799","DOI":"10.1088\/0256-307X\/25\/8\/018","article-title":"Two different bifurcation scenarios in neural firing rhythms discovered in biological experiments by adjusting two parameters","volume":"25","author":"Wu","year":"2008","journal-title":"Chin. Phys. Lett."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1016\/j.physleta.2013.01.015","article-title":"Experimental evidence of a chaotic region in a neural pacemaker","volume":"377","author":"Gu","year":"2013","journal-title":"Phys. Lett. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1007\/s11071-014-1447-5","article-title":"Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models","volume":"78","author":"Gu","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1007\/s11071-019-05260-7","article-title":"Synchronous behavior among different regions of the neural system induced by electromagnetic radiation","volume":"98","author":"Fang","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1650251","DOI":"10.1142\/S0217979216502519","article-title":"Synchronization behaviors of coupled neurons under electromagnetic radiation","volume":"31","author":"Ma","year":"2017","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_48","first-page":"3042","article-title":"Long-range memory effects in a magnetized Hindmarsh-Rose neural network","volume":"84","author":"Tabi","year":"2020","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1023\/A:1016592219341","article-title":"A predictor-corrector approach for the numerical solution of fractional differential equations","volume":"29","author":"Diethelm","year":"2002","journal-title":"Nonlinear Dyn."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0898-1221(91)90220-X","article-title":"A Review of The Decomposition Method and Some Recent Results for Nonlinear Equations","volume":"21","author":"Adomian","year":"1991","journal-title":"Comput. Math. Appl."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"030502","DOI":"10.7498\/aps.63.030502","article-title":"Solution of the fractional-order chaotic system based on Adomian decomposition algorithm and its complexity analysis","volume":"63","author":"He","year":"2014","journal-title":"Acta Phys. Sin."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.neucom.2016.02.002","article-title":"LMI conditions to global Mittag-Leffler stability of fractional-order neural networks with impulses","volume":"193","author":"Wu","year":"2016","journal-title":"Neurocomputing"},{"key":"ref_53","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1979","DOI":"10.1007\/s11071-015-2292-x","article-title":"Exponential synchronization of fractional-order complex networks via pinning impulsive control","volume":"82","author":"Wang","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.1007\/s11071-018-4261-7","article-title":"Phase synchronization dynamics of coupled neurons with coupling phase in the electromagnetic field","volume":"93","author":"Yong","year":"2018","journal-title":"Nonlinear Dyn."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2204\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:32:24Z","timestamp":1760167944000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2204"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,18]]},"references-count":55,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["sym13112204"],"URL":"https:\/\/doi.org\/10.3390\/sym13112204","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,18]]}}}