Abstract
Complex anatomical and physiological structure of an excitable tissue (e.g., cardiac tissue) in the body can represent different electrical activities through normal or abnormal behavior. Abnormalities of the excitable tissue coming from different biological reasons can lead to formation of some defects. Such defects can cause some successive waves that may end up to some additional reorganizing beating behaviors like spiral waves or target waves. In this study, formation of defects and the resulting emitted waves in an excitable tissue are investigated. We have considered a square array network of neurons with nearest-neighbor connections to describe the excitable tissue. Fundamentally, electrophysiological properties of ion currents in the body are responsible for exhibition of electrical spatiotemporal patterns. More precisely, fluctuation of accumulated ions inside and outside of cell causes variable electrical and magnetic field. Considering undeniable mutual effects of electrical field and magnetic field, we have proposed the new Hindmarsh–Rose (HR) neuronal model for the local dynamics of each individual neuron in the network. In this new neuronal model, the influence of magnetic flow on membrane potential is defined. This improved model holds more bifurcation parameters. Moreover, the dynamical behavior of the tissue is investigated in different states of quiescent, spiking, bursting and even chaotic state. The resulting spatiotemporal patterns are represented and the time series of some sampled neurons are displayed, as well.
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Bertram M, Beta C, Pollmann M, Mikhailov AS, Rotermund HH, Ertl G (2003) Pattern formation on the edge of chaos: experiments with CO oxidation on a Pt (110) surface under global delayed feedback. Phys Rev E 67:036208
Beta C, Bertram M, Mikhailov AS, Rotermund HH, Ertl G (2003) Controlling turbulence in a surface chemical reaction by time-delay autosynchronization. Phys Rev E 67:046224
Beta C, Moula MG, Mikhailov AS, Rotermund HH, Ertl G (2004) Excitable CO oxidation on Pt (110) under nonuniform coupling. Phys Rev Lett 93:188302
Bueno-Orovio A, Cherry EM, Fenton FH (2008) Minimal model for human ventricular action potentials in tissue. J Theor Biol 253:544–560
Chen J-X, Peng L, Ma J, Ying H-P (2014a) Liberation of a pinned spiral wave by a rotating electric pulse. Europhys Lett (EPL) 107:38001
Chen J-X, Zhu J-X, Zhao Y-H, Sun W-G, Xu J-R, Ying H-P (2014b) Simulating bistable biochemical systems by means of reactive multiparticle collision dynamics. Commun Nonlinear Sci Numer Simul 19:2505–2512
Cherry EM, Fenton FH (2008) Visualization of spiral and scroll waves in simulated and experimental cardiac tissue. New J Phys 10:125016
Clayton R et al (2011) Models of cardiac tissue electrophysiology: progress, challenges and open questions. Prog Biophys Mol Biol 104:22–48
Davidenko JM, Pertsov AV (1992) Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 355:349
Fenton FH, Cherry EM, Hastings HM, Evans SJ (2002) Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity Chaos: an interdisciplinary. J Nonlinear Sci 12:852–892
Gray RA, Pertsov AM, Jalife J (1998) Correction: spatial and temporal organization during cardiac fibrillation. Nature 393:191
Gu H, Pan B (2015) A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model. Nonlinear Dyn 81:2107–2126
Gu H, Pan B, Chen G, Duan L (2014) Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models. Nonlinear Dyn 78:391–407
Guo S, Xu Y, Wang C, Jin W, Hobiny A, Ma J (2017) Collective response, synapse coupling and field coupling in neuronal network. Chaos Solitons Fractals 105:120–127
Hildebrand M, Bär M, Eiswirth M (1995) Statistics of topological defects and spatiotemporal chaos in a reaction–diffusion system. Phys Rev Lett 75:1503
Hindmarsh J, Rose R (1982) A model of the nerve impulse using two first-order differential equations. Nature 296:162–164
Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544
Hu B, Ma J, Tang J (2013) Selection of multiarmed spiral waves in a regular network of neurons. PLoS ONE 8:e69251
Huang X, Troy WC, Yang Q, Ma H, Laing CR, Schiff SJ, Wu J-Y (2004) Spiral waves in disinhibited mammalian neocortex. J Neurosci 24:9897–9902
Huang X, Xu W, Liang J, Takagaki K, Gao X, Wu J-Y (2010) Spiral wave dynamics in neocortex. Neuron 68:978–990
Jakubith S, Rotermund H, Engel W, Von Oertzen A, Ertl G (1990) Spatiotemporal concentration patterns in a surface reaction: propagating and standing waves, rotating spirals, and turbulence. Phys Rev Lett 65:3013
Jun M, He-Ping Y, Yong L, Shi-Rong L (2009) Development and transition of spiral wave in the coupled Hindmarsh–Rose neurons in two-dimensional space. Chin Phys B 18:98
Lechleiter J, Girard S, Peralta E, Clapham D (1991) Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes. Science 252:123–126
Li B-W, Deng L-Y, Zhang H (2013) Chiral symmetry breaking in a reaction–diffusion system. Phys Rev E 87:042905
Li B-W, Cai M-C, Zhang H, Panfilov AV, Dierckx H (2014) Chiral selection and frequency response of spiral waves in reaction–diffusion systems under a chiral electric field. J Chem Phys 140:184901
Liu T-B, Ma J, Zhao Q, Tang J (2014) Force exerted on the spiral tip by the heterogeneity in an excitable medium. Europhys Lett (EPL) 104:58005
Lv M, Ma J (2016) Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomputing 205:375–381
Lv M, Wang C, Ren G, Ma J, Song X (2016) Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn 85:1–12
Ma J, Tang J (2015) A review for dynamics of collective behaviors of network of neurons. Sci China Technol Sci 58:2038–2045
Ma J, Jia Y, Wang C-N, Li S-R (2008) The instability of the spiral wave induced by the deformation of elastic excitable media. J Phys A Math Theor 41:385105
Ma J, Tang J, Zhang A, Jia Y (2010) Robustness and breakup of the spiral wave in a two-dimensional lattice network of neurons. Sci China Phys Mech Astron 53:672–679
Ma J, Jia Y, Wang C-N, Jin W-Y (2011) Transition of spiral wave in a model of two-dimensional arrays of Hindmarsh–Rose neurons. Int J Mod Phys B 25:1653–1670
Ma J, Huang L, Tang J, Ying H-P, Jin W-Y (2012) Spiral wave death, breakup induced by ion channel poisoning on regular Hodgkin-Huxley neuronal networks. Commun Nonlinear Sci Numer Simul 17:4281–4293
Ma J, Hu B, Wang C, Jin W (2013) Simulating the formation of spiral wave in the neuronal system. Nonlinear Dyn 73:73–83
Ma J, Wu F, Wang C (2016a) Synchronization behaviors of coupled neurons under electromagnetic radiation. Int J Mod Phys B 31:1650251
Ma J, Xu Y, Tang J, Wang C (2016b) Defects formation and wave emitting from defects in excitable media. Commun Nonlinear Sci Numer Simul 34:55–65
Ma J, Wu F, Hayat T, Zhou P, Tang J (2017) Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media. Phys A Stat Mech Appl 486:508–516. https://doi.org/10.1016/j.physa.2017.05.075
Moujahid A, D’Anjou A, Torrealdea F, Torrealdea FJ (2010) Energy cost reduction in the synchronization of a pair of nonidentical coupled Hindmarsh–Rose neurons. In: Trends in practical applications of agents and multiagent systems. Springer, Berlin, pp 657–664
Moujahid A, d’Anjou A, Torrealdea F, Torrealdea F (2011) Efficient synchronization of structurally adaptive coupled Hindmarsh–Rose neurons. Chaos Solitons Fractals 44:929–933
Pan J-t, Cai M-c, Li B-w, Zhang H (2013) Chiralities of spiral waves and their transitions. Phys Rev E 87:062907
Pan D-B, Gao X, Feng X, Pan J-T, Zhang H (2016) Removal of pinned scroll waves in cardiac tissues by electric fields in a generic model of three-dimensional excitable media. Sci Rep 6:21876
Pertsov AM, Davidenko JM, Salomonsz R, Baxter WT, Jalife J (1993) Spiral waves of excitation underlie reentrant activity in isolated cardiac muscle. Circ Res 72:631–650
Qin H, Ma J, Wang C, Chu R (2014) Autapse-induced target wave, spiral wave in regular network of neurons. Sci China Phys Mech Astron 57:1918–1926
Qin H, Wu Y, Wang C, Ma J (2015) Emitting waves from defects in network with autapses. Commun Nonlinear Sci Numer Simul 23:164–174
Rech PC (2012) Dynamics in the parameter space of a neuron model. Chin Phys Lett 29:060506
Schiff SJ, Huang X, Wu J-Y (2007) Dynamical evolution of spatiotemporal patterns in mammalian middle cortex. BMC Neurosci 8:P61
Torrealdea FJ, Sarasola C, d’Anjou A, Moujahid A, de Mendizábal NV (2009) Energy efficiency of information transmission by electrically coupled neurons. Biosystems 97:60–71
Van Der Heide T et al (2010) Spatial self-organized patterning in seagrasses along a depth gradient of an intertidal ecosystem. Ecology 91:362–369
Winfree AT (1972) Spiral waves of chemical activity. Science 175:634–636
Winfree AT (1987) When time breaks down: the three-dimensional dynamics of electrochemical waves and cardiac arrhythmias, vol 14. Princeton University Press, Princeton
Winfree AT (2001) The geometry of biological time, vol 12. Springer, Berlin
Wu J, Xu Y, Ma J (2017) Lévy noise improves the electrical activity in a neuron under electromagnetic radiation. PLoS ONE 12:e0174330
Xin-Lin S, Wu-Yin J, Jun M (2015) Energy dependence on the electric activities of a neuron. Chin Phys B 24:128710
Xu Y, Jin W, Ma J (2015) Emergence and robustness of target waves in a neuronal network. Int J Mod Phys B 29:1550164
Xu Y, Ying H, Jia Y, Ma J, Hayat T (2017) Autaptic regulation of electrical activities in neuron under electromagnetic induction. Sci Rep 7:43452
Zemlin CW, Pertsov AM (2012) Anchoring of drifting spiral and scroll waves to impermeable inclusions in excitable media. Phys Rev Lett 109:038303
Zhang D, Zhang Q, Zhu X (2015) Exploring a type of central pattern generator based on Hindmarsh–Rose model: from theory to application. Int J Neural Syst 25:1450028
Zhao Y-H, Lou Q, Chen J-X, Sun W-G, Ma J, Ying H-P (2013) Emitting waves from heterogeneity by a rotating electric field chaos: an interdisciplinary. J Nonlinear Sci 23:033141
Acknowledgements
Sajad Jafari was supported by Iran National Science Foundation (No. 96000815). The authors would like to thank Dr. Fatemeh Hadaeghi for help and comments which enhanced the quality of this paper.
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Rostami, Z., Jafari, S. Defects formation and spiral waves in a network of neurons in presence of electromagnetic induction. Cogn Neurodyn 12, 235–254 (2018). https://doi.org/10.1007/s11571-017-9472-y
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DOI: https://doi.org/10.1007/s11571-017-9472-y