Abstract
In this paper, a non-autonomous memristive FitzHugh-Nagumo (FHN) circuit is constructed using a second-order memristive diode bridge with LC network. For convenience of circuit implementation, an AC voltage source is adopted to substitute the original AC current stimulus. Stimulated by the slowly varying AC voltage source, the number, locations and stabilities of the equilibrium points slowly evolve with the time, which are thus indicated as the AC equilibrium points. Different sequences of fold and/or Hopf bifurcations are encountered in a full period of time series evolutions, leading to various kinds of chaotic or periodic bursting activities. To figure out the related bifurcation mechanisms, the fold and Hopf bifurcation sets are mathematically formulated to locate the critical bifurcation points. On this basis, the transitions between the resting and repetitive spiking states are clearly illustrated by the time series of the AC equilibrium points and state variables, from which Hopf/subHopf, Hopf/Hopf, and Hopf/fold bursting oscillations are identified in the specified parameter regions. Finally, based on a fabricated hardware circuit, the experimental measurements are executed. The results verify that the presented memristive FHN circuit indeed exhibits complex bursting activities, which enriches the family of memristor-based FHN circuits with bursting dynamics.
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Duan L X, Lu Q S, Wang Q Y. Two-parameter bifurcation analysis of firing activities in the Chay neuronal model. Neurocomputing, 2008, 72: 341–351
Alidousti J, Ghaziani R K. Spiking and bursting of a fractional order of the modified FitzHugh-Nagumo neuron model. Math Model Comput Simul, 2017, 9: 390–403
Ma J, Tang J. A review for dynamics in neuron and neuronal network. Nonlinear Dyn, 2017, 89: 1569–1578
Hodgkin A L, Huxley A F. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol, 1952, 117: 500–544
Hindmarsh J L, Rose R M. A model of the nerve impulse using two first-order differential equations. Nature, 1982, 296: 162–164
Wang H X, Wang Q Y, Zheng Y H. Bifurcation analysis for Hind-marsh-Rose neuronal model with time-delayed feedback control and application to chaos control. Sci China Tech Sci, 2014, 57: 872–878
Morris C, Lecar H. Voltage oscillations in the barnacle giant muscle fiber. Biophys J, 1981, 35: 193–213
Xu L F, Li C D, Chen L. Contrastive analysis of neuron model. Acta Phys Sin, 2016, 65: 240701
Fitzhugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophys J, 1961, 1: 445–466
Nagumo J, Arimoto S, Yoshizawa S. An active pulse transmission line simulating nerve axon. Proc IRE, 1962, 50: 2061–2070
Baltanás J P, Casado J M. Bursting behaviour of the FitzHugh-Nagumo neuron model subject to quasi-monochromatic noise. Physica D, 1998, 122: 231–240
Abbasian A H, Fallah H, Razvan M R. Symmetric bursting behaviors in the generalized FitzHugh-Nagumo model. Biol Cybern, 2013, 107: 465–476
Vaidyanathan S. Global chaos control of the FitzHugh-Nagumo chaotic neuron model via integral sliding mode control. Int J Pharm-Tech Res, 2016, 9: 413–425
Li F, Liu Q, Guo H, et al. Simulating the electric activity of Fitzhugh-Nagumo neuron by using Josephson junction model. Nonlinear Dyn, 2012, 69: 2169–2179
Guo Y F, Wang L J, Wei F, et al. Dynamical behavior of simplified FitzHugh-Nagumo neural system driven by Lévy noise and Gaussian white noise. Chaos Soliton Fract, 2019, 127: 118–126
Wang Z, Campbell S A. Symmetry, Hopf bifurcation, and the emergence of cluster solutions in time delayed neural networks. Chaos, 2017, 27: 114316
Saha A, Feudel U. Extreme events in FitzHugh-Nagumo oscillators coupled with two time delays. Phys Rev E, 2017, 95: 062219
Shepelev I A, Vadivasova T E, Bukh A V, et al. New type of chimera structures in a ring of bistable FitzHugh-Nagumo oscillators with nonlocal interaction. Phys Lett A, 2017, 381: 1398–1404
Masoliver M, Masoller C. Sub-threshold signal encoding in coupled FitzHugh-Nagumo neurons. Sci Rep, 2018, 8: 8276
Hu M, Li H, Chen Y R, et al. Memristor crossbar-based neuromorphic computing system: A case study. IEEE Trans Neural Netw Learning Syst, 2014, 25: 1864–1878
Sheridan P M, Cai F X, Du C, et al. Sparse coding with memristor networks. Nat Nanotech, 2017, 12: 784–789
Wang Z R, Li C, Song W Y, et al. Reinforcement learning with analogue memristor arrays. Nat Electron, 2019, 2: 115–124
Ge M Y, Jia Y, Xu Y, et al. Mode transition in electrical activities of neuron driven by high and low frequency stimulus in the presence of electromagnetic induction and radiation. Nonlinear Dyn, 2018, 91: 515–523
Zhang G, Wang C N, Alzahrani F, et al. Investigation of dynamical behaviors of neurons driven by memristive synapse. Chaos Soliton Fract, 2018, 108: 15–24
Du L, Cao Z L, Lei Y M, et al. Electrical activities of neural systems exposed to sinusoidal induced electric field with random phase. Sci China Tech Sci, 2019, 62: 1141–1150
Ma J, Mi L, Zhou P, et al. Phase synchronization between two neurons induced by coupling of electromagnetic field. Appl Math Comput, 2017, 307: 321–328
Njitacke Z T, Kengne J. Complex dynamics of a 4D Hopfield neural networks (HNNs) with a nonlinear synaptic weight: Coexistence of multiple attractors and remerging Feigenbaum trees. AEU-Int J Electron Commun, 2018, 93: 242–252
Bao H, Liu W B, Hu A H. Coexisting multiple firing patterns in two adjacent neurons coupled by memristive electromagnetic induction. Nonlinear Dyn, 2019, 95: 43–56
Lv M, Ma J, Yao Y G, et al. Synchronization and wave propagation in neuronal network under field coupling. Sci China Tech Sci, 2019, 62: 448–457
Mvogo A, Takembo C N, Ekobena Fouda H P, et al. Pattern formation in diffusive excitable systems under magnetic flow effects. Phys Lett A, 2017, 381: 2264–2271
Ma J, Zhang G, Hayat T, et al. Model electrical activity of neuron under electric field. Nonlinear Dyn, 2019, 95: 1585–1598
Zhang J H, Liao X F. Effects of initial conditions on the synchronization of the coupled memristor neural circuits. Nonlinear Dyn, 2019, 95: 1269–1282
Zhang J H, Liao X F. Synchronization and chaos in coupled memristor-based FitzHugh-Nagumo circuits with memristor synapse. AEU-Int J Electron Commun, 2017, 75: 82–90
Bao H, Liu W B, Chen M. Hidden extreme multistability and dimensionality reduction analysis for an improved non-autonomous memristive FitzHugh-Nagumo circuit. Nonlinear Dyn, 2019, 96: 1879–1894
Chen M, Qi J W, Xu Q, et al. Quasi-period, periodic bursting and bifurcations in memristor-based FitzHugh-Nagumo circuit. AEU-Int J Electron Commun, 2019, 110: 152840
Izhikevich E M. Neural excitability, spiking and bursting. Int J Bifurcat Chaos, 2000, 10: 1171–1266
Guo D Q, Wu S D, Chen M M, et al. Regulation of irregular neuronal firing by autaptic transmission. Sci Rep, 2016, 6: 26096
Lu L L, Jia Y, Xu Y, et al. Energy dependence on modes of electric activities of neuron driven by different external mixed signals under electromagnetic induction. Sci China Tech Sci, 2019, 62: 427–440
Zhang Z D, Li Y Y, Bi Q S. Routes to bursting in a periodically driven oscillator. Phys Lett A, 2013, 377: 975–980
Wu H G, Bao B C, Liu Z, et al. Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator. Nonlinear Dyn, 2016, 83: 893–903
Bi Q S, Li S L, Kurths J, et al. The mechanism of bursting oscillations with different codimensional bifurcations and nonlinear structures. Nonlinear Dyn, 2016, 85: 993–1005
Chen X K, Li S L, Zhang Z D, et al. Relaxation oscillations induced by an order gap between exciting frequency and natural frequency. Sci China Tech Sci, 2017, 60: 289–298
Bao B C, Wu P Y, Bao H, et al. Symmetric periodic bursting behavior and bifurcation mechanism in a third-order memristive diode bridge-based oscillator. Chaos Soliton Fract, 2018, 109: 146–153
Han C Y, Yu S M, Wang G Y. A sinusoidally driven lorenz system and circuit implementation. Math Probl Eng, 2015, 2015: 706902
Xu Q, Zhang Q L, Qian H, et al. Crisis-induced coexisting multiple attractors in a second-order nonautonomous memristive diode bridge-based circuit. Int J Circ Theor Appl, 2018, 46: 1917–1927
Zhao H T, Lin Y P, Dai Y X. Hopf bifurcation and hidden attractor of a modified Chua’s equation. Nonlinear Dyn, 2017, 90: 2013–2021
Xue W, Qi G Y, Mu J J, et al. Hopf bifurcation analysis and circuit implementation for a novel four-wing hyper-chaotic system. Chin Phys B, 2013, 22: 080504
Desroches M, Kaper T J, Krupa M. Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster. Chaos, 2013, 23: 046106
Premraj D, Suresh K, Banerjee T, et al. An experimental study of slow passage through Hopf and pitchfork bifurcations in a parametrically driven nonlinear oscillator. Commun Nonlinear Sci Numer Simul, 2016, 37: 212–221
Wu H G, Ye Y, Chen M, et al. Extremely slow passages in low-pass filter-based memristive oscillator. Nonlinear Dyn, 2019, 97: 2339–2353
Gottwald G A, Melbourne I. On the implementation of the 0–1 test for chaos. SIAM J Appl Dyn Syst, 2009, 8: 129–145
Savi M A, Pereira-Pinto F H I, Viola F M, et al. Using 0–1 test to diagnose chaos on shape memory alloy dynamical systems. Chaos Soliton Fract, 2017, 103: 307–324
Rauber P E, Fadel S G, Falcao A X, et al. Visualizing the hidden activity of artificial neural networks. IEEE Trans Vis Comput Graph, 2016, 23: 101–110
Wang Z, Joshi S, Savel’ev S, et al. Fully memristive neural networks for pattern classification with unsupervised learning. Nat Electron, 2018, 1: 137–145
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Chen, M., Qi, J., Wu, H. et al. Bifurcation analyses and hardware experiments for bursting dynamics in non-autonomous memristive FitzHugh-Nagumo circuit. Sci. China Technol. Sci. 63, 1035–1044 (2020). https://doi.org/10.1007/s11431-019-1458-5
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DOI: https://doi.org/10.1007/s11431-019-1458-5