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Various firing activities and finite-time synchronization of an improved Hindmarsh–Rose neuron model under electric field effect

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Abstract

Nowadays, it is important to realize systems that can model the electrical activity of neurons taking into account almost all the properties of the intracellular and extracellular environment in which they are located. It is in this sense that we propose in this paper, the improved model of Hindmarsh–Rose (HR) which takes into account the fluctuation of the membrane potential created by the variation of the ion concentration in the cell. Considering the effect of the electric field that is produced on the dynamic behavior of neurons, the essential properties of the model such as equilibrium point and its stability, bifurcation diagrams, Lyapunov spectrum, frequency spectra, time series of the membrane potential and phase portraits are thoroughly investigated. We thus prove that Hopf bifurcation occurs in this system when the parameters are chosen appropriately. We also observe that by varying specific parameters of the electric field, the model presents a very rich and striking event, namely hysteresis phenomenon, which justifies the coexistence of multiple attractors. Besides, by applying a suitable sinusoidal excitation current, we prove that the neuron under electric field effect can present several important electrical activities including quiescent, spiking, bursting and even chaos. We propose the improved HR model under electric field effect (mHR) to study the finite-time synchronization between two neurons when performing synapse coupling across the membrane potential and the electric field coupling. As a result, we find that the synchronization between the two neurons is weakly influenced by the variation of the intensity of the electric field coupling while it is strongly impacted when the intensity of the synapse coupling is modified. From these results, it is obvious that the electric field can be another effective bridge connection to encourage the exchange and coding of the signal. Using the finite-time synchronization algorithm, we theoretically quantify the synchronization time between these neurons. Finally, Pspice simulations are presented to show the feasibility of the proposed model as well as that of the developed synchronization strategy.

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References

  • Antonopoulos CG, Martinez EB, Baptista MS (2019) Evaluating performance of neural codes in model neural communication networks. Neural Netw 109:90–102

    Article  PubMed  Google Scholar 

  • Bao BC, Jiang P, Wu HG, Hu FW (2015) Complex transient dynamics in periodically forced memristive chua’s circuit. Nonlinear Dyn 79:2333–2343

    Article  Google Scholar 

  • Bao BC, Hu A, Xu Q, Bao H, Hu W, Chen M (2018) AC-induced coexisting asymetric bursters in the improved Hindmarsh–Rose model. Nonlinear Dyn 92:1695

    Article  Google Scholar 

  • Boccaletti S, Latora V, Moreno Y, Chavez M et al (2006) Complex networks: structure and dynamics. Phys Rep 424:175–308

    Article  Google Scholar 

  • Boccaletti S, Bianconi G, Criado R, Wang Z, Zanin M et al (2014) The structure and dynamics of multilayer networks. Phys Rep 544:1–122

    Article  PubMed  PubMed Central  Google Scholar 

  • Cho YM, Rajamani R (1997) A systematic approach to adaptive observer synthesis for nonlinear systems. IEEE Trans Autom Control 42:534–537

    Article  Google Scholar 

  • Djeundam SRD, Yamapi R, Kofane TC, Azizalaoui MA (2013) Deterministic and stochastic bifurcations in the Hindmarsh–Rose neuronal model. Chaos 23:033125

    Article  Google Scholar 

  • Dong J, Zhang GJ, Xie Y, Yao H, Wang J (2014) Dynamic behavior analysis of fractional-order Hindmarsh–Rose neuronal model. Cogn Neurodyn 8:167–175

    Article  Google Scholar 

  • Estrada E (2012) The structure of complex networks: theory and applications. Oxford University Press, Oxford

    Google Scholar 

  • Fitzhugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1:445

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Fitzhugh R (1969) Mathematical models of excitation and propagation in nerve. In: Schwan HP (ed) Biological engineering. Mc Graw-Hill, New-York

    Google Scholar 

  • Ge M, Jia Y, Xu Y, Yang L (2018) Mode transition in electrical activities of neuron driven by high and low frequency stimulus in the presence of electromagnetic induction and radiation. Nonlinear Dyn 91:515–523

    Article  Google Scholar 

  • González-Miranda JM (2007) Complex bifurcation structures in the Hindmarsh–Rose neuron model. Int J Bifurc Chaos 17:3071–3083

    Article  Google Scholar 

  • Gu HG, Pan BB, Chen GR, Duan LX (2014) Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models. Nonlinear Dyn 78:391–407

    Article  Google Scholar 

  • Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems and bifurcation of vector field. Springer, New York

    Book  Google Scholar 

  • Han C, Yu S, Wang GA (2015) Sinusoidally driven Lorenz system and circuit implementation. Math Prob Eng 2015:706902

    Google Scholar 

  • Hindmarsh JL, Rose RM (1982) A model of the nerve impulse using two first-order differential equations. Nature 296:162–164

    Article  CAS  PubMed  Google Scholar 

  • Hindmarsh JL, Rose RM (1984) A model of neuronal bursting using three coupled first order differential equations. Proc R Soc Lond B Biol Sci 221:87–102

    CAS  PubMed  Google Scholar 

  • 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

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Innocenti G, Genesio R (2009) On the dynamics of chaotic spiking–bursting transition in the Hindmarsh–Rose neuron. Chaos 19:023124

    Article  CAS  PubMed  Google Scholar 

  • Innocenti G, Morelli A, Genesio R, Torcini A (2007) Dynamical phases of the Hindmarsh–Rose neuronal model: studies of the transition from bursting to spiking chaos. Chaos 17:043128

    Article  PubMed  Google Scholar 

  • Jia C, Wang J, Deng B, Wei X, Che Y (2011) Estimating and adjusting abnormal networks with unknown parameters and topology. Chaos 21:013109

    Article  PubMed  Google Scholar 

  • Kaslik E (2017) Analysis of two- and three-dimensional fractional-order Hindmarsh–Rose type neuronal models. Frac Calc Appl Anal 20:623–645

    Google Scholar 

  • Kengne J, Chedjou JC, Kenne G, Kyamakya K, Kom GH (2012) Analog circuit implementation and synchronization of a system consisting of a van der Pol oscillator linearly coupled to a Duffing oscillator. Nonlinear Dyn 70:2163–2173

    Article  Google Scholar 

  • Kengne J, Jafari S, Njitacke ZT, Yousefi AK, Cheukem A (2017) Dynamic analysis and electronic circuit implementation of a novel 3D autonomous system without linear terms. Commun Nonlinear Sci Numer Simul. https://doi.org/10.1016/j.cnsns.2017.04.017

    Article  Google Scholar 

  • Khalil HK (2007) Nonlinear systems, 3rd edn. Prentice Hall, Upper Saddle River

    Google Scholar 

  • Kivelä M, Arenas A, Barthelemy M, Gleeson JP, Moreno Y, Porter MA (2014) Multilayer networks. J Complex Netw 2:203–271

    Article  Google Scholar 

  • Kuznetsov YA (1998) Elements of applied bifurcation theory. Springer, New York

    Google Scholar 

  • Letellier C, Denis F, Aguirre LA (2013) What can be learned from a chaotic cancer model ? J Theor Biol 322:7–16

    Article  CAS  PubMed  Google Scholar 

  • Lopez MJ, Consegliere A, Garcia L, Lorenzo J (2015) Simulation and control of heart rhythm dynamics. Adv Biomed Res 1:509–516

    Google Scholar 

  • Louodop P, Fotsin H, Kountchou M, Bowong S (2013) Finite-time synchronization of Lorenz chaotic systems: theory and circuits. IOP Sci 88:045002

    CAS  Google Scholar 

  • Louodop P, Fotsin H, Kountchou M, Ngouonkadi LBM, Cerdeira HA, Bowong S (2014a) Finite-time synchronization of tunnel-diode-based chaotic oscillators. Phys Rev E 89:032921

    Article  CAS  Google Scholar 

  • Louodop P, Kountchou M, Fotsin H, Bowong S (2014b) Practical finite-time synchronization of Jerk systems: theory and experiment. Nonlinear Dyn 78:597

    Article  Google Scholar 

  • Lu L, Jia Y, Liu W, Yang L (2017) Mixed stimulus-induced mode selection in neural activity driven by high and low frequency current under electromagnetic radiation. Complexity 7628537:1–11

    Google Scholar 

  • Lv M, Ma J (2016) Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomputing 205:375–381

    Article  Google Scholar 

  • Lv M, Wang CN, Ren GD, Ma J (2016) Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn 85:1479–1490

    Article  Google Scholar 

  • Ma J, Xu Y, Wang CN, Jin WY (2016) Pattern selection and self-organization induced by random boundary initial values in a neuronal network. Phys A 461:586–594

    Article  Google Scholar 

  • Ma J, Wu F, Wang C (2017) Synchronization behaviors of coupled neurons under electromagnetic radiation. Int J Mod Phys B 31:1650251

    Article  Google Scholar 

  • Ma J, Fuqiang W, Ahmed A, Jun T (2018) Crack synchronization of chaotic circuits under field coupling. Nonlinear Dyn. https://doi.org/10.1007/s11071-018-4307-x

    Article  Google Scholar 

  • Ma J, Zhang G, Hayat T, Ren GD (2019) Model electrical activity of neuron under electric field. Nonlinear Dyn 95:1585

    Article  Google Scholar 

  • Mondal A, Upadhyay RK, Ma J et al (2019) Bifurcation analysis and diverse firing activities of a modified excitable neuron model. Cogn Neurodyn 13:393–407

    Article  PubMed  PubMed Central  Google Scholar 

  • Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35:193–213

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Negou AN, Kengne J (2018) Dynamic analysis of a unique jerk system with a smoothly adjustable symmetry and nonlinearity: reversals of period doubling, offset boosting and coexisting bifurcations. Int J Electron Commun (AEÜ) 90:1–19

    Article  Google Scholar 

  • Ngouonkadi EB, Fotsin HB, Louodop F (2014) Implementing a memristive Van der Pol oscillator coupled to a linear oscillator: synchronization and application to secure communication. IOP Sci 89:035201

    Google Scholar 

  • Ngouonkadi EBM, Fotsin HB, Fotso PL, Tamba VK, Cerdeira HA (2016) Bifurcations and multistability in the extended Hindmarsh–Rose neuronal oscillator. Chaos, Solitons Fractals 85:151–163

    Article  Google Scholar 

  • Njitacke ZT, Kengne J (2018) Complex dynamics of a 4D Hopfield neural networks (HNNs) with a nonlinear synaptic weight: coexistence of multiple attractors and remerging Feigenbaum trees. Int J Electron Commun (AEÜ) 93:242–252

    Article  Google Scholar 

  • Njitacke ZT, Kengne J, Negou AN (2017) Dynamical analysis and electronic circuit realization of an equilibrium free 3D chaotic system with a large number of coexisting attractors. Optik 130:356–364

    Article  Google Scholar 

  • Njitacke ZT, Kengne J, Fonzin FT, Leutcha PB, Fotsin HB (2018) Dynamical analysis of a novel 4-neurons based Hopfield neural network: emergences of antimonotonicity and coexistence of multiple stable states. Int J Dyn Control. https://doi.org/10.1007/s40435-019-00509-w

    Article  Google Scholar 

  • Paden Brad E, Shankar Sastry (1987) A calculus for computing filippov’s differential inclusion with application to the variable structure control of robot. IEEE Trans Circuit Systems 35:73–82

    Article  Google Scholar 

  • Panahi S, Aram Z, Jafari S, Ma M, Sprott JC (2017) Modeling of epilepsy based on chaotic artificial neural network. Chaos Solitons Fractals 105:150–156

    Article  Google Scholar 

  • Parastesh F, Azarnoush H, Jafari S et al (2019) Synchronizability of two neurons with switching in the coupling. Appl Math Comput 350:217–223

    Google Scholar 

  • Perc M (2009) Optimal spatial synchronization on scale-free networks via noisy chemical synapses. Biophys Chem 141:175–179

    Article  CAS  PubMed  Google Scholar 

  • Ren G, Xu Y, Wang C (2017a) Synchronization behavior of coupled neuron circuits composed of memristors. Nonlinear Dyn 88:893–901

    Article  Google Scholar 

  • Ren GD, Zhou P, Ma J, Cai N, Alsaedi A, Ahmad B (2017b) Dynamical response of electrical activities in digital neuron circuit driven by autapse. Int J Bifurc Chaos 27:1750187

    Article  Google Scholar 

  • Ren G, Xue Y, Li Y, Ma J (2019) Field coupling benefits signal exchange between Colpitts systems. Appl Math Comput 342:45–54

    Google Scholar 

  • Rigatos G, Wira P, Melkikh A (2019) Nonlinear optimal control for the synchronization of biological neurons under time-delays. Cogn Neurodyn 13:89–103

    Article  CAS  PubMed  Google Scholar 

  • Shi X, Wang Z (2012) Adaptive synchronization of time delay Hindmarsh–Rose neuron system via self-feedback. Nonlinear Dyn 69:21472153

    Google Scholar 

  • Spitzer NC (2006) Electrical activity in early neuronal development. Nature 444:707–712

    Article  CAS  PubMed  Google Scholar 

  • Strogatz SH, Friedman M, Mallinckrodt AJ, Mckay S (1994) Nonlinear dynamics and chaos: with applications to physics, biology, chimestry and engineering. Comput Phys 8(5):532

    Article  Google Scholar 

  • Uhhaas PJ, Singer W (2006) Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology. Neuron 52:155–168

    Article  CAS  Google Scholar 

  • Wang Z, Shi X (2020) Electric activities of time delay memristive neuron disturbed by Gaussian white noise. Cogn Neurodyn 14:115–124

    Article  PubMed  Google Scholar 

  • Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Springer, New York

    Book  Google Scholar 

  • Wolf A, Swift JB, Swinney HL, Wastano JA (1985) Determining Lyapunov exponents from time series. Phys D 16:285–317

    Article  Google Scholar 

  • Wouapi KM, Fotsin HB, Feudjio KF, Njitacke ZT (2019) Hopf bifurcation, offset boosting and remerging Feigenbaum trees in an autonomous chaotic system with exponential nonlinearity. SN Appl Sci 1:1715

    Article  Google Scholar 

  • Wu KJ, Luo TQ, Lu HW, Wang Y (2016) Bifurcation study of neuron firing activity of the modified Hindmarsh–Rose model. Neural Comput Appl 27:739–747

    Article  Google Scholar 

  • Wu F, Ma J, Zhang G (2019) A new neuron model under electromagnetic field. Appl Math Comput 347:590–599

    Google Scholar 

  • Xu Q, Zhang QL, Bao BC, Hu YH (2017) Non-autonomous second-order memristive chaotic circuit. IEEE Access 5:21039–21045

    Article  Google Scholar 

  • Zuppa LA, Hernandez CC, Bustos AYA (2002) Finite synchronization of Lorenz-based chaotic systems. www.wseas.us/e-library/conferences/mexico2002/papers/249.pdf. Accessed 4 April 2017

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Authors and Affiliations

  1. Unité de Recherche de Matière Condensée, d’Electronique et de Traitement du Signal (URMACETS), Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon

    K. Marcel Wouapi, B. Hilaire Fotsin & F. Patrick Louodop

  2. Laboratoire d’Energie et des Systemes Electriques et Electroniques, Department of Physics, University of Yaounde I, PO Box 812, Yaoundé, Cameroon

    K. Florent Feudjio

  3. Department of Electrical and Electronic Engineering, College of Technology (COT), University of Buea, P.O. Box 63, Buea, Cameroon

    Z. Tabekoueng Njitacke

  4. Energy and Environmental Technologies Laboratory, Department of Physics, University of Yaounde I, PO Box 812, Yaoundé, Cameroon

    T. Hermann Djeudjo

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  1. K. Marcel Wouapi
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  2. B. Hilaire Fotsin
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  3. F. Patrick Louodop
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Correspondence to K. Marcel Wouapi.

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Wouapi, K.M., Fotsin, B.H., Louodop, F.P. et al. Various firing activities and finite-time synchronization of an improved Hindmarsh–Rose neuron model under electric field effect. Cogn Neurodyn 14, 375–397 (2020). https://doi.org/10.1007/s11571-020-09570-0

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