Complex dynamics of a Fitzhugh-Rinzel neuron model considering the effect of electromagnetic induction

Document Type : Article


1 Shaanxi Engineering Research Center of Controllable Neutron Source, School of Science, Xijing University, Xi'an 710123, China

2 Mathematical Institute, University of Oxford, Andrew Wiles Building, Oxford, UK

3 Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam



Different single computational neuron models have been proposed in the literature. Most of them belong to the Hodgkin–Huxley (H-H) type, in which they can produce complex behavior of the neuron and have efficient computational cost. In this paper, a modified FitzHugh-Rinzel (FH-R) model considering the effect of magnetic induction is proposed. Different features of the model are explored from a complex and nonlinear perspective. For instance, the impact of the magnetic field on the stability of the equilibrium points is studied by stability analysis. Bifurcation analysis reveals that the proposed neuron model has multi-stability. Furthermore, the spatiotemporal behavior of the proposed model is investigated in the complex network consisting of FH-R oscillators, and the effect of external stimuli is explored on wave propagation of the network.


[1] Skarda, C. A., Freeman W.J. , "Chaos and the New Science of the Brain", Concepts in Neuroscience, 1(2), pp. 275-285 (1990).
[2] Kim, J. H., & Stringer, J., "Applied Chaos". New York, NY: John Wiley and Sons, Inc., (1992).
[3] Hodgkin, A. L., & Huxley, A. F., "Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve", J. Physiol, 117(4), pp. 500-544 (1952).
[4] Ma, J., & Tang, J., "A review for dynamics of collective behaviors of network of neurons", Sci. China Technol, 58, pp. 2038-2045 (2015).
[5] Hindmarsh, J. L., & Rose, R. M., "A model of neuronal bursting using three coupled first order differential equations", Proc. Roy. Soc. London B, 221, pp. 87-102 (1984).
[6] Bao, B., Zhu, Y., Ma, J., et al., "Memristive neuron model with an adapting synapse and its hardware experiments", Science China Technological Sciences, pp. 1-11 (2021).
[7] Xu, Y., Ma, J., Zhan, X., et al., "Temperature effect on memristive ion channels", Cognitive neurodynamics, 13(6), pp. 601-611 (2019).
[8] Xu, Y., Guo, Y., Ren, G., et al., "Dynamics and stochastic resonance in a thermosensitive neuron", Applied Mathematics and Computation, 385, 125427 (2020).
[9] Liu, Y., Xu, W.j., Ma, J., et al., "A new photosensitive neuron model and its dynamics", Frontiers of Information Technology & Electronic Engineering, 21, pp. 1387-1396 (2020).
[10] Xu, Y., Liu, M., Zhu, Z., et al., "Dynamics and coherence resonance in a thermosensitive neuron driven by photocurrent", Chinese Physics B, 29(9), 098704 (2020).
[11] Zhou, P., Yao, Z., Ma, J., et al., "A piezoelectric sensing neuron and resonance synchronization between auditory neurons under stimulus", Chaos, Solitons & Fractals, 145, 110751 (2021).
[12] Wang, Y., Ma, J., Xu, Y., et al., "The Electrical Activity of Neurons Subject to Electromagnetic Induction and Gaussian White Noise", International Journal of Bifurcation and Chaos, 27(2), pp. 1750030-1750042 (2017).
[13] FitzHugh, R., "Impulses and Physiological States in Theoretical Models of Nerve Membrane", Biophys. J., 1(6), pp. 445-466 (1961).
[14] Nagumo, J., Arimoto, S., and Yoshizawa, S., "An Active Pulse Transmission Line Simulating Nerve Axon", Proc. of the IRE, 50(10), pp. 2061-2070 (1962).
[15] Pouryahya, S., "Nonlinear dynamics, synchronisation and chaos in coupled FHN cardiac and neural cells," Ph.D., National University of Ireland Maynooth, 2013.
[16] Guckenheimer, J., & Kuehn, C., "Homoclinic orbits of the FitzHugh–Nagumo equation: Bifurcations in the full system", SIAM J. Appl. Dynam. Syst., 9(1), pp. 138-153 (2010).
[17] Hoff, A., dos Santos, J. V., Manchein, C., et al., "Numerical bifurcation analysis of two coupled FitzHugh–Nagumo oscillators", Eur. Phys. J. B 87(7), pp. 1-9 (2014).
[18] Pal, K., Ghosh, D. and Gangopadhyay, G., "Synchronization and metabolic energy consumption in stochastic Hodgkin-Huxley neurons: Patch size and drug blockers", Neurocomputing, 422, pp. 222-234 (2021).
[19] Doss-Bachelet, C., Françoise, J. P., & Piquet, C., "Bursting oscillations in two coupled FitzHugh–Nagumo systems", ComPlexUs, 1(3), pp. 101-111 (2003).
[20] Majhi, S. and Ghosh, D., "Alternating chimeras in networks of ephaptically coupled bursting neurons", Chaos: An Interdisciplinary Journal of Nonlinear Science, 28(8), 083113 (2018).
[21] Bera, B. K., Rakshit, S., Ghosh, D., et al., "Spike chimera states and firing regularities in neuronal hypernetworks", Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(5), 053115 (2019).
[22] Makarov, V. V., Kundu, S., Kirsanov, D. V., et al., "Multiscale interaction promotes chimera states in complex networks", Communications in Nonlinear Science and Numerical Simulation, 71, pp. 118-129 (2019).
[23] Ciszak, M., Euzzor, S., Arecchi, F. T., et al., "Experimental study of firing death in a network of chaotic FitzHugh–Nagumo neurons", Phys. Rev. E 87(2), 022919(022911-022917) (2013).
[24] Gray, C. M., & McCormick, D. A., "Chattering cells: superficial pyramidal neurons contributing to the generation of synchronous oscillations in the visual vortex", Science, 274(5284), pp. 109-113 (1996).
[25] Kudryashov, N. A., "Asymptotic and Exact Solutions of the FitzHugh – Nagumo Model", Regular and Chaotic Dynamics, 23(2), pp. 152-160 (2018).
[26] Tehrani, N. F., & Razvan, M., "Bifurcation structure of two coupled FHN neurons with delay", Mathematical Biosciences 270, pp. 41-56 (2015).
[27] Zemlyanukhin, A. I., & Bochkarev, A. V., "Analytical Properties and Solutions of the FitzHugh – Rinzel Model", Russian Journal of Nonlinear Dynamics, 15(1), pp. 3-12 (2019).
[28] Wojcik, J., & Shilnikov, A., "Voltage Interval Mappings for an Elliptic Bursting Model", 12: Springer, Cham, (2015).
[29] Belykh, V. N., & Pankratova, E. V., "Chaotic Synchronization in Ensembles of Coupled Neurons Modeled by the FitzHugh – Rinzel System", Radiophys. Quantum El., 49(11), pp. 910-921 (2006).
[30] Shima, S. I., & Kuramoto, Y., "Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators", Phys. Rev. E 69, 036213 (2004).
[31] Kuramoto, Y., & Shima, S. I., "Rotating spirals without Phase singularity in Reaction-Diffusion systems", Progr. Theor. Phys. Suppl, 150, 115 (2003).
[32] Kuramoto, Y., Shima, S. I., Battogtokh, D., et al., "Mean-Field Theory Revives in self-oscillatory field with Non-Local coupling", Prog. Theor. Phys. Suppl, 161, 127 (2006).
[33] Brooks, H. A., & Bressloff, P. C., "Quasicycles in the stochastic hybrid Morris-Lecar neural model", Physical Review E, 92, 012704 (2015).
[34] Hou, Z., & Xin, H., "Noise-sustained spiral waves: effect of spatial and temporal memory", Phys Rev Lett 89, 280601 (2002).
[35] Mondal, A., Sharma, S.K., Upadhyay, R.K. et al, "Firing activities of a fractional-order FitzHugh-Rinzel bursting neuron model and its coupled dynamics", Sci Rep 9, 15721 (2019).
[36] Rinzel, J., "A Formal Classification of Bursting Mechanisms in Excitable Systems, in Mathematical Topics in Population Biology", Morphogenesis and Neurosciences, Lecture Notes in Biomathematics, 71, pp. 267-281 (1987).
[37] Rinzel, J., & Troy, W. C., "Bursting phenomena in a simplified Oregonator flow system model", J Chem Phys 76, pp. 1775-1789 (1982).
[38] Lv, M., Wang, C., Ren, G., et al., "Model of electrical activity in a neuron under magnetic flow effect", Nonlinear Dyn., 85, pp. 1479-1490 (2016).
[39] Wolf, A., Swift, J. B., Swinney, H. L., e al., "Determining Lyapunov exponents from a time series", Physica D: Nonlinear Phenomena, 16(3), pp. 285-317 (1985).