Abstract
The dynamical features of a modified Fitzhugh–Nagumo (FHN) nerve model are addressed. The model considered accounts for a relaxation time, induced by diffusion and finite propagation velocities, resulting in a hyperbolic system. Bifurcation analysis of the local kinetic system with the relaxation constant as the principal bifurcation parameter reveals a threshold of the relaxation constant beyond which a supercritical Hopf bifurcation occurs. It is shown that the frequency of the ensuing cycles depends on the relaxation time. The existence of Hopf bifurcations on invariant center manifolds is established using the projection method. Analytical formulas for the critical value of the relaxation constant and the first Lyapunov coefficient are derived; results are confirmed via numerical simulations. The addition of external current, at small values of the relaxation constant, produces excitable behavior consistent with the classical FHN model such as periodic firing, bistability, bursting and canard explosions. At higher values of this constant, chaotic motion and other new dynamical objects such as period-two, -three and -four orbits are observed numerically.
Similar content being viewed by others
References
Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications, vol. 19. Springer, Berlin (2012)
Hassard, B.D., Hassard, B.D., Kazarinoff, N.D., Wan, Y.-H., Wan, Y. W.: Theory and Applications of Hopf Bifurcation, vol. 41. CUP Archive (1981)
Carr, J.: Applications of Centre Manifold Theory, vol. 35. Springer, Berlin (2012)
Kelley, A.: Stability of the center-stable manifold. J. Math. Anal. Appl. 18(2), 336–344 (1967)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer, Berlin (2003)
FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys. 17, 257–278 (1955)
Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos 10, 1171–1266 (2000)
Llinás, R.R.: The intrinsic electrophysiological properties of mammalian neurons: insights into central nervous system function. Science 242, 1654–1664 (1988)
Izhikevich, E.M.: Resonate-and-fire neurons. Neural Netw. 14, 883–894 (2001)
Börgers, C.: An Introduction to Modeling Neuronal Dynamics, vol. 66. Springer, Berlin (2017)
Wechselberger, M.: Canards. Scholarpedia 2, 1356 (2007)
Feingold, M., Gonzalez, D.L., Piro, O., Viturro, H.: Phase locking, period doubling, and chaotic phenomena in externally driven excitable systems. Phys. Rev. A 37, 4060 (1988)
Horikawa, Y.: Period-doubling bifurcations and chaos in the decremental propagation of a spike train in excitable media. Phys. Rev. E 50, 1708 (1994)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. Bull. Math. Biol. 52, 25–71 (1990)
Maugin, G.A., Engelbrecht, J.: A thermodynamical viewpoint on nerve pulse dynamics. J. Non-Equil. Thermodyn. 19, 9–23 (1994)
Cattaneo, C.: Sur une forme de l’equation de la chaleur eliminant la paradoxe d’une propagation instantantée. Comptes Rendus 247, 431–433 (1958)
Compte, A., Metzler, R.: The generalized Cattaneo equation for the description of anomalous transport processes. J. Phys. A Math. Gen. 21, 7277 (1997)
Lewandowska, K.D., Kosztołowicz, T.: Application of generalized cattaneo equation to model subdiffusion impedance. Acta Phys. Polon. B 39, 1211–1220 (2008)
Likus, W., Vsevolod A Vladimirov, V. A. Solitary waves in the model of active media, taking into account effects of relaxation. Rep. Math. Phys. 75, 213–230 (2015)
Gawlik, A., Vladimirov, V., Skurativskyi, S.: Existence of the solitary wave solutions supported by the hyperbolic modification of the Fitzhugh–Nagumo system. arXiv preprint arXiv:1905.02087 (2019)
Gawlik, A., Vladimirov, V., Skurativskyi, S. Solitary wave dynamics governed by the modified Fitzhugh–Nagumo equation. arXiv preprint arXiv:1906.01865 (2019)
Tabi, C.B., Etémé, A.S., Kofané, T.C.: Unstable cardiac multi-spiral waves in a Fitzhugh–Nagumo soliton model under magnetic flow effect. Nonlinear Dyn. 100, 3799–3814 (2020)
Takembo, C.N., Mvogo, A., Fouda, H.P.E., Kofané, T.C.: Effect of electromagnetic radiation on the dynamics of spatiotemporal patterns in memristor-based neuronal network. Nonlinear Dyn. 95, 1067–1078 (2019)
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)
Xu, Y., Guo, Y., Ren, G., Ma, J.: Dynamics and stochastic resonance in a thermosensitive neuron. Appl. Math. Comput. 385, 125427 (2020)
Guo, Y., Zhu, Z., Wang, C., Ren, G.: Coupling synchronization between photoelectric neurons by using memristive synapse. Optik 218, 164993 (2020)
Gopalsamy, K., Leung, I.: Delay induced periodicity in a neural netlet of excitation and inhibition. Phys. D 89, 395–426 (1996)
Plant, R.E.: A Fitzhugh differential-difference equation modeling recurrent neural feedback. SIAM J. Appl. Math. 40(1), 150–162 (1981)
Olien, L., Bélair, J.: Bifurcations, stability, and monotonicity properties of a delayed neural network model. Phys. D 102, 349–363 (1997)
Zhen, B., Xu, J.: Fold-Hopf bifurcation analysis for a coupled Fitzhugh–Nagumo neural system with time delay. Int. J. Bifurc. Chaos 20(12), 3919–3934 (2010)
Din, Q., Khaliq, S.: Flip and Hopf bifurcations of discrete-time Fitzhugh–Nagumo model. Open J. Math. Sci. 2, 209–220 (2018)
Rocsoreanu, C., Georgescu, A., Giurgiteanu, N.: The FitzHugh–Nagumo Model: Bifurcation and Dynamics, vol. 10. Springer, Berlin (2012)
Kuznetsov, Y. A.: Elements of Applied Bifurcation Theory, vol. 112. Springer, Berlin (2013)
El Kahoui, M., Weber, A.: Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. J. Symb. Comput. 30, 161–179 (2000)
Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: Matcont: a Matlab package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Soft. (TOMS) 29, 141–164 (2003)
Hoppensteadt, F.C., Izhikevich, E.M.: Weakly Connected Neural Networks, vol. 126. Springer, Berlin (2012)
Davison, E.N., Aminzare, Z., Dey, B., Leonard, N.E.: Mixed mode oscillations and phase locking in coupled Fitzhugh–Nagumo model neurons. Chaos 29, 033105 (2019)
Saha, P., Strogatz, S.H.: The birth of period three. Math. Mag. 68, 42–47 (1995)
Bechhoefer, J.: The birth of period three, revisited. Math. Mag. 69, 115–118 (1996)
Insperger, T.: On the approximation of delayed systems by Taylor series expansion. J. Comput. Nonlinear Dyn. 10, 024503 (2015)
Swadlow, H.A., Waxman, S.G.: Axonal conduction delays. Scholarpedia 7, 1451 (2012)
Hutt, A.: Generalization of the reaction–diffusion, Swift–Hohenberg, and Kuramoto–Sivashinsky equations and effects of finite propagation speeds. Phys. Rev. E 75, 026214 (2007)
Glass, L., Mackey, M.C.: From Clocks to Chaos: The Rhythms of Life. Princeton University Press, Princeton (1988)
Desmaisons, D., Vincent, J.-D., Lledo, P.-M.: Control of action potential timing by intrinsic subthreshold oscillations in olfactory bulb output neurons. J. Neurosci. 19, 10727–10737 (1999)
V-Ghaffari, B., Kouhnavard, M., Kitajima, T.: Biophysical properties of subthreshold resonance oscillations and subthreshold membrane oscillations in neurons. J. Biol. Syst. 24, 561–575 (2016)
Asl, M.M., Valizadeh, A., Tass, P.A.: Dendritic and axonal propagation delays determine emergent structures of neuronal networks with plastic synapses. Sci. Rep. 7, 39682 (2017)
Acknowledgements
The work by CBT is supported by the Botswana International University of Science and Technology under the Grant DVC/RDI/2/1/16I (25). CBT thanks the Kavli Institute for Theoretical Physics (KITP), University of California Santa Barbara (USA), where this work was supported in part by the National Science Foundation Grant No. NSF PHY-1748958, NIH Grant No. R25GM067110, and the Gordon and Betty Moore Foundation Grant No. 2919.01.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tah, F.A., Tabi, C.B. & Kofané, T.C. Hopf bifurcations on invariant manifolds of a modified Fitzhugh–Nagumo model. Nonlinear Dyn 102, 311–327 (2020). https://doi.org/10.1007/s11071-020-05976-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-020-05976-x