From simple to complex oscillatory behaviour via intermittent chaos in the Rose-Hindmarsh model for neuronal activity. (English) Zbl 0753.92009

Summary: The Rose-Hindmarsh equations [see J. L. Hindmarsh and R. M. Rose, Proc. R. Soc. Lond., Ser. B 221, 87-102 (1984)] are a system of three nonlinear ordinary differential equations that provide a phenomenological model for repetitive, patterned and irregular activity in molluscan neurons. We obtain bifurcation diagrams for this system, and obtain interval maps that reproduce the behaviour of the differential system. These maps are used to explore the bifurcations from simple to complex oscillatory behaviour.


92C20 Neural biology
34C23 Bifurcation theory for ordinary differential equations
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[1] Hindmarsh, J. L.; Rose, R. M., A model of the nerve impulse using two first-order differential equations, Nature, Lond., 296, 162-164 (1982)
[2] Hindmarsh, J. L.; Rose, R. M., A model of neuronal bursting using three coupled first-order differential equations, Proc. R. Soc. Lond., B221, 87-102 (1984)
[3] Kaas-Petersen, C., Bifurcations in the Rose-Hindmarsh and Chay models, (Degen, H.; Holden, A. V.; Olsen, L. F., Chaos in Biological Systems (1987), Plenum: Plenum New York), 183-190
[4] Wang, X. J., Gensis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle (1992), Preprint
[5] Holden, A. V.; Fan, Y. S., From simple to simple bursting oscillatory behaviour via chaos in the Rose-Hindmarsh model for neuronal activity, Chaos, Solitons & Fractals, 2, 221-236 (1992) · Zbl 0766.92006
[6] Buskirk, R. V.; Jeffries, C., Observation of chaotic dynamics of coupled nonlinear oscillators, Phys. Rev., A13, 3332 (1985)
[7] Poincaré, H., Les Méthods Nouvelles de la Mécanique Céleste (1899), Gauthier-Villars: Gauthier-Villars Paris
[8] Collet, P.; Eckman, J. P., Iterated Maps on the Interval as Dynamical Systems (1980), Birkhauser: Birkhauser Basel and Boston · Zbl 0458.58002
[9] Chay, T. R.; Rinzel, J., Bursting, beating, and chaos in an excitable membrane model, Biophys. J., 47, 357-366 (1985)
[10] Decroly, O.; Goldbeter, A., From simple to complex oscillatory behaviour: analysis of bursting in a multiply regulated biochemical system, J. Theor. Biol., 124, 219-250 (1987)
[11] Grebogi, C.; Ott, E.; Yorke, J. A., Critical exponent of chaotic transients in nonlinear dynamical systems, Phys. Rev. Lett., 57, 11, 11284-11287 (1986)
[12] Grebogi, C.; Ott, E.; Romeiras, F.; Yorke, J. A., Critical exponents for crisis-induced intermittency, Phys. Rev., A36, 5365 (1987)
[13] Marsden, J. E.; McCracken, M., The Hopf Bifurcation and its Applications (1976), Springer: Springer New York · Zbl 0346.58007
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