Advertisement

Biological Cybernetics

, Volume 71, Issue 2, pp 153–160 | Cite as

In-phase and antiphase self-oscillations in a model of two electrically coupled pacemakers

  • G. S. Cymbalyuk
  • E. V. Nikolaev
  • R. M. Borisyuk
Article

Abstract

The dynamic behavior of a model of two electrically coupled oscillatory neurons was studied while the external polarizing current was varied. It was found that the system with weak coupling can demonstrate one of five stable oscillatory modes: (1) in-phase oscillations with zero phase shift; (2) antiphase oscillations with halfperiod phase shift; (3) oscillations with any fixed phase shift depending on the value of the external polarizing current; (4) both in-phase and antiphase oscillations for the same current value, where the oscillation type depends on the initial conditions; (5) both in-phase and quasiperiodic oscillations for the same current value. All of these modes were robust, and they persisted despite small variations of the oscillator parameters. We assume that similar regimes, for example antiphase oscillations, can be detected in neurophysiological experiments. Possible applications to central pattern generator models are discussed.

Keywords

Phase Shift Dynamic Behavior Small Variation Weak Coupling Oscillatory Mode 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aronson DG, Ermentrout GB, Kopell N (1990) Amplitude response of coupled oscillators. Physica D 41:403–449CrossRefGoogle Scholar
  2. Arshavsky Yul, Belozerova IN, Orlovsky GN, Panchin YuV, Pavlova GA (1985) Control of locomotion in marine mollusk Clione limacina. III. On the origin of locomotory rhythm. Exp Brain Res 58:263–272PubMedGoogle Scholar
  3. Blehman II (1981) Sychronization in nature and engineering (in Russian). Nauka, MoscowGoogle Scholar
  4. Borisyuk GN, Borisyuk RM, Khibnik AI (1992) Analysis of oscillatory regimes of a coupled neural oscillator system with application to visual cortex modeling. In: Taylor JG, Caianiello RM, Cotterill RMJ (eds) Neural network dynamics. Springer, Berlin Heidelberg New York, pp 208–225Google Scholar
  5. Cazalets JR, Nagy F, Moulins M (1990) Suppressive control of the crustacean pyloric network by a pair of identified interneurons. I. Modulation of the motor pattern. J Neurosci 10:448–457PubMedGoogle Scholar
  6. Collins JJ, Stewart IN (1993) Coupled nonlinear oscillators and the symmetries of animal gaits. J Nonlin Sci 3:349–392Google Scholar
  7. Constantine-Paton M, Cline HT, Debski E (1990) Patterned activity, synaptic convergence, and the NMDA receptor in developing visual pathways. Annu Rev Neurosci 13:129–154CrossRefPubMedGoogle Scholar
  8. Cruse H (1990) What mechanisms coordinate leg movement in walking arthropods? Trends Neurosci 13:15–21CrossRefPubMedGoogle Scholar
  9. Dermietzel R, Spray D (1993) Gap junction in the brain: where, what type, how many and why? TINS 16:186–192PubMedGoogle Scholar
  10. Dongarra JJ, Bunch JR, Moler CB, Stewart GW (1978) LINPACK users guide. SIAM Publications, PhiladelphiaGoogle Scholar
  11. Eisen JS, Marder E (1984) A mechanism for production of phase shifts in a pattern generator. J Neurophysiol 51:1375–1393PubMedGoogle Scholar
  12. Ermentrout GB, Kopell N (1991) Multiple pulse interactions and averaging in systems of coupled neural oscillators. J Math Biol 29:195–217CrossRefGoogle Scholar
  13. Fenichel N (1971) Persistence and smoothness of invariant manifolds for flows. Indiana Univ Mathematics J 21:3Google Scholar
  14. Getting PA (1988) Comparative analysis of invertebrate central pattern generators. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements in vertebrates. Wiley, New York, pp 101–127Google Scholar
  15. Grillner S, Matsushima T (1991) The neural network underlying locomotion in lamprey — synaptic and cellular mechanisms. Neuron 7:1–15CrossRefPubMedGoogle Scholar
  16. Harris-Warrick RM (1988) Chemical modulation of central pattern generators. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements in vertebrates. Wiley, New York, pp 286–331Google Scholar
  17. Hartmann F (1964) Ordinary differential equations. Wiley, New YorkGoogle Scholar
  18. Hindmarsh JL, Rose RM (1982) A model of the nerve impulse using two first-order differential equations. Nature 296:162–164CrossRefPubMedGoogle Scholar
  19. Hindmarsh JL, Rose RM (1984) A model of neuronal bursting using three coupled first order differential equations. Proc R Soc Lond B 221:87–102PubMedGoogle Scholar
  20. Kawato M, Sokabe M, Suzuki R (1979) Synergism and antagonism of neurons caused by an electrical synapse. Biol Cybern 34:81–89CrossRefPubMedGoogle Scholar
  21. Khibnik AI, Borisyuk RM, Roose D (1992) Numerical bifurcation analysis of a model of coupled neural oscillators. Int Ser Num Math 104:215–228Google Scholar
  22. Khibnik AI, Kuznetsov YuA, Levitin VV, Nikolaev EV (1993) Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps. In: Proceedings of Advanced NATO Workshop on Homoclinic Chaos. Physica D V. 62 (sn1–4):360–367Google Scholar
  23. Kopell N (1988) Toward a theory of modelling central pattern generators. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements in vertebrates. Wiley, New York, pp 369–413Google Scholar
  24. Malkin IG (1956) Some problems of the theory of nonlinear oscillations (in Russian). Gostehizdat, MoscowGoogle Scholar
  25. Marder E, Eisen JS (1984) Electrically coupled pacemaker neurons respond differently to same physiological inputs and neurotransmitters. J Neurophysiol 51:1362–1374PubMedGoogle Scholar
  26. Marder E. Hooper SL, Eisen JS (1987) Multiple neurotransmitters provide a mechanism for the production of multiple outputs from a single neuronal circuit. In: Edelman GM, Gall WE, Cowan MW (eds) Synaptic function, NRF. Wiley, New York, pp 305–327Google Scholar
  27. Marder E, Abbot LF, Kepler T, Hooper SL (1992) Modification of oscillator function by electrical coupling to nonoscillatory neurons. In: Basar E, Bullock TH (eds) Induced rhythms in the brain. Birkhauser, Boston, pp 287–296Google Scholar
  28. Müller U, Cruse H (1991) The contralateral coordination of walking legs in the crayfish Astacus leptodactylus. II. Model calculations. Biol Cybern 64:437–446CrossRefGoogle Scholar
  29. Rand RH, Cohen AH, Holmes PJ (1988) Systems of coupled oscillators as models of central pattern generators. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements in vertebrates. Wiley, New York, pp 333–367Google Scholar
  30. Schöner G, Jiang WY, Kelso JAS (1990) A synergetic theory of quadrupedal gaits and gait transitions. J Theor Biol 142:359–391PubMedGoogle Scholar
  31. Selverston AI, Moulins M (1985) Oscillatory neural networks. Annu Rev Physiol 47:29–48CrossRefPubMedGoogle Scholar
  32. Taga G, Yamaguchi Y, Shimizu H (1991) Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment. Biol Cybern 65:147–159CrossRefPubMedGoogle Scholar
  33. Yang XD, Korn H, Faber DS (1990) Long-term potentiation of electrotonic coupling at mixed synapses. Nature 348:542–545CrossRefPubMedGoogle Scholar
  34. Yuste R, Peinado A, Katz LC (1992) Neuronal domains in developing neocortex. Science 257:665–668Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • G. S. Cymbalyuk
    • 1
  • E. V. Nikolaev
    • 1
  • R. M. Borisyuk
    • 1
  1. 1.Institute of Mathematical Problems of Biology, Russian Academy of SciencesPushchinoRussian Federation

Personalised recommendations