Abstract
In this paper, a general approach for studying rings of coupled biological oscillators is presented. This approach, which is group-theoretic in nature, is based on the finding that symmetric ring networks of coupled non-linear oscillators possess generic patterns of phaselocked oscillations. The associated analysis is independent of the mathematical details of the oscillators' intrinsic dynamics and the nature of the coupling between them. The present approach thus provides a framework for distinguishing universal dynamic behaviour from that which depends upon further structure. In this study, the typical oscillation patterns for the general case of a symmetric ring of n coupled non-linear oscillators and the specific cases of three- and five-membered rings are considered. Transitions between different patterns of activity are modelled as symmetry-breaking bifurcations. The effects of one-way coupling in a ring network and the differences between discrete and continuous systems are discussed. The theoretical predictions for symmetric ring networks are compared with physiological observations and numerical simulations. This comparison is limited to two examples: neuronal networks and mammalian intestinal activity. The implications of the present approach for the development of physiologically meaningful oscillator models are discussed.
Similar content being viewed by others
References
Abbott LF, Marder E, Hooper SL (1991) Oscillating networks: control of burst duration by electrically coupled neurons. Neural Comput 3:487–497
Alexander JC (1986) Patterns at primary Hopf bifurcations of a plexus of identical oscillators. SIAM J Appl Math 46:199–221
Arnold VI (1983) Geometrical methods in the theory of ordinary differential equations. Springer, Berlin Heidelberg New York
Ashwin P (1990) Symmetric chaos in systems of three and four forced oscillators. Nonlinearity 3:603–617
Ashwin P, King GP, Swift JW (1990) Three identical oscillators with symmetric coupling. Nonlinearity 3:585–602
Ashwin P, Swift J (1992) The dynamics of n weakly coupled identical oscillators. J Nonlinear Sci 2:69–108
Atiya A, Baldi P (1989) Oscillations and synchronizations in neural networks: an exploration of the labelling hypothesis. Int J Neural Syst 1:103–124
Baesens C, Guckenheimer J, Kim S, MacKay RS (1991) Three coupled oscillators: mode-locking, global bifurcations, and toroidal chaos. Physica 49D:387–475
Bardakjian BL, Sarna SK (1980) A computer model of human colonic electrical control activity (ECA). IEEE Trans Biomed Eng 27:193–202
Bass P, Code CF, Lambert EH (1961) Motor and electric activity of the duodenum. Am J Physiol 201:287–291
Bay JS, Hemami H (1987) Modeling of a neural pattern generator with coupled nonlinear oscillators. IEEE Trans Biomed Eng 34:297–306
Bullock TH (1976) In search of principles in neural integration. In: Fentress JD (eds) Simple networks and behavior. Sinauer, Sunderland, MA, pp 52–60
Cohen AH, Rossignol S, Grillner S (eds) (1988) Neural control of rhythmic movements in vertebrates. Wiley, New York
Collins JJ, Stewart IN (1992) Symmetry-breaking bifurcation: a possible mechanism for 2∶1 frequency-locking in animal locomotion. J Math Biol 30:827–838
Collins JJ, Stewart IN (1993a) Coupled nonlinear oscillators and the symmetries of animal gaits. J Nonlinear Sci 3:349–392
Collins JJ, Stewart I (1993b) Hexapodal gaits and coupled nonlinear oscillator models. Biol Cybern 68:287–298
Cowan JD (1980) Symmetry breaking in embryology and in neurobiology. In: Gruber B, Millman RS (eds) Symmetries in science. Plenum, New York, pp 459–474
Cowan JD (1982) Spontaneous symmetry breaking in large scale nervous activity. Int J Quantum Chem 22:1059–1082
Dunin-Barkovskii VL (1970) Fluctuations in the level of activity in simple closed neurone chains. Biophysics 15:396–401
Elias SA, Grossberg S (1975) Pattern formation, contrast control, and oscillations in short term memory of shunting on-center off-surround networks. Biol Cybern 20:69–98
Endo T, Mori S (1978) Mode analysis of a ring of a large number of mutually coupled van der Pol oscillators. IEEE Trans Circuits Syst 25:7–18
Ermentrout GB (1985) The behavior of rings of coupled oscillators. J Math Biol 23:55–74
Friesen WO, Stent GS (1977) Generation of a locomotory rhythm by a neural network with recurrent cyclic inhibition. Biol Cybern 28:27–40
Friesen WO, Stent GS (1978) Neural circuits for generating rhythmic movements. Annu Rev Biophys Bioeng 7:37–61
Glass L, Mackey MC (1988) From clocks to chaos. Princeton University Press, Princeton
Glass L, Young RE (1979) Structure and dynamics of neural network oscillators. Brain Res 179:207–218
Golubitsky M, Stewart IN (1985) Hopf bifurcation in the presence of symmetry. Arch Rational Mech Anal 87:107–165
Golubitsky M, Stewart IN (1986) Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators. In: Golubitsky M, Guckenheimer J (eds) Multiparameter bifurcation theory. (Contemporary Mathematics 56) American Mathematical Society, Providence RI, pp 131–173
Golubitsky M, Stewart IN, Schaeffer DG (1988) Singularities and groups in bifurcation theory, vol II. Springer, Berlin Heidelberg New York
Grasman J, Jansen MJW (1979) Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology. J Math Biol 7:171–197
Grillner S, Wallén P (1985) Central pattern generators for locomotion, with special reference to vertebrates. Annu Rev Neurosci 8:233–261
Gurfinkel VS, Shik ML (1973) The control of posture and locomotion. In: Gydikov AA, Tankov NT, Kosarov DS (eds) Motor control. Plenum, New York, pp 217–234
Hadley P, Beasley MR, Wiesenfeld K (1988) Phase locking of Josephson-junction series of arrays. Phys Rev B 38:8712–8719
Hassard BD, Kazarinoff ND, Wan Y-H (1981) Theory and applications of Hopf bifurcation. (London Mathematical Society Lecture Note Series 41) Cambridge University Press, Cambridge, UK
Hoppensteadt FC (1986) An introduction to the mathematics of neurons. Cmbridge University Press Cambridge, UK
Kling U, Székely G (1968) Simulation of rhythmic nervous activities. I. Function of networks with cyclic inhibitions. Kybernetik 5:89–103
Kouda A, Mori S (1981) Analysis of a ring of mutually coupled van der Pol oscillators with coupling delay. IEEE Trans Circuits Syst 28:247–253
Linkens DA (1974) Analytical solution of large numbers of mutually coupled nearly sinusoidal oscillators. IEEE Trans Circuits Syst 21:294–300
Linkens DA (1976) Stability of entrainment conditions for a particular form of mutually coupled van der Pol oscillators. IEEE Trans Circuits Syst 23:113–121
Linkens DA (1980) Pulse synchronization of intestinal myoelectrical models. IEEE Trans Biomed Eng 27:177–186
Linkens DA, Taylor I, Duthie HL (1976) Mathematical modeling of the colorectal myoelectrical activity in humans. IEEE Trans Biomed Eng 23:101–110
Marsden JE, McCracken M (1976) The Hopf bifurcation and its applications (Applied Mathematics in Science 19) Springer, Berlin Heidelberg New York
Morishita I, Yajima A (1972) Analysis and simulation of networks of mutually inhibiting neurons. Kybernetik 11:154–165
Murray JD (1989) Mathematical biology. (Biomathematics Texts 19) Springer, Berlin Heidelberg New York
Pozin NV, Shulpin YA (1970) Analysis of the work of auto-oscillatory neurone junctions. Biophysics 15:162–171
Selverston AI (1988) A consideration of invertebrate central pattern generators as computational data bases. Neural Networks 1:109–117
Székely G (1965) Logical network for controlling limb movements in urodela. Acta Physiol Acad Sci Hung 27:285–289
Thompson JMT, Stewart HB (1986) Nonlinear dynamics and chaos. Wiley, New York
Tsutsumi K, Matsumoto H (1984a) A synaptic modification algorithm in consideration of the generation of rhythmic oscillation in a ring neural network. Biol Cybern 50:419–430
Tsutsumi K, Matsumoto H (1984b) Ring neural network qua a generator of rhythmic oscillation with period control mechanism. Biol Cybern 51:181–194
Turing AM (1952) The chemical basis of morphogenesis. Phil Trans R Soc Lond [Biol] 237:37–72
Winfree AT (1980) The geometry of biological time. Springer, Berlin Heidelberg New York
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Collins, J.J., Stewart, I. A group-theoretic approach to rings of coupled biological oscillators. Biol. Cybern. 71, 95–103 (1994). https://doi.org/10.1007/BF00197312
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00197312