Phase models and oscillators with time delayed coupling

Sue Ann Campbell; Ilya Kobelevskiy

doi:10.3934/dcds.2012.32.2653

Article Contents

2012, Volume 32, Issue 8: 2653-2673. Doi: 10.3934/dcds.2012.32.2653

This issue Previous Article Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations Next Article Numerical recipes for investigating endemic equilibria of age-structured SIR epidemics

Phase models and oscillators with time delayed coupling

Sue Ann Campbell^1, and
Ilya Kobelevskiy^2,

1.
Department of Applied Mathematics, University of Waterloo, Waterloo ON N2L 3G1
2.
Imaging Research, Sunnybrook Health Sciences Centre, Toronto ON M4N 3M5

Received: June 2011

Revised: September 2011

Published: August 2012

Abstract / Introduction Related Papers Cited by

Abstract

We consider two identical oscillators with time delayed coupling, modelled by a system of delay differential equations. We reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. By analyzing the phase model, we show how the time delay affects the stability of phase-locked periodic solutions and causes stability switching of in-phase and anti-phase solutions as the delay is increased. In particular, we show how the phase model can predict when the phase-flip bifurcation will occur in the original delay differential equation model. The results of the phase model analysis are applied to pairs of Morris-Lecar oscillators with diffusive or synaptic coupling and compared with numerical studies of the full system of delay differential equations.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Related Papers

Cited by

References

[1]	D. Hansel, G. Mato and C. Meunier, Phase dynamics for weakly coupled Hodgkin-Huxley neurons, Europhys. Lett., 23 (1993), 367-372.doi: 10.1209/0295-5075/23/5/011.
[2]	N. Kopell and G. B. Ermentrout, Coupled oscillators and the design of central pattern generators, Math. Biosci., 90 (1988), 87-109.doi: 10.1016/0025-5564(88)90059-4.
[3]	H. G. Winful and S. S. Wang, Dynamics of phase-locked semiconductor laser arrays, Appl. Phys. Lett., 52 (1988), 1774-1776.doi: 10.1063/1.99622.
[4]	H. G. Winful and S. S. Wang, Stability of phase locking in coupled semiconductor laser arrays, Applied Physics Letters, 53 (1988), 1894-1896.doi: 10.1063/1.100363.
[5]	R. E. Mirollo and S. H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), 1645-1662.doi: 10.1137/0150098.
[6]	C. S. Peskin, "Mathematical Aspects of Heart Physiology,'' Notes based on a course given at New York University during the year 1973/74, Courant Institute of Mathematical Sciences, New York University, New York, 1975.
[7]	A. Takamatsu, T. Fujii and I. Endo, Time delay effect in a living coupled oscillator system with plasmodium of physarum polycephalum, Phys. Rev. E, 85 (2000), 2026-2029.
[8]	S. M. Crook, G. B. Ermentrout, M. C. Vanier and J. M. Bower, The role of axonal delay in synchronization of networks of coupled cortical oscillators, J. Comp. Neurosci., 4 (1997), 161-172.doi: 10.1023/A:1008843412952.
[9]	N. Kopell and G. B. Ermentrout, Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators, in "Handbook of Dynamical Systems, Vol. 2'' (ed. B Fiedler), North Holland, Amsterdam, (2002), 3-54.doi: 10.1016/S1874-575X(02)80022-4.
[10]	P. Bressloff and S. Coombes, Symmetry and phase-locking in a ring of pulse-coupled oscillators with distributed delays, Physica D, 126 (1999), 99-122.doi: 10.1016/S0167-2789(98)00264-4.
[11]	P. Kitanov, "Normal Form Analysis for Bifurcations with Huygens Symmetry,'' Ph.D thesis, University of Guelph, Canada, 2011.
[12]	A. Prasad, S. Kumar Dana, R. Karnatak, J. Kurths, B. Blasius and R. Ramaswamy, Universal occurrence of the phase-flip bifurcation in time-delay coupled systems, CHAOS, 18 (2008), 023111.doi: 10.1063/1.2905146.
[13]	A. Prasad, J. Kurths, S. Kumar Dana and R. Ramaswamy, Phase-flip bifurcation induced by time delay, Phys. Rev. E (3), 74 (2006), 035204.doi: 10.1103/PhysRevE.74.035204.
[14]	J. M. Cruz, J. Escalona, P. Parmananda, R. Karnatak, A. Prasad and R. Ramaswamy, Phase-flip transition in coupled electrochemical cells, Phys. Rev. E, 81 (2010), 046213.doi: 10.1103/PhysRevE.81.046213.
[15]	R. G. Carson, W. D. Byblow and D. Goodman, The dynamical substructure of bimanual coordination, in "Interlimb Coordination: Neural, Dynamical and Cognitive Constraints'' (eds. S Swinnen, H Heuer, J Massion and P Casaer), Academic Press, (1994), 319-337.
[16]	K. J. Jantzen and J. A. S. Kelso, Neural coordination dynamics of human sensorimotor behaviour: A review, in "Handbook of Brain Connectivity'' (eds. R. McIntosh and V. K. Jirsa), Underst. Complex Syst., Springer, Berlin, (2007), 421-461.
[17]	J. A. S. Kelso, Phase transitions and critical behaviour in human bimanual coordination, Am. J. Physiol. Reg. I, 15 (1984), R1000-R1004.
[18]	J. A. S. Kelso, K. G. Holt, P. Rubin and P. N. Kugler, Patterns of human interlimb coordination emerge from nonlinear limit cycle oscillatory processes: Theory and data, J. Motor Behav., 13 (1981), 226-261.
[19]	H. Haken, J. A. S. Kelso and H. Bunz, A theoretical model of phase transitions in human hand movements, Biol. Cybern., 51 (1985), 347-356.doi: 10.1007/BF00336922.
[20]	A. K. Sen and R. Rand, A numerical investigation of the dynamics of a system of two time-delayed coupled relaxation oscillators, Comm. Pur. Appl. Math., 2 (2003), 567-577.
[21]	S. Wirkus and R. Rand, The dynamics of two coupled van der Pol oscillators with delay coupling, Nonlinear Dynam., 30 (2002), 205-221.doi: 10.1023/A:1020536525009.
[22]	N. Burić and D. Todorović, Dynamics of Fitzhugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E (3), 67 (2003), 066222.
[23]	N. Burić and D. Todorović, Bifurcations due to small time-lag in coupled excitable systems, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 15 (2005), 1775-1785.doi: 10.1142/S0218127405012831.
[24]	S. A. Campbell, R. Edwards and P. van den Dreissche, Delayed coupling between two neural network loops, SIAM J. Appl. Math., 65 (2004), 316-335.doi: 10.1137/S0036139903434833.
[25]	M. A. Dahlem, G. Hiller, A. Panchuk and E. Schöll, Dynamics of delay-coupled excitable neural systems, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 19 (2009), 745-753.doi: 10.1142/S0218127409023111.
[26]	E. Schöll, G. Hiller, P. Hövel and M. A. Dahlem, Time-delay feedback in neurosystems, Philos. T. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 1079-1096.
[27]	Y. Kuramoto, Cooperative dynamics of oscillator community. A study based on lattice of rings, Prog. Theor. Phys. Suppl., (1984), 223-240.
[28]	Y. Kuramoto and I Nishikawa, Statistical macrodynamics of large dynamical systems. Case of phase transition in oscillator communities, J. Stat. Phys., 49 (1987), 569-605.doi: 10.1007/BF01009349.
[29]	K. Okuda, Variety and generality of clustering in globally coupled oscillators, Physica D, 63 (1993), 424-436.doi: 10.1016/0167-2789(93)90121-G.
[30]	Y.-X. Li, Y.-Q. Wang and R. Miura, Clustering in small networks of excitatory neurons with heterogeneous coupling strengths, J. Comp. Neurosci., 14 (2003), 139-159.doi: 10.1023/A:1021902717424.
[31]	G. B. Ermentrout, Type I membranes, phase resetting curves, and synchrony, Neural Comput., 8 (1996), 979-1001.doi: 10.1162/neco.1996.8.5.979.
[32]	D. Hansel, G. Mato and C. Meunier, Synchrony in excitatory neural networks, Neural Comput., 7 (1995), 307-337.doi: 10.1162/neco.1995.7.2.307.
[33]	J. G. Mancilla, T. J. Lewis, D. J. Pinto, J. Rinzel and B. W. Connors, Synchronization of electrically coupled pairs of inhibitory interneurons in neocortex, J. Neurosci., 27 (2007), 2058-2073.doi: 10.1523/JNEUROSCI.2715-06.2007.
[34]	T. Zahid and F. K. Skinner, Predicting synchronous and asynchronous network groupings of hippocampal interneurons coupled with dendritic gap junctions, Brain Res., 1262 (2009), 115-129.doi: 10.1016/j.brainres.2008.12.068.
[35]	S. Kim, S. H. Park and C. S. Ryu, Multistability in coupled oscillator systems with time delay, Phys. Rev. Lett., 79 (1997), 2911-2914.doi: 10.1103/PhysRevLett.79.2911.
[36]	T. Luzyanina, Synchronization in an oscillator neural network model with time-delayed coupling, Network: Comput. Neural Sys., 6 (1995), 43-59.doi: 10.1088/0954-898X/6/1/003.
[37]	E. Niebur, H. G. Schuster and D. M. Kammen, Collective frequencies and metastability in networks of limit-cycle oscillators with time delay, Phys. Rev. Lett., 67 (1991), 2753-2756.doi: 10.1103/PhysRevLett.67.2753.
[38]	H. G. Schuster and P. Wagner, Mutual entrainment of two limit cycle oscillators with time delayed coupling, Prog. Theor. Phys., 81 (1989), 939-945.doi: 10.1143/PTP.81.939.
[39]	M. K. S. Yeung and S. H. Strogatz, Time delay in the Kuramoto model of coupled oscillators, Phys. Rev. Lett., 82 (1999), 648-651.doi: 10.1103/PhysRevLett.82.648.
[40]	G. B. Ermentrout, An introduction to neural oscillators, in "Neural Modelling and Neural Networks'' (eds. F. Ventriglia), Pergamon, 1994, 79-110.
[41]	E. M. Izhikevich, Phase models with explicit time delays, Phys. Rev. E, 58 (1998), 905-908.doi: 10.1103/PhysRevE.58.905.
[42]	P. Bressloff and S. Coombes, Travelling waves in chains of pulse-coupled integrate-and-fire oscillators with distributed delays, Physica D, 130 (1999), 232-254.doi: 10.1016/S0167-2789(99)00013-5.
[43]	G. B. Ermentrout and N. Kopell, Frequency plateaus in a chain of weakly coupled oscillators, I, SIAM J. Math. Anal., 15 (1984), 215-237.doi: 10.1137/0515019.
[44]	G. B. Ermentrout and N. Kopell, Multiple pulse interactions and averaging in coupled neural oscillators, J. Math. Biol., 29 (1991), 195-217.doi: 10.1007/BF00160535.
[45]	G. B. Ermentrout and D. H. Terman, "Mathematical Foundations of Neuroscience,'' Interdisciplinary Applied Mathematics, 35, Springer, New York, 2010.
[46]	F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'' Applied Mathematical Sciences, 126, Springer-Verlag, New York, 1997.
[47]	C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fibre, Biophys. J., 35 (1981), 193-213.doi: 10.1016/S0006-3495(81)84782-0.
[48]	J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and neural oscillations, in "Methods in Neuronal Modeling: From Synapses to Networks'' (eds. C. Koch and I. Segev), MIT Press, 1989.
[49]	K. Tsumoto, H. Kitajima, T. Yoshinaga, K. Aihara and H. Kawakami, Bifurcations in the Morris-Lecar neuron model, Neurocomputing, 69 (2006), 293-316.doi: 10.1016/j.neucom.2005.03.006.
[50]	N. Burić, I. Grozdanović and N. Vasović, Type I vs. type II excitable systems with delayed coupling, Chaos Solitons Fract., 23 (2005), 1221-1233.
[51]	G. B. Ermentrout, "Simulating, Analyzing and Animating Dynamical Systems: A Guide to XPPAUT for Researcher and Students,'' Software, Environments, and Tools, 14, SIAM, Philadelphia, PA, 2002.
[52]	K. Engelborghs, T. Luzyanina and G. Samaey, "DDE-BIFTOOL V. 2.00: A MATLAB Package for Bifurcation Analysis of Delay Differential Equations,'' Technical Report TW-330, Department of Computer Science, K.U. Leuven, Leuven, Belgium, 2001.