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Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations
August  2012, 32(8): 2653-2673. doi: 10.3934/dcds.2012.32.2653

Phase models and oscillators with time delayed coupling

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

Received  June 2011 Revised  September 2011 Published  March 2012

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.
Citation: Sue Ann Campbell, Ilya Kobelevskiy. Phase models and oscillators with time delayed coupling. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2653-2673. doi: 10.3934/dcds.2012.32.2653
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.

show all references

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.
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