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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 & Continuous Dynamical Systems - A, 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. doi: 10.1209/0295-5075/23/5/011. Google Scholar

[2]

N. Kopell and G. B. Ermentrout, Coupled oscillators and the design of central pattern generators,, Math. Biosci., 90 (1988), 87. doi: 10.1016/0025-5564(88)90059-4. Google Scholar

[3]

H. G. Winful and S. S. Wang, Dynamics of phase-locked semiconductor laser arrays,, Appl. Phys. Lett., 52 (1988), 1774. doi: 10.1063/1.99622. Google Scholar

[4]

H. G. Winful and S. S. Wang, Stability of phase locking in coupled semiconductor laser arrays,, Applied Physics Letters, 53 (1988), 1894. doi: 10.1063/1.100363. Google Scholar

[5]

R. E. Mirollo and S. H. Strogatz, Synchronization of pulse-coupled biological oscillators,, SIAM J. Appl. Math., 50 (1990), 1645. doi: 10.1137/0150098. Google Scholar

[6]

C. S. Peskin, "Mathematical Aspects of Heart Physiology,'', Notes based on a course given at New York University during the year 1973/74, (1973). Google Scholar

[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. Google Scholar

[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. doi: 10.1023/A:1008843412952. Google Scholar

[9]

N. Kopell and G. B. Ermentrout, Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators,, in, (2002), 3. doi: 10.1016/S1874-575X(02)80022-4. Google Scholar

[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. doi: 10.1016/S0167-2789(98)00264-4. Google Scholar

[11]

P. Kitanov, "Normal Form Analysis for Bifurcations with Huygens Symmetry,'', Ph.D thesis, (2011). Google Scholar

[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). doi: 10.1063/1.2905146. Google Scholar

[13]

A. Prasad, J. Kurths, S. Kumar Dana and R. Ramaswamy, Phase-flip bifurcation induced by time delay,, Phys. Rev. E (3), 74 (2006). doi: 10.1103/PhysRevE.74.035204. Google Scholar

[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). doi: 10.1103/PhysRevE.81.046213. Google Scholar

[15]

R. G. Carson, W. D. Byblow and D. Goodman, The dynamical substructure of bimanual coordination,, in, (1994), 319. Google Scholar

[16]

K. J. Jantzen and J. A. S. Kelso, Neural coordination dynamics of human sensorimotor behaviour: A review,, in, (2007), 421. Google Scholar

[17]

J. A. S. Kelso, Phase transitions and critical behaviour in human bimanual coordination,, Am. J. Physiol. Reg. I, 15 (1984). Google Scholar

[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. Google Scholar

[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. doi: 10.1007/BF00336922. Google Scholar

[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. Google Scholar

[21]

S. Wirkus and R. Rand, The dynamics of two coupled van der Pol oscillators with delay coupling,, Nonlinear Dynam., 30 (2002), 205. doi: 10.1023/A:1020536525009. Google Scholar

[22]

N. Burić and D. Todorović, Dynamics of Fitzhugh-Nagumo excitable systems with delayed coupling,, Phys. Rev. E (3), 67 (2003). Google Scholar

[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. doi: 10.1142/S0218127405012831. Google Scholar

[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. doi: 10.1137/S0036139903434833. Google Scholar

[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. doi: 10.1142/S0218127409023111. Google Scholar

[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. Google Scholar

[27]

Y. Kuramoto, Cooperative dynamics of oscillator community. A study based on lattice of rings,, Prog. Theor. Phys. Suppl., (1984), 223. Google Scholar

[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. doi: 10.1007/BF01009349. Google Scholar

[29]

K. Okuda, Variety and generality of clustering in globally coupled oscillators,, Physica D, 63 (1993), 424. doi: 10.1016/0167-2789(93)90121-G. Google Scholar

[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. doi: 10.1023/A:1021902717424. Google Scholar

[31]

G. B. Ermentrout, Type I membranes, phase resetting curves, and synchrony,, Neural Comput., 8 (1996), 979. doi: 10.1162/neco.1996.8.5.979. Google Scholar

[32]

D. Hansel, G. Mato and C. Meunier, Synchrony in excitatory neural networks,, Neural Comput., 7 (1995), 307. doi: 10.1162/neco.1995.7.2.307. Google Scholar

[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. doi: 10.1523/JNEUROSCI.2715-06.2007. Google Scholar

[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. doi: 10.1016/j.brainres.2008.12.068. Google Scholar

[35]

S. Kim, S. H. Park and C. S. Ryu, Multistability in coupled oscillator systems with time delay,, Phys. Rev. Lett., 79 (1997), 2911. doi: 10.1103/PhysRevLett.79.2911. Google Scholar

[36]

T. Luzyanina, Synchronization in an oscillator neural network model with time-delayed coupling,, Network: Comput. Neural Sys., 6 (1995), 43. doi: 10.1088/0954-898X/6/1/003. Google Scholar

[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. doi: 10.1103/PhysRevLett.67.2753. Google Scholar

[38]

H. G. Schuster and P. Wagner, Mutual entrainment of two limit cycle oscillators with time delayed coupling,, Prog. Theor. Phys., 81 (1989), 939. doi: 10.1143/PTP.81.939. Google Scholar

[39]

M. K. S. Yeung and S. H. Strogatz, Time delay in the Kuramoto model of coupled oscillators,, Phys. Rev. Lett., 82 (1999), 648. doi: 10.1103/PhysRevLett.82.648. Google Scholar

[40]

G. B. Ermentrout, An introduction to neural oscillators,, in, (1994), 79. Google Scholar

[41]

E. M. Izhikevich, Phase models with explicit time delays,, Phys. Rev. E, 58 (1998), 905. doi: 10.1103/PhysRevE.58.905. Google Scholar

[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. doi: 10.1016/S0167-2789(99)00013-5. Google Scholar

[43]

G. B. Ermentrout and N. Kopell, Frequency plateaus in a chain of weakly coupled oscillators, I,, SIAM J. Math. Anal., 15 (1984), 215. doi: 10.1137/0515019. Google Scholar

[44]

G. B. Ermentrout and N. Kopell, Multiple pulse interactions and averaging in coupled neural oscillators,, J. Math. Biol., 29 (1991), 195. doi: 10.1007/BF00160535. Google Scholar

[45]

G. B. Ermentrout and D. H. Terman, "Mathematical Foundations of Neuroscience,'', Interdisciplinary Applied Mathematics, 35 (2010). Google Scholar

[46]

F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'', Applied Mathematical Sciences, 126 (1997). Google Scholar

[47]

C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fibre,, Biophys. J., 35 (1981), 193. doi: 10.1016/S0006-3495(81)84782-0. Google Scholar

[48]

J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and neural oscillations,, in, (1989). Google Scholar

[49]

K. Tsumoto, H. Kitajima, T. Yoshinaga, K. Aihara and H. Kawakami, Bifurcations in the Morris-Lecar neuron model,, Neurocomputing, 69 (2006), 293. doi: 10.1016/j.neucom.2005.03.006. Google Scholar

[50]

N. Burić, I. Grozdanović and N. Vasović, Type I vs. type II excitable systems with delayed coupling,, Chaos Solitons Fract., 23 (2005), 1221. Google Scholar

[51]

G. B. Ermentrout, "Simulating, Analyzing and Animating Dynamical Systems: A Guide to XPPAUT for Researcher and Students,'', Software, 14 (2002). Google Scholar

[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, (2001). Google Scholar

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. doi: 10.1209/0295-5075/23/5/011. Google Scholar

[2]

N. Kopell and G. B. Ermentrout, Coupled oscillators and the design of central pattern generators,, Math. Biosci., 90 (1988), 87. doi: 10.1016/0025-5564(88)90059-4. Google Scholar

[3]

H. G. Winful and S. S. Wang, Dynamics of phase-locked semiconductor laser arrays,, Appl. Phys. Lett., 52 (1988), 1774. doi: 10.1063/1.99622. Google Scholar

[4]

H. G. Winful and S. S. Wang, Stability of phase locking in coupled semiconductor laser arrays,, Applied Physics Letters, 53 (1988), 1894. doi: 10.1063/1.100363. Google Scholar

[5]

R. E. Mirollo and S. H. Strogatz, Synchronization of pulse-coupled biological oscillators,, SIAM J. Appl. Math., 50 (1990), 1645. doi: 10.1137/0150098. Google Scholar

[6]

C. S. Peskin, "Mathematical Aspects of Heart Physiology,'', Notes based on a course given at New York University during the year 1973/74, (1973). Google Scholar

[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. Google Scholar

[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. doi: 10.1023/A:1008843412952. Google Scholar

[9]

N. Kopell and G. B. Ermentrout, Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators,, in, (2002), 3. doi: 10.1016/S1874-575X(02)80022-4. Google Scholar

[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. doi: 10.1016/S0167-2789(98)00264-4. Google Scholar

[11]

P. Kitanov, "Normal Form Analysis for Bifurcations with Huygens Symmetry,'', Ph.D thesis, (2011). Google Scholar

[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). doi: 10.1063/1.2905146. Google Scholar

[13]

A. Prasad, J. Kurths, S. Kumar Dana and R. Ramaswamy, Phase-flip bifurcation induced by time delay,, Phys. Rev. E (3), 74 (2006). doi: 10.1103/PhysRevE.74.035204. Google Scholar

[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). doi: 10.1103/PhysRevE.81.046213. Google Scholar

[15]

R. G. Carson, W. D. Byblow and D. Goodman, The dynamical substructure of bimanual coordination,, in, (1994), 319. Google Scholar

[16]

K. J. Jantzen and J. A. S. Kelso, Neural coordination dynamics of human sensorimotor behaviour: A review,, in, (2007), 421. Google Scholar

[17]

J. A. S. Kelso, Phase transitions and critical behaviour in human bimanual coordination,, Am. J. Physiol. Reg. I, 15 (1984). Google Scholar

[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. Google Scholar

[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. doi: 10.1007/BF00336922. Google Scholar

[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. Google Scholar

[21]

S. Wirkus and R. Rand, The dynamics of two coupled van der Pol oscillators with delay coupling,, Nonlinear Dynam., 30 (2002), 205. doi: 10.1023/A:1020536525009. Google Scholar

[22]

N. Burić and D. Todorović, Dynamics of Fitzhugh-Nagumo excitable systems with delayed coupling,, Phys. Rev. E (3), 67 (2003). Google Scholar

[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. doi: 10.1142/S0218127405012831. Google Scholar

[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. doi: 10.1137/S0036139903434833. Google Scholar

[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. doi: 10.1142/S0218127409023111. Google Scholar

[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. Google Scholar

[27]

Y. Kuramoto, Cooperative dynamics of oscillator community. A study based on lattice of rings,, Prog. Theor. Phys. Suppl., (1984), 223. Google Scholar

[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. doi: 10.1007/BF01009349. Google Scholar

[29]

K. Okuda, Variety and generality of clustering in globally coupled oscillators,, Physica D, 63 (1993), 424. doi: 10.1016/0167-2789(93)90121-G. Google Scholar

[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. doi: 10.1023/A:1021902717424. Google Scholar

[31]

G. B. Ermentrout, Type I membranes, phase resetting curves, and synchrony,, Neural Comput., 8 (1996), 979. doi: 10.1162/neco.1996.8.5.979. Google Scholar

[32]

D. Hansel, G. Mato and C. Meunier, Synchrony in excitatory neural networks,, Neural Comput., 7 (1995), 307. doi: 10.1162/neco.1995.7.2.307. Google Scholar

[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. doi: 10.1523/JNEUROSCI.2715-06.2007. Google Scholar

[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. doi: 10.1016/j.brainres.2008.12.068. Google Scholar

[35]

S. Kim, S. H. Park and C. S. Ryu, Multistability in coupled oscillator systems with time delay,, Phys. Rev. Lett., 79 (1997), 2911. doi: 10.1103/PhysRevLett.79.2911. Google Scholar

[36]

T. Luzyanina, Synchronization in an oscillator neural network model with time-delayed coupling,, Network: Comput. Neural Sys., 6 (1995), 43. doi: 10.1088/0954-898X/6/1/003. Google Scholar

[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. doi: 10.1103/PhysRevLett.67.2753. Google Scholar

[38]

H. G. Schuster and P. Wagner, Mutual entrainment of two limit cycle oscillators with time delayed coupling,, Prog. Theor. Phys., 81 (1989), 939. doi: 10.1143/PTP.81.939. Google Scholar

[39]

M. K. S. Yeung and S. H. Strogatz, Time delay in the Kuramoto model of coupled oscillators,, Phys. Rev. Lett., 82 (1999), 648. doi: 10.1103/PhysRevLett.82.648. Google Scholar

[40]

G. B. Ermentrout, An introduction to neural oscillators,, in, (1994), 79. Google Scholar

[41]

E. M. Izhikevich, Phase models with explicit time delays,, Phys. Rev. E, 58 (1998), 905. doi: 10.1103/PhysRevE.58.905. Google Scholar

[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. doi: 10.1016/S0167-2789(99)00013-5. Google Scholar

[43]

G. B. Ermentrout and N. Kopell, Frequency plateaus in a chain of weakly coupled oscillators, I,, SIAM J. Math. Anal., 15 (1984), 215. doi: 10.1137/0515019. Google Scholar

[44]

G. B. Ermentrout and N. Kopell, Multiple pulse interactions and averaging in coupled neural oscillators,, J. Math. Biol., 29 (1991), 195. doi: 10.1007/BF00160535. Google Scholar

[45]

G. B. Ermentrout and D. H. Terman, "Mathematical Foundations of Neuroscience,'', Interdisciplinary Applied Mathematics, 35 (2010). Google Scholar

[46]

F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'', Applied Mathematical Sciences, 126 (1997). Google Scholar

[47]

C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fibre,, Biophys. J., 35 (1981), 193. doi: 10.1016/S0006-3495(81)84782-0. Google Scholar

[48]

J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and neural oscillations,, in, (1989). Google Scholar

[49]

K. Tsumoto, H. Kitajima, T. Yoshinaga, K. Aihara and H. Kawakami, Bifurcations in the Morris-Lecar neuron model,, Neurocomputing, 69 (2006), 293. doi: 10.1016/j.neucom.2005.03.006. Google Scholar

[50]

N. Burić, I. Grozdanović and N. Vasović, Type I vs. type II excitable systems with delayed coupling,, Chaos Solitons Fract., 23 (2005), 1221. Google Scholar

[51]

G. B. Ermentrout, "Simulating, Analyzing and Animating Dynamical Systems: A Guide to XPPAUT for Researcher and Students,'', Software, 14 (2002). Google Scholar

[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, (2001). Google Scholar

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Richard H. Rand, Asok K. Sen. A numerical investigation of the dynamics of a system of two time-delay coupled relaxation oscillators. Communications on Pure & Applied Analysis, 2003, 2 (4) : 567-577. doi: 10.3934/cpaa.2003.2.567

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