September  2014, 34(9): 3703-3745. doi: 10.3934/dcds.2014.34.3703

Spatially structured networks of pulse-coupled phase oscillators on metric spaces

1. 

Institute of Applied Mathematics, University of British Columbia, 121-1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

2. 

Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany

Received  November 2012 Revised  January 2014 Published  March 2014

The Winfree model describes finite networks of phase oscillators. Oscillators interact by broadcasting pulses that modulate the frequencies of connected oscillators. We study a generalization of the model and its fluid-dynamical limit for networks, where oscillators are distributed on some abstract $\sigma$-finite Borel measure space over a separable metric space. We give existence and uniqueness statements for solutions to the continuity equation for the oscillator phase densities. We further show that synchrony in networks of identical oscillators is locally asymptotically stable for finite, strictly positive measures and under suitable conditions on the oscillator response function and the coupling kernel of the network. The conditions on the latter are a generalization of the strong connectivity of finite graphs to abstract coupling kernels.
Citation: Stilianos Louca, Fatihcan M. Atay. Spatially structured networks of pulse-coupled phase oscillators on metric spaces. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3703-3745. doi: 10.3934/dcds.2014.34.3703
References:
[1]

J. Acebrón, L. Bonilla, C. Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137. doi: 10.1103/RevModPhys.77.137.

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, Birkhäuser, 2008.

[3]

J. T. Ariaratnam, Collective Dynamics of the Winfree Model of Coupled Nonlinear Oscillators, PhD thesis, Cornell University, 2002.

[4]

J. T. Ariaratnam and S. H. Strogatz, Phase diagram for the Winfree model of coupled nonlinear oscillators, Phys. Rev. Lett., 86 (2001), 4278-4281.

[5]

V. Arnold, Ordinary Differential Equations, Second printing of the 1992 edition. Universitext. Springer-Verlag, Berlin, 2006.

[6]

K. Athreya and S. Lahiri, Measure Theory and Probability Theory, Springer, 2006.

[7]

L. Basnarkov and V. Urumov, Critical exponents of the transition from incoherence to partial oscillation death in the Winfree model, J. Stat. Mech., 2009 (2009), P10014. doi: 10.1088/1742-5468/2009/10/P10014.

[8]

R. Bellman, The stability of solutions of linear differential equations, Duke. Math. J., 10 (1943), 643-647. doi: 10.1215/S0012-7094-43-01059-2.

[9]

R. Beurle, Properties of a mass of cells capable of regenerating pulses, Philos. T. R. Soc. B, 240 (1956), 55-94. doi: 10.1098/rstb.1956.0012.

[10]

V. Bogachev, Measure Theory, Vol. I, II. Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.

[11]

A. P. Boris Makarov, Real Analysis: Measures, Integrals and Applications, Springer, 2013. doi: 10.1007/978-1-4471-5122-7.

[12]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, P. Am. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.

[13]

J. D. Crawford and K. Davies, Synchronization of globally coupled phase oscillators: Singularities and scaling for general couplings, Physica D, 125 (1999), 1-46. doi: 10.1016/S0167-2789(98)00235-8.

[14]

D. Drachman, Do we have brain to spare? Neurology, 64 (2005), 2004-2005. doi: 10.1212/01.WNL.0000166914.38327.BB.

[15]

B. Eckhardt, E. Ott, S. H. Strogatz, D. M. Abrams and A. McRobie, Modeling walker synchronization on the millennium bridge, Phys. Rev. E, 75 (2007), 021110. doi: 10.1103/PhysRevE.75.021110.

[16]

T. Eisner, Stability of Operators and Operator Semigroups, Birkhäuser, 2010.

[17]

F. Giannuzzi, D. Marinazzo, G. Nardulli, M. Pellicoro and S. Stramaglia, Phase diagram of a generalized Winfree model, Phys. Rev. E, 75 (2007), 051104. doi: 10.1103/PhysRevE.75.051104.

[18]

P. Goel and B. Ermentrout, Synchrony, stability, and firing patterns in pulse-coupled oscillators, Physica D, 163 (2002), 191-216. doi: 10.1016/S0167-2789(01)00374-8.

[19]

J. Griffith, A field theory of neural nets: I: Derivation of field equations, B. Math. Biol., 25 (1963), 111-120. doi: 10.1007/BF02477774.

[20]

J. J. Grobler, A note on the theorems of Jentzsch-Perron and Frobenius, in Indagationes Mathematicae (Proceedings), 90 (1987), 381-391.

[21]

P. Hertel, Continuum Physics, Springer, 2012. doi: 10.1007/978-3-642-29500-3.

[22]

R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1990.

[23]

G. Katriel, Stability of synchronized oscillations in networks of phase-oscillators, Discret. Contin. Dyn. B, 5 (2005), 353-364. doi: 10.3934/dcdsb.2005.5.353.

[24]

K. Kowalski, Methods of Hilbert Spaces in the Theory of Nonlinear Dynamical Systems, World Scientific, 1994. doi: 10.1142/9789814354127.

[25]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Springer, 39 (1975), 420-422.

[26]

Y. Kuramoto and H. Arakai, Chemical Oscillations, Waves and Turbulence, Springer, 1984. doi: 10.1007/978-3-642-69689-3.

[27]

C. Laing, The dynamics of chimera states in heterogeneous Kuramoto networks, Physica D, 238 (2009), 1569-1588. doi: 10.1016/j.physd.2009.04.012.

[28]

S. Lang, Real and Functional Analysis, Third edition. Graduate Texts in Mathematics, 142. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0897-6.

[29]

W. S. Lee, J. G. Restrepo, E. Ott and T. M. Antonsen, Dynamics and pattern formation in large systems of spatially-coupled oscillators with finite response times, Chaos, 21 (2011), 023122. doi: 10.1063/1.3596697.

[30]

K. Mardia and P. Jupp, Directional Statistics, Wiley, 2000.

[31]

L. Nicolaescu, Lectures on the Geometry of Manifolds, World Scientific, 2007. doi: 10.1142/9789814261012.

[32]

D. D. Quinn, R. H. Rand and S. H. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Phys. Rev. E, 75 (2007), 036218. doi: 10.1103/PhysRevE.75.036218.

[33]

H. Schaefer, Banach Lattices and Positive Operators, Springer, 1974.

[34]

Y. Shiogai and Y. Kuramoto, Wave propagation in nonlocally coupled oscillators with noise, Prog. Theor. Phys. Supp., 150 (2003), 435-438. doi: 10.1143/PTPS.150.435.

[35]

R. M. Smeal, G. B. Ermentrout and J. A. White, Phase-response curves and synchronized neural networks, Philos. T. Roy. Soc. B, 365 (2010), 2407-2422. doi: 10.1098/rstb.2009.0292.

[36]

S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion Press, 2003.

[37]

S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt and E. Ott, Crowd synchrony on the Millennium Bridge, Nature, 438 (2005), 43-44. doi: 10.1038/43843a.

[38]

B. R. Trees, V. Saranathan and D. Stroud, Synchronization in disordered Josephson junction arrays: Small-world connections and the Kuramoto model, Phys. Rev. E, 71 (2005), 016215. doi: 10.1103/PhysRevE.71.016215.

[39]

K. Wiesenfeld, P. Colet and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E, 57 (1998), 1563-1569. doi: 10.1103/PhysRevE.57.1563.

[40]

A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. doi: 10.1016/0022-5193(67)90051-3.

[41]

A. Winfree, The Geometry of Biological Time, Second edition. Interdisciplinary Applied Mathematics, 12. Springer-Verlag, New York, 2001.

[42]

A. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-60637-3.

show all references

References:
[1]

J. Acebrón, L. Bonilla, C. Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137. doi: 10.1103/RevModPhys.77.137.

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, Birkhäuser, 2008.

[3]

J. T. Ariaratnam, Collective Dynamics of the Winfree Model of Coupled Nonlinear Oscillators, PhD thesis, Cornell University, 2002.

[4]

J. T. Ariaratnam and S. H. Strogatz, Phase diagram for the Winfree model of coupled nonlinear oscillators, Phys. Rev. Lett., 86 (2001), 4278-4281.

[5]

V. Arnold, Ordinary Differential Equations, Second printing of the 1992 edition. Universitext. Springer-Verlag, Berlin, 2006.

[6]

K. Athreya and S. Lahiri, Measure Theory and Probability Theory, Springer, 2006.

[7]

L. Basnarkov and V. Urumov, Critical exponents of the transition from incoherence to partial oscillation death in the Winfree model, J. Stat. Mech., 2009 (2009), P10014. doi: 10.1088/1742-5468/2009/10/P10014.

[8]

R. Bellman, The stability of solutions of linear differential equations, Duke. Math. J., 10 (1943), 643-647. doi: 10.1215/S0012-7094-43-01059-2.

[9]

R. Beurle, Properties of a mass of cells capable of regenerating pulses, Philos. T. R. Soc. B, 240 (1956), 55-94. doi: 10.1098/rstb.1956.0012.

[10]

V. Bogachev, Measure Theory, Vol. I, II. Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.

[11]

A. P. Boris Makarov, Real Analysis: Measures, Integrals and Applications, Springer, 2013. doi: 10.1007/978-1-4471-5122-7.

[12]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, P. Am. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.

[13]

J. D. Crawford and K. Davies, Synchronization of globally coupled phase oscillators: Singularities and scaling for general couplings, Physica D, 125 (1999), 1-46. doi: 10.1016/S0167-2789(98)00235-8.

[14]

D. Drachman, Do we have brain to spare? Neurology, 64 (2005), 2004-2005. doi: 10.1212/01.WNL.0000166914.38327.BB.

[15]

B. Eckhardt, E. Ott, S. H. Strogatz, D. M. Abrams and A. McRobie, Modeling walker synchronization on the millennium bridge, Phys. Rev. E, 75 (2007), 021110. doi: 10.1103/PhysRevE.75.021110.

[16]

T. Eisner, Stability of Operators and Operator Semigroups, Birkhäuser, 2010.

[17]

F. Giannuzzi, D. Marinazzo, G. Nardulli, M. Pellicoro and S. Stramaglia, Phase diagram of a generalized Winfree model, Phys. Rev. E, 75 (2007), 051104. doi: 10.1103/PhysRevE.75.051104.

[18]

P. Goel and B. Ermentrout, Synchrony, stability, and firing patterns in pulse-coupled oscillators, Physica D, 163 (2002), 191-216. doi: 10.1016/S0167-2789(01)00374-8.

[19]

J. Griffith, A field theory of neural nets: I: Derivation of field equations, B. Math. Biol., 25 (1963), 111-120. doi: 10.1007/BF02477774.

[20]

J. J. Grobler, A note on the theorems of Jentzsch-Perron and Frobenius, in Indagationes Mathematicae (Proceedings), 90 (1987), 381-391.

[21]

P. Hertel, Continuum Physics, Springer, 2012. doi: 10.1007/978-3-642-29500-3.

[22]

R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1990.

[23]

G. Katriel, Stability of synchronized oscillations in networks of phase-oscillators, Discret. Contin. Dyn. B, 5 (2005), 353-364. doi: 10.3934/dcdsb.2005.5.353.

[24]

K. Kowalski, Methods of Hilbert Spaces in the Theory of Nonlinear Dynamical Systems, World Scientific, 1994. doi: 10.1142/9789814354127.

[25]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Springer, 39 (1975), 420-422.

[26]

Y. Kuramoto and H. Arakai, Chemical Oscillations, Waves and Turbulence, Springer, 1984. doi: 10.1007/978-3-642-69689-3.

[27]

C. Laing, The dynamics of chimera states in heterogeneous Kuramoto networks, Physica D, 238 (2009), 1569-1588. doi: 10.1016/j.physd.2009.04.012.

[28]

S. Lang, Real and Functional Analysis, Third edition. Graduate Texts in Mathematics, 142. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0897-6.

[29]

W. S. Lee, J. G. Restrepo, E. Ott and T. M. Antonsen, Dynamics and pattern formation in large systems of spatially-coupled oscillators with finite response times, Chaos, 21 (2011), 023122. doi: 10.1063/1.3596697.

[30]

K. Mardia and P. Jupp, Directional Statistics, Wiley, 2000.

[31]

L. Nicolaescu, Lectures on the Geometry of Manifolds, World Scientific, 2007. doi: 10.1142/9789814261012.

[32]

D. D. Quinn, R. H. Rand and S. H. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Phys. Rev. E, 75 (2007), 036218. doi: 10.1103/PhysRevE.75.036218.

[33]

H. Schaefer, Banach Lattices and Positive Operators, Springer, 1974.

[34]

Y. Shiogai and Y. Kuramoto, Wave propagation in nonlocally coupled oscillators with noise, Prog. Theor. Phys. Supp., 150 (2003), 435-438. doi: 10.1143/PTPS.150.435.

[35]

R. M. Smeal, G. B. Ermentrout and J. A. White, Phase-response curves and synchronized neural networks, Philos. T. Roy. Soc. B, 365 (2010), 2407-2422. doi: 10.1098/rstb.2009.0292.

[36]

S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion Press, 2003.

[37]

S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt and E. Ott, Crowd synchrony on the Millennium Bridge, Nature, 438 (2005), 43-44. doi: 10.1038/43843a.

[38]

B. R. Trees, V. Saranathan and D. Stroud, Synchronization in disordered Josephson junction arrays: Small-world connections and the Kuramoto model, Phys. Rev. E, 71 (2005), 016215. doi: 10.1103/PhysRevE.71.016215.

[39]

K. Wiesenfeld, P. Colet and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E, 57 (1998), 1563-1569. doi: 10.1103/PhysRevE.57.1563.

[40]

A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. doi: 10.1016/0022-5193(67)90051-3.

[41]

A. Winfree, The Geometry of Biological Time, Second edition. Interdisciplinary Applied Mathematics, 12. Springer-Verlag, New York, 2001.

[42]

A. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-60637-3.

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