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 & Continuous Dynamical Systems - A, 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). doi: 10.1103/RevModPhys.77.137. Google Scholar

[2]

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

[3]

J. T. Ariaratnam, Collective Dynamics of the Winfree Model of Coupled Nonlinear Oscillators,, PhD thesis, (2002). Google Scholar

[4]

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

[5]

V. Arnold, Ordinary Differential Equations,, Second printing of the 1992 edition. Universitext. Springer-Verlag, (1992). Google Scholar

[6]

K. Athreya and S. Lahiri, Measure Theory and Probability Theory,, Springer, (2006). Google Scholar

[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). doi: 10.1088/1742-5468/2009/10/P10014. Google Scholar

[8]

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

[9]

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

[10]

V. Bogachev, Measure Theory,, Vol. I, (2007). doi: 10.1007/978-3-540-34514-5. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[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). doi: 10.1103/PhysRevE.75.021110. Google Scholar

[16]

T. Eisner, Stability of Operators and Operator Semigroups,, Birkhäuser, (2010). Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

R. Horn and C. Johnson, Matrix Analysis,, Cambridge University Press, (1990). Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

S. Lang, Real and Functional Analysis,, Third edition. Graduate Texts in Mathematics, (1993). doi: 10.1007/978-1-4612-0897-6. Google Scholar

[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). doi: 10.1063/1.3596697. Google Scholar

[30]

K. Mardia and P. Jupp, Directional Statistics,, Wiley, (2000). Google Scholar

[31]

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

[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). doi: 10.1103/PhysRevE.75.036218. Google Scholar

[33]

H. Schaefer, Banach Lattices and Positive Operators,, Springer, (1974). Google Scholar

[34]

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

[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. doi: 10.1098/rstb.2009.0292. Google Scholar

[36]

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

[37]

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

[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). doi: 10.1103/PhysRevE.71.016215. Google Scholar

[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. doi: 10.1103/PhysRevE.57.1563. Google Scholar

[40]

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

[41]

A. Winfree, The Geometry of Biological Time,, Second edition. Interdisciplinary Applied Mathematics, (2001). Google Scholar

[42]

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

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). doi: 10.1103/RevModPhys.77.137. Google Scholar

[2]

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

[3]

J. T. Ariaratnam, Collective Dynamics of the Winfree Model of Coupled Nonlinear Oscillators,, PhD thesis, (2002). Google Scholar

[4]

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

[5]

V. Arnold, Ordinary Differential Equations,, Second printing of the 1992 edition. Universitext. Springer-Verlag, (1992). Google Scholar

[6]

K. Athreya and S. Lahiri, Measure Theory and Probability Theory,, Springer, (2006). Google Scholar

[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). doi: 10.1088/1742-5468/2009/10/P10014. Google Scholar

[8]

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

[9]

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

[10]

V. Bogachev, Measure Theory,, Vol. I, (2007). doi: 10.1007/978-3-540-34514-5. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[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). doi: 10.1103/PhysRevE.75.021110. Google Scholar

[16]

T. Eisner, Stability of Operators and Operator Semigroups,, Birkhäuser, (2010). Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

R. Horn and C. Johnson, Matrix Analysis,, Cambridge University Press, (1990). Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

S. Lang, Real and Functional Analysis,, Third edition. Graduate Texts in Mathematics, (1993). doi: 10.1007/978-1-4612-0897-6. Google Scholar

[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). doi: 10.1063/1.3596697. Google Scholar

[30]

K. Mardia and P. Jupp, Directional Statistics,, Wiley, (2000). Google Scholar

[31]

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

[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). doi: 10.1103/PhysRevE.75.036218. Google Scholar

[33]

H. Schaefer, Banach Lattices and Positive Operators,, Springer, (1974). Google Scholar

[34]

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

[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. doi: 10.1098/rstb.2009.0292. Google Scholar

[36]

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

[37]

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

[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). doi: 10.1103/PhysRevE.71.016215. Google Scholar

[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. doi: 10.1103/PhysRevE.57.1563. Google Scholar

[40]

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

[41]

A. Winfree, The Geometry of Biological Time,, Second edition. Interdisciplinary Applied Mathematics, (2001). Google Scholar

[42]

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

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