American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The patch recovery for finite element approximation of elasticity problems under quadrilateral meshes
  • DCDS-B Home
  • This Issue
  • Next Article
    Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity
January  2008, 9(1): 145-162. doi: 10.3934/dcdsb.2008.9.145

Phase-locking and Arnold coding in prototypical network topologies

Stefan Martignoli 1, and  Ruedi Stoop 2,

1. 

Institute for Neuroinformatics UZH/ETHZ, Winterthurerstrasse 190, 8057 Zürich, Switzerland
2. 

Institute of Neuroinformatics UZH/ETHZ, Winterthurerstrasse 190, 8057 Zürich, Switzerland

Received  August 2006 Revised  July 2007 Published  October 2007

Phase-and-frequency-locking phenomena among coupled biological oscillators are a topic of current interest, in particular to neuroscience. In the case of mono-directionally pulse-coupled oscillators, phase-locking is well understood, where the phenomenon is globally described by Arnold tongues. Here, we develop the tools that allow corresponding investigations to be made for more general pulse-coupled networks. For two bi-directionally coupled oscillators, we prove the existence of three-dimensional Arnold tongues that mediate from the mono- to the bi-directional coupling topology. Under this transformation, the coupling strength at which the onset of chaos is observed is invariant. The developed framework also allows us to compare information transfer in feedforward versus recurrent networks. We find that distinct laws govern the propagation of phase-locked spike-time information, indicating a qualitative difference between classical artificial vs. biological computation.
Keywords: Arnold tongues, recurrent topology., phase-locked oscillators.
Mathematics Subject Classification: Primary: 37E10; Secondary: 37E4.
Citation: Stefan Martignoli, Ruedi Stoop. Phase-locking and Arnold coding in prototypical network topologies. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 145-162. doi: 10.3934/dcdsb.2008.9.145
[1]

Ryotaro Tsuneki, Shinji Doi, Junko Inoue. Generation of slow phase-locked oscillation and variability of the interspike intervals in globally coupled neuronal oscillators. Mathematical Biosciences & Engineering, 2014, 11 (1) : 125-138. doi: 10.3934/mbe.2014.11.125
[2]

Jeremias Epperlein, Stefan Siegmund. Phase-locked trajectories for dynamical systems on graphs. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1827-1844. doi: 10.3934/dcdsb.2013.18.1827
[3]

Seung-Yeal Ha, Jinyeong Park, Sang Woo Ryoo. Emergence of phase-locked states for the Winfree model in a large coupling regime. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3417-3436. doi: 10.3934/dcds.2015.35.3417
[4]

Yuming Chen. Synchronized and nonsymmetric phase-locked periodic solutions in a neteork of neurons with McCulloch-Pitts nonlinearity. Conference Publications, 2001, 2001 (Special) : 102-108. doi: 10.3934/proc.2001.2001.102
[5]

Victor S. Kozyakin, Alexander M. Krasnosel’skii, Dmitrii I. Rachinskii. Arnold tongues for bifurcation from infinity. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 107-116. doi: 10.3934/dcdss.2008.1.107
[6]

Victor Kozyakin, Alexander M. Krasnosel’skii, Dmitrii Rachinskii. Asymptotics of the Arnold tongues in problems at infinity. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 989-1011. doi: 10.3934/dcds.2008.20.989
[7]

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

Guy Katriel. Stability of synchronized oscillations in networks of phase-oscillators. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 353-364. doi: 10.3934/dcdsb.2005.5.353
[9]

Da-Peng Li. Phase transition of oscillators and travelling waves in a class of relaxation systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2601-2614. doi: 10.3934/dcdsb.2016063
[10]

William F. Thompson, Rachel Kuske, Yue-Xian Li. Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2971-2995. doi: 10.3934/dcds.2012.32.2971
[11]

Hayato Chiba. Continuous limit and the moments system for the globally coupled phase oscillators. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1891-1903. doi: 10.3934/dcds.2013.33.1891
[12]

Stilianos Louca, Fatihcan M. Atay. Spatially structured networks of pulse-coupled phase oscillators on metric spaces. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3703-3745. doi: 10.3934/dcds.2014.34.3703
[13]

Xiaoxue Zhao, Zhuchun Li, Xiaoping Xue. Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring. Networks & Heterogeneous Media, 2018, 13 (2) : 323-337. doi: 10.3934/nhm.2018014
[14]

Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451
[15]

Heesung Shin, Jiang Zeng. More bijections for Entringer and Arnold families. Electronic Research Archive, 2021, 29 (2) : 2167-2185. doi: 10.3934/era.2020111
[16]

Bruce Hughes. Geometric topology of stratified spaces. Electronic Research Announcements, 1996, 2: 73-81.

[17]

Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929
[18]

Fengbo Hang, Fanghua Lin. Topology of Sobolev mappings IV. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1097-1124. doi: 10.3934/dcds.2005.13.1097
[19]

Juan Campos, Rafael Obaya, Massimo Tarallo. Recurrent equations with sign and Fredholm alternative. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 959-977. doi: 10.3934/dcdss.2016036
[20]

Lyudmila Grigoryeva, Juan-Pablo Ortega. Dimension reduction in recurrent networks by canonicalization. Journal of Geometric Mechanics, 2021, 13 (4) : 647-677. doi: 10.3934/jgm.2021028

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (141)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy