Global synchronization and asymptotic phases for a ring of identical cells with delayed coupling

Chih-Wen Shih*, Jui Pin Tseng

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


We consider a neural network which consists of a ring of identical neurons coupled with their nearest neighbors and is subject to self-feedback delay and transmission delay. We present an iteration scheme to analyze the synchronization and asymptotic phases for the system. Delay-independent, delay-dependent, and scale-dependent criteria are formulated for the global synchronization and global convergence. Under this setting, the possible asymptotic dynamics include convergence to single equilibrium, multiple equilibria, and synchronous oscillation. The study aims at elucidating the effects from the scale of network, self-decay, self-feedback strength, coupling strength, and delay magnitudes upon synchrony, convergent dynamics, and oscillation of the network. The disparity between the contents of synchrony induced by distinct factors is investigated. Two different types of multistable dynamics are distinguished. Moreover, oscillation and desynchronization induced by delays are addressed. We answer two conjectures in the literature.

Original languageEnglish
Pages (from-to)1667-1697
Number of pages31
JournalSIAM Journal on Mathematical Analysis
Issue number4
StatePublished - 19 Sep 2011


  • Delay
  • Multistability
  • Neural network
  • Oscillation
  • Synchronization

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