Biological Cybernetics

, Volume 109, Issue 3, pp 333–347 | Cite as

Synchrony, waves and ripple in spatially coupled Kuramoto oscillators with Mexican hat connectivity

  • Stewart HeitmannEmail author
  • G. Bard Ermentrout
Original Paper


Spatiotemporal waves of synchronized activity are known to arise in oscillatory neural networks with lateral inhibitory coupling. How such patterns respond to dynamic changes in coupling strength is largely unexplored. The present study uses analysis and simulation to investigate the evolution of wave patterns when the strength of lateral inhibition is varied dynamically. Neural synchronization was modeled by a spatial ring of Kuramoto oscillators with Mexican hat lateral coupling. Broad bands of coexisting stable wave solutions were observed at all levels of inhibition. The stability of these waves was formally analyzed in both the infinite ring and the finite ring. The broad range of multi-stability predicted hysteresis in transitions between neighboring wave solutions when inhibition is slowly varied. Numerical simulation confirmed the predicted transitions when inhibition was ramped down from a high initial value. However, non-wave solutions emerged from the uniform solution when inhibition was ramped upward from zero. These solutions correspond to spatially periodic deviations of phase that we call ripple states. Numerical continuation showed that stable ripple states emerge from synchrony via a supercritical pitchfork bifurcation. The normal form of this bifurcation was derived analytically, and its predictions compared against the numerical results. Ripple states were also found to bifurcate from wave solutions, but these were locally unstable. Simulation also confirmed the existence of hysteresis and ripple states in two spatial dimensions. Our findings show that spatial synchronization patterns can remain structurally stable despite substantial changes in network connectivity.


Kuramoto oscillators Synchrony Waves Ripple  Lateral inhibition Mexican hat 

Supplementary material

Supplementary material 1 (mpg 6570 KB)

Supplementary material 2 (mpg 8821 KB)


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics and the Center for the Neural Basis of CognitionUniversity of PittsburghPittsburghUSA

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