WIAS Preprint No. 2514, (2018)

Leap-frog patterns in systems of two coupled FitzHugh--Nagumo units


  • Eydam, Sebastian
  • Franović, Igor
  • Wolfrum, Matthias

2010 Mathematics Subject Classification

  • 34C15 34C60

2008 Physics and Astronomy Classification Scheme

  • 05.45.Xt 87.19.lj


  • Coupled oscillator systems, mutltiple timescale dynamics, FitzHugh--Nagumo model




We study a system of two identical FitzHugh-Nagumo units with a mutual linear coupling in the fast variables. While an attractive coupling always leads to synchronous behavior, a repulsive coupling can give rise to dynamical regimes with alternating spiking order, called leap-frogging. We analyze various types of periodic and chaotic leap-frogging regimes, using numerical pathfollowing methods to investigate their emergence and stability, as well as to obtain the complex bifurcation scenario which organizes their appearance in parameter space. In particular, we show that the stability region of the simplest periodic leap-frog pattern has the shape of a locking cone pointing to the canard transition of the uncoupled system. We also discuss the role of the timescale separation in the coupled FitzHugh-Nagumo system and the relation of the leap-frog solutions to the theory of mixed-mode oscillations in multiple timescale systems.

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