Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models

Albert Granados*, Gemma Huguet

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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In this work we consider a general class of 2-dimensional hybrid systems. Assuming that the system possesses an attracting equilibrium point, we show that, when periodically driven with a square-wave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semi-rigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrate-and fire model with a dynamic threshold. We use the stroboscopic map, which in this context is a 2-dimensional piecewise-smooth discontinuous map. For some parameter values we are able to show that the map is a quasi-contraction possessing a (locally) unique maximin periodic orbit. We complement our analysis using advanced numerical techniques to provide a complete portrait of the dynamics as parameters are varied. We find that for some regions of the parameter space the model undergoes a cascade of gluing bifurcations, while for others the model shows multistability between orbits of different periods.
Original languageEnglish
JournalCommunications in Nonlinear Science and Numerical Simulation
Pages (from-to)48-73
Publication statusPublished - 2019


  • Integrate-and-fire
  • Hybrid systems
  • Piecewise smooth 2d maps
  • Quasi-contractions

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