### Abstract

Original language | English |
---|---|

Journal | S I A M Journal on Applied Dynamical Systems |

Volume | 13 |

Issue number | 4 |

Pages (from-to) | 1387-1416 |

ISSN | 1536-0040 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

### Cite this

*S I A M Journal on Applied Dynamical Systems*,

*13*(4), 1387-1416. https://doi.org/10.1137/13094637X

}

*S I A M Journal on Applied Dynamical Systems*, vol. 13, no. 4, pp. 1387-1416. https://doi.org/10.1137/13094637X

**Border Collision Bifurcations of Stroboscopic Maps in Periodically Driven Spiking Models.** / Granados, Albert; Krupa, M.; Clément, F.

Research output: Contribution to journal › Journal article › Research › peer-review

TY - JOUR

T1 - Border Collision Bifurcations of Stroboscopic Maps in Periodically Driven Spiking Models

AU - Granados, Albert

AU - Krupa, M.

AU - Clément, F.

PY - 2014

Y1 - 2014

N2 - In this work we consider a general nonautonomous hybrid system based on the integrate-and-fire model, widely used as simplified version of neuronal models and other types of excitable systems. Our assumptions are that the system is monotonic, possesses an attracting subthreshold equilibrium point, and is forced by means of a periodic pulsatile (square wave) function. In contrast to classical methods, in our approach we use the stroboscopic map (time-$T$ return map) instead of the so-called firing map. It becomes a discontinuous map potentially defined in an infinite number of partitions. By applying theory for piecewise-smooth systems, we avoid relying on particular computations, and we develop a novel approach that can be easily extended to systems with other topologies (expansive dynamics) and higher dimensions. More precisely, we rigorously study the bifurcation structure in the two-dimensional parameter space formed by the amplitude of the pulse and the ratio between $T$ and the duration of the pulse (duty cycle). We show that it is covered by regions of existence of periodic orbits given by period adding structures. The period adding structures completely describe not only all the possible spiking asymptotic dynamics but also the behavior of the firing rate, which is a devil's staircase as a function of the parameters.

AB - In this work we consider a general nonautonomous hybrid system based on the integrate-and-fire model, widely used as simplified version of neuronal models and other types of excitable systems. Our assumptions are that the system is monotonic, possesses an attracting subthreshold equilibrium point, and is forced by means of a periodic pulsatile (square wave) function. In contrast to classical methods, in our approach we use the stroboscopic map (time-$T$ return map) instead of the so-called firing map. It becomes a discontinuous map potentially defined in an infinite number of partitions. By applying theory for piecewise-smooth systems, we avoid relying on particular computations, and we develop a novel approach that can be easily extended to systems with other topologies (expansive dynamics) and higher dimensions. More precisely, we rigorously study the bifurcation structure in the two-dimensional parameter space formed by the amplitude of the pulse and the ratio between $T$ and the duration of the pulse (duty cycle). We show that it is covered by regions of existence of periodic orbits given by period adding structures. The period adding structures completely describe not only all the possible spiking asymptotic dynamics but also the behavior of the firing rate, which is a devil's staircase as a function of the parameters.

U2 - 10.1137/13094637X

DO - 10.1137/13094637X

M3 - Journal article

VL - 13

SP - 1387

EP - 1416

JO - S I A M Journal on Applied Dynamical Systems

JF - S I A M Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 4

ER -