# Border collision bifurcations of stroboscopic maps in periodically driven spiking models

Abstract : In this work we consider a general non-autonomous hybrid system based on the integrate-and-fire model, widely used as simplified version of neuronal models and other types of excitable systems. Our unique assumption is that the system is monotonic, possesses an attracting subthreshold equilibrium point and is forced by means of 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 and the duty cycle of the pulse. We show that it is covered by regions of existence of periodic orbits given by period adding structures. They do not only completely describe 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.
Type de document :
Pré-publication, Document de travail
2013

https://hal.inria.fr/hal-00910277
Contributeur : Frederique Clement <>
Soumis le : mercredi 27 novembre 2013 - 16:57:46
Dernière modification le : vendredi 25 mai 2018 - 12:02:05

### Identifiants

• HAL Id : hal-00910277, version 1
• ARXIV : 1310.1054

### Citation

Albert Granados, Martin Krupa, Frédérique Clement. Border collision bifurcations of stroboscopic maps in periodically driven spiking models. 2013. 〈hal-00910277〉

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