Global analysis of firing maps

Alexandre Mauroy, J.M. Hendrickx, Alexander Megretski, Rodolphe Sepulchre

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Abstract

— In this paper, we study the behavior of pulsecoupled integrate-and-fire oscillators. Each oscillator is characterized by a state evolving between two threshold values. As the state reaches the upper threshold, it is reset to the lower threshold and emits a pulse which increments by a constant value the state of every other oscillator. The behavior of the system is described by the so-called firing map: depending on the stability of the firing map, an important dichotomy characterizes the behavior of the oscillators (synchronization or clustering). The firing map is the composition of a linear map with a scalar nonlinearity. After briefly discussing the case of the scalar firing map (corresponding to two oscillators), the stability analysis is extended to the general n-dimensional firing map (for n + 1 oscillators). Different models are considered (leaky oscillators, quadratic oscillators,. . . ), with a particular emphasis on the persistence of the dichotomy in higher dimensions.
Original languageEnglish
Title of host publicationProceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS)
Pages1775-1782
Number of pages8
Publication statusPublished - Jul 2010

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oscillators
dichotomies
thresholds
scalars
synchronism
nonlinearity
pulses

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Mauroy, A., Hendrickx, J. M., Megretski, A., & Sepulchre, R. (2010). Global analysis of firing maps. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS) (pp. 1775-1782)
Mauroy, Alexandre ; Hendrickx, J.M. ; Megretski, Alexander ; Sepulchre, Rodolphe. / Global analysis of firing maps. Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS). 2010. pp. 1775-1782
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Mauroy, A, Hendrickx, JM, Megretski, A & Sepulchre, R 2010, Global analysis of firing maps. in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS). pp. 1775-1782.

Global analysis of firing maps. / Mauroy, Alexandre; Hendrickx, J.M.; Megretski, Alexander; Sepulchre, Rodolphe.

Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS). 2010. p. 1775-1782.

Research output: Contribution in Book/Catalog/Report/Conference proceedingConference contribution

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N2 - — In this paper, we study the behavior of pulsecoupled integrate-and-fire oscillators. Each oscillator is characterized by a state evolving between two threshold values. As the state reaches the upper threshold, it is reset to the lower threshold and emits a pulse which increments by a constant value the state of every other oscillator. The behavior of the system is described by the so-called firing map: depending on the stability of the firing map, an important dichotomy characterizes the behavior of the oscillators (synchronization or clustering). The firing map is the composition of a linear map with a scalar nonlinearity. After briefly discussing the case of the scalar firing map (corresponding to two oscillators), the stability analysis is extended to the general n-dimensional firing map (for n + 1 oscillators). Different models are considered (leaky oscillators, quadratic oscillators,. . . ), with a particular emphasis on the persistence of the dichotomy in higher dimensions.

AB - — In this paper, we study the behavior of pulsecoupled integrate-and-fire oscillators. Each oscillator is characterized by a state evolving between two threshold values. As the state reaches the upper threshold, it is reset to the lower threshold and emits a pulse which increments by a constant value the state of every other oscillator. The behavior of the system is described by the so-called firing map: depending on the stability of the firing map, an important dichotomy characterizes the behavior of the oscillators (synchronization or clustering). The firing map is the composition of a linear map with a scalar nonlinearity. After briefly discussing the case of the scalar firing map (corresponding to two oscillators), the stability analysis is extended to the general n-dimensional firing map (for n + 1 oscillators). Different models are considered (leaky oscillators, quadratic oscillators,. . . ), with a particular emphasis on the persistence of the dichotomy in higher dimensions.

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Mauroy A, Hendrickx JM, Megretski A, Sepulchre R. Global analysis of firing maps. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS). 2010. p. 1775-1782