Biological Cybernetics

, Volume 102, Issue 1, pp 1–8 | Cite as

Unfolding an electronic integrate-and-fire circuit

Original Paper

Abstract

Many physical and biological phenomena involve accumulation and discharge processes that can occur on significantly different time scales. Models of these processes have contributed to understand excitability self-sustained oscillations and synchronization in arrays of oscillators. Integrate-and-fire (I+F) models are popular minimal fill-and-flush mathematical models. They are used in neuroscience to study spiking and phase locking in single neuron membranes, large scale neural networks, and in a variety of applications in physics and electrical engineering. We show here how the classical first-order I+F model fits into the theory of nonlinear oscillators of van der Pol type by demonstrating that a particular second-order oscillator having small parameters converges in a singular perturbation limit to the I+F model. In this sense, our study provides a novel unfolding of such models and it identifies a constructible electronic circuit that is closely related to I+F.

Keywords

Integrate-and-fire van der Pol neon bulb oscillator Singular perturbation 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Laboratorio de Dinámica no Lineal, Facultad de Ciencias y Centro de Ciencias de la ComplejidadUniversidad Nacional Autónoma de MéxicoMexicoMexico
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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