Biological Cybernetics

, Volume 102, Issue 1, pp 1–8

Unfolding an electronic integrate-and-fire circuit

Original Paper

DOI: 10.1007/s00422-009-0358-x

Cite this article as:
Carrillo, H. & Hoppensteadt, F. Biol Cybern (2010) 102: 1. doi:10.1007/s00422-009-0358-x

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 

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