Biological Cybernetics

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

Unfolding an electronic integrate-and-fire circuit

Original Paper


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.


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


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  1. Andronov AA, Vitt AA, Khaikin SE (1987) Theory of oscillators. Dover Publications, New York (Original published in 1937 in Russian, in English in 1966)Google Scholar
  2. Carrillo H, Ongay F (2001) On the firing maps of a general class of forced integrate and fire systems. J Math Biosci 172: 33–53CrossRefGoogle Scholar
  3. Carrillo H, Mendoza M, Ongay F (2004) Integrate-and-fire neurons and circle maps. WSEAS Trans Biol Biomed 1(2): 287–293Google Scholar
  4. Cole JD, Kevorkian JK (1981) Perturbation methods in applied mathematics. Springer, BerlinGoogle Scholar
  5. Eckhaus W (1979) Asymptotic analysis of singular perturbation problems. Elsevier/North-Holland, New YorkGoogle Scholar
  6. Fitzhugh R (1961) Impulses and physiological states in theoretical models of nerve membranes. Biophys J 1: 445–466CrossRefPubMedGoogle Scholar
  7. Flaherty JE, Hoppensteadt FC (1978) Frequency entrainment of a forced van der Pol oscillator. Stud Appl Math 59: 5–15Google Scholar
  8. Hoppensteadt F (1971) Properties of solutions of ordinary differential equations with small parameters. Comm Pure Appl Math XXIV: 807–840CrossRefGoogle Scholar
  9. Hoppensteadt F (2000) Analysis and simulation of chaotic systems. Springer, New YorkGoogle Scholar
  10. Hoppensteadt F (1969) On systems of ordinary differential equations with several small parameters multiplying the derivatives. J Differ Equ 5: 106–116CrossRefGoogle Scholar
  11. Hoppensteadt FC, Izhikevich EM (1997) Weakly connected neural networks. Springer, New YorkGoogle Scholar
  12. Hoppensteadt FC (1997) Introduction to the mathematics of neurons: modeling in the frequency domain, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  13. Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. The MIT Press, CambridgeGoogle Scholar
  14. Izhikevich E (2003) Simple model of spiking neurons. IEEE Trans Neural Netw 14(6):1569ff. Scholarpedia,
  15. Keener JP, Hoppensteadt FC, Rinzel J (1981) Integrate and fire models of nerve membrane response to oscillatory input. SIAM J Appl Math 41: 503–517CrossRefGoogle Scholar
  16. Kennedy M, Chua L (1986) Van der Pol and Chaos. IEEE Trans Circuits Syst 33(l): 974–980CrossRefGoogle Scholar
  17. Knight BW (1972) Dynamics of encoding in a population of neurons. J Gen Physiol 59: 734–766CrossRefPubMedGoogle Scholar
  18. Landauer R (1977) Poor man’s chaos. IBM Technical ReportGoogle Scholar
  19. Levi M (1981) Qualitative analysis of the periodically forced relaxation oscillations. Memoirs, 32 (244), American Mathematical Society, Providence, RIGoogle Scholar
  20. Levi M (1990) A period adding phenomenon. SIAM J Appl Math 50(4): 943–955CrossRefGoogle Scholar
  21. Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50: 2061–2070CrossRefGoogle Scholar
  22. Peskin CS, Neuroscience lecture notes,
  23. Petrani ML, Karakotsou CH, Kyprianidis IM, Anagnostopoulos AN (1994) Characterization of the attractor governing the neon bulb RC relaxation oscillator. Phys Rev E 49(6): 5863–5866CrossRefGoogle Scholar
  24. Stoker JJ (1951) Nonlinear vibrations. Wiley-Interscience, New YorkGoogle Scholar
  25. van der Pol B, van der Mark J (1927) Frequency demultiplication. Nature, 120:363–364. (See also van der Pol B (1960) Selected scientific papers (Bremmer H, Bouwkamp CJ, eds) North Holland Publ., Amsterdam)Google Scholar
  26. Winfree AS (2000) The geometry of biological time, 2nd edn. Springer, New YorkGoogle Scholar
  27. Yang T, Kiehl R, Chua L (2001) Tunneling phase logic cellular nonlinear networks. Int J Bifurcation Chaos Appl Sci 11: 2895–2912CrossRefGoogle Scholar

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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