Journal of Mathematical Biology

, Volume 67, Issue 3, pp 609–655 | Cite as

Integrate and fire neural networks, piecewise contractive maps and limit cycles

  • Eleonora Catsigeras
  • Pierre GuiraudEmail author


We study the global dynamics of integrate and fire neural networks composed of an arbitrary number of identical neurons interacting by inhibition and excitation. We prove that if the interactions are strong enough, then the support of the stable asymptotic dynamics consists of limit cycles. We also find sufficient conditions for the synchronization of networks containing excitatory neurons. The proofs are based on the analysis of the equivalent dynamics of a piecewise continuous Poincaré map associated to the system. We show that for efficient interactions the Poincaré map is piecewise contractive. Using this contraction property, we prove that there exist a countable number of limit cycles attracting all the orbits dropping into the stable subset of the phase space. This result applies not only to the Poincaré map under study, but also to a wide class of general n-dimensional piecewise contractive maps.


Integrate and fire neural networks Piecewise contractive maps Limit cycles Synchronization 

Mathematics Subject Classification (2000)

37N25 92B20 34C15 34D05 54H20 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Instituto de MatemáticaUniversidad de la RepúblicaMontevideoUruguay
  2. 2.Centro de Investigación y Modelamiento de Fenómenos Aleatorios—Valparaíso, Facultad de IngenieraUniversidad de ValparaísoValparaísoChile

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