Acta Biotheoretica

, Volume 64, Issue 4, pp 375–402 | Cite as

Hopf Bifurcations in Directed Acyclic Networks of Linearly Coupled Hindmarsh–Rose Systems

  • N. Corson
  • V. LanzaEmail author
  • N. Verdière
Regular Article


This paper addresses the existence of Hopf bifurcations in a directed acyclic network of neurons, each of them being modeled by a Hindmarsh–Rose (HR) neuronal model. The bifurcation parameter is the small parameter corresponding to the ratio of time scales between the fast and the slow dynamics. We first prove that, under certain hypotheses, the single uncoupled neuron can undergo a Hopf bifurcation. Hopf bifurcation occurrences in a directed acyclic network of HR neurons are then discussed. Numerical simulations are carried out to observe these bifurcations and to illustrate the theoretical results.


Hindmarsh–Rose system Hopf bifurcation Coupled systems 



The project is co-financed by the European Union with the European regional development fund (ERDF).


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCNLe HavreFrance

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