Regular and Chaotic Dynamics

, Volume 15, Issue 2–3, pp 146–158 | Cite as

Burst-duration mechanism of in-phase bursting in inhibitory networks

  • I. BelykhEmail author
  • S. Jalil
  • A. Shilnikov
L. P. Shilnikov-75 Special Issue


We study the emergence of in-phase and anti-phase synchronized rhythms in bursting networks of Hodgkin-Huxley-type neurons connected by inhibitory synapses. We show that when the state of the individual neuron composing the network is close to the transition from bursting into tonic spiking, the appearance of the network’s synchronous rhythms becomes sensitive to small changes in parameters and synaptic coupling strengths. This bursting-spiking transition is associated with codimension-one bifurcations of a saddle-node limit cycle with homoclinic orbits, first described and studied by Leonid Pavlovich Shilnikov. By this paper, we pay tribute to his pioneering results and emphasize their importance for understanding the cooperative behavior of bursting neurons. We describe the burst-duration mechanism of inphase synchronized bursting in a network with strong repulsive connections, induced by weak inhibition. Through the stability analysis, we also reveal the dual property of fast reciprocal inhibition to establish in- and anti-phase synchronized bursting.

Key words

Bursting neurons synchronization inhibitory networks burst duration 

MSC2000 numbers

34D06 34C23 92B25 37H20 37G15 


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Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsGeorgia State UniversityAtlantaUSA
  2. 2.Neuroscience InstituteGeorgia State UniversityAtlantaUSA

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