Skip to main content
  • Oral presentation
  • Open access
  • Published:

A dynamical study of pulse-coupled oscillators in the brain

In 1967, Winfree [1] proposed a novel mathematical approach to describe phenomena of collective synchrony in nature (i.e. flashing of fireflies, clapping in a theatre, alpha rhythms, etc.) using a large coupled network of phase oscillators with a diversity of natural frequencies. By analyzing this large heterogeneous network from a “mean field” approach, the spontaneous synchrony can be understood as a critical phase transition similar to most statistical mechanical systems.

In this work, we employ this approach to model the phase transitions and bifurcation structures of a large network of pulse-coupled theta neurons [2] by appropriate choice of Winfree's "response" and "influence" functions, the latter of which is parameterized by a "sharpness" parameter n [1]. As this parameter increases, the influence function approximates the behavior of a pulse-coupled synapse. Assuming a Lorentzian distribution of natural frequencies of width Δ and mean value ω 0 , taking the thermodynamic limit, and employing the Ott-Antonsen reduction method [3], the collective dynamics of the pulse-coupled network can be analytically reduced to a single low-dimensional dynamical equation for the mean field parameter z(t).

We analyze the bifurcation diagrams for different values of the sharpness parameter n. We find that more complex behavior is apparent with increasing sharpness of the influence function, and that equilibria outside the physically relevant region (within the unit circle) affect the transient dynamics of z(t) inside the circle. Further, we find that some level of coherence always exists in the network for non-zero coupling, in contrast to other mean field coupled phase oscillator networks [1, 4]. Most interestingly, heterogeneity is observed to suppress complexity in the collective/macroscopic behavior. As the network becomes more homogeneous (∆→0), more complex dynamic including aperiodic and multistability emerge from the macroscopic mean field.

Figure 1
figure 1

A sample bifurcation diagram showing the complex structure of fixed points at various locations in parameter space, for a sharpness parameter of (n = 7). Several representative phase portraits from several distinct region of parameter space are included.


  1. Ariaratnam JT, Strogatz SH: Phase diagram for the Winfree model of coupled oscillators. Phys Rev Lett. 2001, 86: 4278-4281. 10.1103/PhysRevLett.86.4278.

    Article  CAS  PubMed  Google Scholar 

  2. Ermentrout B: Type I membranes, phase resetting curves, and synchrony. Neural Comput. 1996, 8 (5): 979-1001. 10.1162/neco.1996.8.5.979.

    Article  CAS  PubMed  Google Scholar 

  3. Ott E, Antonsen TM: Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators. Chaos. 2008, 18: 037113-10.1063/1.2930766.

    Article  PubMed  Google Scholar 

  4. Ott E, Antonsen TM: Long time evolution of phase oscillator system. Chaos. 2009, 19: 023117-10.1063/1.3136851.

    Article  PubMed  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Tanushree Luke.

Rights and permissions

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Luke, T., Barreto, E. & So, P. A dynamical study of pulse-coupled oscillators in the brain. BMC Neurosci 13 (Suppl 1), O12 (2012).

Download citation

  • Published:

  • DOI: