Journal of Computational Neuroscience

, Volume 31, Issue 2, pp 401–418

Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

Article

DOI: 10.1007/s10827-011-0315-2

Cite this article as:
Woodman, M.M. & Canavier, C.C. J Comput Neurosci (2011) 31: 401. doi:10.1007/s10827-011-0315-2

Abstract

Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.

Keywords

SynchronyPulse-couplingOscillatorConduction delays

Supplementary material

10827_2011_315_Fig10_ESM.jpg (123 kb)
Figure S1

Firing pattern for scheme B in Fig. 2. A. Firing pattern for the lowest possible k value for this scheme, which is 2. B. General firing pattern for arbitrary j1 and j2 (JPEG 123 kb)

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High resolution image (EPS 1.37 mb)
10827_2011_315_Fig11_ESM.jpg (119 kb)
Figure S2

Firing pattern for scheme C in Fig. 2. A. Firing pattern for the lowest possible k value for this scheme, which is 2. B. General firing pattern for arbitrary j1 and j2 (JPEG 119 kb)

10827_2011_315_MOESM2_ESM.eps (1.4 mb)
High resolution image (EPS 1.37 mb)
10827_2011_315_Fig12_ESM.jpg (140 kb)
Figure S3

Figure S1. Firing pattern for scheme D in Fig. 2. A. Firing pattern for the lowest possible k value for this scheme, which is 3. B. General firing pattern for arbitrary j1 and j2 (JPEG 140 kb)

10827_2011_315_MOESM3_ESM.eps (1.4 mb)
High resolution image (EPS 1.38 mb)

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Michael Marmaduke Woodman
    • 1
    • 2
  • Carmen C. Canavier
    • 1
  1. 1.Neuroscience Center of ExcellenceLouisiana State University Health Sciences CenterNew OrleansUSA
  2. 2.Université de la MéditerranéeInstitut des Sciences du Mouvement, UMR6233 CNRSMarseilleFrance