Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators |
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Authors: | Michael Marmaduke Woodman Carmen C Canavier |
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Institution: | 1.Neuroscience Center of Excellence,Louisiana State University Health Sciences Center,New Orleans,USA;2.Université de la Méditerranée,Institut des Sciences du Mouvement, UMR6233 CNRS,Marseille,France |
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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. |
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