**Bifurcation
analysis of a periodically forced relaxation oscillator:**

**Differential model versus
phase-resetting map resetting.**** **

full
article

(*Phys Rev E* **79**:016209,
2009)

in Adobe Acrobat (PDF) format:

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**MOVIES**

1) Movie files “phiI0133tu.avi” and “phiI0135tu.avi”

These two movies show the evolution, as the
phase of stimulation `phi`` `is varied from 0 to 1, of the *u*-waveform of the FHN oscillator over [0,
3T_{0}] (blue curve). This
waveform deforms continuously as `phi`
is varied, but because there is a dramatic change over a very narrow range of
phases, it was necessary to use a continuation method to reveal the continuity.
This is also why `phi` does not vary uniformly over the
course of the movie, but spends a lot of time in the critical region of phases
around `phi` = 0.8. The black curve shows the *u*-waveform of an unperturbed oscillator.
For I_{0 }= 0.133, the
stimulus is able to cause a long delay of the following action potential, but
not to elicit an additional one; for I_{0 }= 0.135, it is able to
elicit an additional action potential.

2) Movie files “phiI0133uv.avi” and “phiI0135uv.avi”

These two movies show the evolution, in the phase plane, of the position of
the point corresponding to the new phase` phi`’
(approximated by the point reached at t = t_{c}+2T_{0}, where t_{c} is the coupling interval), represented by a
circle (o), as the point corresponding to the old phase` phi`, represented by a cross (x), winds
once around the limit cycle. The net number of turns made by the circle symbol
around the limit cycle over the course of the movie is the topological degree
(1 for I_{0 }= 0.133, 0 for I_{0 }= 0.135).

In each frame of the movies, the blue curve is the complete trajectory
followed by the state-point from t = 0 to t = 3T_{0} (it corresponds to
the *u*-waveform shown in the
animations “phiI0133tu.avi” and “phiI0135tu.avi”). For
the reason explained above, `phi` does
not vary uniformly over the course of the movie, but spends a lot of time in a
very narrow range of phases, giving the impression that the cross is “stuck”
at the same point of the limit-cycle during about 1/3 of the movie. The dotted
black curves represent the nullclines.

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