Bifurcation analysis of a periodically forced relaxation oscillator:
Differential model versus phase-resetting map resetting.
(Phys Rev E 79:016209, 2009)
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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, 3T0] (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 I0 = 0.133, the stimulus is able to cause a long delay of the following action potential, but not to elicit an additional one; for I0 = 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 = tc+2T0, where tc 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 I0 = 0.133, 0 for I0 = 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 = 3T0 (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.