Seminar in applied mathematics

SPEAKER:  Antoni Guillamon (Polytechnic University of Catalonia, Barcelona)

TITLE:  Dynamics of phase-amplitude response maps in transient oscillatory dynamics

Phase response curves (PRC) have been profusely used in computational neuroscience to study the effect of an external stimulus (perturbation) on the phase of a neuron with regular tonic firing (limit cycle). It requires assuming that all the dynamics can be explained by the phase variable. However, factors like the rate of convergence to the oscillator, strong forcing or high stimulation frequency may weaken the above assumption thus leading to the question of how is the phase variation away from the attractor. The concept of isochron turns out to be crucial to answer this question; from it, we have proposed an extension of advancement functions during the transient states by defining both the Phase Response Function (PRF) and the Amplitude Response Function (ARF) in order to monitor changes both in phase and in some transversal variable. Based on the knowledge of both the PRF and the ARF, we applyy pulse-train periodic stimuli and compare the predictions given by the PRC-approach (a 1D map) with those given by the PRF-ARF-approach (a 2D map); we observe differences up to two orders of magnitude in favor of the 2D predictions, especially when the stimulation frequency is high or the strength of the stimulus is large.  We identify the role of the hyperbolicity of the limit cycle as well as geometric aspects of the isochrons as possible explanations of these differences. We further investigate the attractors underlying the dynamics of the two approaches (1D and 2D) and we compare them by computing the corresponding Arnold tongues in terms of the amplitude and the frequency of stimulation. We %come up with remarkable differences in the corresponding bifurcations. Summing up, we aim at enlightening the contribution of transient effects in predicting the phase response and showing the limits of the phase reduction approach in order to avoid wrong predictions concerning synchronization. This lecture is based on joint work with Oriol Castejón and Gemma Huguet.

Tea and chocolate will be served in room 04.3.15 after the seminar.