A Geometric Approach to Phase Response Curves and Its Numerical Computation Through the Parameterization Method
The phase response curve (PRC) is a tool used in neuroscience that measures the phase shift experienced by an oscillator due to a perturbation applied at different phases of the limit cycle. In this paper, we present a new approach to PRCs based on the parameterization method. The underlying idea relies on the construction of a periodic system whose corresponding stroboscopic map has an invariant curve. We demonstrate the relationship between the internal dynamics of this invariant curve and the PRC, which yields a method to numerically compute the PRCs. Moreover, we link the existence properties of this invariant curve as the amplitude of the perturbation is increased with changes in the PRC waveform and with the geometry of isochrons. The invariant curve and its dynamics will be computed by means of the parameterization method consisting of solving an invariance equation. We show that the method to compute the PRC can be extended beyond the breakdown of the curve by means of introducing a modified invariance equation. The method also computes the amplitude response functions (ARCs) which provide information on the displacement away from the oscillator due to the effects of the perturbation. Finally, we apply the method to several classical models in neuroscience to illustrate how the results herein extend the framework of computation and interpretation of the PRC and ARC for perturbations of large amplitude and not necessarily pulsatile.
KeywordsPhase response curves Isochrons Phase equation Parameterization method NHIM Synchronization
Mathematics Subject Classification37D10 92B25 65P99 37N30
This work has been partially funded by the Grants MINECO-FEDER MTM2015-65715-P, MDM-2014-0445, PGC2018-098676-B-100 AEI/FEDER/UE, the Catalan Grant 2017SGR1049, (GH, AP, TS), the MINECO-FEDER-UE MTM-2015-71509-C2-2-R (GH), and the Russian Scientific Foundation Grant 14-41-00044 (TS). GH acknowledges the RyC project RYC-2014-15866. TS is supported by the Catalan Institution for research and advanced studies via an ICREA academia price 2018. AP acknowledges the FPI Grant from project MINECO-FEDER-UE MTM2012-31714. We thank C. Bonet for providing us valuable references to prove Theorem 3.2 and A. Granados for numeric support. We also acknowledge the use of the UPC Dynamical Systems group’s cluster for research computing (https://dynamicalsystems.upc.edu/en/computing/).
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