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Phase reduction is a well-established method to study weakly driven and weakly perturbed oscillators. Traditional phase-reduction approaches characterize the perturbed system dynamics solely in terms of the timing of the oscillations. In the case of large perturbations, the introduction of amplitude (isostable) coordinates improves the accuracy of the phase description by providing a sense of distance from the underlying limit cycle. Importantly, phase-amplitude coordinates allow for the study of both the timing and shape of system oscillations. A parallel tool is the infinitesimal shape response curve (iSRC), a variational method that characterizes the shape change of a limit-cycle oscillator under sustained perturbation. Despite the importance of oscillation amplitude in a wide range of physical systems, systematic studies on the shape change of oscillations remain scarce. Both phase-amplitude coordinates and the iSRC represent methods to analyze oscillation shape change, yet a relationship between the two has not been previously explored. In this work, we establish the iSRC and phase-amplitude coordinates as complementary tools to study oscillation amplitude. We extend existing iSRC theory and specify conditions under which a general class of systems can be analyzed by the joint iSRC phase-amplitude approach. We show that the iSRC takes on a dramatically simple form in phase-amplitude coordinates, and directly relate the phase and isostable response curves to the iSRC. We apply our theory to weakly perturbed single oscillators, and to study the synchronization and entrainment of coupled oscillators.
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Cycle-by-cycle analysis of neural oscillations
Neural oscillations are widely studied using methods based on the Fourier transform, which models data as sums of sinusoids. This has successfully uncovered numerous links between oscillations and cognition or disease. However, neural data are nonsinusoidal, and these nonsinusoidal features are increasingly linked to a variety of behavioral and cognitive states, pathophysiology, and underlying neuronal circuit properties. Here, we present a new analysis framework-one that is complementary to existing Fourier- and Hilbert-transform based approaches-that quantifies oscillatory features in the time domain, on a cycle-by-cycle basis. We have released this cycle-by-cycle analysis suite as bycycle, a fully documented, open-source Python package with detailed tutorials and troubleshooting cases. This approach performs tests to assess whether an oscillation is present at any given moment and, if so, quantifies each oscillatory cycle by its amplitude, period, and waveform symmetry, the latter of which is missed using conventional approaches. In a series of simulated event-related studies, we show how conventional Fourier- and Hilbert-transform approaches can conflate event-related changes in oscillation burst duration as increased oscillatory amplitude and as a change in the oscillation frequency, even though those features were unchanged in simulation. Our approach avoids these errors. Further, we validate this approach in simulation and against experimental recordings of patients with Parkinson's disease, who are known to have nonsinusoidal beta (12-30 Hz) oscillations.
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- Award ID(s):
- 1736028
- PAR ID:
- 10101280
- Date Published:
- Journal Name:
- Journal of Neurophysiology
- ISSN:
- 0022-3077
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1152/jn.00273.2019
