Phase Description of Oscillations in Singularly-Perturbed Systems
Final Report Abstract
We studied the rhythmic behavior of oscillators that are close to a stall. Such oscillators are often found in biological rhythm-pattern generators because close to stalemate such rhythm generators are easily controlled by the organism through alteration of a closeness parameter. First, we studied pulmonary respiration as a prime example of irregular oscillations with highly variable amplitude, i.e. respiratory depth, and frequency. By generalizing the mathematical notion of phase to oscillations that are irregular, we were able to cast irregular respiration as an oscillatory process continuously recurring to well-described phases. Our approach has a broad applicability especially in biomedical data analysis, where a rigorous phase reduction method had been missing and may now be found in our publication in Physical Review Letters. Second, we investigated how neural circuits can robustly exert multiple functions without having to undergo structural alteration to switch from one operational mode to another. Our model implements such multistability using ‘contradictory connections’: for example, we imagine three rhythmically active neurons connected in such a way that each one inhibits the activities of the other two. Consequently, either two neurons can be forced into a state of synchrony by the third, despite their force to mutually inhibit each other. From this mechanism one observes three circuit rhythms, one for each inhibiting two others, as well as two rhythms in which the neurons take turns of activity. Each of such rhythms could exert a different function in the organism, among which rapid switching in response to a changing environment is possible. In real triplets of neurons, the symmetry is not perfect. However, our investigations show that even with imperfect symmetry the mechanism of multistability can be operative. We also analyzed how such circuits generate coexistent rhythms without accidentally switching due to small perturbations. We found that rhythm robustness is irreconcilable with the requirement that the circuit should be close to a stall. Furthermore, one would typically expect that strengthening the circuit connections could reinforce the rhythms. However, we found the paradoxical effect that such enhancements made the circuit more vulnerable to accidental switching between rhythms.
Publications
- Phase description of stochastic oscillations. Phys. Rev. Lett., 110(20):204102, 2013
Justus T. C. Schwabedal and Arkady Pikovsky
(See online at https://doi.org/10.1103/PhysRevLett.110.204102) - Key bifurcations of bursting polyrhythms in 3-cell central pattern generators. PloS one, 9(4):e92918, 2014
Jeremy Wojcik, Justus Schwabedal, Robert Clewley, and Andrey L Shilnikov
(See online at https://doi.org/10.1371/journal.pone.0092918) - Parametric phase diffusion analysis of irregular oscillations. Europhys. Lett., 107(6):68001, 2014
Justus TC Schwabedal
(See online at https://doi.org/10.1209/0295-5075/107/68001) - Robust design of polyrhythmic neural circuits. Physical Review E, 90(2):022715, 2014
Justus TC Schwabedal, Alexander B Neiman, and Andrey L Shilnikov
(See online at https://doi.org/10.1103/PhysRevE.90.022715)
