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Analysis of input-induced oscillations using the isostable coordinate framework. (English) Zbl 1458.34069

Summary: Many reduced order modeling techniques for oscillatory dynamical systems are only applicable when the underlying system admits a stable periodic orbit in the absence of input. By contrast, very few reduction frameworks can be applied when the oscillations themselves are induced by coupling or other exogenous inputs. In this work, the behavior of such input-induced oscillations is considered. By leveraging the isostable coordinate framework, a high-accuracy reduced set of equations can be identified and used to predict coupling-induced bifurcations that precipitate stable oscillations. Subsequent analysis is performed to predict the steady state phase-locking relationships. Input-induced oscillations are considered for two classes of coupled dynamical systems. For the first, stable fixed points of systems with parameters near Hopf bifurcations are considered so that the salient dynamical features can be captured using an asymptotic expansion of the isostable coordinate dynamics. For the second, an adaptive phase-amplitude reduction framework is used to analyze input-induced oscillations that emerge in excitable systems. Examples with relevance to circadian and neural physiology are provided that highlight the utility of the proposed techniques.
©2021 American Institute of Physics

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
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[1] Abrams, D. M.; Strogatz, S. H., Chimera states for coupled oscillators, Phys. Rev. Lett., 93, 17, 174102 (2004) · doi:10.1103/PhysRevLett.93.174102
[2] Aton, S. J.; Colwell, C. S.; Harmar, A. J.; Waschek, J.; Herzog, E. D., Vasoactive intestinal polypeptide mediates circadian rhythmicity and synchrony in mammalian clock neurons, Nat. Neurosci., 8, 4, 476-483 (2005) · doi:10.1038/nn1419
[3] Bressloff, P. C.; MacLaurin, J. N., A variational method for analyzing stochastic limit cycle oscillators, SIAM J. Appl. Dyn. Syst., 17, 3, 2205-2233 (2018) · Zbl 1409.60101 · doi:10.1137/17M1155235
[4] Brown, E.; Moehlis, J.; Holmes, P., On the phase reduction and response dynamics of neural oscillator populations, Neural Comput., 16, 4, 673-715 (2004) · Zbl 1054.92006 · doi:10.1162/089976604322860668
[5] Brunton, S. L.; Brunton, B. W.; Proctor, J. L.; Kutz, N. J., Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control, PLoS One, 11, 2, e0150171 (2016) · doi:10.1371/journal.pone.0150171
[6] Budišić, M.; Mohr, R.; Mezić, I., Applied Koopmanism, Chaos, 22, 4, 047510 (2012) · Zbl 1319.37013 · doi:10.1063/1.4772195
[7] Castejón, O.; Guillamon, A., Phase-amplitude dynamics in terms of extended response functions: Invariant curves and arnold tongues, Commun. Nonlinear Sci. Numer. Simul., 81, 105008 (2020) · Zbl 1467.37078 · doi:10.1016/j.cnsns.2019.105008
[8] Castejón, O.; Guillamon, A.; Huguet, G., Phase-amplitude response functions for transient-state stimuli, J. Math. Neurosci., 3, 13 (2013) · Zbl 1291.92026 · doi:10.1186/2190-8567-3-13
[9] Cui, J.; Canavier, C. C.; Butera, R. J., Functional phase response curves: A method for understanding synchronization of adapting neurons, J. Neurophysiol., 102, 1, 387-398 (2009) · doi:10.1152/jn.00037.2009
[10] Diekman, C. O.; Bose, A., Entrainment maps: A new tool for understanding properties of circadian oscillator models, J. Biol. Rhythms, 31, 6, 598-616 (2016) · doi:10.1177/0748730416662965
[11] Dörfler, F.; Chertkov, M.; Bullo, F., Synchronization in complex oscillator networks and smart grids, Proc. Natl. Acad. Sci., 110, 6, 2005-2010 (2013) · Zbl 1292.94185 · doi:10.1073/pnas.1212134110
[12] Ermentrout, B., Linearization of FI curves by adaptation, Neural Comput., 10, 7, 1721-1729 (1998) · doi:10.1162/089976698300017106
[13] Ermentrout, G. B.; Terman, D. H., Mathematical Foundations of Neuroscience, 35 (2010), Springer: Springer, New York · Zbl 1320.92002
[14] FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1, 6, 445-466 (1961) · doi:10.1016/S0006-3495(61)86902-6
[15] Gonze, D.; Bernard, S.; Waltermann, C.; Kramer, A.; Herzel, H., Spontaneous synchronization of coupled circadian oscillators, Biophys. J., 89, 1, 120-129 (2005) · doi:10.1529/biophysj.104.058388
[16] Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 42 (1983), Springer Verlag: Springer Verlag, New York · Zbl 0515.34001
[17] Hafner, M.; Koeppl, H.; Gonze, D., Effect of network architecture on synchronization and entrainment properties of the circadian oscillations in the suprachiasmatic nucleus, PLoS Comput. Biol., 8, 3, e1002419 (2012) · doi:10.1371/journal.pcbi.1002419
[18] Kawamura, Y.; Nakao, H.; Arai, K.; Kori, H.; Kuramoto, Y., Collective phase sensitivity, Phys. Rev. Lett., 101, 2, 024101 (2008) · doi:10.1103/PhysRevLett.101.024101
[19] Ko, T. W.; Ermentrout, G. B., Phase-response curves of coupled oscillators, Phys. Rev. E, 79, 1, 016211 (2009) · doi:10.1103/PhysRevE.79.016211
[20] Kopell, N. J.; Gritton, H. J.; Whittington, M. A.; Kramer, M. A., Beyond the connectome: The dynome, Neuron, 83, 6, 1319-1328 (2014) · doi:10.1016/j.neuron.2014.08.016
[21] Kotani, K.; Yamaguchi, I.; Yoshida, L.; Jimbo, Y.; Ermentrout, G. B., Population dynamics of the modified theta model: Macroscopic phase reduction and bifurcation analysis link microscopic neuronal interactions to macroscopic gamma oscillation, J. R. Soc. Interface, 11, 95, 20140058 (2014) · doi:10.1098/rsif.2014.0058
[22] Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence (1984), Springer-Verlag: Springer-Verlag, Berlin · Zbl 0558.76051
[23] Kvalheim, M. D.; Revzen, S.
[24] Letson, B.; Rubin, J. E., LOR for analysis of periodic dynamics: A one-stop shop approach, SIAM J. Appl. Dyn. Syst., 19, 1, 58-84 (2020) · Zbl 1461.34065 · doi:10.1137/19M1258529
[25] Levnajić, Z.; Pikovsky, A., Phase resetting of collective rhythm in ensembles of oscillators, Phys. Rev. E, 82, 5, 056202 (2010) · doi:10.1103/PhysRevE.82.056202
[26] Lohmiller, W.; Slotine, J. J. E., On contraction analysis for non-linear systems, Automatica, 34, 6, 683-696 (1998) · Zbl 0934.93034 · doi:10.1016/S0005-1098(98)00019-3
[27] Mauroy, A.; Mezić, I., Global computation of phase-amplitude reduction for limit-cycle dynamics, Chaos, 28, 7, 073108 (2018) · Zbl 1401.37034 · doi:10.1063/1.5030175
[28] Mauroy, A.; Mezić, I.; Moehlis, J., Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics, Physica D, 261, 19-30 (2013) · Zbl 1284.37047 · doi:10.1016/j.physd.2013.06.004
[29] Monga, B.; Wilson, D.; Matchen, T.; Moehlis, J., Phase reduction and phase-based optimal control for biological systems: A tutorial, Biol. Cybern., 113, 1-2, 11-46 (2019) · Zbl 1411.92122 · doi:10.1007/s00422-018-0780-z
[30] Moore, R. Y.; Speh, J. C.; Leak, R. K., Suprachiasmatic nucleus organization, Cell Tissue Res., 309, 1, 89-98 (2002) · doi:10.1007/s00441-002-0575-2
[31] Nabi, A.; Stigen, T.; Moehlis, J.; Netoff, T., Minimum energy control for in vitro neurons, J. Neural Eng., 10, 3, 036005 (2013) · doi:10.1088/1741-2560/10/3/036005
[32] Pecora, L. M.; Carroll, T. L., Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80, 10, 2109 (1998) · doi:10.1103/PhysRevLett.80.2109
[33] Pecora, L. M.; Sorrentino, F.; Hagerstrom, A. M.; Murphy, T. E.; Roy, R., Cluster synchronization and isolated desynchronization in complex networks with symmetries, Nat. Commun., 5, 1, 1-8 (2014) · doi:10.1038/ncomms5079
[34] Pietras, B.; Daffertshofer, A., Network dynamics of coupled oscillators and phase reduction techniques, Phys. Rep., 819, 1-105 (2019) · doi:10.1016/j.physrep.2019.06.001
[35] Pikovsky, A.; Rosenblum, M.; Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences (2001), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 0993.37002
[36] Reppert, S. M.; Weaver, D. R., Coordination of circadian timing in mammals, Nature, 418, 6901, 935 (2002) · doi:10.1038/nature00965
[37] Rogers, J.; McCulloch, A., A collocation-Galerkin finite element model of cardiac action potential propagation, IEEE Trans. Biomed. Eng., 41, 743-757 (1994) · doi:10.1109/10.310090
[38] Della Rossa, F.; Pecora, L.; Blaha, K.; Shirin, A.; Klickstein, I.; Sorrentino, F., Symmetries and cluster synchronization in multilayer networks, Nat. Commun., 11, 1, 1-17 (2020) · doi:10.1038/s41467-020-16343-0
[39] Sanders, J. A.; Verhulst, F.; Murdock, J., Averaging Methods in Nonlinear Dynamical Systems (2007), Springer-Verlag: Springer-Verlag, New York · Zbl 1128.34001
[40] Serkh, K.; Forger, D. B., Optimal schedules of light exposure for rapidly correcting circadian misalignment, PLoS Comput. Biol., 10, 4, e1003523 (2014) · doi:10.1371/journal.pcbi.1003523
[41] Shirasaka, S.; Kurebayashi, W.; Nakao, H., Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems, Chaos, 27, 2, 023119 (2017) · Zbl 1387.37045 · doi:10.1063/1.4977195
[42] Sootla, A.; Mauroy, A., Geometric properties of isostables and basins of attraction of monotone systems, IEEE Trans. Automat. Control, 62, 12, 6183-6194 (2017) · doi:10.1109/TAC.2017.2707660
[43] Strogatz, S. H.; Abrams, D. M.; McRobie, A.; Eckhardt, B.; Ott, E., Theoretical mechanics: Crowd synchrony on the millennium bridge, Nature, 438, 7064, 43 (2005) · doi:10.1038/438043a
[44] To, T. L.; Henson, M. A.; Herzog, E. D.; Doyle III, F. J., A molecular model for intercellular synchronization in the mammalian circadian clock, Biophys. J., 92, 11, 3792-3803 (2007) · doi:10.1529/biophysj.106.094086
[45] Wang, X. J.; Buzsáki, G., Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model, J. Neurosci., 16, 20, 6402-6413 (1996) · doi:10.1523/JNEUROSCI.16-20-06402.1996
[46] Webb, A. B.; Angelo, N.; Huettner, J. E.; Herzog, E. D., Intrinsic, nondeterministic circadian rhythm generation in identified mammalian neurons, Proc. Natl. Acad. Sci., 106, 38, 16493-16498 (2009) · doi:10.1073/pnas.0902768106
[47] Wedgwood, K. C. A.; Lin, K. K.; Thul, R.; Coombes, S., Phase-amplitude descriptions of neural oscillator models, J. Math. Neurosci., 3, 1, 2 (2013) · Zbl 1291.92052 · doi:10.1186/2190-8567-3-2
[48] Wilson, D., “Data-driven inference of high-accuracy isostable-based dynamical models in response to external inputs,” arxiv.org/abs/2102.04526.
[49] Wilson, D., “Optimal control of oscillation timing and entrainment using large magnitude inputs: An adaptive phase-amplitude-coordinate-based approach,” arxiv.org/abs/2102.04535.
[50] Wilson, D., Isostable reduction of oscillators with piecewise smooth dynamics and complex floquet multipliers, Phys. Rev. E, 99, 2, 022210 (2019) · doi:10.1103/PhysRevE.99.022210
[51] Wilson, D.
[52] Wilson, D., A data-driven phase and isostable reduced modeling framework for oscillatory dynamical systems, Chaos, 30, 1, 013121 (2020) · Zbl 1433.37077 · doi:10.1063/1.5126122
[53] Wilson, D., Phase-amplitude reduction far beyond the weakly perturbed paradigm, Phys. Rev. E, 101, 2, 022220 (2020) · doi:10.1103/PhysRevE.101.022220
[54] Wilson, D. and Djouadi, S., “Isostable reduction and boundary feedback control for nonlinear convective flow,” in Proceedings of the 58th IEEE Conference on Decision and Control (IEEE, 2019).
[55] Wilson, D.; Ermentrout, B., Greater accuracy and broadened applicability of phase reduction using isostable coordinates, J. Math. Biol., 76, 1-2, 37-66 (2018) · Zbl 1392.92007 · doi:10.1007/s00285-017-1141-6
[56] Wilson, D.; Ermentrout, B., An operational definition of phase characterizes the transient response of perturbed limit cycle oscillators, SIAM J. Appl. Dyn. Syst., 17, 4, 2516-2543 (2018) · Zbl 1404.70040 · doi:10.1137/17M1153261
[57] Wilson, D.; Ermentrout, B., Augmented phase reduction of (not so) weakly perturbed coupled oscillators, SIAM Rev., 61, 2, 277-315 (2019) · Zbl 1419.70007 · doi:10.1137/18M1170558
[58] Wilson, D.; Holt, A. B.; Netoff, T. I.; Moehlis, J., Optimal entrainment of heterogeneous noisy neurons, Front. Neurosci., 9, 192 (2015) · doi:10.3389/fnins.2015.00192
[59] Wilson, D.; Moehlis, J., Isostable reduction with applications to time-dependent partial differential equations, Phys. Rev. E, 94, 1, 012211 (2016) · doi:10.1103/PhysRevE.94.012211
[60] Wilson, D. D., An optimal framework for nonfeedback stability control of chaos, SIAM J. Appl. Dyn. Syst., 18, 4, 1982-1999 (2019) · Zbl 1443.34060 · doi:10.1137/18M1229146
[61] Winfree, A., The Geometry of Biological Time (2001), Springer Verlag: Springer Verlag, New York · Zbl 1014.92001
[62] Zlotnik, A.; Chen, Y.; Kiss, I. Z.; Tanaka, H. A.; Li, J. S., Optimal waveform for fast entrainment of weakly forced nonlinear oscillators, Phys. Rev. Lett., 111, 2, 024102 (2013) · doi:10.1103/PhysRevLett.111.024102
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