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The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems

Published online by Cambridge University Press:  02 April 2018

Y. PARK ,
K. M. SHAW ,
H. J. CHIEL  and
P. J. THOMAS
Y. PARK
Affiliation:
Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA email: yop6@pitt.edu, pjthomas@case.edu Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
K. M. SHAW
Affiliation:
Department of Biology, Case Western Reserve University, Cleveland, OH 44106, USA e-mail: hjc@case.edu Department of Anesthesia, Critical Care, Small and Pain Medicine, Massachusetts General Hospital, Boston, MA 02114, USA email: kmshaw@partners.org
H. J. CHIEL
Affiliation:
Department of Biology, Case Western Reserve University, Cleveland, OH 44106, USA e-mail: hjc@case.edu
P. J. THOMAS
Affiliation:
Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA email: yop6@pitt.edu, pjthomas@case.edu
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Abstract

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The asymptotic phase θ of an initial point x in the stable manifold of a limit cycle (LC) identifies the phase of the point on the LC to which the flow φt(x) converges as t → ∞. The infinitesimal phase response curve (iPRC) quantifies the change in timing due to a small perturbation of a LC trajectory. For a stable LC in a smooth dynamical system, the iPRC is the gradient ∇x(θ) of the phase function, which can be obtained via the adjoint of the variational equation. For systems with discontinuous dynamics, the standard approach to obtaining the iPRC fails. We derive a formula for the iPRCs of LCs occurring in piecewise smooth (Filippov) dynamical systems of arbitrary dimension, subject to a transverse flow condition. Discontinuous jumps in the iPRC can occur at the boundaries separating subdomains, and are captured by a linear matching condition. The matching matrix, M, can be derived from the saltation matrix arising in the associated variational problem. For the special case of linear dynamics away from switching boundaries, we obtain an explicit expression for the iPRC. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). We apply the iPRCs obtained to study synchronization and phase-locking in piecewise smooth LC systems in which synchronization arises solely due to the crossing of switching manifolds.

Keywords

Mathematical biologyNonsmooth analysisPhase plane analysisNonlinear oscillators

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 5: Theory and applications of nonsmooth dynamical systems , October 2018 , pp. 905 - 940
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†This work was supported in part by NSF grant DMS-1413770 and NSF grant DMS-1010434.

References

[1] Acary, V., De Jong, H. & Brogliato, B. (2014) Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems. Physica D: Nonlinear Phenom. 269, 103–119.Google Scholar
[2] Aizerman, M. & Gantmakher, F. (1958) On the stability of periodic motions. J. Appl. Math. Mech. 22 (6), 1065–1078.Google Scholar
[3] Bizzarri, F., Linaro, D. & Storace, M. (2007) PWL approximation of the Hindmarsh-Rose neuron model in view of its circuit implementation. In: Proceedings of the 18th European Conference on Circuit Theory and Design, 2007, IEEE, pp. 878–881.Google Scholar
[4] Brown, E., Moehlis, J. & Holmes, P. (2004) On the phase reduction and response dynamics of neural oscillator populations. Neural Comput. 16 (4), 673–715.Google Scholar
[5] Carmona, V., Fernández-García, S., Freire, E. & Torres, F. (2013) Melnikov theory for a class of planar hybrid systems. Physica D: Nonlinear Phenom. 248, 44–54. http://www.sciencedirect.com/science/article/pii/S0167278913000067Google Scholar
[6] Cheng, Y. (2013) Bifurcation of limit cycles of a class of piecewise linear differential systems in with three zones. Discrete Dyn. Nat. Soc. 2013, Art. no. 385419.Google Scholar
[7] Coombes, S. (2001) Phase locking in networks of synaptically coupled McKean relaxation oscillators. Physica D: Nonlinear Phenom. 160 (3), 173–188.Google Scholar
[8] Coombes, S. (2008) Neuronal networks with gap junctions: A study of piecewise linear planar neuron models. SIAM J. Appl. Dyn. Syst. 7, 1101–1129. http://link.aip.org/link/?SJA/7/1101/1Google Scholar
[9] Coombes, S. & Thul, R. (2016) Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function. Eur. J. Appl. Math. 27 (6), 904–922.Google Scholar
[10] Coombes, S., Thul, R. & Wedgwood, K. (2012) Nonsmooth dynamics in spiking neuron models. Physica D: Nonlinear Phenom. 241 (22), 2042–2057.Google Scholar
[11] Di Bernardo, M., Budd, C., Champneys, A. R. & Kowalczyk, P. (2008) Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, Vol. 163, Springer Science & Business Media, London.Google Scholar
[12] Edwards, R. & Gill, P. (2003) On synchronization and cross-talk in parallel networks. Dynamics of Continuous Discrete and Impulsive Systems Series B 10, 287–300.Google Scholar
[13] Ermentrout, B. (1996) Type I membranes, phase resetting curves, and synchrony. Neural Comput. 8 (5), 979–1001.Google Scholar
[14] Ermentrout, G. B. & Terman, D. H. (2010) Foundations of Mathematical Neuroscience, Springer, Berlin, Germany.Google Scholar
[15] Ermentrout, G. & Kopell, N. (1991) Multiple pulse interactions and averaging in systems of coupled neural oscillators. J. Math. Biol. 29 (3), 195–217. http://dx.doi.org/10.1007/BF00160535Google Scholar
[16] Field, R. J. & Noyes, R. M. (1974) Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys. 60 (5), 1877–1884.Google Scholar
[17] Filippov, A. F. (1988) Differential Equations with Discontinuous Righthand Sides, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar
[18] Fruth, F., Jülicher, F. & Lindner, B. (2014) An active oscillator model describes the statistics of spontaneous otoacoustic emissions. Biophys. J. 107 (4), 815–824.Google Scholar
[19] Glass, L. & Pasternack, J. S. (1978) Stable oscillations in mathematical models of biological control systems. J. Math. Biol. 6, 207–223.Google Scholar
[20] Glass, L. & Pérez, R. (1974) Limit cycle oscillations in compartmental chemical systems. J. Chem. Phys. 61 (12), 5242–5249.Google Scholar
[21] Guckenheimer, J. (1975) Isochrons and phaseless sets. J. Math. Biol. 1, 259–273. 10.1007/BF01273747. http://dx.doi.org/10.1007/BF01273747Google Scholar
[22] Guckenheimer, J. & Holmes, P. (1990) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 3rd ed., Vol. 42, Springer-Verlag, Berlin, Germany.Google Scholar
[23] Hawkins, J. (1887) Automatic regulators for heating apparatus. Trans. Amer. Soc. Mech. Eng. 9, 432.Google Scholar
[24] Huan, S.-M. & Yang, X.-S. (2012) On the number of limit cycles in general planar piecewise linear systems. Discrete Continuous Dyn. Syst. 32 (6), 2147–2164.Google Scholar
[25] Ijspeert, A. J. (2008) Central pattern generators for locomotion control in animals and robots: A review. Neural Netw. 21 (4), 642–53.Google Scholar
[26] Izhikevich, E. M. (2000) Phase equations for relaxation oscillators. SIAM J. Appl. Math. 60 (5), 1789–1804.Google Scholar
[27] Izhikevich, E. M. (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Computational Neuroscience, MIT Press, Cambridge, Massachusetts.Google Scholar
[28] Izhikevich, E. M. & Ermentrout, B. (2008) Phase model. Scholarpedia 3 (10), 1487.Google Scholar
[29] Johansson, K. H., Barabanov, A. E. & Åström, K. J. (2002) Limit cycles with chattering in relay feedback systems. IEEE Trans. Autom. Control 47 (9), 1414–1423.Google Scholar
[30] Kelso, J. S., Holt, K. G., Rubin, P. & Kugler, P. N. (1981) Patterns of human interlimb coordination emerge from the properties of non-linear, limit cycle oscillatory processes: theory and data. J. Motor Behavior 13 (4), 226–261.Google Scholar
[31] Kuramoto, Y. (2003) Chemical Oscillations Waves and Turbulence, Dover, Mineola, New York, USA.Google Scholar
[32] Leine, R. & Nijmeijer, H. (2004) Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Vol. 18, Springer-Verlag, Heidelberg, Germany.Google Scholar
[33] Lin, C., Wang, Q.-G. & Lee, T. H. (2003) Local stability of limit cycles for MIMO relay feedback systems. J. Math. Anal. Appl. 288 (1), 112–123.Google Scholar
[34] Lin, H. & Antsaklis, P. J. (2009) Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54 (2), 308–322.Google Scholar
[35] Lin, K. K., Wedgwood, K. C. A., Coombes, S. & Young, L.-S. (2012) Limitations of perturbative techniques in the analysis of rhythms and oscillations. J. Math. Biol 66, 139–161.Google Scholar
[36] Llibre, J. & Ponce, E. (2012) Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 19 (3), 325–335.Google Scholar
[37] Lyttle, D. N., Gill, J. P., Shaw, K. M., Thomas, P. J. & Chiel, H. J. (2017) Robustness, flexibility, and sensitivity in a multifunctional motor control model. Biol. Cybern. 111 (1), 25–47.Google Scholar
[38] Ma, Y., Yuan, R., Li, Y., Ao, P. & Yuan, B. (2013) Lyapunov functions in piecewise linear systems: From fixed point to limit cycle. arXiv:1306.6880.Google Scholar
[39] Machina, A. & Ponosov, A. (2011) Filippov solutions in the analysis of piecewise linear models describing gene regulatory networks. Nonlinear Anal.: Theor. Methods Appl. 74 (3), 882–900.Google Scholar
[40] McKean, H. (1970) Nagumo's equation. Adv. Math. 4, 209–223.Google Scholar
[41] Modolo, J., Henry, J. & Beuter, A. (2008) Dynamics of the subthalamo-pallidal complex in Parkinsonõs disease during deep brain stimulation. J. Biol. Phys. 34 (3–4), 251–266.Google Scholar
[42] Morrison, K., Degeratu, A., Itskov, V. & Curto, C. (2016) Diversity of emergent dynamics in competitive threshold-linear networks: A preliminary report. arXiv:1605.04463.Google Scholar
[43] Morse, A. (1997) Control Using Logic Based Switching, Lecture Notes in Control and Information Sciences, Vol. 222.Google Scholar
[44] Nowotny, T. & Rabinovich, M. I. (2007) Dynamical origin of independent spiking and bursting activity in neural microcircuits. Phys. Rev. Lett. 98, 128106.Google Scholar
[45] Osinga, H. M. & Moehlis, J. (2010) Continuation-based computation of global isochrons. SIAM J. Appl. Dyn. Syst. 9 (4), 1201–1228.Google Scholar
[46] Park, Y. (2013) Infinitesimal Phase Response Curves for Piecewise Smooth Dynamical Systems, Master's Thesis, Case Western Reserve University.Google Scholar
[47] Park, Y. & Ermentrout, B. (2016) Weakly coupled oscillators in a slowly varying world. J. Comput. Neurosci. 40 (3), 269–281.Google Scholar
[48] Pettit, N. B. (1996) Analysis of Piecewise Linear Dynamical Systems, John Wiley & Sons, New York, New York, USA.Google Scholar
[49] Pettit, N. B. & Wellstead, P. E. (1995) Analyzing piecewise linear dynamical systems. Control Systems, IEEE 15 (5), 43–50.Google Scholar
[50] Poggi, T., Sciutto, A. & Storace, M. (2009) Piecewise linear implementation of nonlinear dynamical systems: From theory to practice. Electron. Lett. 45 (19), 966–967.Google Scholar
[51] Ponce, E., Ros, J. & Vela, E. (2013) The focus-center-limit cycle bifurcation in discontinuous planar piecewise linear systems without sliding. In: Ibáñez, S., Pérez del Río, J. S., Pumariño, A. & Rodríguez, J. Á. (editors), Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics and Statistics, Vol. 54, Springer, Berlin Heidelberg, pp. 335–349. http://dx.doi.org/10.1007/978-3-642-38830-9_21Google Scholar
[52] Rohden, M., Sorge, A., Timme, M. & Witthaut, D. (2012) Self-organized synchronization in decentralized power grids. Phys. Rev. Lett. 109 (6), 064101.Google Scholar
[53] Schwemmer, M. A. & Lewis, T. J. (2012) The Theory of Weakly Coupled Oscillators. In: Phase Response Curves in Neuroscience Theory, Experiment, and Analysis, Springer Series in Computational Neuroscience, Vol. 6, Springer, pp. 3–31.Google Scholar
[54] Shaw, K. M., Lyttle, D. N., Gill, J. P., Cullins, M. J., McManus, J. M., Lu, H., Thomas, P. J. & Chiel, H. J. (2015) The significance of dynamical architecture for adaptive responses to mechanical loads during rhythmic behavior. J. Comput. Neurosci. 38 (1), 25–51.Google Scholar
[55] Shaw, K. M., Park, Y., Chiel, H. J. & Thomas, P. J. (2012) Phase resetting in an asymptotically phaseless system: On the phase response of limit cycles verging on a heteroclinic orbit. SIAM J. Appl. Dyn. Syst. 11 (1), 350–391.Google Scholar
[56] Shirasaka, S., Kurebayashi, W. & Nakao, H. (2017) Phase reduction theory for hybrid nonlinear oscillators. Phys. Rev. E 95 (1), 012212.Google Scholar
[57] Simpson, D. J. W. & Jeffrey, M. R. (2016) Fast phase randomization via two-folds. Proc. R. Soc. A: Math. Phys. Eng. Sci. 472 (2186), 20150782.Google Scholar
[58] So, P., Francis, J. T., Netoff, T. I., Gluckman, B. J. & Schiff, S. J. (1998) Periodic orbits: a new language for neuronal dynamics. Biophys. J. 74 (6), 2776–2785.Google Scholar
[59] Stensby, J. L. (1997) Phase-Locked Loops: Theory and Applications, CRC Press, Boca Raton, Florida.Google Scholar
[60] Storace, M. & De Feo, O. (2004) Piecewise-linear approximation of nonlinear dynamical systems. IEEE Trans. Circuits Syst. I: Regular Papers 51 (4), 830–842.Google Scholar
[61] Tsypkin, I. Z. (1984) Relay Control Systems, Cambridge University Press, Cambridge, UK.Google Scholar
[62] G.aiko, V. A. & van Horssen, W. T. (2009) Global analysis of a piecewise linear Liénard-type dynamical system. Int. J. Dyn. Syst. Differ. Equ. 2 (1–2), 115–128.Google Scholar
[63] Walsh, J., Widiasih, E., Hahn, J. & McGehee, R. (2016) Periodic orbits for a discontinuous vector field arising from a conceptual model of glacial cycles. Nonlinearity 29 (6), 1843.Google Scholar
[64] Zinovik, I., Chebiryak, Y. & Kroening, D. (2010) Periodic orbits and equilibria in Glass models for gene regulatory networks. IEEE Trans. Inform. Theory 56 (2), 805–820.Google Scholar