Abstract
A new numerical algorithm for computation of phase response curves of stable limit cycle oscillators is proposed. The idea of the algorithm originates from a direct method that is based on computation of the oscillator response to short finite pulses delivered at different phases of oscillations. Here we adapt the direct method to the case of infinitesimal perturbations and compare our algorithm with the standard algorithm based on the backward integration of the adjoint equations. In contrast to the standard algorithm, our algorithm does not require any backward integration and it is easier to program since a necessity of numerical interpolation for the Jacobian matrix is avoided. In addition, we demonstrate by examples that our algorithm is faster than the standard algorithm and this advantage is especially notable for weakly stable limit cycle oscillators.
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References
Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Dover, New York (2003)
Pikovsky, A., Rosemblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Science. Cambridge University Press, Cambridge (2001)
Winfree, A.: The Geometry of Biological Time. Springer, New York (2001)
Hastings, J.W., Sweeney, B.M.: A persistent diurnal rhythm of luminescence in Gonyaulax polyedra. Biol. Bull. 115, 440–458 (1958)
Johnson, C.H.: Forty years of PRC—what have we learned? Chronobiol. Int. 16, 711–743 (1999)
Winfree, A.T.: When Time Breaks Down. Princeton University Press, Princeton (1987)
Ikeda, N.: Model of bidirectional interaction between myocardial pacemakers based on the phase response curve. Biol. Cybern. 43, 157–167 (1982)
Tsalikakisa, D.G., Zhangb, H.G., Fotiadisa, D.I., Kremmydasa, G.P., Michalis, L.K.: Phase response characteristics of sinoatrial node cells. Comput. Biol. Med. 37, 8–20 (2007)
Ermentrout, G.B., Kopell, N.: Multiple pulse interactions and averaging in systems of coupled neural oscillators. J. Math. Biol. 29, 195–217 (1991)
Ermentrout, G.B.: Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM J. Appl. Math. 52, 1665–1687 (1992)
Ermentrout, B.: Type I membranes, phase resetting curves, and synchrony. Neural Comput. 8, 979–1001 (1996)
Izhikevich, E.M.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, Cambridge (2007)
Tass, P.: Phase Resetting in Medicine and Biology. Springer, Berlin (1999)
Stiger, T., Danzl, P., Moehlis, J., Netoff, T.I.: Linear control of neuronal spike timing using phase response curves. J. Med. Devices 4, 027533 (2010)
Reyes, A.D., Fetz, E.E.: Two modes of interspike interval shortening by brief transient depolarizations in cat neocortical neurons. J. Neurophysiol. 69, 1661–1672 (1993)
Galan, R.F., Ermentrout, G.B., Urban, N.N.: Efficient estimation of phase-resetting curves in real neurons and its significance for neural-network modeling. Phys. Rev. Lett. 94, 158101 (2005)
Tateno, T., Robinson, H.P.C.: Phase resetting curves and oscillatory stability in interneurons of rat somatosensory cortex. Biophys. J. 92, 683–695 (2007)
Brown, E., Moehlis, J., Holmes, P.: On the phase reduction and response dynamics of neural oscillator populations. Neural Comput. 16, 673–715 (2004)
Kuznetsov, A.P., Stankevich, N.V., Turukina, L.V.: Coupled van der Pol–Duffing oscillators: phase dynamics and structure of synchronization tongues. Physica D 238, 1203–1215 (2009)
Sherwood, E.W., Guckenheimer, J.: Dissecting the phase response of a model bursting neuron. SIAM J. Appl. Dyn. Syst. 9, 659–703 (2010)
Guillamon, A., Huguet, G.: A computational and geometric approach to phase resetting curves and surfaces. SIAM J. Appl. Dyn. Syst. 8, 1005–1042 (2009)
Guevara, M.R., Glass, L., Mackey, M.C., Shier, A.: Chaos in neurobiology. IEEE Trans. Syst. Man Cybern. 13(5), 790–798 (1983)
Ermentrout, G.: Simulating, Analysing, and Animating Dynamical Systems: A Guide to XXPAUT for Researchers and Students. SIAM, Philadelphia (2002)
Govaerts, W., Sautois, B.: Computation of the phase response curve: a direct numerical approach. Neural Comput. 18, 817–847 (2006)
Malkin, I.G.: Methods of Poincaré and Lyapunov in Theory of Non-linear Oscillations. Gostexizdat, Moscow (1949) (in Russian: Metodi Puankare i Liapunova v teorii nelineinix kolebanii)
Malkin, I.G.: Some Problems in Nonlinear Oscillation Theory. Gostexizdat, Moscow (1956) (in Russian: Nekotorye zadachi teorii nelineinix kolebanii)
Hoppensteadt, F.C., Izhikevich, E.M.: Weakly Connected Neural Networks. Springer, New York (1997)
Winfree, A.: Patterns of phase compromise in biological cycles. J. Math. Biol. 1, 73–95 (1974)
Guckenheimer, J.: Isochrons and phaseless sets. J. Math. Biol. 1, 259–273 (1975)
Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193–213 (1981)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)
Hindmarsh, J.L., Rose, R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B, Biol. Sci. 221, 87–102 (1984)
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Novičenko, V., Pyragas, K. Computation of phase response curves via a direct method adapted to infinitesimal perturbations. Nonlinear Dyn 67, 517–526 (2012). https://doi.org/10.1007/s11071-011-0001-y
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DOI: https://doi.org/10.1007/s11071-011-0001-y