A. T. Winfree, The Geometry of Biological Time (Biomathematics, vol. 8), 1st ed. New York: Springer-Verlag, 1980.
Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer Series in Synergetics, vol. 19), 1st ed. Berlin Heidelberg, Germany: Springer-Verlag, 1984.
L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life. Princeton, NJ: Princeton Univ. Press, 1988.
A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour. Cambridge, U.K.: Cambridge Univ. Press, 1996.
A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge Nonlinear Science Series, vol. 12). Cambridge, U.K.: Cambridge Univ. Press, 2001.
E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Cambridge, MA: MIT Press, 2007.
R. Sepulchre, "Oscillators as systems and synchrony as a design principle," in Current Trends in Nonlinear Systems and Control: In Honor of Petar Kokotović and Turi Nicosia, L. Menini, L. Zaccarian, and C. T. Abdallah, Eds. Boston, MA: Birkhäuser, 2006, pp. 123-141.
G. Zames and A. K. El-Sakkary, "Unstable systems and feedback: The gap metric," in Proc. 16th Allerton Conf., Oct. 1980, pp. 380-385.
A. K. El-Sakkary, "The gap metric: Robustness of stabilization of feedback systems," IEEE Trans. Automat. Control, vol. 30, no. 3, pp. 240-247, Mar. 1985.
G. Vinnicombe, Uncertainty and Feedback: H∞ Loop-Shaping and the o-Gap Metric. London, U.K.: Imperial College Press, 2000.
T. T. Georgiou, "Distances and Riemannian metrics for spectral density functions," IEEE Trans. Signal Process., vol. 55, no. 8, pp. 3995-4003, Aug. 2007. (Pubitemid 47261522)
P. J. DeCoursey, "Daily light sensitivity rhythm in a rodent," Science, vol. 131, no. 3392, pp. 33-35, Jan. 1960.
A. Goldbeter, "Computational approaches to cellular rhythms," Nature, vol. 420, no. 6912, pp. 238-245, Nov. 2002. (Pubitemid 35340136)
N. W. Schultheiss, A. A. Prinz, and R. J. Butera, Eds., Phase Response Curves in Neuroscience: Theory, Experiment, and Analysis (Springer Series in Computational Neuroscience, vol. 6). New York: Springer, 2012.
A. T. Winfree, "Biological rhythms and the behavior of populations of coupled oscillators," J. Theor. Biol., vol. 16, no. 1, pp. 15-42, July 1967.
Y. Kuramoto, "Self-entrainment of a population of coupled non-linear oscillators," in Proc. Int. Symp. Mathematical Problems in Theoretical Physics, 1975, pp. 420-422.
S. H. Strogatz, "From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators," Physica D, vol. 143, nos. 1-4, pp. 1-20, Sept. 2000.
S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order. New York: Hyperion, 2003.
F. Dörfler and F. Bullo. (2013, Apr.). Synchronization in complex oscillator networks: A survey. Automatica [Online]. Available: http://motion. me.ucsb.edu/pdf/2013b-db.pdf
P. Sacré, "Systems analysis of oscillator models in the space of phase response curves," Ph.D. dissertation, Univ. Liège, Dept. Elec. Eng. Comp. Sci., Belgium, Sept. 2013.
P. Sacré and R. Sepulchre, "Matching an oscillator model to a phase response curve," in Proc. 50th IEEE Conf. Decision Control and European Control Conf., Orlando, Florida, Dec. 2011, pp. 3909-3914.
P. Sacré and R. Sepulchre. (2012, Nov.). Sensitivity analysis of circadian entrainment in the space of phase response curves. [Online]. Available: arXiv:1211.7317
M. Farkas, Periodic Motions (Applied Mathematical Sciences, vol. 104). New York: Springer-Verlag, 1994.
B. Pfeuty, Q. Thommen, and M. Lefranc, "Robust entrainment of circadian oscillators requires specific phase response curves," Biophys. J., vol. 100, no. 11, pp. 2557-2565, June 2011.
S. A. Oprisan, V. Thirumalai, and C. C. Canavier, "Dynamics from a time series: Can we extract the phase resetting curve from a time series?" Biophys. J., vol. 84, no. 5, pp. 2919-2928, May 2003. (Pubitemid 36534463)
S. Achuthan and C. C. Canavier, "Phase-resetting curves determine synchronization, phase locking, and clustering in networks of neural oscillators," J. Neurosci., vol. 29, no. 16, pp. 5218-5233, Apr. 2009.
S. Wang, M. M. Musharoff, C. C. Canavier, and S. Gasparini, "Hippocampal CA1 pyramidal neurons exhibit type 1 phase-response curves and type 1 excitability," J. Neurophysiol., vol. 109, no. 11, pp. 2757-2766, Mar. 2013.
G. B. Ermentrout and N. Kopell, "Frequency plateaus in a chain of weakly coupled oscillators, I," SIAM. J. Math. Anal., vol. 15, no. 2, pp. 215-237, 1984.
R. E. Mirollo and S. H. Strogatz, "Synchronization of pulse-coupled biological oscillators," SIAM J. Appl. Math., vol. 50, no. 6, pp. 1645-1662, 1990.
A. Mauroy, P. Sacré, and R. Sepulchre, "Kick synchronization versus diffusive synchronization," in Proc. 51st IEEE Conf. Decision Control, Maui, Hawaii, Dec. 2013, pp. 7171-7183.
F. Dörfler and F. Bullo, "Exploring synchronization in complex oscillator networks," in Proc. 51st IEEE Conf. Decision Control, Maui, Hawaii, Oct. 2012, pp. 7157-7170.
F. C. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks (Applied Mathematical Sciences, vol. 126). New York: Springer-Verlag, 1997.
Y. Kuramoto, "Phase-and center-manifold reductions for large populations of coupled oscillators with application to non-locally coupled systems," Int. J. Bifurcat. Chaos, vol. 7, no. 4, pp. 789-805, Apr. 1997. (Pubitemid 127338618)
E. T. Brown, J. Moehlis, and P. Holmes, "On the phase reduction and response dynamics of neural oscillator populations," Neural Comput., vol. 16, no. 4, pp. 673-715, Apr. 2004. (Pubitemid 38318129)
U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Englewood Cliffs, NJ: Prentice-Hall, Feb. 1988.
R. Seydel, Practical Bifurcation and Stability Analysis, 3rd ed. (Interdisciplinary Applied Mathematics, vol. 5). New York: Springer-Verlag, 2010.
I. G. Malkin, The Methods of Lyapunov and Poincare in the Theory of Nonlinear Oscillations. Moscow, Russia: Gostexizdat, 1949.
I. G. Malkin, Some Problems in the Theory of Nonlinear Oscillations. Moscow, Russia: Gostexizdat, 1956.
J. C. Neu, "Coupled chemical oscillators," SIAM J. Appl. Math., vol. 37, no. 2, pp. 307-315, 1979.
W. Govaerts and B. Sautois, "Computation of the phase response curve: A direct numerical approach," Neural Comput., vol. 18, no. 4, pp. 817-847, Apr. 2006. (Pubitemid 43543823)
A. T. Winfree, "Patterns of phase compromise in biological cycles," J. Math. Biol., vol. 1, no. 1, pp. 73-95, 1974.
S. R. Taylor, R. Gunawan, L. R. Petzold, and F. J. Doyle III, "Sensitivity measures for oscillating systems: Application to mammalian circadian gene network," IEEE Trans. Automat. Control, vol. 53, pp. 177-188, Jan. 2008. (Pubitemid 351314548)
A. Goh and R. Vidal, "Unsupervised Riemannian clustering of probability density functions," in Machine Learning and Knowledge Discovery in Databases. Berlin Heidelberg, Germany: Springer-Verlag, 2008, pp. 377-392.
M. A. Kramer, H. Rabitz, and J. M. Calo, "Sensitivity analysis of oscillatory systems," Appl. Math. Model., vol. 8, no. 5, pp. 328-340, Oct. 1984.
E. Rosenwasser and R. Yusupov, Sensitivity of Automatic Control Systems. Boca Raton, FL: CRC Press, 1999.
B. P. Ingalls, "Autonomously oscillating biochemical systems: Parametric sensitivity of extrema and period," Syst. Biol., vol. 1, no. 1, pp. 62-70, June 2004.
A. K. Wilkins, B. Tidor, J. White, and P. I. Barton, "Sensitivity analysis for oscillating dynamical systems," SIAM J. Sci. Comput., vol. 31, no. 4, pp. 2706-2732, 2009.
I. Vytyaz, D. C. Lee, P. K. Hanumolu, U.-K. Moon, and K. Mayaram, "Sensitivity analysis for oscillators," IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 27, no. 9, pp. 1521-1534, Sept. 2008.
J. Stelling, E. D. Gilles, and F. J. Doyle III, "Robustness properties of circadian clock architectures," Proc. Natl. Acad. Sci. USA, vol. 101, no. 36, pp. 13210-13215, Sept. 2004. (Pubitemid 39209644)
A. K. Wilkins, P. I. Barton, and B. Tidor, "The Per2 negative feedback loop sets the period in the mammalian circadian clock mechanism," PLoS Comput. Biol., vol. 3, no. 12, Article e242, Dec. 2007.
N. Bagheri, J. Stelling, and F. J. Doyle III, "Quantitative performance metrics for robustness in circadian rhythms," Bioinformatics, vol. 23, no. 3, pp. 358-364, Feb. 2007. (Pubitemid 46323159)
R. Gunawan and F. J. Doyle III, "Phase sensitivity analysis of circadian rhythm entrainment," J. Biol. Rhythms, vol. 22, no. 2, pp. 180-194, Apr. 2007. (Pubitemid 46568653)
D. Gonze, J. Halloy, and A. Goldbeter, "Robustness of circadian rhythms with respect to molecular noise," Proc. Natl. Acad. Sci. USA, vol. 99, no. 2, pp. 673-678, Jan. 2002. (Pubitemid 34106570)
M. Hafner, P. Sacré, L. Symul, R. Sepulchre, and H. Koeppl, "Multiple feedback loops in circadian cycles: Robustness and entrainment as selection criteria," in Proc. 7th Int. Workshop Computational Systems Biology, Luxembourg, June 2010, pp. 51-54.
J.-C. Leloup and A. Goldbeter, "Toward a detailed computational model for the mammalian circadian clock," Proc. Natl. Acad. Sci. USA, vol. 100, no. 12, pp. 7051-7056, June 2003. (Pubitemid 36706391)
J.-C. Leloup and A. Goldbeter, "Modeling the mammalian circadian clock: Sensitivity analysis and multiplicity of oscillatory mechanisms," J. Theoret. Biol., vol. 230, no. 4, pp. 541-562, Oct. 2004. (Pubitemid 39195087)
B. C. Goodwin, "Oscillatory behavior in enzymatic control processes," Adv. Enzyme Regul., vol. 3, pp. 425-438, 1965.
J.-C. Leloup, D. Gonze, and A. Goldbeter, "Limit cycle models for circadian rhythms based on transcriptional regulation in Drosophila and Neurospora," J. Biol. Rhythms, vol. 14, no. 6, pp. 433-448, Dec. 1999. (Pubitemid 30086851)
J.-C. Leloup and A. Goldbeter, "Modeling the molecular regulatory mechanism of circadian rhythms in Drosophila," Bio Essays, vol. 22, no. 1, pp. 84-93, Jan. 2000. (Pubitemid 30057137)
D. Hansel, G. Mato, and C. Meunier, "Synchrony in excitatory neural networks," Neural Comput., vol. 7, no. 2, pp. 307-337, Mar. 1995.
R. F. Galán, N. Fourcaud-Trocmé, G. B. Ermentrout, and N. N. Urban, "Correlation-induced synchronization of oscillations in olfactory bulb neurons," J. Neurosci., vol. 26, no. 14, pp. 3646-3655, Apr. 2006.
R. F. Galán, G. B. Ermentrout, and N. N. Urban, "Reliability and stochastic synchronization in type I vs. type II neural oscillators," Neurocomputing, vol. 70, pp. 2102-2106, June 2007. (Pubitemid 46657018)
R. F. Galán, G. B. Ermentrout, and N. N. Urban, "Stochastic dynamics of uncoupled neural oscillators: Fokker-Planck studies with the finite element method," Phys. Rev. E, vol. 76, no. 5 pt. 2, Article 056110, Nov. 2007.
S. Marella and G. B. Ermentrout, "Class-II neurons display a higher degree of stochastic synchronization than class-I neurons," Phys. Rev. E, vol. 77, no. 4, pt. 1, Article 041918, Apr. 2008.
A. Abouzeid and G. B. Ermentrout, "Type-II phase resetting curve is optimal for stochastic synchrony," Phys. Rev. E, vol. 80, no. 1, Article 011911, July 2009.
S. Hata, K. Arai, R. F. Galán, and H. Nakao, "Optimal phase response curves for stochastic synchronization of limit-cycle oscillators by common Poisson noise," Phys. Rev. E, vol. 84, no. 1, Article 016229, July 2011.
G. B. Ermentrout, "Type I membranes, phase resetting curves, and synchrony," Neural Comput., vol. 8, no. 5, pp. 979-1001, July 1996.
G. B. Ermentrout, L. Glass, and B. E. Oldeman, "The shape of phaseresetting curves in oscillators with a saddle node on an invariant circle bifurcation," Neural Comput., vol. 24, no. 12, pp. 3111-3125, Dec. 2012.
C. Morris and H. Lecar, "Voltage oscillations in the barnacle giant muscle fiber," Biophys. J., vol. 35, no. 1, pp. 193-213, July 1981. (Pubitemid 11079105)
J. Rinzel and G. B. Ermentrout, "Analysis of neural excitability and oscillations," in Methods in Neuronal Modeling: From Ions to Networks. Cambridge, MA: MIT Press, 1998, pp. 251-291.
K. Tsumoto, H. Kitajima, T. Yoshinaga, K. Aihara, and H. Kawakami, "Bifurcations in Morris-Lecar neuron model," Neurocomputing, vol. 69, nos. 4-6, pp. 293-316, Jan. 2006. (Pubitemid 43009116)
T. Kreuz, D. Chicharro, C. Houghton, R. G. Andrzejak, and F. Mormann, "Monitoring spike train synchrony," J. Neurophysiol., vol. 109, no. 5, pp. 1457-1472, Mar. 2013.
L. Trotta, E. Bullinger, and R. Sepulchre, "Global analysis of dynamical decision-making models through local computation around the hidden saddle," PloS ONE, vol. 7, no. 3, Article e33110, Mar. 2012.
C. H. Johnson, "An atlas of phase responses curves for circadian and circatidal rhythms," Ph.D. dissertation, Dept. Biol., Vanderbilt Univ., Nashville, TN, 1990.
F. X. Kaertner, "Determination of the correlation spectrum of oscillators with low noise," IEEE Trans. Microwave Theory Tech., vol. 37, no. 1, pp. 90-101, Jan. 1989.
F. X. Kaertner, "Analysis of white and f-α noise in oscillators," Int. J. Circuit Theor. Appl., vol. 18, no. 5, pp. 485-519, Sept. 1990.
A. Demir, A. Mehrotra, and J. Roychowdhury, "Phase noise in oscillators: A unifying theory and numerical methods for characterization," IEEE Trans. Circuits Syst. I, vol. 47, no. 5, pp. 655-674, May 2000.
A. Demir, "Phase noise and timing jitter in oscillators with colorednoise sources," IEEE Trans. Circuits Syst. I, vol. 49, no. 12, pp. 1782-1791, Dec. 2002.
A. Demir, "Computing timing jitter from phase noise spectra for oscillators and phase-locked loops with white and 1=f noise," IEEE Trans. Circuits Syst. I, vol. 53, no. 9, pp. 1869-1884, Sept. 2006. (Pubitemid 44500794)
I. Vytyaz, D. C. Lee, P. K. Hanumolu, U.-K. Moon, and K. Mayaram, "Automated design and optimization of low-noise oscillators," IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 28, no. 5, pp. 609-622, May 2009.
J. Guckenheimer, "Isochrons and phaseless sets," J. Math. Biol., vol. 1, no. 3, pp. 259-273, 1975.
A. Guillamon and G. Huguet, "A computational and geometric approach to phase resetting curves and surfaces," SIAM J. Appl. Dyn. Syst., vol. 8, no. 3, pp. 1005-1042, 2009.
H. M. Osinga and J. Moehlis, "Continuation-based computation of global isochrons," SIAM J. Appl. Dyn. Syst., vol. 9, no. 4, pp. 1201-1228, 2010.
W. E. Sherwood and J. Guckenheimer, "Dissecting the phase response of a model bursting neuron," SIAM J. Appl. Dyn. Syst., vol. 9, no. 3, pp. 659-703, 2010.
A. Mauroy and I. Mezić, "On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics," Chaos, vol. 22, no. 3, Article 033112, 2012.
A. Mauroy, I. Mezić, and J. Moehlis. (2013, Jan.). Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics [Online]. Available: arXiv:1302.0032
P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds. Princeton, NJ: Princeton Univ. Press, 2008.
H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002.
U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Classics in Applied Mathematics, vol. 13). Philadelphia, PA: Soc. Ind. Appl. Math., 1995.
K. Lust, "Improved numerical Floquet multipliers," Int. J. Bifurcat. Chaos, vol. 11, no. 9, pp. 2389-2410, Sept. 2001.
J. Guckenheimer and B. Meloon, "Computing periodic orbits and their bifurcations with automatic differentiation," SIAM J. Sci. Comput., vol. 22, no. 3, pp. 951-985, 2000. (Pubitemid 32548796)
