# American Institute of Mathematical Sciences

August  2012, 32(8): 2971-2995. doi: 10.3934/dcds.2012.32.2971

## Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators

 1 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2, Canada, Canada 2 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2

Received  April 2011 Revised  September 2011 Published  March 2012

We consider a pair of uncoupled conditional oscillators near a subcritical Hopf bifurcation that are driven by two weak white noise sources, one intrinsic and one common. In this context the noise drives oscillations in a setting where the underlying deterministic dynamics are quiescent. Synchronization of these noise-driven oscillations is considered, where the noise is also driving synchronization. We first derive the envelope equations of the noise-driven oscillations using a stochastic multiple scales method, providing access to phase and amplitude information. Using both a linearized approximation and an asymptotic analysis of the nonlinear system, we obtain approximations for the probability density of the phase difference of the oscillators. It is found that common noise increases the degree of synchrony in the pair of oscillators, which can be characterized by the ratio of intrinsic to common noise. Asymptotic expressions for the phase difference density provide explicit parametric expressions for the probability of observing different phase dynamics: in-phase synchronization, phase shifted oscillations, and non-synchronized states. Computational results are provided to support analytical conclusions.
Citation: William F. Thompson, Rachel Kuske, Yue-Xian Li. Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2971-2995. doi: 10.3934/dcds.2012.32.2971
##### References:
 [1] L. Arnold and N. Sri Namachchivaya and K. Schenk-Hoppé, Toward an understanding of the stochastic Hopf bifurcation: A case study, Int. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 1947-1975. doi: 10.1142/S0218127496001272. [2] P. H. Baxendale, Stochastic Averaging and asymptotic behavior of the stochastic Duffing-van der Pol equation, Stoch. Proc. and Appl., 113 (2004), 235-272. doi: 10.1016/j.spa.2004.05.001. [3] P. H. Baxendale and P. E. Greenwood, Sustained oscillations for density dependent Markov processes, J. Math. Bio., 63 (2011), 433-457. doi: 10.1007/s00285-010-0376-2. [4] R. Brette and E. Guigon, Reliability of spike timing is a general property of spiking model neurons, Neural Comp., 15 (2003), 279-308. doi: 10.1162/089976603762552924. [5] H. L. Bryant and J. P. Segundo, Spike initiation by transmembrane current: A white-noise analysis, J. Physiol., 260 (1976), 279-314. [6] T. Caraballo and P. E. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch. Dyn., 8 (2008), 139-154. doi: 10.1142/S0219493708002184. [7] M. G. Earl and S. H. Strogatz, Synchronization in oscillator networks with delayed coupling: A stability criterion, Phys. Rev E., 67 (2003), 036204. doi: 10.1103/PhysRevE.67.036204. [8] B. Ermentrout and T.-W. Ko, Delays and weakly coupled neuronal oscillators, Phil. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 1097-1115. doi: 10.1098/rsta.2008.0259. [9] R. F. Galán, G. B. Ermentrout and N. N. Urban, Optimal time scale for spike-time reliability, J. Neurophysiol., 99 (2008), 277-283. [10] R. F. Galán, N. Fourcaud-Trocmé, G. B. Ermentrout and N. N. Urban, Correlation-induced synchronization of oscillations in olfactory bulb neurons, Jour. Neurosci., 26 (2006), 3646-3655. doi: 10.1523/JNEUROSCI.4605-05.2006. [11] D. García-Alvarez, A. Bahraminasab, A. Stefanovska and P. V. E. McClintock, Competition between noise and coupling in the induction of synchronisation, Europhys. Lett., 88 (2009), 30005. doi: 10.1209/0295-5075/88/30005. [12] C. W. Gardiner, "Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences,'' Springer Series in Synergetics, 13, Springer-Verlag, Berlin, 1983. [13] D. S. Goldobin and A. Pikovsky, Synchronization and desynchronization of self-sustained oscillators by common noise, Phys. Rev. E (3), 71 (2005), 045201, 4 pp. [14] D. S. Goldobin, J. Teramae, H. Nakao and G. B. Ermentrout, Dynamics of limit-cycle oscillators subject to general noise, Phys. Rev. Lett., 105 (2010), 154101. doi: 10.1103/PhysRevLett.105.154101. [15] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544. [16] R. P. Kanwal, "Generalized Functions. Theory and Technique,'' Second edition, Birkhäuser Boston, Inc., Boston, MA, 1998. [17] J. Kevorkian and J. D. Cole, "Perturbation Methods in Applied Mathematics,'' Applied Mathematical Sciences, 34, Springer-Verlag, New York-Berlin, 1981. [18] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. [19] M. M. Klosek and R. Kuske, Multiscale analysis of stochastic delay differential equations, Mult. Model. Sim., 3 (2005), 706-729. doi: 10.1137/030601375. [20] Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,'' Dover, 2003. [21] R. Kuske, Multi-scale analysis of noise-sensitivity near a bifurcation, in "IUTAM Symposium on Nonlinear Stochastic Dynamics" (eds. Namachchivaya and Lin), Solid Mech. Appl., 110, Kluwer Acad. Publ., Dordrecht, (2003), 147-156. [22] R. Kuske, P. Greenwood and L. Gordillo, Sustained oscillations via coherence resonance in SIR, J. Theor. Bio., 245 (2007), 459-469. doi: 10.1016/j.jtbi.2006.10.029. [23] B. Lindner, J. Garcia-Ojalvo, A. Neiman and L. Schimansky-Geier, Effects of noise in excitable systems, Phys. Rep., 392 (2004), 321-424. doi: 10.1016/j.physrep.2003.10.015. [24] C. Ly and G. B. Ermentrout, Synchronization dynamics of two coupled neural oscillators receiving shared and unshared noisy stimuli, J. Comput. Neurosci., 26 (2009), 425-443. doi: 10.1007/s10827-008-0120-8. [25] Z. F. Mainen and T. J. Sejnowski, Reliability of spike timing in neocortical neurons, Science, 268 (1995), 1503-1506. [26] L. Markus and A. Weerasinghe, Stochastic oscillators, J. Diff. Eq., 71 (1988), 288-314. [27] L. Markus and A. Weerasinghe, Stochastic non-linear oscillators, Adv. Appl. Prob., 25 (1993), 649-666. doi: 10.2307/1427528. [28] C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophys. J., 35 (1981), 193-213. doi: 10.1016/S0006-3495(81)84782-0. [29] K. H. Nagai and H. Kori, Noise-induced synchronization of a large population of globally coupled nonidentical oscillators, Phys. Rev. E, 81 (2010), 065202. doi: 10.1103/PhysRevE.81.065202. [30] H. Nakao, K. Arai and Y. Kawamura, Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators, Phys. Rev. Lett., 98 (2007), 184101. doi: 10.1103/PhysRevLett.98.184101. [31] A. B. Neiman and D. F. Russell, Synchronization of noise-induced bursts in noncoupled sensory neurons, Phys. Rev. Lett., 88 (2002). [32] J. C. Neu, Coupled chemical oscillators, SIAM J. Appl. Math., 37 (1979), 307-315. doi: 10.1137/0137022. [33] A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences,'' Cambridge Nonlinear Science Series, 12, Cambridge University Press, Cambridge, 2001. [34] A. S. Pikovsky and J. Kurths, Coherence resonance in a noise-driven excitable system, Phys. Rev. Lett., 78 (1997), 775-778. doi: 10.1103/PhysRevLett.78.775. [35] J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, in "Methods in Neuronal Modeling: From Synapses to Networks," MIT Press, (1989), 135-169. [36] J. Teramae, H. Nakao and G. B. Ermentrout, Stochastic phase reduction for a general class of noisy limit cycle oscillators, Phys. Rev. Lett., 102 (2009), 194102. doi: 10.1103/PhysRevLett.102.194102. [37] J. Teramae and D. Tanaka, Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators, Phys. Rev. Lett., 93 (2004), 204103. doi: 10.1103/PhysRevLett.93.204103. [38] K. Yoshimura and K. Arai, Phase reduction of stochastic limit cycle oscillators, Phys. Rev. Lett., 101 (2008), 154101. doi: 10.1103/PhysRevLett.101.154101. [39] N. Yu, R. Kuske and Y.-X. Li, Stochastic phase dynamics: Multiscale behavior and coherence measures, Phys. Rev. E, 73 (2006), 056205. doi: 10.1103/PhysRevE.73.056205. [40] N. Yu, R. Kuske and Y.-X. Li, A computational study of spike time reliability in two cases of threshold dynamics,, preprint., ().

show all references

##### References:
 [1] L. Arnold and N. Sri Namachchivaya and K. Schenk-Hoppé, Toward an understanding of the stochastic Hopf bifurcation: A case study, Int. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 1947-1975. doi: 10.1142/S0218127496001272. [2] P. H. Baxendale, Stochastic Averaging and asymptotic behavior of the stochastic Duffing-van der Pol equation, Stoch. Proc. and Appl., 113 (2004), 235-272. doi: 10.1016/j.spa.2004.05.001. [3] P. H. Baxendale and P. E. Greenwood, Sustained oscillations for density dependent Markov processes, J. Math. Bio., 63 (2011), 433-457. doi: 10.1007/s00285-010-0376-2. [4] R. Brette and E. Guigon, Reliability of spike timing is a general property of spiking model neurons, Neural Comp., 15 (2003), 279-308. doi: 10.1162/089976603762552924. [5] H. L. Bryant and J. P. Segundo, Spike initiation by transmembrane current: A white-noise analysis, J. Physiol., 260 (1976), 279-314. [6] T. Caraballo and P. E. Kloeden and A. Neuenkirch, Synchronization of systems with multiplicative noise, Stoch. Dyn., 8 (2008), 139-154. doi: 10.1142/S0219493708002184. [7] M. G. Earl and S. H. Strogatz, Synchronization in oscillator networks with delayed coupling: A stability criterion, Phys. Rev E., 67 (2003), 036204. doi: 10.1103/PhysRevE.67.036204. [8] B. Ermentrout and T.-W. Ko, Delays and weakly coupled neuronal oscillators, Phil. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 1097-1115. doi: 10.1098/rsta.2008.0259. [9] R. F. Galán, G. B. Ermentrout and N. N. Urban, Optimal time scale for spike-time reliability, J. Neurophysiol., 99 (2008), 277-283. [10] R. F. Galán, N. Fourcaud-Trocmé, G. B. Ermentrout and N. N. Urban, Correlation-induced synchronization of oscillations in olfactory bulb neurons, Jour. Neurosci., 26 (2006), 3646-3655. doi: 10.1523/JNEUROSCI.4605-05.2006. [11] D. García-Alvarez, A. Bahraminasab, A. Stefanovska and P. V. E. McClintock, Competition between noise and coupling in the induction of synchronisation, Europhys. Lett., 88 (2009), 30005. doi: 10.1209/0295-5075/88/30005. [12] C. W. Gardiner, "Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences,'' Springer Series in Synergetics, 13, Springer-Verlag, Berlin, 1983. [13] D. S. Goldobin and A. Pikovsky, Synchronization and desynchronization of self-sustained oscillators by common noise, Phys. Rev. E (3), 71 (2005), 045201, 4 pp. [14] D. S. Goldobin, J. Teramae, H. Nakao and G. B. Ermentrout, Dynamics of limit-cycle oscillators subject to general noise, Phys. Rev. Lett., 105 (2010), 154101. doi: 10.1103/PhysRevLett.105.154101. [15] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544. [16] R. P. Kanwal, "Generalized Functions. Theory and Technique,'' Second edition, Birkhäuser Boston, Inc., Boston, MA, 1998. [17] J. Kevorkian and J. D. Cole, "Perturbation Methods in Applied Mathematics,'' Applied Mathematical Sciences, 34, Springer-Verlag, New York-Berlin, 1981. [18] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. [19] M. M. Klosek and R. Kuske, Multiscale analysis of stochastic delay differential equations, Mult. Model. Sim., 3 (2005), 706-729. doi: 10.1137/030601375. [20] Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,'' Dover, 2003. [21] R. Kuske, Multi-scale analysis of noise-sensitivity near a bifurcation, in "IUTAM Symposium on Nonlinear Stochastic Dynamics" (eds. Namachchivaya and Lin), Solid Mech. Appl., 110, Kluwer Acad. Publ., Dordrecht, (2003), 147-156. [22] R. Kuske, P. Greenwood and L. Gordillo, Sustained oscillations via coherence resonance in SIR, J. Theor. Bio., 245 (2007), 459-469. doi: 10.1016/j.jtbi.2006.10.029. [23] B. Lindner, J. Garcia-Ojalvo, A. Neiman and L. Schimansky-Geier, Effects of noise in excitable systems, Phys. Rep., 392 (2004), 321-424. doi: 10.1016/j.physrep.2003.10.015. [24] C. Ly and G. B. Ermentrout, Synchronization dynamics of two coupled neural oscillators receiving shared and unshared noisy stimuli, J. Comput. Neurosci., 26 (2009), 425-443. doi: 10.1007/s10827-008-0120-8. [25] Z. F. Mainen and T. J. Sejnowski, Reliability of spike timing in neocortical neurons, Science, 268 (1995), 1503-1506. [26] L. Markus and A. Weerasinghe, Stochastic oscillators, J. Diff. Eq., 71 (1988), 288-314. [27] L. Markus and A. Weerasinghe, Stochastic non-linear oscillators, Adv. Appl. Prob., 25 (1993), 649-666. doi: 10.2307/1427528. [28] C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophys. J., 35 (1981), 193-213. doi: 10.1016/S0006-3495(81)84782-0. [29] K. H. Nagai and H. Kori, Noise-induced synchronization of a large population of globally coupled nonidentical oscillators, Phys. Rev. E, 81 (2010), 065202. doi: 10.1103/PhysRevE.81.065202. [30] H. Nakao, K. Arai and Y. Kawamura, Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators, Phys. Rev. Lett., 98 (2007), 184101. doi: 10.1103/PhysRevLett.98.184101. [31] A. B. Neiman and D. F. Russell, Synchronization of noise-induced bursts in noncoupled sensory neurons, Phys. Rev. Lett., 88 (2002). [32] J. C. Neu, Coupled chemical oscillators, SIAM J. Appl. Math., 37 (1979), 307-315. doi: 10.1137/0137022. [33] A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences,'' Cambridge Nonlinear Science Series, 12, Cambridge University Press, Cambridge, 2001. [34] A. S. Pikovsky and J. Kurths, Coherence resonance in a noise-driven excitable system, Phys. Rev. Lett., 78 (1997), 775-778. doi: 10.1103/PhysRevLett.78.775. [35] J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, in "Methods in Neuronal Modeling: From Synapses to Networks," MIT Press, (1989), 135-169. [36] J. Teramae, H. Nakao and G. B. Ermentrout, Stochastic phase reduction for a general class of noisy limit cycle oscillators, Phys. Rev. Lett., 102 (2009), 194102. doi: 10.1103/PhysRevLett.102.194102. [37] J. Teramae and D. Tanaka, Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators, Phys. Rev. Lett., 93 (2004), 204103. doi: 10.1103/PhysRevLett.93.204103. [38] K. Yoshimura and K. Arai, Phase reduction of stochastic limit cycle oscillators, Phys. Rev. Lett., 101 (2008), 154101. doi: 10.1103/PhysRevLett.101.154101. [39] N. Yu, R. Kuske and Y.-X. Li, Stochastic phase dynamics: Multiscale behavior and coherence measures, Phys. Rev. E, 73 (2006), 056205. doi: 10.1103/PhysRevE.73.056205. [40] N. Yu, R. Kuske and Y.-X. Li, A computational study of spike time reliability in two cases of threshold dynamics,, preprint., ().
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