Synchronization dynamics of two coupled neural oscillators receiving shared and unshared noisy stimuli

  • Cheng Ly
  • G. Bard Ermentrout


The response of neurons to external stimuli greatly depends on the intrinsic dynamics of the network. Here, the intrinsic dynamics are modeled as coupling and the external input is modeled as shared and unshared noise. We assume the neurons are repetitively firing action potentials (i.e., neural oscillators), are weakly and identically coupled, and the external noise is weak. Shared noise can induce bistability between the synchronous and anti-phase states even though the anti-phase state is the only stable state in the absence of noise. We study the Fokker-Planck equation of the system and perform an asymptotic reduction ρ 0. The ρ 0 solution is more computationally efficient than both the Monte Carlo simulations and the 2D Fokker-Planck solver, and agrees remarkably well with the full system with weak noise and weak coupling. With moderate noise and coupling, ρ 0 is still qualitatively correct despite the small noise and coupling assumption in the asymptotic reduction. Our phase model accurately predicts the behavior of a realistic synaptically coupled Morris-Lecar system.


Neural oscillators Shared noise Weak coupling Weak noise Bistability 



We thank Brent Doiron for useful discussions. CL is supported by an NSF Postdoctoral Fellowship # DMS0703502. GBE is supported by an NSF grant # DMS0513500.


  1. Apfaltrer, F., Ly, C., & Tranchina, D. (2006). Population density methods for stochastic neurons with a 2-D state space: Application to neural networks with realistic synaptic kinetics. Network: Computation in Neural Systems, 17, 373–418.CrossRefGoogle Scholar
  2. Arieli, A., Sterkin, A., Grinvald, A., & Aertsen, A. (1996). Dynamics of ongoing activity: Explanation of the large variability in evoked cortical responses. Science, 273, 1868.PubMedCrossRefGoogle Scholar
  3. Doiron, B., Chacron, M., Maler, L., Longtin, A., & Bastain, J. (2003). Inhibitory feedback required for network oscillatory responses to communication but not prey stimuli. Nature, 421, 539–543.PubMedCrossRefGoogle Scholar
  4. Doiron, B., Lindner, B., Longtin, A., Maler, L., & Bastian, J. (2004). Oscillatory activity in electrosensory neurons increases with spatial correlation of the stochastic input stimulus. Physical Review Letters, 93, 048101.PubMedCrossRefGoogle Scholar
  5. Ermentrout, G., & Kopell, N. (1991). Multiple pulse interactions and averaging in systems of coupled neural oscillators. Journal of Mathematical Biology, 29, 195–217.CrossRefGoogle Scholar
  6. Ermentrout, G. B. (1996). Type I membranes, phase-resetting curves, and synchrony. Neural Computation, 8, 979–1001.PubMedCrossRefGoogle Scholar
  7. Ermentrout, G. B. (2002). Simulating, analyzing, and animating dynamical systems: A guide to XPPAUT for researchers and students. Philadelphia: SIAM.Google Scholar
  8. Ferster, D., & Miller, K. (2000). Neural mechanisms of orientation selectivity in the visual cortex. Annual Review of Neuroscience, 23, 441–471.PubMedCrossRefGoogle Scholar
  9. Galán, R., Ermentrout, G. B., & Urban, N. N. (2007). Stochastic dynamics of uncoupled neural oscillators: Fokker-planck studies with the finite element method. Physical Review E, 76, 056110.CrossRefGoogle Scholar
  10. Galán, R., Fourcaud, N., Ermentrout, G. B., & Urban, N. N. (2006). Correlation-induced synchronization of oscillations in olfactory bulb neurons. Journal of Neuroscience, 26, 3646–3655.PubMedCrossRefGoogle Scholar
  11. Gardiner, C. W. (2004). Handbook of stochastic methods. New York: Springer Complexity.Google Scholar
  12. Gutkin, B., Laing, C., Colby, C., Chow, C., & Ermentrout, G. B. (2001). Turning on and off with excitation: The role of spike-timing asynchrony and synchrony in sustained neural activity. Journal of Computational Neuroscience, 11, 121–134.PubMedCrossRefGoogle Scholar
  13. Haider, B., Duque, A., Hasenstaub, A., Yu, Y., & McCormick, D. (2007). Enhancement of visual responsiveness by spontaneous local network activity in vivo. Journal of Neurophysiology, 97, 4186.PubMedCrossRefGoogle Scholar
  14. Hinrichsen, D., & Pritchard, A. (2005). Mathematical systems theory I, texts in applied mathematics (Vol. 48, pp. 340–372). New York: Springer.Google Scholar
  15. Hoppensteadt, F., & Izhekevich, I. (1997). Weakly connected neural networks. New York: Springer.Google Scholar
  16. Kanamaru, T. (2006). Analysis of Synchronization between two modules of pulse neural networks with excitatory and inhibitory connections. Neural Computation, 18, 1111–1131.PubMedCrossRefGoogle Scholar
  17. Kanamaru, T., & Sekine, M. (2005). Synchronized firings in the networks of class 1 excitable neurons with excitatory and inhibitory connections and their dependences on the forms of interactions. Neural Computation, 17, 1315–1338.PubMedCrossRefGoogle Scholar
  18. Kuramoto, Y. (1984). Chemical oscillations, waves and turbulence. New York: Springer.Google Scholar
  19. Ly, C., & Tranchina, D. (2007). Critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling. Neural Computation, 19, 2032–2092.PubMedCrossRefGoogle Scholar
  20. MacLean, J., Watson, B., Aaron, G., & Yuste, R. (2005). Internal dynamics determine the cortical response to thalamic stimulation. Neuron, 48, 811–823.PubMedCrossRefGoogle Scholar
  21. Marella, S., & Ermentrout, G. B. (2008). Class-II neurons display a higher degree of stochastic synchronization than class-I neurons. Physical Review E, 77, 41918.CrossRefGoogle Scholar
  22. Morris, C., & Lecar, H. (1981). Voltage oscillations int he barnacle giant muscle fiber. Biophysical Journal, 193, 193–213.CrossRefGoogle Scholar
  23. Nakao, H., Arai, K., & Kawamura, Y. (2007). Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators. Physical Review Letters, 98, 184101.PubMedCrossRefGoogle Scholar
  24. Papanicolaou, G. (1977). Introduction to asymptotic analysis of stochastic equations. In R. C. DiPrima (Ed.), Modern modeling of continuum phenomena (Vol. 16, pp. 109–148). Providence: American Mathematical Society.Google Scholar
  25. Pfeuty, B., Mato, G., Golomb, D., & Hansel, D. (2005). The combined effects of inhibitory and electrical synapses in synchrony. Neural Computation, 17, 633–670.PubMedCrossRefGoogle Scholar
  26. Singer, W., & Strategies, T. (1999). Neuronal synchrony: A versatile code review for the definition of relations? Neuron, 24, 49–65.PubMedCrossRefGoogle Scholar
  27. Teramae, J., & Tanaka, D. (2004). Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators. Physical Review Letters, 93, 204103.PubMedCrossRefGoogle Scholar
  28. Tsodyks, M., Kenet, T., Grinvald, A., & Arieli, A. (1999). Linking spontaneous activity of single cortical neurons and the underlying functional architecture. Science, 286, 1943.PubMedCrossRefGoogle Scholar
  29. Winfree, A. (1974). Patteras of phase compromise in biological cycles. Journal of Mathematical Biology, 1, 73–93.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.University Professor of Computational Biology, Professor, Department of MathematicsUniversity of PittsburghPittsburghUSA

Personalised recommendations