Journal of Computational Neuroscience

, Volume 30, Issue 2, pp 427–445 | Cite as

Stability of two cluster solutions in pulse coupled networks of neural oscillators

  • Lakshmi Chandrasekaran
  • Srisairam Achuthan
  • Carmen C. Canavier
Article

Abstract

Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.

Keywords

Neuronal networks Synchronization Clustering Phase response curves Pulse coupled oscillators 

References

  1. Achuthan, S., & Canavier, C. C. (2009a). Phase-resetting curves determine synchronization, phase locking and clustering in networks of neural oscillators. Journal of Neuroscience, 29, 5218–5233.PubMedCrossRefGoogle Scholar
  2. Achuthan, S., & Canavier, C. C. (2009b). Prediction of phase locked neuronal network activity in the presence of heterogeneity and noise using phase resetting curves. In Society for Neuroscience, Abstract no. 321.5.Google Scholar
  3. Achuthan, S., Sieling, F., Prinz, A., & Canavier, C. C. (2011). Phase resetting curves in presence of heterogeneity and noise. In D. Glanzman, & M. Ding (Eds.), Neuronal variability and its functional significance. New York: Oxford University Press.Google Scholar
  4. Acker, C. D., Kopell, N., & White, J. A. (2003). Synchronization of strongly coupled excitatory neurons: Relating network behavior to biophysics. Journal of Computational Neuroscience, 15, 71–90.PubMedCrossRefGoogle Scholar
  5. Bartos, M., Vida, I., Frotscher, M., Geiger, J. R. P., & Jonas, P. (2001). Rapid signaling at inhibitory synapses in a dentate gyrus interneuron network. Journal of Neuroscience, 21, 2687–2698.PubMedGoogle Scholar
  6. Borgers, C., & Kopell, N. (2003). Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity. Neural Computation, 15, 509–538.PubMedCrossRefGoogle Scholar
  7. Bendels, M. H. K., & Leibold, C. (2007). Generation of theta oscillations by weakly coupled neural oscillators in the presence of noise. Journal of Computational Neuroscience, 22, 173–189.PubMedCrossRefGoogle Scholar
  8. Buzsaki, G. (2006). Rhythms of the brain. Oxford University Press.Google Scholar
  9. Canavier, C. C., Butera, R. J., Dror, R. O., Baxter, D. A., Clark, J. W., & Byrne, J. H. (1997). Phase response characteristics of model neurons determine which patterns are expressed in a ring circuit model of gait generation. Biological Cybernetics, 77, 367–380.PubMedCrossRefGoogle Scholar
  10. Cunningham, M. O., Halliday, D. M., Davies, C. H., Traub, R. D., Buhl, E. H., & Whittington, M. A. (2004). Coexistence of gamma and high-frequency oscillations in the medial entorhinal cortex in vitro. Journal of Physiology, 559, 347–353.PubMedCrossRefGoogle Scholar
  11. Dror, R. O., Canavier, C. C., Butera, R. J., Clark, J. W., & Byrne, J. H. (1999). A mathematical criterion based on phase response curves for stability in a ring of coupled oscillators. Biological Cybernetics, 80, 11–23.PubMedCrossRefGoogle Scholar
  12. Ermentrout, B. (1996). Type I membranes, phase resetting curves, and synchrony. Neural Computation, 8, 979–1002.PubMedCrossRefGoogle Scholar
  13. Ermentrout, G. B., & Kopell, N. (1991). Multiple pulse interactions and averaging in systems of coupled neural oscillators. Journal of Mathematical Biology, 29, 195–217.CrossRefGoogle Scholar
  14. Glass, L., & Mackey, M. C. (1988). From clocks to chaos: The rhythms of life. Princeton: Princeton University Press.Google Scholar
  15. Goel, P., & Ermentrout, G. B. (2002). Synchrony, stability and firing patterns in pulse coupled oscillators. Physica D, 63(3–4), 191–216.CrossRefGoogle Scholar
  16. Goldsztein, G., & Strogatz, S. H. (1995). Stability of synchronization in a network of digital phase locked loops. International Journal of Bifurcations and Chaos, 5, 983–990.CrossRefGoogle Scholar
  17. Golomb, D., & Rinzel, J. (1994). Clustering in globally coupled inhibitory neurons. Physica D, 72, 259–282.CrossRefGoogle Scholar
  18. Hairer, E., & Wanner, G. (1991). Solving ordinary differential equations II. In Stiff and differential-algebraic problems. Springer series in computational mathematics. Berlin: Springer.Google Scholar
  19. Hansel, D., & Mato, G. (2003). Asynchronous states and emergence of synchrony in large networks of interacting excitatory and inhibitory neurons. Neural Computation, 15, 1–56.PubMedCrossRefGoogle Scholar
  20. Hansel, D., Mato, G., & Meunier, C. (1993). Clustering and slow switching in globally coupled phase oscillators. Physical Review E, 48(5), 3470–3477.CrossRefGoogle Scholar
  21. Hansel, D., Mato, G., & Meunier, C. (1995). Synchrony in excitatory neural networks. Neural Computation, 7, 307–337.PubMedCrossRefGoogle Scholar
  22. Hodgkin, A. L. (1948). The local electric changes associated with repetitive action in a non-medullated axon. Journal of Physiology, 107, 165–181.PubMedGoogle Scholar
  23. Janczewski, W. A., & Feldman, J. L. (2006). Distinct rhythm generators for inspiration and expiration in the juvenile rat. Journal of Physiology, 570, 407–420.PubMedGoogle Scholar
  24. Joseph, I. M. P., & Butera, R. J. (2005). A simple model of dynamic interactions between respiratory centers. In Y. T. Zhang & N. J. Piscataway (Eds.), IEEE, Proceedings of the 2005 IEEE engineering in medicine and biology 27th annual conference, Shanghai, China.Google Scholar
  25. Li, Y., Wang, Y., & Miura, R. (2003). Clustering in small networks of excitatory neurons with heterogenous coupling strengths. Journal of Computational Neuroscience, 14, 139–159.PubMedCrossRefGoogle Scholar
  26. Marder, E., & Calabrese, R. (1996). Principles of rhythmic motor pattern generation. Physiological Reviews, 76, 687–717.PubMedGoogle Scholar
  27. Mellen, N. M., Janczweski, W. A., Bocchiaro, C. M., & Feldman, J. L. (2003). Opioid-induced quantal slowing reveals dual networks for respiratory rhythm generation. Neuron, 37, 821–826.PubMedCrossRefGoogle Scholar
  28. Mirollo, R. E., & Strogatz, S. H. (1990). Synchronization of pulse coupled biological oscillators. SIAM Journal on Applied Mathematics, 50, 1645–1662.CrossRefGoogle Scholar
  29. Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35, 193–213.PubMedCrossRefGoogle Scholar
  30. Netoff, T. I., Banks, M., Dorval, A., Acker, C., Haas, J., Kopell, N. J., et. al. (2005). Synchronization in hybrid neuronal networks of the hippocampal formation. Journal of Neurophysiology, 93, 1197–1208.PubMedCrossRefGoogle Scholar
  31. Oprisan, S., & Canavier, C. C. (2002). The influence of limit cycle topology on the phase resetting curve. Neural Computation, 14, 1027–1057.PubMedCrossRefGoogle Scholar
  32. Oprisan, S. A., & Canavier, C. C. (2001). Stability analysis of rings of pulse coupled oscillators: The effect of phase-resetting in the second cycle after the pulse is important at synchrony and for long pulses. Differential Equations and Dynamical Systems, 9, 243–258.Google Scholar
  33. Oprisan, S. A., Prinz, A. A., & Canavier, C. C. (2004). Phase resetting and phase locking in hybrid circuits of one model and one biological neuron. Biophysical Journal, 87, 2283–2298.PubMedCrossRefGoogle Scholar
  34. Orosz, G., Moehlis, J., & Ashwin, P. (2009). Designing the dynamics of globally coupled oscillators. Progress of Theoretical Physics, 122(3), 611–630.CrossRefGoogle Scholar
  35. Pervouchine, D. D., Netoff, T. I., Rotstein, H. G., White, J. A., Cunningham, M. O., Whittington, W. A., et. al. (2006). Low-dimensional maps encoding dynamics in entorhinal cortex and hippocampus. Neural Computation, 18, 1–34.CrossRefGoogle Scholar
  36. Peskin, C. (1975). Mathematical aspects of heart physiology. Courant Institute of Mathematical Sciences, NYU, New York, 5, 268–278.Google Scholar
  37. Rinzel, J., & Ermentrout, B. (1998). Analysis of neural excitability and oscillations. In Methods in neuronal modeling from ions to networks. Cambridge: MIT.Google Scholar
  38. Rotstein, H. G., Pervouchine, D. D., Acker, C. D., Gillies, M. J., White, J. A., Buhl, E. H., et al. (2005). Slow and fast inhibition and an H-current interact to create a theta rhythm in a model of CA1 interneuron network. Journal of Neurophysiology, 94, 1509–1518.PubMedCrossRefGoogle Scholar
  39. Rubin, J., & Terman, D. (2000). Analysis of clustered firing patterns in synaptically coupled networks of oscillators. Journal of Mathematical Biology, 41, 513–545.PubMedCrossRefGoogle Scholar
  40. Sieling, F., Canavier, C. C., & Prinz, A. A. (2009). Predictions of phase-locking in excitatory hybrid networks: Excitation does not promote phase-locking in pattern generating networks as reliably as inhibition. Journal of Neurophysiology, 102, 69–84.PubMedCrossRefGoogle Scholar
  41. Tass, P. A. (2007). Phase resetting in Medicine and Biology. Berlin: Springer.Google Scholar
  42. Terman, D., Kopell, N., & Bose, A. (1996). Functional reorganization in thalamocortical networks: Transition between spindling and delta sleep rhythms. Proceedings of the National Academy of Sciences of the United States of America, 93(26), 15417–15422.PubMedCrossRefGoogle Scholar
  43. Velasquez, J. L. P., Galan, R. F., Dominguez, L. G., Leshchenko, Y., Lo, S., Belkas, J., et al. (2007). Phase response curves in the characterization of epileptiform activity. Physical Review E, 76, 061912.CrossRefGoogle Scholar
  44. Wang, X. J., & Buzsaki, G. (1996). Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. Journal of Neuroscience, 16, 6402–6413.PubMedGoogle Scholar
  45. Winfree, A. T. (1967). Biological rhythms and the behavior of populations of coupled oscillators. Journal of Theoretical Biology, 16, 15–42.PubMedCrossRefGoogle Scholar
  46. Winfree, A. T. (2001). The geometry of biological time. New York: Springer.Google Scholar
  47. Wu, W., & Chen, T. (2007). Desynchronization of pulse coupled oscillators with delayed excitatory coupling. Nonlinearity, 20, 789–808.CrossRefGoogle Scholar

Copyright information

© US Government 2010

Authors and Affiliations

  • Lakshmi Chandrasekaran
    • 1
  • Srisairam Achuthan
    • 1
  • Carmen C. Canavier
    • 1
    • 2
  1. 1.Neuroscience Center of ExcellenceLSU Health Sciences CenterNew OrleansUSA
  2. 2.Department of OphthalmologyLSU Health Sciences CenterNew OrleansUSA

Personalised recommendations