Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities

Article

Abstract

We investigate how synchrony can be generated or induced in networks of electrically coupled integrate-and-fire neurons subject to noisy and heterogeneous inputs. Using analytical tools, we find that in a network under constant external inputs, synchrony can appear via a Hopf bifurcation from the asynchronous state to an oscillatory state. In a homogeneous net work, in the oscillatory state all neurons fire in synchrony, while in a heterogeneous network synchrony is looser, many neurons skipping cycles of the oscillation. If the transmission of action potentials via the electrical synapses is effectively excitatory, the Hopf bifurcation is supercritical, while effectively inhibitory transmission due to pronounced hyperpolarization leads to a subcritical bifurcation. In the latter case, the network exhibits bistability between an asynchronous state and an oscillatory state where all the neurons fire in synchrony. Finally we show that for time-varying external inputs, electrical coupling enhances the synchronization in an asynchronous network via a resonance at the firing-rate frequency.

Keywords

Gap junctions Oscillations Neural networks 

References

  1. Abbott, L. F., & van Vreeswijk, C. (1993). Asynchronous states in a network of pulse-coupled oscillators. Physical Review, E, 48, 1483–1490.CrossRefGoogle Scholar
  2. Amit, D., & Brunel, N. (1997). Model of global spontaneous activity and local structured delay activity during delay periods in the cerebral cortex. Cerebral Cortex, 7, 237–252.PubMedCrossRefGoogle Scholar
  3. Beierlein, M., Gibson, J. R., & Connors, B. (2000). A network of electrically coupled interneurons drives synchronized inhibition in neocortex. Nature Neuroscience, 3, 904–909PubMedCrossRefGoogle Scholar
  4. Bem, T., Feuvre, Y. L., Rinzel, J., & Meyrand, P. (2005). Electrical coupling induces bistability of rhythms in networks of inhibitory spiking neurons. European Journal of Neuroscience, 22, 2661–2668.PubMedCrossRefGoogle Scholar
  5. Bennett, M., & Zukin, R. (2004). Electrical coupling and neuronal synchronization in the mammalian brain. Neuron, 41, 495–511.PubMedCrossRefGoogle Scholar
  6. Brunel, N. (2000). Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience, 8, 183–208.PubMedCrossRefGoogle Scholar
  7. Brunel, N., Chance, F., Fourcaud, N., & Abbott, L. (2001). Effects of synaptic noise and filtering on the frequency response of spiking neurons. Physical Review Letters, 86, 2186–2189.PubMedCrossRefGoogle Scholar
  8. Brunel, N., & Hakim, V. (1999). Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Computation, 11, 162–1671.CrossRefGoogle Scholar
  9. Brunel, N., & Hakim, V. (2008). Sparsely synchronized neuronal oscillations. Chaos, 18, 015113.PubMedCrossRefGoogle Scholar
  10. Brunel, N., & Hansel, D. (2006). How noise affects the synchronization properties of reccurent networks of inhibitory neurons. Neural Computation, 18, 1066–1110.PubMedCrossRefGoogle Scholar
  11. Chow, C. C., & Kopell, N. (2000). Dynamics of spiking neurons with electrical coupling. Neural Computation, 12, 1643–1678.PubMedCrossRefGoogle Scholar
  12. Connors, B., & Long, M. (2004). Electrical synapses in the mammalian brain. Annual Review of Neuroscience, 27, 393–418.PubMedCrossRefGoogle Scholar
  13. Coombes, S., & Zachariou, M. (2008). Gap junctions and emergent rhythms. In J. Rubin, R. Matias (Ed.), Coherent behavior in neuronal networks, computational neuroscience series. New York: Springer.Google Scholar
  14. Coombes, S. (2008). Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models. SIAM Journal on Applied Dynamical Systems, 7, 1101–1129.CrossRefGoogle Scholar
  15. Draguhn, A., Traub, R., Schmitz, D., & Jefferys, J. (1998). Electrical coupling underlies high-frequency oscillations in the hippocampus in vitro. Nature, 394, 189–192.PubMedCrossRefGoogle Scholar
  16. Dugué G., Brunel N., Hakim V., Schwartz E., Chat M., Levesque M., et al. (2008). Electrical coupling mediates tunable low-frequency oscillations and resonance in the cerebellar Golgi network. Neuron (in press).Google Scholar
  17. Fourcaud-Trocmé, N., Hansel, D., van Vreeswijk, C., & Brunel, N. (2003). How spike generation mechanisms determine the neuronal response to fluctuating inputs. Journal of Neuroscience, 23, 11628–11640.PubMedGoogle Scholar
  18. Fukuda, T., & Kosaka, T. (2000). Gap-junction coupling linking the dendritic network of gabaergic neurons in the hippocampus. Journal of Neuroscience, 20, 1519–1528.PubMedGoogle Scholar
  19. Galarreta, M., Erdelyi, F., Szabo, G., & Hestrin, S. (2004). Electrical coupling among irregular-spiking gabaergic interneurons expressing cannabinoid receptors. Journal of Neuroscience, 24, 9770–9778.PubMedCrossRefGoogle Scholar
  20. Galarreta, M., & Hestrin, S. (1999). A network of fast-spiking cells in the neocortex connected by electrical synapses. Nature, 402, 72–75.PubMedCrossRefGoogle Scholar
  21. Galarreta, M., & Hestrin, S. (2001a). Electrical synapses between gaba-releasing interneurons. Nature Reviews, Neuroscience, 2, 425–433.CrossRefGoogle Scholar
  22. Galarreta, M., & Hestrin, S. (2001b). Spike transmission and synchrony detection in networks of gabaergic interneurons. Science, 292, 2295–2299.PubMedCrossRefGoogle Scholar
  23. Galarreta, M., & Hestrin, S. (2002). Electrical and chemical synapses among parvalbumin fast-spiking gabaergic interneurons in adult mouse neocortex. Proceedings of the National Academy of Sciences of the United States of America, 00, 12438–12443.CrossRefGoogle Scholar
  24. Gerstner, W., & van Hemmen, J. L. (1993). Coherence and incoherence in a globally coupled ensemble of pulse-emitting units. Physical Review Letters, 71, 312–315.PubMedCrossRefGoogle Scholar
  25. Gibson, J. R., Beierlein, M., & Connors, B. (1999). Two networks of electrically coupled inhibitory neurons in neocortex. Nature, 402, 75–79.PubMedCrossRefGoogle Scholar
  26. Hestrin, S., & Galarreta, M. (2005). Electrical synapses define networks of neocortical gabaergic neurons. Trends in Neuroscience, 28, 304–309.CrossRefGoogle Scholar
  27. Kopell, N., & Ermentrout, B. (2004). Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks. Proceedings of the National Academy of Sciences of the United States of America, 101, 15482–15487.PubMedCrossRefGoogle Scholar
  28. Landisman, C., Long, M., Beierlein, M., Deans, M., Paul, D. & Connors, B. (2002). Electrical synapses in the thalamic reticular nucleus. Journal of Neuroscience, 22, 1002–1009.PubMedGoogle Scholar
  29. LeBeau, F., Traub, R., Monyer, H., Whittington, M., & Buhl, E. (2003). The role of electrical signaling via gap junctions in the generation of fast network oscillations. Brain Research Bulletin, 62, 3–13.PubMedCrossRefGoogle Scholar
  30. Lewis, T. J., & Rinzel, J. (2003). Dynamics of spiking neurons connected by both inhibitory and electrical coupling. Journal of Computational Neuroscience, 14, 283–309.PubMedCrossRefGoogle Scholar
  31. Mann-Metzer, P., & Yarom, Y. (1999). Electrotonic coupling interacts with intrinsic properties to generate synchronized activity in cerebellar networks of inhibitory interneurons. Journal of Neuroscience, 19, 3298–3306.PubMedGoogle Scholar
  32. Pfeuty, B., Mato, G., Golomb, D., & Hansel, D. (2003). Electrical synapses and synchrony: The role of intrinsic currents. Journal of Neuroscience, 23, 6280–6294.PubMedGoogle Scholar
  33. 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
  34. Risken, H. (1984). The Fokker Planck equation: methods of solution and applications. New York: Springer.Google Scholar
  35. Schneider A. R., Lewis T. J., & Rinzel J. (2006). Effects of correlated input and electrical coupling on synchrony in fast-spiking cell networks. Neurocomputing, 69, 1125–1129CrossRefGoogle Scholar
  36. Sherman, A., & Rinzel, J. (1992). Proceedings of the National Academy of Sciences of the United States of America, 89, 2471–2474.PubMedCrossRefGoogle Scholar
  37. Skinner F. K., Zhang L., Perez Velazquez J.L., & Carlen P. L. (1999). Bursting in Inhibitory Interneuronal Networks: A Role for Gap-Junctional Coupling. Journal of Neurophysiology, 81, 1274–1283.PubMedGoogle Scholar
  38. Tamas, G., Buhl, E., Lorincz, A., & Somogyi, P. (2000). Proximally targeted gabaergic synapses and gap junctions synchronize cortical interneurons. Nature Neuroscience, 3, 366–371.PubMedCrossRefGoogle Scholar
  39. Timme, M., Wolf, F., & Geisel, T. (2002). Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators. Physical Review Letters, 89, 258701.PubMedCrossRefGoogle Scholar
  40. Traub, R., Kopell, N., Bibbig, A., Buhl, E., LeBeau, F., & Whittington, M. (2001). Gap junctions between interneuron dendrites can enhance synchrony of gamma oscillations in distributed networks. Journal of Neuroscience, 21, 9478–86.PubMedGoogle Scholar
  41. Tuckwell, H. (1988). Introduction to theoretical neurobiology. Cambridge: Cambridge University Press.Google Scholar
  42. Venance, L., Rozov, A., Blatow, M., Burnashev, N., Feldmeyer, D., & Monyer, H. (2000). Connexin expression in electrically coupled postnatal rat brain neurons. Proceedings of the National Academy of Sciences of the United States of America, 97, 10260–10265.PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Laboratoire de Physique Statistique, CNRS UMR 8550Ecole Normale SupérieureParis Cedex 05France
  2. 2.Laboratory of Neurophysics and Physiology, CNRS UMR 8119Université Paris DescartesParisFrance

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