# A coarse-grained framework for spiking neuronal networks: between homogeneity and synchrony

- 352 Downloads
- 1 Citations

## Abstract

Homogeneously structured networks of neurons driven by noise can exhibit a broad range of dynamic behavior. This dynamic behavior can range from homogeneity to synchrony, and often incorporates brief spurts of collaborative activity which we call multiple-firing-events (MFEs). These multiple-firing-events depend on neither structured architecture nor structured input, and are an emergent property of the system. Although these MFEs likely play a major role in the neuronal avalanches observed in culture and *in vivo*, the mechanisms underlying these MFEs cannot easily be captured using current population-dynamics models. In this work we introduce a coarse-grained framework which illustrates certain dynamics responsible for the generation of MFEs. By using a new kind of ensemble-average, this coarse-grained framework can not only address the nucleation of MFEs, but can also faithfully capture a broad range of dynamic regimes ranging from homogeneity to synchrony.

## Keywords

Spiking neurons Synchrony Homogeneity Multiple firing events Partitioned-ensemble-average Dynamical systems## Notes

### Acknowledgments

JZ is partially supported by NSF Grant DMS-1009575. AR is supported by NSF Grants DMS-0914827 and DMS/NIGMS-1162548. DZ is supported by Shanghai Pujiang Program (Grant No. 10PJ1406300), NSFC (Grant No. 11101275 and No. 91230202) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars from State Education Ministry in China. DC is supported by NSF Grant DMS-1009575. DZ and DC are supported by New York University Abu Dhabi Research Grant G1301.

## Conflict of interests

The authors declare that they have no conflict of interest.

## Supplementary material

## References

- Battaglia, D., & Hansel, D. (2011). Synchronous chaos and broad band gamma rhythm in a minimal multi-layer model of primary visual cortex.
*PLoS Computational Biology*, 7.Google Scholar - Beggs, J.M., & Plenz, D. (2003). Neuronal avalanches in neocortical circuits.
*Journal of Neuroscience*,*23*(35), 11167–11177.PubMedGoogle Scholar - Bornholdt, S., & Rohl, T. (2003). Self-organized critical neural networks.
*Physical Review E*,*67*, 066118.CrossRefGoogle Scholar - Brunel, N. (2000). Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons.
*Journal of Comparative Neuroscience*,*8*, 183–208.CrossRefGoogle Scholar - Buice, M.A., & Chow, C.C. (2007). Correlations, fluctuations, and stability of a finite-size network of coupled oscillators.
*Physical Review E*,*76*, 031118.1–031118.25.CrossRefGoogle Scholar - Cai, D., Tao, L., Shelley, M., McLaughlin, D. (2004). An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex.
*Proceedings of the Natural Academy Sciences*,*101*(20), 7757–7762.CrossRefGoogle Scholar - Cai, D., Tao, L., Rangan, A., McLaughlin, D. (2006). Kinetic theory for neuronal network dynamics.
*Committee in Mathematical Science*,*4*, 97–127.CrossRefGoogle Scholar - Cardanobile, S., & Rotter, S. (2010). Multiplicatively interacting point processes and applications to neural modeling.
*Journal of Computational Neuroscience*,*28*, 267–284.PubMedCrossRefGoogle Scholar - Churchland, M.M., et al. (2010). Stimulus onset quenches neural variability: A widespread cortical phenomenon.
*Nature Neuroscience*,*13*(3), 369–378.PubMedCentralPubMedCrossRefGoogle Scholar - Csicsvari, J., Hirase, H., Mamiya, A., Buzsaki, G. (2000). Ensemble patterns of hippocampal ca3-ca1 neurons during sharp wave-associated population events.
*Neuron*,*28*, 585–594.PubMedCrossRefGoogle Scholar - Dehghani, N., Hatsopoulos, N.G., Haga, N.G., Parker, R.A., Greger, B., Halgren, E., Cash, S.S., Destexhe, A. (2012). Avalanche analysis from multi-electrode ensemble recordings in cat, monkey and human cerebral cortex during wakefulness and sleep. arXiv:1203.0738v4:[q–bio.NC].
- Erdős, P., & Rényi (1959). On random graphs. I.
*Publicationes Mathematiques (Debr.)*,*6*, 290–297.Google Scholar - Erdős, P., & Rényi (1960). On the evolution of random graphs.
*Magy. Tud. Akad, Mat. Kut. Intéz. Közl.*,*5*, 17–61.Google Scholar - Hahn, G., Petermann, T., Havenith, M.N., Yu, S., Singer, W., Plenz, D., Nikolic, D. (2010). Neuronal avalanches in spontaneous activity
*in vivo*.*Journal of Neurophysiology*,*104*, 3313–3322.CrossRefGoogle Scholar - Hatsopoulos, N.G., Ojakangas, C.L., Paniniski, L., Donoghue, J.P. (1998). Information about movement direction obtained from synchronous activity of motor cortical neurons.
*Proceedings of the National Academy of Science*,*95*, 15706–15711.CrossRefGoogle Scholar - Hu, Y., Trousdale, J., Josic, K., Shea-Brown, E. (2013). Motif statistics and spike correlations in neuronal networks.
*Journal of Statistical Mechanics*,*P03012*, 1–51.Google Scholar - Kenet, T., Bibitchkov, D., Tsodyks, M., Grinvald, A., Arieli, A. (2003). Spontaneously emerging cortical representations of visual attributes.
*Nature*,*425*, 954–956.PubMedCrossRefGoogle Scholar - Kohn, A., & Smith, M.A. (2005). Stimulus dependence of neuronal correlation in primary visual cortex of the macaque.
*Journal of Neuroscience*,*25*, 3661–73.PubMedCrossRefGoogle Scholar - Lampl, I., Reichova, I., Ferster, D. (1999). Synchronous membrane potential fluctuations in neurons of the cat visual cortex.
*Neuron*,*22*, 361–374.PubMedCrossRefGoogle Scholar - Ledoux, E., & Brunel, N. (2011). Dynamics of networks of excitatory and inhibitory neurons in response to time-dependent inputs.
*Frontiers in Computational Neuroscience*,*5*(25), 1–17.Google Scholar - Lee DeVille, R.E., & Peskin, C.S. (2012). Synchrony and asynchrony for neuronal dynamics defined on complex networks.
*Bulletin of Mathematical Biology*,*74*, 769–802.CrossRefGoogle Scholar - Leinekugel, X., Khazipov, R., Cannon, R., Hirase, H., Ben-Ari, Y., Buzsaki, G. (2002). Correlated bursts of activity in the neonatal hippocampus
*in vivo*.*Science*,*296*, 2049–2052.PubMedCrossRefGoogle Scholar - Mazzoni, A., Broccard, F.D., Garcia-Perez, E., Bonifazi, P., Ruaro, M.E., Torre, V. (2007). On the dynamics of the spontaneous activity in neuronal networks.
*PLoS One*,*5*, e439.CrossRefGoogle Scholar - Newhall, K., Kovacic, G., Kramer, P., Zhou, D., Rangan, A.V., Cai, D. (2010). Dynamics of current-based, poisson driven, integrate-and-fire neuronal networks.
*Committee in Mathematical Science*,*8*(2), 541–600.CrossRefGoogle Scholar - Ostojic, S., & Brunel, N. (2011). From spiking neuron models to linear-nonlinear models.
*PLoS Computational Biology*,*7*(1), e1001056.PubMedCentralPubMedCrossRefGoogle Scholar - Petermann, T., Thiagarajan, T.C., Lebedev, M.A., Nicolelis, M.A.L., Chailvo, D.R., Plenz, D. (2009). Spontaneous cortical activity in awake monkeys composed of neuronal avalanches.
*Proceedings of the National Academy of Science*,*106*(37), 15921–15926.CrossRefGoogle Scholar - Plenz, D., Stewart, C.V., Shew, W., Yang, H., Klaus, A., Bellay, T. (2011). Multi-electrode array recordings of neuronal avalanches in organotypic cultures.
*Journal of Visualized Experiments*,*54*, 2949.PubMedGoogle Scholar - Poil, S.S., Hardstone, R., Mansvelder, H.D., Linkenkaer-Hansen, K. (2012). Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks.
*Journal of Neuroscience*,*33*, 9817–9823.CrossRefGoogle Scholar - Rangan, A.V. (2009). Diagrammatic expansion of pulse-coupled network dynamics.
*Physical Review Letters*,*102*, 158101.PubMedCrossRefGoogle Scholar - Rangan, A.V., & Young, L.S. (2013a). Dynamics of spiking neurons: between homogeneity and synchrony.
*Journal of Computational Neuroscience*. doi: 10.1007/s10827-012-0429-1. - Rangan, A.V., & Young, L.S. (2013b). Emergent dynamics in a model of visual cortex.
*Journal of Computational Neuroscience*. doi: 10.1007/s10827-013-0445-9. - Rangan, A.V., & Cai, D. (2006). Maximum-entropy closures for kinetic theories of neuronal network dynamics.
*Physical Review Letters*,*96*, 178101.PubMedCrossRefGoogle Scholar - Roxin, A., Brunel, N., Hansel, D., Mongillo, G., Vreeswijk, C.V. (2011). On the distribution of firing rates in networks of cortical neurons.
*Journal of Neuroscience*,*31*(45), 16217–16226.PubMedCrossRefGoogle Scholar - Sakata, S., & Harris, K.D. (2009). Laminar structure of spontaneous and sensory-evoked population activity in auditory cortex.
*Neuron*,*12*(3), 404–418.CrossRefGoogle Scholar - Samonds, J.M., Zhou, Z., Bernard, M.R., Bonds, A.B. (2005). Synchronous activity in cat visual cortex encodes collinear and cocircular contours.
*Journal of Neurophysiology*,*95*(4), 2602–2616.PubMedCrossRefGoogle Scholar - Shew, S., Yang, H., Yu, S., Roy, R., Plenz, D. (2011). Information capacity and transmission are maximized in balanced cortical networks with neuronal avalanches.
*Journal of Neuroscience*,*31*, 55–63.PubMedCentralPubMedCrossRefGoogle Scholar - Sirovich, L., Omurtag, A., Knight, B. (2000). Dynamics of neuronal populations; the equilibrium solution.
*SIAM Journal on Applied Mathematics*,*60*, 2009–2028.CrossRefGoogle Scholar - Vogels, T.P., & Abbott, L.F. (2005). Signal propagation and logic gating in networks of integrate-and-fire neurons.
*Journal of Neuroscience*,*25*, 10786–95.PubMedCrossRefGoogle Scholar - Werner, G. (2007). Metastability, criticality and phase transitions in brain and its models.
*BioSystems*,*90*, 496–508.PubMedCrossRefGoogle Scholar - Yu, Y., & Ferster, D. (2010). Membrane potential synchrony in primary visual cortex during sensory stimulation.
*Neuron*,*68*, 1187–1201.PubMedCentralPubMedCrossRefGoogle Scholar - Yu, S., Yang, H., Nakahara, H., Santos, G.S., Nikolic, D., Plenz, D. (2011). Higher-order interactions characterized in cortical activity.
*Journal of Neuroscience*,*31*, 17514–17526.PubMedCrossRefGoogle Scholar - Zhang, J.W., Newhall, K., Zhou, D., Rangan, A.V. (2013). Distribution of correlated spiking events in a population-based approach for integrate-and-fire networks.
*Journal of Computational Neuroscience*. 1–17. doi: 10.1007/s10827-013-0472-6. - Zhao, L.Q., Beverlin, B., Netoff, T., Nykamp, D.Q. (2011). Synchronization from second order network connectivity statistics.
*Frontiers in Computational Neuroscience*,*5*(28), 1-16. doi: 10.3389/fncom.2011.00028.