Journal of Computational Neuroscience

, Volume 46, Issue 2, pp 211–232 | Cite as

A coarse-graining framework for spiking neuronal networks: from strongly-coupled conductance-based integrate-and-fire neurons to augmented systems of ODEs

  • Jiwei Zhang
  • Yuxiu Shao
  • Aaditya V. Rangan
  • Louis TaoEmail author


Homogeneously structured, fluctuation-driven networks of spiking neurons can exhibit a wide variety of dynamical behaviors, ranging from homogeneity to synchrony. We extend our partitioned-ensemble average (PEA) formalism proposed in Zhang et al. (Journal of Computational Neuroscience, 37(1), 81–104, 2014a) to systematically coarse grain the heterogeneous dynamics of strongly coupled, conductance-based integrate-and-fire neuronal networks. The population dynamics models derived here successfully capture the so-called multiple-firing events (MFEs), which emerge naturally in fluctuation-driven networks of strongly coupled neurons. Although these MFEs likely play a crucial role in the generation of the neuronal avalanches observed in vitro and in vivo, the mechanisms underlying these MFEs cannot easily be understood using standard population dynamic models. Using our PEA formalism, we systematically generate a sequence of model reductions, going from Master equations, to Fokker-Planck equations, and finally, to an augmented system of ordinary differential equations. Furthermore, we show that these reductions can faithfully describe the heterogeneous dynamic regimes underlying the generation of MFEs in strongly coupled conductance-based integrate-and-fire neuronal networks.


Spiking neurons Synchrony Homogeneity Multiple firing events Partitioned-ensemble-average Maximum entropy principle Coarse-graining method 



This work was partially supported by the Natural Science Foundation of China through grants 11771035 (J.Z.), 91430216 (J.Z.), U1530401 (J.Z.), 31771147 (Y.S., L.T.) and 91232715 (L.T.), by the Open Research Fund of the State Key Laboratory of Cognitive Neuroscience and Learning grant CNLZD1404 (Y.S., L.T.), and by the Beijing Municipal Science andTechnology Commission under contract Z151100000915070 (Y.S., L.T.).

Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.


  1. Abbott, L.F., & van Vreeswijk, C.A. (1993). Asynchronous states in networks of pulse-coupled neurons. Physical Review E, 48, 1483–1488.Google Scholar
  2. Anderson, J., Lampl, I., Reichova, I., Carandini, M., Ferster, D. (2000). Stimulus dependence of two-state fluctuations of membrane potential in cat visual cortex. Nature Neuroscience, 3(6), 617–621.Google Scholar
  3. Bak, P., Tang, C., Wiesenfeld, K. (1987). Self-organized criticality: an explanation of 1/f noise. Physical Review Letters, 59(4), 381–384.Google Scholar
  4. 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
  5. Buzsaki, G., & Wang, X.J. (2012). Mechanisms of gamma oscillations. Annual Reviews in the Neurosciences, 35, 203–225.Google Scholar
  6. Bornholdt, S., & Rohl, T. (2003). Self-organized critical neural networks. Physical Review E, 67, 066118.Google Scholar
  7. Bressloff, P.C. (2015). Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks. Journal of Mathematical Neuroscience, 5, 4.Google Scholar
  8. Brunel, N. (2000). Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. The Journal of Comparative Neurology, 8, 183–208.Google Scholar
  9. Brunel, N., & Hakim, V. (1999). Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Computation, 11, 1621–1671.Google Scholar
  10. Bruzsaki, G., & Draguhn, A. (2004). Neuronal oscillations in cortical networks. Science, 304, 1926–1929.Google Scholar
  11. 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.Google Scholar
  12. Buice, M.A., Cowan, J.D., Chow, C.C. (2010). Systematic fluctuation expansion for neural network activity equations. Neural Computation, 22(2), 377–426.Google Scholar
  13. 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 National Academy of Sciences of the United States of America, 101(20), 7757–7762.Google Scholar
  14. Cai, D., Tao, L., Rangan, A., McLaughlin, D. (2006). Kinetic theory for neuronal network dynamics. Communication in Mathematical Sciences, 4, 97–127.Google Scholar
  15. Cardanobile, S., & Rotter, S. (2010). Multiplicatively interacting point processes and applications to neural modeling. Journal of Computational Neuroscience, 28, 267–284.Google Scholar
  16. Churchland, M.M., & et al. (2010). Stimulus onset quenches neural variability: a widespread cortical phenomenon. Nature Neuroscience, 13, 3:369–378.Google Scholar
  17. 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.Google Scholar
  18. Destexhe, A., & Pare, D. (1999). Impact of network activity on the integrative properties of neocortical pyramidal neurons in vivo. Journal of Neurophysiology, 81, 1531–1547.Google Scholar
  19. Destexhe, A., Rudolph, M., Pare, D. (2003). The high-conductance state of neocortical neurons in vivo. Nature Reviews. Neuroscience, 4, 739–751.Google Scholar
  20. 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. Frontiers in Physiology, 3.Google Scholar
  21. DeVille, L., & Zheng, Y. (2014). Synchrony and periodicity in excitable neural networks with multiple subpopulations. SIAM Journal on Applied Dynamical Systems, 13(3), 1060–1081.Google Scholar
  22. El Boustani, S., & Destexhe, A. (2009). A master equation formalism for macroscopic modeling of asynchronous irregular activity states. Neural Computation, 21(1), 46–100.Google Scholar
  23. Fourcaud, N., & Brunel, N. (2002). Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Computation, 14, 2057–2110.Google Scholar
  24. Fries, P. (2009). Neuronal gamma-band synchronization as a fundamental process in cortical computation. Annual Review of Neuroscience, 32, 209–24.Google Scholar
  25. Grill-Spector, K., & Weiner, K. (2014). The functional architecture of the ventral temporal cortex and its role in categorization. Nature Reviews in the Neurosciences, 15, 536–548.Google Scholar
  26. 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.Google Scholar
  27. Hansel, D., & Sompolinsky, H. (1996). Chaos and synchrony in a model of a hypercolumn in visual cortex. Journal of Computational Neuroscience, 3, 7–34.Google Scholar
  28. 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 Sciences, 95, 15706–15711.Google Scholar
  29. Helias, M., Deger, M., Rotter, S., Diesmann, M. (2010). Instantaneous nonlinear processing by pulse-coupled threshold units. PLoS Computational Biology, 6(9), e1000929.Google Scholar
  30. Hertz, A.V.M., & Hopfield, J.J. (1995). Earthquake cycles and neural reverberations: collective oscillations in systems with pulse-coupled threshold elements. Physical Review Letters, 75(6), 1222–1225.Google Scholar
  31. 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
  32. Kenet, T., Bibitchkov, D., Tsodyks, M., Grinvald, A., Arieli, A. (2003). Spontaneously emerging cortical representations of visual attributes. Nature, 425, 954–956.Google Scholar
  33. Knight, B. (1972). The relationship between the firing rate of a single neuron and the level of activity in a population of neurons. The Journal of General Physiology, 59, 734.Google Scholar
  34. Koch, C. (1999). Biophysics of computation. Oxford: Oxford University Press.Google Scholar
  35. Kohn, A., & Smith, M.A. (2005). Stimulus dependence of neuronal correlation in primary visual cortex of the macaque. Journal of Neuroscience, 25, 3661–73.Google Scholar
  36. Kriener, B., Tetzlaff, T., Aertsen, A., Diesmann, M., Rotter, S. (2008). Correlatilons and population dynamics in cortical networks. Neural Computation, 20, 2185–2226.Google Scholar
  37. 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
  38. 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.Google Scholar
  39. Litwin-Kumar, A., & Doiron, B. (2012). Slow dynamics and high variability in balanced cortical networks with clustered connections. Nature Neuroscience, 15(11), 1498–1505.Google Scholar
  40. 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.Google Scholar
  41. Nykamp, D. (2000). A population density approach that facilitates large scale modeling of neural networks: analysis and application to orientation tuning. Journal of Computational Neuroscience, 8, 19–50.Google Scholar
  42. Newhall, K. A., Kovac̆ic̆, G., Kramer, P.R., et al. (2010). Cascade-induced synchrony in stochastically driven neuronal networks. Physical Review E, 82(1), 041903.Google Scholar
  43. Ohira, T., & Cowan, J.D. (1993). Master-equation approach to stochastic neurodynamics. Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 48(3), 2259–2266.Google Scholar
  44. Omurtag, A., Kaplan, E., Knight, B., Sirovich, L. (2000). A population approach to cortical dynamics with an application to orientation tuning. Network: Computation in Neural Systems, 11(4), 247–260.Google Scholar
  45. Ostojic, S., & Brunel, N. (2011). From spiking neuron models to linear-nonlinear models. PLoS Computational Biology, 7, 1:e1001056.Google Scholar
  46. Percival, D.B., & Walden, A.T. (1993). Spectral analysis for physical applications. Cambridge: Cambridge University Press.Google Scholar
  47. 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 Sciences, 106, 37:15921–15926.Google Scholar
  48. 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.Google Scholar
  49. 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. The Journal of Neuroscience, 33, 9817–9823.Google Scholar
  50. Rangan, A.V. (2009). Diagrammatic expansion of pulse-coupled network dynamics. Physical Reviews Letters, 102, 158101.Google Scholar
  51. Rangan, A.V., & Cai, D. (2006). Maximum-entropy closures for kinetic theories of neuronal network dynamics. Physical Review Letters, 96, 178101.Google Scholar
  52. Rangan, A.V., & Young, L.S. (2013a). Dynamics of spiking neurons: between homogeneity and synchrony. Journal of Computational Neuroscience, 34(3), 433–460.Google Scholar
  53. Rangan, A.V., & Young, L.S. (2013b). Emergent dynamics in a model of visual cortex. Journal of Computational Neuroscience, 35(2), 155–167.Google Scholar
  54. Richardson, M.J. (1918). Effects of synaptic conductance on the voltage distribution and firing rate of spiking neurons. Physical Review E, 69(05), 2004.Google Scholar
  55. Robert, P., & Touboul, J. (2016). On the dynamics of random networks. Journal of Statistical Physics, 165, 545–584.Google Scholar
  56. Roopum, A.K., Kramer, M.A., Carracedo, L.M., Kaiser, M., Davies, C.H., Traub, R.D., Kopell, N.J., Whittington, M.A. (2008). Temporal interactions between cortical rhythms. Frontiers in Neuroscience, 2, 145–154.Google Scholar
  57. Roxin, A., Brunel, N., Hansel, D., Mongillo, G., Vreeswijk, C.V. (2011). On the distribution of firing rates in networks of cortical neurons. The Journal of Neuroscience, 31(45), 16217–16226.Google Scholar
  58. Sakata, S., & Harris, K.D. (2009). Laminar structure of spontaneous and sensory-evoked population activity in auditory cortex. Neuron, 12(3), 404–418.Google Scholar
  59. 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.Google Scholar
  60. Seejnowski, T.J., & Paulsen, O. (2006). Network oscillations: emerging computational principles. The Journal of Neuroscience, 26, 1673–1676.Google Scholar
  61. Singer, W. (1999). Neuronal synchrony: a versatile code for the definition of relations? Neuron, 24, 49–65.Google Scholar
  62. Sirovich, L., Omurtag, A., Knight, B. (2000). Dynamics of neuronal populations; the equilibrium solution. SIAM Journal on Applied Mathematics, 60, 2009–2028.Google Scholar
  63. Shelley, M., McLaughlin, D., Shapley, R., Wielaard, J. (2002). States of high conductance in a large-scale model of the visual cortex. Journal of Computational Neuroscience, 13, 93–109.Google Scholar
  64. Shew, S., Yang, H., Yu, S., Roy, R., Plenz, D. (2011). Information capacity and transmission are maximized in balanced cortical networks with neuronal avalanches. The Journal of Neuroscience, 31, 55–63.Google Scholar
  65. Stern, E.A., Kincaid, A.E., Wilson, C.J. (1997). Spontaneous subthreshold membrane potential fluctuations and action potential variability of rat corticostriatal and striatal neurons in vivo. Journal of Neurophysiology, 77, 1697–1715.Google Scholar
  66. Storch, H., & Zwiers, F.W. (2001). Statistical analysis in climate research. Cambridge University Press.Google Scholar
  67. Touboul, J. (2014). Propagation of chaos in neural fields. Annals of Applied Probability, 24, 1298–1328.Google Scholar
  68. Traub, R.D., Jeffreys, J., Whittington, M. (1999). Fast oscillations in cortical circuits. Cambridge: MIT Press.Google Scholar
  69. Vogels, T.P., & Abbott, L.F. (2005). Signal propagation and logic gating in networks of integrate-and-fire neurons. The Journal of Neuroscience, 25, 10786–95.Google Scholar
  70. Werner, G. (2007). Metastability, criticality and phase transitions in brain and its models. BioSystems, 90, 496–508.Google Scholar
  71. Xiao, Z.C., Zhang, J.W., Sornborger, A.T., Tao, L. (2308). Cusps enable line attractors for neural computation. Physical Review E, 96(05), 2017.Google Scholar
  72. Yu, Y., & Ferster, D. (2010). Membrane potential synchrony in primary visual cortex during sensory stimulation. Neuron, 68, 1187–1201.Google Scholar
  73. Yu, S., Yang, H., Nakahara, H., Santos, G.S., Nikolic, D., Plenz, D. (2011). Higher-order interactions characterized in cortical activity. The Journal of Neuroscience, 31, 17514–17526.Google Scholar
  74. Zerlaut, Y., Chemla, S., Chavane, F., Destexhe, A. (2018). Modeling mesoscopic cortical dynamics using a mean-field model of conductance-based networks of adaptive exponential integrate-and-fire neurons. Journal of Computational Neuroscience, 44, 45–61.Google Scholar
  75. Zhang, J., & Rangan, A.V. (2015). A reduction for spiking integrate-and-fire network dynamics ranging from homogeneity to synchrony. Journal of Computational Neuroscience, 38(2), 355–404.Google Scholar
  76. Zhang, J.W., Zhou, D., Cai, D., Rangan, A.V. (2014a). A coarse-grained framework for spiking neuronal networks: between homogeneity and synchrony. Journal of Computational Neuroscience, 37(1), 81–104.Google Scholar
  77. Zhang, J.W., Newhall, K., Zhou, D., Rangan, A.V. (2014b). Distribution of correlated spiking events in a population-based approach for integrate-and-fire networks. Journal of Computational Neuroscience, 36(2), 279–295.Google Scholar
  78. Zhao, L.Q., Beverlin, B., Netoff, T., Nykamp, D.Q. (2011). Synchronization from second order network connectivity statistics. Frontiers in Computational Neuroscience, 5(28).Google Scholar

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina
  2. 2.Hubei Key Laboratory of Computational ScienceWuhan UniversityWuhanChina
  3. 3.Center for Bioinformatics, National Laboratory of Protein Engineering and Plant Genetic Engineering, School of Life SciencesPeking UniversityBeijingChina
  4. 4.Center for Quantitative BiologyPeking UniversityBeijingChina
  5. 5.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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