Journal of Computational Neuroscience

, Volume 44, Issue 1, pp 45–61 | Cite as

Modeling mesoscopic cortical dynamics using a mean-field model of conductance-based networks of adaptive exponential integrate-and-fire neurons

  • Yann ZerlautEmail author
  • Sandrine Chemla
  • Frederic Chavane
  • Alain DestexheEmail author


Voltage-sensitive dye imaging (VSDi) has revealed fundamental properties of neocortical processing at macroscopic scales. Since for each pixel VSDi signals report the average membrane potential over hundreds of neurons, it seems natural to use a mean-field formalism to model such signals. Here, we present a mean-field model of networks of Adaptive Exponential (AdEx) integrate-and-fire neurons, with conductance-based synaptic interactions. We study a network of regular-spiking (RS) excitatory neurons and fast-spiking (FS) inhibitory neurons. We use a Master Equation formalism, together with a semi-analytic approach to the transfer function of AdEx neurons to describe the average dynamics of the coupled populations. We compare the predictions of this mean-field model to simulated networks of RS-FS cells, first at the level of the spontaneous activity of the network, which is well predicted by the analytical description. Second, we investigate the response of the network to time-varying external input, and show that the mean-field model predicts the response time course of the population. Finally, to model VSDi signals, we consider a one-dimensional ring model made of interconnected RS-FS mean-field units. We found that this model can reproduce the spatio-temporal patterns seen in VSDi of awake monkey visual cortex as a response to local and transient visual stimuli. Conversely, we show that the model allows one to infer physiological parameters from the experimentally-recorded spatio-temporal patterns.


Recurrent network dynamics Mean-field description Adex model Voltage-sensitive dye imaging 



Research supported by the CNRS, the ICODE excellence network, the European Community: Human Brain Project H2020-720270 and a Flag-Era JTC (SLOW-DYN) to A.D., FET Grant BrainScaleS FP7-269921 to A.D. and F.C and the ANR BalaV1 and Trajectory (ANR-13-BSV4-0014-02 to F.C). Y.Z. was supported by fellowships from the Initiative d’Excellence Paris-Saclay and the Fondation pour la Recherche Médicale (FDT 20150532751).

Compliance with Ethical Standards

Experimental protocols have been approved by the Marseille Ethical Committee in Neuroscience (approval A10/01/13, official national registration French Ministry of Research). All procedures complied with the French and European regulations for animal research, as well as the guidelines from the Society for Neuroscience.

Conflict of interests

The authors declare that they have no conflict of interest.


  1. Angelucci, A., Levitt, J.B., Walton, E.J., Hupe, J.M., Bullier, J., Lund, J.S. (2002). Circuits for local and global signal integration in primary visual cortex. Journal of Neuroscience, 22(19), 8633–8646.PubMedGoogle Scholar
  2. Amit, D.J., & Brunel, N. (1997). Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex. Cerebral Cortex, 7, 237–252.CrossRefPubMedGoogle Scholar
  3. Arieli, A., Sterkin, A., Grinvald, A., Aertsen, A., An, J.H. (1996). Dynamics of ongoing activity: explanation of the large variability in evoked cortical responses. Science (New York, N.Y.), 273, 1868–71.CrossRefGoogle Scholar
  4. Arieli, A., Grinvald, A., Slovin, H. (2002). Dural substitute for long-term imaging of cortical activity in behaving monkeys and its clinical implications. Journal of Neuroscience Methods, 114, 119–133.CrossRefPubMedGoogle Scholar
  5. Augustin, M., Ladenbauer, J., Baumann, F., Obermayer, K. (2016). Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: comparison and implementation. arXiv:1611.07999.
  6. Berger, T., Borgdorff, A., Crochet, S., Neubauer, F.B., Lefort, S., Fauvet, B., Ferezou, I., Carleton, A., Lüscher, H.R., Petersen, C.C.H. (2007). Combined voltage and calcium epifluorescence imaging in vitro and in vivo reveals subthreshold and suprathreshold dynamics of mouse barrel cortex. Journal of Neurophysiology, 97, 3751–3762.CrossRefPubMedGoogle Scholar
  7. Brette, R., & Gerstner, W. (2005). Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. Journal of Neurophysiology 3637–3642.Google Scholar
  8. Bringuier, V., Chavane, F., Glaeser, L., Fregnac, Y. (1999). Horizontal propagation of visual activity in the synaptic integration field of area 17 neurons. Science, 283, 695–699.CrossRefPubMedGoogle Scholar
  9. Brunel, N. (2000). Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience, 8, 183–208.CrossRefPubMedGoogle Scholar
  10. Brunel, N., & Hakim, V. (1999). Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Computation, 11, 1621–1671.CrossRefPubMedGoogle Scholar
  11. Brunel, N., & Wang, X.J. (2003). What determines the frequency of fast network oscillations with irregular neural discharges? I. Synaptic dynamics and excitation-inhibition balance. Journal of Neurophysiology, 90, 415–430.CrossRefPubMedGoogle Scholar
  12. Buzás, P., Kovács, K., Ferecskó, A.S., Budd, J.M.L., Eysel, U.T., Kisvárday, Z.F. (2006). Model-based analysis of excitatory lateral connections in the visual cortex. The Journal of Comparative Neurology, 499, 861–81.CrossRefPubMedGoogle Scholar
  13. Chemla, S., & Chavane, F. (2010). A biophysical cortical column model to study the multi-component origin of the VSDI signal. NeuroImage, 53, 420–438.CrossRefPubMedGoogle Scholar
  14. Chemla, S., & Chavane, F. (2016). Effects of gabaa kinetics on cortical population activity: computational studies and physiological confirmations. Journal of Neurophysiology, 115, 2867–2879.CrossRefPubMedPubMedCentralGoogle Scholar
  15. Chen, Y., Geisler, W.S., Seidemann, E. (2006). Optimal decoding of correlated neural population responses in the primate visual cortex. Nature Neuroscience, 9, 1412–1420.CrossRefPubMedPubMedCentralGoogle Scholar
  16. Chen, Y., Geisler, W.S., Seidemann, E. (2008). Optimal temporal decoding of neural population responses in a reaction-time visual detection task. Journal of Neurophysiology, 99, 1366–1379.CrossRefPubMedPubMedCentralGoogle Scholar
  17. Civillico, E.F., & Contreras, D. (2012). Spatiotemporal properties of sensory responses in vivo are strongly dependent on network context. Frontiers in Systems Neuroscience, 6, 25.CrossRefPubMedPubMedCentralGoogle Scholar
  18. Contreras, D., & Llinas, R. (2001). Voltage-sensitive dye imaging of neocortical spatiotemporal dynamics to afferent activation frequency. The Journal of Neuroscience: the Official Journal of the Society for Neuroscience, 21, 9403–9413.Google Scholar
  19. Daley, D.J., & Vere-Jones, D. (2007). An introduction to the theory of point processes: volume II: general theory and structure, vol. 2. Springer Science & Business Media.Google Scholar
  20. Destexhe, A., Rudolph, M., Paré, D. (2003). The high-conductance state of neocortical neurons in vivo. Nature Reviews Neuroscience, 4, 739–751.CrossRefPubMedGoogle Scholar
  21. El Boustani, S., & Destexhe, A. (2009). A master equation formalism for macroscopic modeling of asynchronous irregular activity states. Neural Computation, 21, 46–100.CrossRefPubMedGoogle Scholar
  22. Ferezou, I., Bolea, S., Petersen, C.C.H. (2006). Visualizing the cortical representation of whisker touch: voltage-sensitive dye imaging in freely moving mice. Neuron, 50, 617–629.CrossRefPubMedGoogle Scholar
  23. Gawne, T., McClurkin, J., Richmond, B., Optican, L. (1991). Lateral geniculate neurons in behaving primates. III. Response predictions of a channel model with multiple spatial-to-temporal filters. Journal of Neurophysiology, 66, 809–823.CrossRefPubMedGoogle Scholar
  24. Gilad, A., & Slovin, H. (2015). Population responses in v1 encode different figures by response amplitude. Journal of Neuroscience, 35, 6335–6349.CrossRefPubMedGoogle Scholar
  25. Girard, P., Hupé, J. M., Bullier, J. (2001). Feedforward and feedback connections between areas V1 and V2 of the monkey have similar rapid conduction velocities. Journal of Neurophysiology, 85(3), 1328–1331.CrossRefPubMedGoogle Scholar
  26. Goodman, D.F.M., & Brette, R. (2009). The brian simulator. Frontiers in Neuroscience, 3, 192–197.CrossRefPubMedPubMedCentralGoogle 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.CrossRefPubMedGoogle Scholar
  28. Jancke, D., Chavane, F., Naaman, S., Grinvald, A. (2004). Imaging cortical correlates of illusion in early visual cortex. Nature, 428, 423–426.CrossRefPubMedGoogle Scholar
  29. Kuhn, A., Aertsen, A., Rotter, S. (2004). Neuronal integration of synaptic input in the fluctuation-driven regime. The Journal of Neuroscience: the Official Journal of the Society for Neuroscience, 24, 2345–56.CrossRefGoogle Scholar
  30. Kumar, A., Schrader, S., Aertsen, A., Rotter, S. (2008). The high-conductance state of cortical networks. Neural Computation, 20, 1–43.CrossRefPubMedGoogle Scholar
  31. Latham, P.E., Richmond, B.J., Nelson, P.G., Nirenberg, S. (2000). Intrinsic dynamics in neuronal networks. I. Theory. Journal of Neurophysiology, 83, 808–827.CrossRefPubMedGoogle Scholar
  32. 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.CrossRefPubMedPubMedCentralGoogle Scholar
  33. Markram, H., Toledo-Rodriguez, M., Wang, Y., Gupta, A., Silberberg, G., Wu, C. (2004). Interneurons of the neocortical inhibitory system. Nature Reviews. Neuroscience, 5, 793–807.CrossRefPubMedGoogle Scholar
  34. Markram, H., Muller, E., Ramaswamy, S., Reimann, M. (2015). Reconstruction and simulation of neocortical microcircuitry. Cell, 163, 456–492.CrossRefPubMedGoogle Scholar
  35. McCormick, D.A., Connors, B.W., Lighthall, J.W., Da, Prince. (1985). Comparative electrophysiology of pyramidal and sparsely spiny stellate neurons of the neocortex. Journal of Neurophysiology, 54, 782–806.CrossRefPubMedGoogle Scholar
  36. Meirovithz, E., Ayzenshtat, I., Bonneh, Y.S., Itzhack, R., Werner-Reiss, U., Slovin, H. (2009). Population response to contextual influences in the primary visual cortex. Cerebral Cortex, 20, 1293–1304.CrossRefPubMedGoogle Scholar
  37. Muller, L., Reynaud, A., Chavane, F., Destexhe, A. (2014). The stimulus-evoked population response in visual cortex of awake monkey is a propagating wave. Nature Communications, 5, 3675.PubMedPubMedCentralGoogle Scholar
  38. Papoulis, A. (1991). Probability, random variables and stochastic processes. Mcgraw-Hill.Google Scholar
  39. Platkiewicz, J., & Brette, R. (2010). A threshold equation for action potential initiation. PLoS Computational Biology, 6, e1000850.CrossRefPubMedPubMedCentralGoogle Scholar
  40. Reinhold, K., Lien, A.D., Scanziani, M. (2015). Distinct recurrent versus afferent dynamics in cortical visual processing. Nature Neuroscience, 18.Google Scholar
  41. Renart, A., Brunel, N., Wang, X.J. (2004). Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks. Computational Neuroscience: A Comprehensive Approach 431–490.Google Scholar
  42. Reynaud, A., Masson, G.S., Chavane, F. (2012). Dynamics of local input normalization result from balanced short- and long-range intracortical interactions in area v1. The Journal of Neuroscience: the Official Journal of the Society for Neuroscience, 32, 12558–69.CrossRefGoogle Scholar
  43. Shoham, D., Glaser, D.E., Arieli, A., Kenet, T., Wijnbergen, C., Toledo, Y., Hildesheim, R., Grinvald, A. (1999). Imaging cortical dynamics at high spatial and temporal resolution with novel blue voltage-sensitive dyes. Neuron, 24, 791–802.CrossRefPubMedGoogle Scholar
  44. Stettler, D. D., Das, A., Bennett, J., Gilbert, C. D. (2002). Lateral connectivity and contextual interactions in macaque primary visual cortex. Neuron, 36(4), 739–750.CrossRefPubMedGoogle Scholar
  45. Steriade, M., Timofeev, I., Grenier, F. (2001). Natural waking and sleep states: a view from inside neocortical neurons. Journal of Neurophysiology, 85, 1969–1985.CrossRefPubMedGoogle Scholar
  46. Tan, A.Y., Chen, Y., Scholl, B., Seidemann, E., Priebe, N.J. (2014). Sensory stimulation shifts visual cortex from synchronous to asynchronous states. Nature, 509, 226–229.CrossRefPubMedPubMedCentralGoogle Scholar
  47. van Vreeswijk, C., & Sompolinsky, H. (1996). Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science (New York, N.Y.), 274, 1724–6.CrossRefGoogle Scholar
  48. Vogels, T.P., & Abbott, L.F. (2005). Signal propagation and logic gating in networks of integrate-and-fire neurons. The Journal of Neuroscience, 25, 10786–10795.CrossRefPubMedGoogle Scholar
  49. Yger, P., El Boustani, S., Destexhe, A., Yves, F. (2011). Topologically invariant macroscopic statistics in balanced networks of conductance-based integrate-and-fire neurons. Journal of Computational Neuroscience, 31, 229–245.CrossRefPubMedGoogle Scholar
  50. Zerlaut, Y., Telenczuk, B., Deleuze, C., Bal, T., Ouanounou, G., Destexhe, A. (2016). Heterogeneous firing response of mice layer V pyramidal neurons in the fluctuation-driven regime. The Journal of Physiology, 594, 3791–808.CrossRefPubMedPubMedCentralGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Unité de Neurosciences, Information et ComplexitéCentre National de la Recherche ScientifiqueGif sur YvetteFrance
  2. 2.Neural Coding laboratory, Center for Neuroscience and Cognitive Systems @UniTnIstituto Italiano di TecnologiaRoveretoItaly
  3. 3.Centre de Recherche Cerveau et CognitionUMR 5549 CNRS & Université Paul Sabatier Toulouse IIIToulouseFrance
  4. 4.Institut de Neurosciences de la Timone (INT)UMR 7289 CNRS & Aix-Marseille UniversitéMarseille Cedex 05France
  5. 5.European Institute for Theoretical NeuroscienceParisFrance

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