The relationship between the dynamics of neural networks and their patterns of connectivity is far from clear, despite its importance for understanding functional properties. Here, we have studied sparsely-connected networks of conductance-based integrate-and-fire (IF) neurons with balanced excitatory and inhibitory connections and with finite axonal propagation speed. We focused on the genesis of states with highly irregular spiking activity and synchronous firing patterns at low rates, called slow Synchronous Irregular (SI) states. In such balanced networks, we examined the “macroscopic” properties of the spiking activity, such as ensemble correlations and mean firing rates, for different intracortical connectivity profiles ranging from randomly connected networks to networks with Gaussian-distributed local connectivity. We systematically computed the distance-dependent correlations at the extracellular (spiking) and intracellular (membrane potential) levels between randomly assigned pairs of neurons. The main finding is that such properties, when they are averaged at a macroscopic scale, are invariant with respect to the different connectivity patterns, provided the excitatory-inhibitory balance is the same. In particular, the same correlation structure holds for different connectivity profiles. In addition, we examined the response of such networks to external input, and found that the correlation landscape can be modulated by the mean level of synchrony imposed by the external drive. This modulation was found again to be independent of the external connectivity profile. We conclude that first and second-order “mean-field” statistics of such networks do not depend on the details of the connectivity at a microscopic scale. This study is an encouraging step toward a mean-field description of topological neuronal networks.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
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(3), 237–252.
Arieli, A., Sterkin, A., Grinvald, A., & Aertsen, A. (1996). Dynamics of ongoing activity: Explanation of the large variability in evoked cortical responses. Science, 273(5283), 1868–1871.
Benucci, A., Frazor, R. A., & Carandini, M. (2007). Standing waves and traveling waves distinguish two circuits in visual cortex. Neuron, 55(1), 103–117.
Berger, D., Warren, D., Normann, R., Arieli, A., & Grün, S. (2007). Spatially organized spike correlation in cat visual cortex. Neurocomputing, 70(10–12), 2112–2116.
Bienenstock, E. (1996). On the dimensionality of cortical graphs. Journal of Physiology (Paris), 90(3–4), 251–256.
Braitenberg, V, & Schüz, A. (1998). Cortex: Statistics and geometry of neuronal connectivity. Berlin: Springer.
Bringuier, V., Chavane, F., Glaeser, L., & Frégnac, Y. (1999). Horizontal propagation of visual activity in the synaptic integration field of area 17 neurons. Science, 283(5402), 695–699.
Brunel, N. (2000). Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience, 8(3), 183–208.
Cessac, B., & Viéville, T. (2008). On dynamics of integrate-and-fire neural networks with conductance based synapses. Frontiers in Computational Neuroscience, 2, 2. doi:10.3389/neuro.10.002.2008.
Contreras, D. (2007). Propagating waves in visual cortex. Neuron, 55(1), 3–5.
Davison, A. P., Bruederle, D., Eppler, J., Kremkow, J., Muller, E., Pecevski, D., et al. (2009) PyNN: A common interface for neuronal network simulators. Front Neuroinformatics 2, 11. doi:10.3389/neuro.11.011.2008.
de la Rocha, J., Doiron, B., Shea-Brown, E., Josi, K., & Reyes, A. (2007). Correlation between neural spike trains increases with firing rate. Nature, 448(7155), 802–806.
Destexhe, A, & Paré, D. (1999). Impact of network activity on the integrative properties of neocortical pyramidal neurons in vivo. Journal of Neurophysiology, 81(4), 1531–1547.
Diesmann, M., & Gewaltig, M. (2001). NEST: An environment for neural systems simulations. Forschung und wisschenschaftliches Rechnen, Beitrage zum Heinz-Biling-Preis 58, 43–70.
El Boustani, S., & Destexhe, A. (2009). A master equation formalism for macroscopic modeling of asynchronous irregular activity states. Neural Computation, 21(1), 46–100.
El Boustani, S., Marre, O., Béhuret, S., Baudot, P., Yger, P., Bal, T., et al. (2009). Network-state modulation of power-law frequency-scaling in visual cortical neurons. PLoS Computational Biology, 5(9), e1000,519.
Gil, Z., Connors, B. W., & Amitai, Y. (1999). Efficacy of thalamocortical and intracortical synaptic connections: Quanta, innervation, and reliability. Neuron, 23(2), 385–397.
Gilbert, C. D., & Wiesel, T. N. (1983). Clustered intrinsic connections in cat visual cortex. Journal of Neuroscience, 3(5), 1116–1133.
Göbel, W., Kampa, B. M., & Helmchen, F. (2007). Imaging cellular network dynamics in three dimensions using fast 3d laser scanning. Nature Methods, 4(1), 73–79.
Gonzlez-Burgos, G., Barrionuevo, G., & Lewis, D. A. (2000). Horizontal synaptic connections in monkey prefrontal cortex: An in vitro electrophysiological study. Cerebral Cortex, 10(1), 82–92.
Greenberg, D. S., Houweling, A. R., & Kerr, J. N. D. (2008). Population imaging of ongoing neuronal activity in the visual cortex of awake rats. Nature Neuroscience, 11(7), 749–751.
Grinvald, A., Lieke, E. E., Frostig, R. D., & Hildesheim, R. (1994). Cortical point-spread function and long-range lateral interactions revealed by real-time optical imaging of macaque monkey primary visual cortex. Journal of Neuroscience, 14(5 Pt 1), 2545–2568.
Han, F., Caporale, N., & Dan, Y. (2008). Reverberation of recent visual experience in spontaneous cortical waves. Neuron, 60(2), 321–327.
Hellwig, B. (2000). A quantitative analysis of the local connectivity between pyramidal neurons in layers 2/3 of the rat visual cortex. Biological Cybernetics, 82(2), 111–121.
Izhikevich, E. M., Gally, J. A., & Edelman, G. M. (2004). Spike-timing dynamics of neuronal groups. Cerebral Cortex, 14(8), 933–944.
Kitano, K., & Fukai, T. (2007). Variability v.s. synchronicity of neuronal activity in local cortical network models with different wiring topologies. Journal of Computational Neuroscience, 23(2), 237–250.
Kohn, A., & Smith, M. A. (2005). Stimulus dependence of neuronal correlation in primary visual cortex of the macaque. Journal of Neuroscience, 25(14), 3661–3673.
Kriener, B., Tetzlaff, T., Aertsen, A., Diesmann, M., & Rotter, S. (2008). Correlations and population dynamics in cortical networks. Neural Computation, 20(9), 2185–2226.
Kriener, B., Helias, M., Aertsen, A., & Rotter, S. (2009). Correlations in spiking neuronal networks with distance dependent connections. Journal of Computational Neuroscience.
Kuhn, A., Aertsen, A., & Rotter, S. (2003). Higher-order statistics of input ensembles and the response of simple model neurons. Neural Computation, 15(1), 67–101.
Kumar, A., Schrader, S., Aertsen, A., & Rotter, S. (2008), The high-conductance state of cortical networks. Neural Computation, 20(1), 1–43.
Larkum, M. E., Zhu, J. J., & Sakmann, B. (2001). Dendritic mechanisms underlying the coupling of the dendritic with the axonal action potential initiation zone of adult rat layer 5 pyramidal neurons. Journal of Physiology, 533(Pt 2), 447–466.
Liu, C. Y., & Nykamp, D. Q. (2009). A kinetic theory approach to capturing interneuronal correlation: The feedforward case. Journal of Computational Neuroscience, 26, 339–368.
Marre, O., Yger, P., Davison, A., & Frégnac, Y. (2009). Reliable recall of spontaneous activity patterns in cortical networks. Journal of Neuroscience, 29(46), 14596–14606.
Mehring, C., Hehl, U., Kubo, M., Diesmann, M., & Aertsen, A. (2003). Activity dynamics and propagation of synchronous spiking in locally connected random networks. Biological Cybernetics, 88(5), 395–408.
Mitchell, J. F., Sundberg, K. A., & Reynolds, J. H. (2009). Spatial attention decorrelates intrinsic activity fluctuations in macaque area v4. Neuron, 63(6), 879–888.
Nauhaus, I., Busse, L., Carandini, M., & Ringach, D. L. (2009). Stimulus contrast modulates functional connectivity in visual cortex. Nature. Neuroscience, 12(1), 70–76.
Nirenberg, S., & Latham, P. E. (2003). Decoding neuronal spike trains: How important are correlations? Proceedings of the National Academy of Sciences of the United States of America, 100(12), 7348–7353.
Oswald, A. M. M., & Reyes, A. D. (2008). Maturation of intrinsic and synaptic properties of layer 2/3 pyramidal neurons in mouse auditory cortex. Journal of Neurophysiology, 99(6), 2998–3008.
Renart, A., de la Rocha, J., Bartho, P., Hollender, L., Parga, N., Reyes, A., et al. (2010). The asynchronous state in cortical circuits. Science, 327(5965), 587–590.
Roxin, A., Brunel, N., & Hansel, D. (2005). Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks. Physical Review Letters, 94(23), 238103.
Schwarz, C., & Bolz, J. (1991). Functional specificity of a long-range horizontal connection in cat visual cortex: A cross-correlation study. Journal of Neuroscience, 11(10), 2995–3007.
Shea-Brown, E., Josi, K., de la Rocha, J., & Doiron, B. (2008). Correlation and synchrony transfer in integrate-and-fire neurons: Basic properties and consequences for coding. Physical Review Letters, 100(10), 108102.
Singer, W., & Gray, C. M. (1995). Visual feature integration and the temporal correlation hypothesis. Annual Review of Neuroscience, 18, 555–586.
Smith, M. A., & Kohn, A. (2008). Spatial and temporal scales of neuronal correlation in primary visual cortex. Journal of Neuroscience, 28(48), 12591–12603.
Usher, M., Stemmler, M., Koch, C., & Olami, Z. (1994). Network amplification of local fluctuations causes high spike rate variability, fractal firing patterns and oscillatory local field potentials. Neural Computation, 6(5), 795–836.
van Vreeswijk, C., & Sompolinsky, H. (1996). Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science, 274(5293), 1724–1726.
van Vreeswijk, C., & Sompolinsky, H. (1998). Chaotic balanced state in a model of cortical circuits. Neural Computation, 10(6), 1321–1371.
Vogels, T. P., & Abbott, L. F. (2005). Signal propagation and logic gating in networks of integrate-and-fire neurons. Journal of Neuroscience, 25(46), 10786–10795.
Voges, N., Aertsen, A., & Rotter, S. (2007). Statistical analysis of spatially embedded networks: From grid to random node positions. Neurocomputing, 70(10–12), 1833–1837.
Zohary, E., Shadlen, M. N., & Newsome, W. T. (1994). Correlated neuronal discharge rate and its implications for psychophysical performance. Nature, 370(6485), 140–143.
We thank Olivier Marre for helpful discussions and Andrew Davison for comments on the manuscript. P.Y. was supported by a MENRT bursary from the University of Paris XI, S.E.B was supported by a FRM fellowship. Research supported by the CNRS, ANR (NATSTATS and HR-CORTEX) and the European Commission (FACETS FP6-2004-IST-FETPI 15879 and Brain-i-Nets FP7-ICT-2007-C 243914).
Action Editor: Mark van Rossum
Pierre Yger and Sami El Boustani contributed equally.
About this article
Cite this article
Yger, P., El Boustani, S., Destexhe, A. et al. Topologically invariant macroscopic statistics in balanced networks of conductance-based integrate-and-fire neurons. J Comput Neurosci 31, 229–245 (2011). https://doi.org/10.1007/s10827-010-0310-z