Emergence of Small-World Structure in Networks of Spiking Neurons Through STDP Plasticity

  • Gleb Basalyga
  • Pablo M. Gleiser
  • Thomas Wennekers
Part of the Advances in Experimental Medicine and Biology book series (AEMB, volume 718)


In this work, we use a complex network approach to investigate how a neural network structure changes under synaptic plasticity. In particular, we consider a network of conductance-based, single-compartment integrate-and-fire excitatory and inhibitory neurons. Initially the neurons are connected randomly with uniformly distributed synaptic weights. The weights of excitatory connections can be strengthened or weakened during spiking activity by the mechanism known as spike-timing-dependent plasticity (STDP). We extract a binary directed connection matrix by thresholding the weights of the excitatory connections at every simulation step and calculate its major topological characteristics such as the network clustering coefficient, characteristic path length and small-world index. We numerically demonstrate that, under certain conditions, a nontrivial small-world structure can emerge from a random initial network subject to STDP learning.


Cluster Coefficient Random Network Synaptic Weight Excitatory Synapse Connection Probability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported by an EPSRC research grant (Ref. EP/C010841/1).


  1. 1.
    Sporns, O., Chialvo, D.R., Kaiser, M., Hilgetag, C.C.: Organization, development and function of complex brain networks. Trends Cogn. Sci. 8, 418–425 (2004) PubMedCrossRefGoogle Scholar
  2. 2.
    Reijneveld, J.C., Ponten, S.C., Berendse, H.W., Stam, C.J.: The application of graph theoretical analysis to complex networks in the brain. Clin. Neurophysiol. 118, 2317–2331 (2007) PubMedCrossRefGoogle Scholar
  3. 3.
    Bullmore, E., Sporns, O.: Complex brain networks: graph theoretical analysis of structural and functional systems. Nat. Rev., Neurosci. 10(3), 186–198 (2009) CrossRefGoogle Scholar
  4. 4.
    Gomez Portillo, I.J., Gleiser, P.M.: An adaptive complex network model for brain functional networks. PLoS ONE 4(9), e6863 (2009). doi: 10.1371/journal.pone.0006863 PubMedCrossRefGoogle Scholar
  5. 5.
    Sporns, O., Honey, C.J.: Small worlds inside big brains. Proc. Natl. Acad. Sci. USA 103(51), 19219–19220 (2006) PubMedCrossRefGoogle Scholar
  6. 6.
    Yu, S., Huang, D., Singer, W., Nikolic, D.: A small world of neuronal synchrony. Cereb. Cortex 18(12), 2891–2901 (2008) PubMedCrossRefGoogle Scholar
  7. 7.
    Bassett, D.S., Bullmore, E.: Small-world brain networks. Neuroscientist 10, 512–523 (2006) CrossRefGoogle Scholar
  8. 8.
    Sporns, O., Tononi, G., Edelman, G.M.: Connectivity and complexity: the relationship between neuroanatomy and brain dynamics. Neural Netw. 13(8–9), 909–922 (2000) PubMedCrossRefGoogle Scholar
  9. 9.
    Tononi, G., Edelman, G.M., Sporns, O.: Complexity and coherency: integrating information in the brain. Trends Cogn. Sci. 2, 474–484 (1998) PubMedCrossRefGoogle Scholar
  10. 10.
    Tononi, G.: An information integration theory of consciousness. BMC Neurosci. 5, 42 (2004) PubMedCrossRefGoogle Scholar
  11. 11.
    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.-U.: Complex networks: Structure and dynamics. Phys. Rep. 424, 175–308 (2006) CrossRefGoogle Scholar
  12. 12.
    Rubinov, M., Kotter, R., Hagmann, P., Sporns, O.: Brain connectivity toolbox: a collection of complex network measurements and brain connectivity datasets. NeuroImage 47(Suppl 1), 39–41 (2009) Google Scholar
  13. 13.
    Rubinov, M., Sporns, O.: Complex network measures of brain connectivity: Uses and interpretations. NeuroImage 52(3), 1059–1069 (2010) PubMedCrossRefGoogle Scholar
  14. 14.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998) PubMedCrossRefGoogle Scholar
  15. 15.
    Fagiolo, G.: Clustering in complex directed networks. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 76(2), 026107 (2007) CrossRefGoogle Scholar
  16. 16.
    Humphries, M.D., Gurney, K.: Network ‘small-world-ness’: A quantitative method for determining canonical network equivalence. PLoS ONE 3(4), 0002051 (2008). doi: 10.1371/journal.pone.0002051 CrossRefGoogle Scholar
  17. 17.
    Maslov, S., Sneppen, K.: Specificity and stability in topology of protein networks. Science 296(5569), 910–913 (2002) PubMedCrossRefGoogle Scholar
  18. 18.
    Shin, C.-W., Kim, S.: Self-organized criticality and scale-free properties in emergent functional neural networks. Phys. Rev. E 74(4), 45101 (2006) CrossRefGoogle Scholar
  19. 19.
    Kato, H., Kimura, T., Ikeguchi, T.: Self-organized neural network structure depending on the STDP learning rules. In: Visarath, X., et al. (eds.) Applications of Nonlinear Dynamics. Model and Design of Complex Systems. Understanding Complex Systems, pp. 413–416. Springer, Berlin (2009) Google Scholar
  20. 20.
    Kato, H., Ikeguchi, T., Aihara, K.: Structural analysis on STDP neural networks using complex network theory. In: Artificial Neural Networks—ICANN 2009. Lecture Notes in Computer Science, vol. 5768, pp. 306–314. Springer, Berlin (2009) CrossRefGoogle Scholar
  21. 21.
    Bi, G., Poo, M.: Synaptic modifications in cultured hippocampal neurons: Dependence on spike timing, synaptic strength, and postsynaptic cell type. J. Neurosci. 18, 10464–10472 (1998) PubMedGoogle Scholar
  22. 22.
    Song, S., Miller, K.D., Abbott, L.F.: Competitive Hebbian learning through spike-timing-dependent synaptic plasticity. Nat. Neurosci. 3, 919–926 (2000) PubMedCrossRefGoogle Scholar
  23. 23.
    Gleeson, P., Steuber, V., Silver, R.A.: neuroConstruct: a tool for modeling networks of neurons in 3D space. Neuron 54(2), 219–235 (2007) PubMedCrossRefGoogle Scholar
  24. 24.
    Gleeson, P., Crook, S., Cannon, R.C., Hines, M.L., Billings, G.O., Farinella, M., Morse, T.M., Davison, A.P., Ray, S., Bhalla, U.S., Barnes, S.R., Dimitrova, Y.D., Silver, R.A.: NeuroML: A language for describing data driven models of neurons and networks with a high degree of biological detail. PLoS Comput. Biol. 6(6), 1000815 (2010). doi: 10.1371/journal.pcbi.1000815 CrossRefGoogle Scholar
  25. 25.
    Brette, R., Rudolph, M., Carnevale, T., Hines, M., Beeman, D., Bower, J., Diesmann, M., Morrison, A., Goodman, P., Harris, F., Zirpe, M., Natschlager, T., Pecevski, D., Ermentrout, B., Djurfeldt, M., Lansner, A., Rochel, O., Vieville, T., Muller, E., Davison, A., El Boustani, S., Destexhe, A.: Simulation of networks of spiking neurons: A review of tools and strategies. J. Comput. Neurosci. 23, 349–398 (2007) PubMedCrossRefGoogle Scholar
  26. 26.
    Billings, G., van Rossum, M.C.W.: Memory retention and spike-timing-dependent plasticity. J. Neurophysiol. 101, 2775–2788 (2009) PubMedCrossRefGoogle Scholar
  27. 27.
    Carnevale, T., Hines, M.: The NEURON Book. Cambridge University Press, Cambridge (2006) Google Scholar
  28. 28.
    Tononi, G., Sporns, O.: Measuring information integration. BMC Neurosci. 4, 31–51 (2003) PubMedCrossRefGoogle Scholar
  29. 29.
    Balduzzi, D., Tononi, G.: Integrated information in discrete dynamical systems: Motivation and theoretical framework. PLoS Comput. Biol. 4(6), 1000091 (2008). doi: 10.1371/journal.pcbi.1000091 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Gleb Basalyga
    • 1
  • Pablo M. Gleiser
    • 2
  • Thomas Wennekers
    • 3
  1. 1.Centre for Robotics and Neural Systems (CRNS)University of PlymouthPlymouthUK
  2. 2.CONICETCentro Atomico BarilocheBarilocheArgentina
  3. 3.Centre for Robotics and Neural SystemsUniversity of PlymouthDevonUK

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