Robust emergence of small-world structure in networks of spiking neurons

  • Hoi Fei Kwok
  • Peter Jurica
  • Antonino Raffone
  • Cees van Leeuwen
Research Article


Spontaneous activity in biological neural networks shows patterns of dynamic synchronization. We propose that these patterns support the formation␣of a small-world structure—network connectivity␣optimal for distributed information processing. We␣present numerical simulations with connected Hindmarsh–Rose neurons in which, starting from random connection distributions, small-world networks evolve as a result of applying an adaptive rewiring rule. The rule connects pairs of neurons that tend fire in synchrony, and disconnects ones that fail to synchronize. Repeated application of the rule leads to small-world structures. This mechanism is robustly observed for bursting and irregular firing regimes.


Self-organization Spiking neuron Modularity Neural network 



The authors like to thank Dr Pulin Gong and the three anonymous reviewers for valuable advice and comments.


  1. Alexander JC, Cai D (1991) On the dynamics of bursting systems. J Math Biol 29:405–23CrossRefPubMedGoogle Scholar
  2. Antonini A, Stryker MP (1993) Development of individual geniculocortical arbors in cat straite cortex and effects of binocular impulse blockade. J Neurosci 13:3549–573PubMedGoogle Scholar
  3. Babloyantz A, Destexhe A (1986) Low-dimensional chaos in an instance of epilepsy. Proc Natl Acad Sci USA 83:3513–517CrossRefPubMedGoogle Scholar
  4. Barabasi AL, Albert R (1999) Emergence of scaling in random networks. Science 286:509–12CrossRefPubMedGoogle Scholar
  5. Barbour B, Hausser M (1997) Intersynaptic diffusion of neurotransmitter. Trend Neurosci 20:377–84CrossRefPubMedGoogle Scholar
  6. Barrett HC (2005) Enzymatic computation and cognitive modularity. Mind Lang 20:259–87Google Scholar
  7. Barrett HC, Kurzban R (2006) Modularity in cognition: framing the debate. Psychol Rev 113(3):628–47Google Scholar
  8. Breakspear M, Roberts JA, Terry JR, Rodrigues S, Mahant N, Robinson PA (2005) A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. Cereb Cortex 16(9):1296–313Google Scholar
  9. Bucolo M, Fortuna L, la Rosa M (2002) Network self-organization through “small-worlds–topologies. Chaos Solitons Fractals 14:1059–064CrossRefGoogle Scholar
  10. Carruthers P (2003) Is the mind a system of modules shaped by natural selection? In: Hitchcock C (ed) Contemporary debates in the philosophy of science. BlackwellGoogle Scholar
  11. Carruthers P (2005a) The case for massively modular models of mind. In: Stainton R (ed) Contemporary debates in cognitive science. BlackwellGoogle Scholar
  12. Carruthers P (2005b) Distinctively human thinking: modular precursors and components. In: Carruthers P, Laurence S, Stich S (eds) The innate mind: structure and content. Oxford University PressGoogle Scholar
  13. Catalano SM, Shatz CJ (1998) Activity-dependent cortical target selection by thalamic axons. Science 281:559–62CrossRefPubMedGoogle Scholar
  14. Changeux JP, Danchin A (1976) Selective stabilisation of developing synapses as a mechanism for the specification of neuronal networks. Nature 264:705–12CrossRefPubMedGoogle Scholar
  15. Edelman GM (1987) Neural Darwinism: the theory of neuronal group selection. Basic Books, New YorkGoogle Scholar
  16. Feller MB (1999) Spontaneous correlated activity in developing neural circuits. Neuron 22:653–56CrossRefPubMedGoogle Scholar
  17. Fodor JA (1983) The modularity of mind. MIT Press, Cambridge, MAGoogle Scholar
  18. Gade PM, Hu C-K (2000) Synchronous chaos in coupled map␣lattices with small-world interactions. Phys Rev E 62:6409–413CrossRefGoogle Scholar
  19. Gong P, van Leeuwen C (2003) Emergence of scale-free network with chaotic units. Physica A 321:679–88CrossRefGoogle Scholar
  20. Gong P, van Leeuwen C (2004) Evolution to a small-world network with chaotic units. Europhys Lett 67:328–33CrossRefGoogle Scholar
  21. Hansel D, Sompolinsky H (1992) Synchronization and computation in a chaotic neural network. Phys Rev Lett 68:718–21CrossRefPubMedGoogle Scholar
  22. Hindmarsch J, Rose RM (1984) A model of neuronal bursting using three coupled first order differential equations. Trans R Soc Lond B 221:87–02Google Scholar
  23. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–44PubMedGoogle Scholar
  24. Izhikevich EM (2004) Which model to use for cortical spiking neurons? IEEE Trans Neural Netw 155:1063–070CrossRefGoogle Scholar
  25. Kaas-Petersen C (1987) Bifurcations in the Rose–Hindmarch model and the Chay model. In: Degn H, Holden AV, Olsen LF (eds) Chaos in biological systems. Plenum, New York, pp 183–90Google Scholar
  26. Kaneko K (1989) Pattern dynamics in spatiotemporal chaos: pattern selection, diffusion of defect and pattern competition intermittency. Physica D 34:1–1CrossRefGoogle Scholar
  27. Kaneko K (1990) Globally coupled chaos violates the law of large numbers but not the central-limit theorem. Phys Rev Lett 65:1391–394CrossRefPubMedGoogle Scholar
  28. Katz LC, Shatz CJ (1996) Synaptic activity and the construction of cortical circuits. Science 274:1133–138CrossRefPubMedGoogle Scholar
  29. Kwon O, Moon H (2002) Coherence resonance in small-world networks of excitable cells. Phys Lett A 298:319–24CrossRefGoogle Scholar
  30. Lago-Fernandez LF, Huerta R, Corbacho F, Siguenza JA (2000) Fast response and temporal coherent oscillations in small-world networks. Phys Rev Lett 8412:2758–761CrossRefGoogle Scholar
  31. Latora V, Marchiori M (2001) Efficient behavior of small-world networks. Phys Rev Lett 87:198701-1–98701-4Google Scholar
  32. Maeda E, Kuroda Y, Robinson HPC, Kawana A (1998) Modification of parallel activity elicited by propagating bursts in developing networks of rat cortical neurons. Eur J Neurosci 10:488–96CrossRefPubMedGoogle Scholar
  33. Manrubia SC, Mikhailov AS (1999) Mutual synchronization and clustering in randomly coupled chaotic dynamical networks. Phys Rev E 60:1579–589CrossRefGoogle Scholar
  34. Mason C, Erskine L (2000) Growth cone form, behavior, and interactions in vivo: retinal axon pathfinding as a model. J Neurobiol 44:260–70CrossRefPubMedGoogle Scholar
  35. Masuda N, Aihara K (2004) Global and local synchrony of coupled neurons in small-world networks. Biol Cybern 90:302–09CrossRefPubMedGoogle Scholar
  36. McCormick DA (1999) Spontaneous activity: signal or noise? Science 285:541–54CrossRefPubMedGoogle Scholar
  37. Menendez de la Prida L, Sanchez-Andres JV (2000) Heterogeneous populations of cells mediate spontaneous synchronous bursting in the developing hippocampus through␣a frequency-dependent mechanism. Neuroscience 97:227–41CrossRefPubMedGoogle Scholar
  38. Mountcastle VB (1997) The columnar organization of the neocortex. Brain 120:701–22CrossRefPubMedGoogle Scholar
  39. Nakatani H, Khalilov I, Gong P, van Leeuwen C (2003) Nonlinearity in giant depolarizing potentials. Phys Lett A 319:167–72CrossRefGoogle Scholar
  40. Nishiyama M, Hong K, Mikoshiba K, Poo M, Kato K (2000) Calcium stores regulate the polarity and input specificity of synaptic modification. Nature 408:584–88CrossRefPubMedGoogle Scholar
  41. Otsu Y, Kimura F, Tsumoto T (1995) Hebbian induction of LTP in visual cortex: perforated patch-clamp study in cultured neurons. J Neurophysiol 746:2437–444Google Scholar
  42. Penn AA, Riquelme PA, Feller MB, Shatz CJ (1998) Competition in retinogeniculate patterning generated by spontaneous activity. Science 279:2108–112CrossRefPubMedGoogle Scholar
  43. Pike FG, Meredith RM, Olding AWA, Paulsen O (1999) Postsynaptic bursting is essential for ‘Hebbian–induction of associative long-term potentiation at excitatory synapses in rat hippocampus. J Physiol 518:571–76CrossRefPubMedGoogle Scholar
  44. Quartz SR (1999) The constructivist brain. Trend Cogn Sci 3:48–7CrossRefGoogle Scholar
  45. Raffone A, van Leeuwen C (2003) Dynamic synchronization and chaos in an associative neural network with multiple active memories. Chaos 13:1090–104CrossRefPubMedGoogle Scholar
  46. Ravasz E, Somera AL, Mongru DA, Oltvai ZN, Barabasi AL (2002) Hierarchical organization of modularity in metabolic networks. Science 297:1551–555CrossRefPubMedGoogle Scholar
  47. Rose RM, Hindmarsh JL (1989) The assembly of ionic currents in a thalamic neuron. I. The three-dimensional model. Proc R Soc Lond B 237:267–88CrossRefPubMedGoogle Scholar
  48. Roxin A, Riecke H, Solla SA (2004) Self-sustained activity and failure in small-world networks of excitable neurons. Phys Rev Lett 92 198101-1–98101-4CrossRefGoogle Scholar
  49. Sem’yanov AV (2005) Diffusional extrasynaptic neurotransmission via glutamate and GABA. Neurosci Behav Physiol 35:253–66Google Scholar
  50. Shefi O, Golding I, Segev R, Ben-Jacob E, Ayali A (2002) Morphological characterization of in vitro neuronal networks. Phys Rev E 66 021905-1–21905-5CrossRefGoogle Scholar
  51. Stam CJ (2004) Functional connectivity patterns of human magnetoencephalographic recordings: a ‘small-world–network? Neurosci Lett 355:25–8CrossRefPubMedGoogle Scholar
  52. Van den Berg D, van Leeuwen C (2004) Adaptive wiring in chaotic networks renders small-world connectivity with consistent clusters. Europhys Lett 654:459–64CrossRefGoogle Scholar
  53. Van Pelt J, Corner MA, Wolters PS, Rutten WLC, Ramakers GJA (2004) Longterm stability and developmental changes in spontaneous network burst firing patterns in dissociated rat cerebral cortex cell cultures on multielectrode arrays. Neurosci Lett 361:86–9CrossRefPubMedGoogle Scholar
  54. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world–networks. Nature 393:440–42CrossRefPubMedGoogle Scholar
  55. Zhang LI, Poo M (2001) Electrical activity and development of neural circuits. Nat Neurosci 4:1207–214CrossRefPubMedGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Hoi Fei Kwok
    • 1
    • 2
    • 3
  • Peter Jurica
    • 1
  • Antonino Raffone
    • 1
    • 2
  • Cees van Leeuwen
    • 1
    • 2
  1. 1.Laboratory for Perceptual DynamicsRIKEN Brain Science InstituteSaitamaJapan
  2. 2.Department of PsychologySunderland UniversitySunderlandUK
  3. 3.University of BirminghamBirminghamUK

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