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

Cognitive Neurodynamics volume 1, Article number: 39 (2007) Cite this article

Abstract

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.

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References

  • Alexander JC, Cai D (1991) On the dynamics of bursting systems. J Math Biol 29:405–23

    Article  CAS  PubMed  Google Scholar 

  • Antonini A, Stryker MP (1993) Development of individual geniculocortical arbors in cat straite cortex and effects of binocular impulse blockade. J Neurosci 13:3549–573

    CAS  PubMed  Google Scholar 

  • Babloyantz A, Destexhe A (1986) Low-dimensional chaos in an instance of epilepsy. Proc Natl Acad Sci USA 83:3513–517

    Article  CAS  PubMed  Google Scholar 

  • Barabasi AL, Albert R (1999) Emergence of scaling in random networks. Science 286:509–12

    Article  PubMed  Google Scholar 

  • Barbour B, Hausser M (1997) Intersynaptic diffusion of neurotransmitter. Trend Neurosci 20:377–84

    Article  CAS  PubMed  Google Scholar 

  • Barrett HC (2005) Enzymatic computation and cognitive modularity. Mind Lang 20:259–87

    Google Scholar 

  • Barrett HC, Kurzban R (2006) Modularity in cognition: framing the debate. Psychol Rev 113(3):628–47

    Google Scholar 

  • 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):1296–313

    Google Scholar 

  • Bucolo M, Fortuna L, la Rosa M (2002) Network self-organization through “small-worlds–topologies. Chaos Solitons Fractals 14:1059–064

    Article  Google Scholar 

  • 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. Blackwell

  • Carruthers P (2005a) The case for massively modular models of mind. In: Stainton R (ed) Contemporary debates in cognitive science. Blackwell

  • 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 Press

  • Catalano SM, Shatz CJ (1998) Activity-dependent cortical target selection by thalamic axons. Science 281:559–62

    Article  CAS  PubMed  Google Scholar 

  • Changeux JP, Danchin A (1976) Selective stabilisation of developing synapses as a mechanism for the specification of neuronal networks. Nature 264:705–12

    Article  CAS  PubMed  Google Scholar 

  • Edelman GM (1987) Neural Darwinism: the theory of neuronal group selection. Basic Books, New York

    Google Scholar 

  • Feller MB (1999) Spontaneous correlated activity in developing neural circuits. Neuron 22:653–56

    Article  CAS  PubMed  Google Scholar 

  • Fodor JA (1983) The modularity of mind. MIT Press, Cambridge, MA

    Google Scholar 

  • Gade PM, Hu C-K (2000) Synchronous chaos in coupled map␣lattices with small-world interactions. Phys Rev E 62:6409–413

    Article  CAS  Google Scholar 

  • Gong P, van Leeuwen C (2003) Emergence of scale-free network with chaotic units. Physica A 321:679–88

    Article  Google Scholar 

  • Gong P, van Leeuwen C (2004) Evolution to a small-world network with chaotic units. Europhys Lett 67:328–33

    Article  CAS  Google Scholar 

  • Hansel D, Sompolinsky H (1992) Synchronization and computation in a chaotic neural network. Phys Rev Lett 68:718–21

    Article  PubMed  Google Scholar 

  • Hindmarsch J, Rose RM (1984) A model of neuronal bursting using three coupled first order differential equations. Trans R Soc Lond B 221:87–02

    Google Scholar 

  • Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–44

    CAS  PubMed  Google Scholar 

  • Izhikevich EM (2004) Which model to use for cortical spiking neurons? IEEE Trans Neural Netw 155:1063–070

    Article  Google Scholar 

  • 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 183–90

    Google Scholar 

  • Kaneko K (1989) Pattern dynamics in spatiotemporal chaos: pattern selection, diffusion of defect and pattern competition intermittency. Physica D 34:1–1

    Article  Google Scholar 

  • Kaneko K (1990) Globally coupled chaos violates the law of large numbers but not the central-limit theorem. Phys Rev Lett 65:1391–394

    Article  PubMed  Google Scholar 

  • Katz LC, Shatz CJ (1996) Synaptic activity and the construction of cortical circuits. Science 274:1133–138

    Article  CAS  PubMed  Google Scholar 

  • Kwon O, Moon H (2002) Coherence resonance in small-world networks of excitable cells. Phys Lett A 298:319–24

    Article  CAS  Google Scholar 

  • Lago-Fernandez LF, Huerta R, Corbacho F, Siguenza JA (2000) Fast response and temporal coherent oscillations in small-world networks. Phys Rev Lett 8412:2758–761

    Article  Google Scholar 

  • Latora V, Marchiori M (2001) Efficient behavior of small-world networks. Phys Rev Lett 87:198701-1–98701-4

    Google Scholar 

  • 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 10:488–96

    Article  CAS  PubMed  Google Scholar 

  • Manrubia SC, Mikhailov AS (1999) Mutual synchronization and clustering in randomly coupled chaotic dynamical networks. Phys Rev E 60:1579–589

    Article  CAS  Google Scholar 

  • Mason C, Erskine L (2000) Growth cone form, behavior, and interactions in vivo: retinal axon pathfinding as a model. J Neurobiol 44:260–70

    Article  CAS  PubMed  Google Scholar 

  • Masuda N, Aihara K (2004) Global and local synchrony of coupled neurons in small-world networks. Biol Cybern 90:302–09

    Article  PubMed  Google Scholar 

  • McCormick DA (1999) Spontaneous activity: signal or noise? Science 285:541–54

    Article  CAS  PubMed  Google Scholar 

  • 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 97:227–41

    Article  CAS  PubMed  Google Scholar 

  • Mountcastle VB (1997) The columnar organization of the neocortex. Brain 120:701–22

    Article  PubMed  Google Scholar 

  • Nakatani H, Khalilov I, Gong P, van Leeuwen C (2003) Nonlinearity in giant depolarizing potentials. Phys Lett A 319:167–72

    Article  CAS  Google Scholar 

  • Nishiyama M, Hong K, Mikoshiba K, Poo M, Kato K (2000) Calcium stores regulate the polarity and input specificity of synaptic modification. Nature 408:584–88

    Article  CAS  PubMed  Google Scholar 

  • Otsu Y, Kimura F, Tsumoto T (1995) Hebbian induction of LTP in visual cortex: perforated patch-clamp study in cultured neurons. J Neurophysiol 746:2437–444

    Google Scholar 

  • Penn AA, Riquelme PA, Feller MB, Shatz CJ (1998) Competition in retinogeniculate patterning generated by spontaneous activity. Science 279:2108–112

    Article  CAS  PubMed  Google Scholar 

  • 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 518:571–76

    Article  CAS  PubMed  Google Scholar 

  • Quartz SR (1999) The constructivist brain. Trend Cogn Sci 3:48–7

    Article  Google Scholar 

  • Raffone A, van Leeuwen C (2003) Dynamic synchronization and chaos in an associative neural network with multiple active memories. Chaos 13:1090–104

    Article  PubMed  Google Scholar 

  • Ravasz E, Somera AL, Mongru DA, Oltvai ZN, Barabasi AL (2002) Hierarchical organization of modularity in metabolic networks. Science 297:1551–555

    Article  CAS  PubMed  Google Scholar 

  • Rose RM, Hindmarsh JL (1989) The assembly of ionic currents in a thalamic neuron. I. The three-dimensional model. Proc R Soc Lond B 237:267–88

    Article  CAS  PubMed  Google Scholar 

  • Roxin A, Riecke H, Solla SA (2004) Self-sustained activity and failure in small-world networks of excitable neurons. Phys Rev Lett 92 198101-1–98101-4

    Article  Google Scholar 

  • Sem’yanov AV (2005) Diffusional extrasynaptic neurotransmission via glutamate and GABA. Neurosci Behav Physiol 35:253–66

    Google Scholar 

  • Shefi O, Golding I, Segev R, Ben-Jacob E, Ayali A (2002) Morphological characterization of in vitro neuronal networks. Phys Rev E 66 021905-1–21905-5

    Article  Google Scholar 

  • Stam CJ (2004) Functional connectivity patterns of human magnetoencephalographic recordings: a ‘small-world–network? Neurosci Lett 355:25–8

    Article  CAS  PubMed  Google Scholar 

  • Van den Berg D, van Leeuwen C (2004) Adaptive wiring in chaotic networks renders small-world connectivity with consistent clusters. Europhys Lett 654:459–64

    Article  Google Scholar 

  • 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 361:86–9

    Article  PubMed  Google Scholar 

  • Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world–networks. Nature 393:440–42

    Article  CAS  PubMed  Google Scholar 

  • Zhang LI, Poo M (2001) Electrical activity and development of neural circuits. Nat Neurosci 4:1207–214

    Article  CAS  PubMed  Google Scholar 

Download references

Acknowledgements

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

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Author notes

  1. Hoi Fei Kwok

    Present address: University of Birmingham, Birmingham, UK

Affiliations

  1. Laboratory for Perceptual Dynamics, RIKEN Brain Science Institute, 2-1 Hirosawa, Wako-shi, Saitama, 351-0198, Japan

    Hoi Fei Kwok, Peter Jurica, Antonino Raffone & Cees van Leeuwen

  2. Department of Psychology, Sunderland University, St Peter’s Campus, Sunderland, SR6 0DD, UK

    Hoi Fei Kwok, Antonino Raffone & Cees van Leeuwen

Authors
  1. Hoi Fei Kwok
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  2. Peter Jurica
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  3. Antonino Raffone
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  4. Cees van Leeuwen
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Corresponding author

Correspondence to Cees van Leeuwen.

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Kwok, H.F., Jurica, P., Raffone, A. et al. Robust emergence of small-world structure in networks of spiking neurons. Cogn Neurodyn 1, 39 (2007). https://doi.org/10.1007/s11571-006-9006-5

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  • DOI: https://doi.org/10.1007/s11571-006-9006-5

Keywords

  • Self-organization
  • Spiking neuron
  • Modularity
  • Neural network