Advertisement

Biological Cybernetics

, Volume 108, Issue 4, pp 381–396 | Cite as

Aspects of randomness in neural graph structures

  • Michelle Rudolph-LilithEmail author
  • Lyle E. Muller
Original Paper

Abstract

In the past two decades, significant advances have been made in understanding the structural and functional properties of biological networks, via graph-theoretic analysis. In general, most graph-theoretic studies are conducted in the presence of serious uncertainties, such as major undersampling of the experimental data. In the specific case of neural systems, however, a few moderately robust experimental reconstructions have been reported, and these have long served as fundamental prototypes for studying connectivity patterns in the nervous system. In this paper, we provide a comparative analysis of these “historical” graphs, both in their directed (original) and symmetrized (a common preprocessing step) forms, and provide a set of measures that can be consistently applied across graphs (directed or undirected, with or without self-loops). We focus on simple structural characterizations of network connectivity and find that in many measures, the networks studied are captured by simple random graph models. In a few key measures, however, we observe a marked departure from the random graph prediction. Our results suggest that the mechanism of graph formation in the networks studied is not well captured by existing abstract graph models in their first- and second-order connectivity.

Keywords

Graph theory Network structure Random graphs  Scale-free graphs Mammalian brain C. elegans Network models 

Notes

Acknowledgments

The authors wish to thank OD Little for inspiring comments and A Destexhe for continuing support. Work supported by the CNRS and the European Community (BrainScales project, FP7-269921). LM is a PhD fellow from École des Neurosciences de Paris (ENP).

Supplementary material

422_2014_606_MOESM1_ESM.pdf (917 kb)
Supplementary material 1 (pdf 917 KB)

References

  1. Aiello W, Chung F, Lu L (2000) A random graph model for massive graphs. In: Proceedings of the 32nd annual ACM symposium on theory of computing, association of computing machinery. New York, pp 171–180Google Scholar
  2. Albert R, Barabási A-L (2002) The statistical mechanics of complex networks. Rev Mod Phys 74:47–97CrossRefGoogle Scholar
  3. Albert R, Jeong H, Barabási A-L (1999) Diameter of the world-wide web. Nature 401:130–131CrossRefGoogle Scholar
  4. Alm E, Arkin A (2003) Biological networks. Curr Opinion Struct Biol 13:193–202CrossRefGoogle Scholar
  5. Alon U (2003) Biological networks: the tinkerer as engineer. Science 301:1866–1867PubMedCrossRefGoogle Scholar
  6. Amaral LAN, Ottino JM (2004) Complex networks. Eur Phys J B38:147–162CrossRefGoogle Scholar
  7. Artzy-Randrup Y, Fleishman SJ, Ben-Tal N, Stone L (2004) Comment on “Network motifs: simple building blocks of complex networks” and “Superfamilies of evolved and designed networks”. Science 305:1107PubMedCrossRefGoogle Scholar
  8. Babadi B, Abbott LF (2013) Pairwise analysis can account for network structures arising from spike-timing dependent plasticity. PLoS Comput Biol 9:e1002906PubMedCentralPubMedCrossRefGoogle Scholar
  9. Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286:509–512PubMedCrossRefGoogle Scholar
  10. Barabási A-L, Bonabeau E (2003) Scale-free networks. Scientific American, pp 50–59Google Scholar
  11. Barabási A-L, Oltvai Z (2004) Network biology: understanding the cell’s functional organization. Nature Rev Gen 5:101–113CrossRefGoogle Scholar
  12. Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D-U (2006) Complex networks: structure and dynamics. Phys Rep 424:175–308CrossRefGoogle Scholar
  13. Bourjaily MA, Miller P (2011) Excitatory, inhibitory, and structural plasticity produce correlated connectivity in random networks trained to solve paired-stimulus tasks. Front Comput Neurosci 5:37PubMedCentralPubMedCrossRefGoogle Scholar
  14. Braitenberg V, Shüz (1998) Cortex: statistics and geometry of neuronal connectivity (revised, 2nd edition of anatomy of the cortex—statistics and geometry, 1998). Springer, BerlinGoogle Scholar
  15. Bray D (2003) Molecular networks: the top–down view. Science 301:1864–1865PubMedCrossRefGoogle Scholar
  16. Bullmore E, Sporns O (2009) Complex brain networks: graph theoretical analysis of structural and functional systems. Nat Rev Neuro 10:186–198CrossRefGoogle Scholar
  17. Caldarelli G, Capocci A, De Los Rios P, Muñoz MA (2002) Scale-free networks from varying vertex intrinsic fitness. Phys Rev Lett 89:258702PubMedCrossRefGoogle Scholar
  18. Chen BL, Hall DH, Chklovskii DB (2006) Wiring optimization can relate neuronal structure and function. PNAS 103:4723–4728PubMedCentralPubMedCrossRefGoogle Scholar
  19. Choe Y, McCormick BH, Koh W (2004) Network connectivity analysis on the temporally augmented C. elegans web: a pilot study. Soc Neurosci Abstr 30:921.9Google Scholar
  20. Chung F, Lu L (2002) The average distances in random graphs with given expected degrees. Proc Natl Acad Sci USA 99:15879–15882PubMedCentralPubMedCrossRefGoogle Scholar
  21. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661703CrossRefGoogle Scholar
  22. Clopath C, Büsing L, Vasilaki E, Gerstner W (2010) Connectivity reflects coding: a model of voltage-based STDP with homeostasis. Nature Neurosci 13:344–352PubMedCrossRefGoogle Scholar
  23. Costa LF, Rodrigues FA, Travieso G, Villas Boas PR (2007) Characterization of complex networks: a survey of measurements. Adv Phys 56:167–242CrossRefGoogle Scholar
  24. de Solla Price DJ (1965) Networks of scientific papers. Science 149:510–515CrossRefGoogle Scholar
  25. Del Genio CI, Kim H, Toroczkai Z, Bassler KE (2010) Efficient and exact sampling of simple graphs with given arbitrary degree sequence. PLoS One 5:e10012PubMedCentralPubMedCrossRefGoogle Scholar
  26. Diestel R (2000) Graph theory. Springer, New YorkGoogle Scholar
  27. Dorogovtsev SN, Mendes JFF (2001) Giant strongly connected component of directed networks. Phys Rev E 63:025101CrossRefGoogle Scholar
  28. Dorogovtsev SN, Mendes J (2002) Evolution of networks. Adv Phys 51:1079–1187CrossRefGoogle Scholar
  29. Dorogovtsev SN, Mendes JFF, Samukhin AN (2000) Structure of growing networks: exact solution of the Barabási-Albert model. Phys Rev Lett 85:4633–4636PubMedCrossRefGoogle Scholar
  30. Eguíluz VM, Chialvo DR, Cecchi GA, Baliki M, Apkarian AV (2005) Scale-free brain functional networks. Phys Rev Lett 94:018102PubMedCrossRefGoogle Scholar
  31. Felleman DJ, Van Essen DC (1991) Distributed hierarchical processing in the primate cerebral cortex. Cereb Cortex 1:1–47PubMedCrossRefGoogle Scholar
  32. Foster JG, Foster DV, Grassberger P, Paczuski M (2010) Edge direction and the structure of networks. Proc Natl Acad Sci USA 107:10815PubMedCentralPubMedCrossRefGoogle Scholar
  33. Garlaschelli D, Loffredo MI (2004) Patterns of link reciprocity in directed networks. Phys Rev Lett 93:268701PubMedCrossRefGoogle Scholar
  34. Gilson M, Burkitt AN, Grayden DB, Thomas DA, van Hemmen JA (2009) Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks IV: structuring synaptic pathways among recurrent connections. Biol Cybern 101:427–444PubMedCrossRefGoogle Scholar
  35. Goh KI, Kahng B, Kim D (2001) Universal behavior of load distribution in scale-free networks. Phys Rev Lett 87:278701PubMedCrossRefGoogle Scholar
  36. Goh KI, Kahng B, Kim D (2002) Fluctuation-driven dynamics of the internet topology. Phys Rev Lett 88:108701PubMedCrossRefGoogle Scholar
  37. Hennequin G, Vogels TP, Gerstner W (2012) Non-normal amplification in random balanced neuronal networks. Phys Rev E 86:011909CrossRefGoogle Scholar
  38. Honey CJ, Kötter R, Breakspear M, Sporns O (2007) Network structure of cerebral cortex shapes functional connectivity on multiple time scales. Proc Natl Acad Sci USA 104:10240–10245PubMedCentralPubMedCrossRefGoogle Scholar
  39. Hu Y, Trousdale J, Krešimir J, Shea-Brown E (2013) Motif statistics and spike correlations in neuronal networks. J Stat Mech P03012Google Scholar
  40. Johnson S, Torres JJ, Marro J, Muñoz MA (2010) Entropic origin of disassortativity in complex networks. Phys Rev Lett 104:108702PubMedCrossRefGoogle Scholar
  41. Kaiser M, Hilgetag CC (2006) Non-optimal component placement, but short processing paths, due to long-distance projections in neural systems. PLoS Comput Biol 2:e95PubMedCentralPubMedCrossRefGoogle Scholar
  42. Ko H, Hofer SB, Pichler B, Buchanan KA, Sjöström PJ, Mrsic-Flogel TD (2011) Functional specificity of local synaptic connections in neocortical networks. Nature 473:87–91PubMedCentralPubMedCrossRefGoogle Scholar
  43. Kim H, Del Genio CI, Bassler KE, Toroczkai Z (2012) Constructing and sampling directed graphs with given degree sequences. New J Phys 14:023012CrossRefGoogle Scholar
  44. Klemm K, Eguíluz VM (2002) Highly clustered scale-free networks. Phys Rev E65:36123CrossRefGoogle Scholar
  45. Kötter R (2004) Online retrieval, processing, and visualization of primate connectivity data from the CoCoMac database. Neuroinformatics 2:127–144PubMedCrossRefGoogle Scholar
  46. Lima-Mendez G, van Helden J (2009) The powerful law of the power law and other myths in network biology. Mol BioSyst 5:1482–1493PubMedCrossRefGoogle Scholar
  47. Merton RK (1968) The Matthew effect in science. Science 159:56–63CrossRefGoogle Scholar
  48. Milgram S (1967) The small World problem. Psychology today, May 1967, pp 60–67Google Scholar
  49. Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U (2002) Network motifs: simple building blocks of complex networks. Science 298:824–827PubMedCrossRefGoogle Scholar
  50. Modhaa DA, Singh R (2010) Network architecture of the long-distance pathways in the macaque brain. PNAS 107:1348513490CrossRefGoogle Scholar
  51. Newman MEJ (2002) Assortative mixing in networks. Phys Rev Lett 89:208701PubMedCrossRefGoogle Scholar
  52. Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167–256CrossRefGoogle Scholar
  53. Newman MEJ (2010) An introduction., NetworksOxford University Press, OxfordGoogle Scholar
  54. Newman MEJ, Forrest S, Balthrop J (2002) Email networks and the spread of computer viruses. Phys Rev E 66:035101CrossRefGoogle Scholar
  55. Pernice V, Deger M, Cardanobile S, Rotter S (2013) The relevance of network micro-structure for neural dynamics. Front Comput Neurosci 7:72PubMedCentralPubMedCrossRefGoogle Scholar
  56. Ravasz E, Barabási AL (2003) Hierarchical organization in complex networks. Phys Rev E 67:026112CrossRefGoogle Scholar
  57. Roxin A (2011) The role of degree distribution in shaping the dynamics in networks of sparsely connected spiking neurons. Font Comput Neurosci 5:8Google Scholar
  58. Rudolph-Lilith M, Destexhe A, Muller LE (2012) Structual Vulnerability of the Nematode Worm Neural Graph. arXiv:1208.3383v1 [cond-mat.dis-nn]. Available: http://arxiv.org/abs/1208.3383v1
  59. Scannell JW, Burns GA, Hilgetag CC, O’Neil MA, Young MP (1999) The connectional organization of the cortico-thalamic system of the cat. Cereb Cortex 9:277–299PubMedCrossRefGoogle Scholar
  60. Serrano MA, Boguñá M (2003) Topology of the world trade web. Phys Rev E 68:015101CrossRefGoogle Scholar
  61. Song S, Sjöström PJ, Reigl M, Nelson A, Chklovskii DB (2005) Highly nonrandom features of synaptic connectivity in local cortical circuits. PLoS Biol 3:e68PubMedCentralPubMedCrossRefGoogle Scholar
  62. Sporns O (2002) Graph theory methods for the analysis of neural connectivity patterns. In: Kotter R (ed) Neuroscience databases. A practical guide. Kluwer, New York, p 171186Google Scholar
  63. Sporns O, Ktter R (2004) Motifs in brain networks. PLoS Biol 2:19101918CrossRefGoogle Scholar
  64. Sporns O, Tononi G (2002) Classes of network connectivity and dynamics. Complexity 7:28–38CrossRefGoogle Scholar
  65. Sporns O, Tononi G, Edelman GM (2000) Theoretical neuroanatomy: relating anatomical and functional connectivity in graphs and cortical connection matrices. Cereb Cortex 10:127–141PubMedCrossRefGoogle Scholar
  66. Sporns O, Zwi J (2004) The small world of the cerebral cortex. Neuroinformatics 2:145–162PubMedCrossRefGoogle Scholar
  67. Stephan KE, Kamper L, Bozkurt A, Burns GA, Young MP, Kötter R (2001) Advanced database methodology for the collation of connectivity data on the Macaque brain (CoCoMac). Philos Trans R Soc Lond B Biol Sci 356:1159–1186PubMedCentralPubMedCrossRefGoogle Scholar
  68. Stumpf MPH, Porter MA (2012) Critical truths about power laws. Science 335:665–666PubMedCrossRefGoogle Scholar
  69. Van Essen DC (2005) Corticocortical and thalamocortical information flow in the primate visual system. Prog Brain Res 149:173–185PubMedCrossRefGoogle Scholar
  70. Varshney LR, Chen BL, Paniaqua E, Hall DH, Chklovskii DB (2011) Structural properties of the C. elegans neuronal network. PLoS Comput Biol 7:e1001066PubMedCentralPubMedCrossRefGoogle Scholar
  71. Vasilaki E, Giugliano M (2012) Emergence of connectivity patterns from long-term and short-term plasticities. In: Villa AEP et al (eds) Artificial neural networks and machine learning ICANN 2012. Springer, New York, pp 193–200Google Scholar
  72. Vázquez A, Flammini A, Maritan A, Vespignani A (2003) Modeling of protein interaction networks. Complexus 1:38–44 Google Scholar
  73. Wasserman S, Faust K (1994) Social network analysis. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  74. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–442PubMedCrossRefGoogle Scholar
  75. White JG, Southgate E, Thompson JN, Brenner S (1986) The structure of the nervous system of the nematode caenorhabditis elegans. Phil Trans R Soc Lond 314:1–340CrossRefGoogle Scholar
  76. Young MP (1993) The organization of neural systems in the primate cerebral cortex. Proc R Soc Lond B 252:13–18CrossRefGoogle Scholar
  77. Zhao L, Beverlin B II, Netoff T, Nykamp DQ (2011) Synchronization from second order network connectivity statistics. Front Comput Neurosci 5:28Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.CNRSUnité de Neurosciences, Information et Complexité (UNIC)Gif-sur-YvetteFrance

Personalised recommendations