Biological Cybernetics

, Volume 106, Issue 3, pp 191–200 | Cite as

Developing structural constraints on connectivity for biologically embedded neural networks

Original Paper

Abstract

In this article, we analyse under which conditions an abstract model of connectivity could actually be embedded geometrically in a mammalian brain. To this end, we adopt and extend a method from circuit design called Rent’s Rule to the highly branching structure of cortical connections. Adding on recent approaches, we introduce the concept of a limiting Rent characteristic that captures the geometrical constraints of a cortical substrate on connectivity. We derive this limit for the mammalian neocortex, finding that it is independent of the species qualitatively as well as quantitatively. In consequence, this method can be used as a universal descriptor for the geometrical restrictions of cortical connectivity. We investigate two widely used generic network models: uniform random and localized connectivity, and show how they are constrained by the limiting Rent characteristic. Finally, we discuss consequences of these restrictions on the development of cortex-size models.

Keywords

Neural networks Rent’s Rule Network connectivity Multi-point nets Network analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abeles M (1991) Corticonics. Neural circuits of the cerebral cortex. Cambridge University Press, New YorkCrossRefGoogle Scholar
  2. Bassett D, Bullmore E (2006) Small-world brain networks. Neuroscientist 12(6): 512–523PubMedCrossRefGoogle Scholar
  3. Bassett D, Greenfield D, Meyer-Lindenberg A, Weinberger D, Moore S, Bullmore E (2010) Efficient physical embedding of topologically complex information processing networks in brains and computer circuits. PLoS Comput Biol 6(4): e1000748PubMedCrossRefGoogle Scholar
  4. Beiu V, Madappuram B, Kelly P, McDaid L (2009) On two-layer brain-inspired hierarchical topologies—a Rent’s Rule approach. In: LNCS transactions on high-performance embedded architecture and compilers, vol 4(4), pp 1–22Google Scholar
  5. Binzegger T, Douglas R, Martin K (2004) A quantitative map of the circuit of cat primary visual cortex. J Neurosci 24(39): 8441–8453PubMedCrossRefGoogle Scholar
  6. Braitenberg V (2001) Brain size and number of neurons: an exercise in synthetic neuroanatomy. J Comput Neurosci 10: 71–77PubMedCrossRefGoogle Scholar
  7. Braitenberg V, Schüz A (1998) cortex: statistics and geometry of neuronal connectivity. Springer, New YorkGoogle Scholar
  8. Brunel N (2000) Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J Comput Neurosci 8: 183–208PubMedCrossRefGoogle Scholar
  9. Changizi M (2007) Scaling the brain and its connections. In: Kaas J (ed) Evolution of nervous systems. Elsevier, OxfordGoogle Scholar
  10. Chklovskii D (2004) Exact solution for the optimal neuronal layout problem. Neural Comput 16: 2067–2078PubMedCrossRefGoogle Scholar
  11. Christie P, Stroobandt D (2000) The interpretation and application of Rent’s Rule. IEEE Trans VLSI Syst 8(6): 639–648CrossRefGoogle Scholar
  12. Donath W (1979) Placement and average interconnection lengths of computer logic. IEEE Trans Circ Syst 26(4): 272–277CrossRefGoogle Scholar
  13. Hagen L, Kahng A, Fadi J, Ramachandran C (1994) On the intrinsic Rent parameter and spectra-based partitioning methodologies. IEEE Trans Comput Aid Des Integr Circ Syst 13: 27–37CrossRefGoogle Scholar
  14. Harrison K, Hof P, Wang SH (2002) Scaling laws in the mammalian neocortex: does form provide clues to function. J Neurocytol 31: 289–298PubMedCrossRefGoogle Scholar
  15. Häusler S, Schuch K, Maass W (2009) Motif distribution, dynamical properties, and computational performance of two data-based cortical microcircuit templates. J Physiol 103: 73–87Google Scholar
  16. Hellwig B (2000) A quantitative analysis of the local connectivity between pyramidal neurons in layers 2/3 of the rat visual cortex. Biol Cybern 82: 111–121PubMedCrossRefGoogle Scholar
  17. Karypis G, Kumar V (2000) Multilevel k-way hypergraph partitioning. VLSI Des 11(3): 285–300CrossRefGoogle Scholar
  18. Kremkow J, Kumar A, Rotter S, Aertsen A (2007) Emergence of population synchrony in a layered network of the cat visual cortex. Neurocomputing 70: 2069–2073CrossRefGoogle Scholar
  19. Landman B, Russo R (1971) On a pin versus block relationship for partitions of logic graphs. IEEE Trans Comput C 20(12): 1469–1479CrossRefGoogle Scholar
  20. Lanzerotti M, Fiorenza G, Rand R (2004) Interpretation of Rent’s Rule for ultralarge-scale integrated circuit designs, with an application to wirelength distribution models. IEEE Trans VLSI Syst 12(12): 1330–1347CrossRefGoogle Scholar
  21. Mehring C, Hehl U, Kubo M, Diesmann M, Aertsen A (2003) Activity dynamics and propagation of synchronous spiking in locally connected random networks. Biol Cybern 88: 395–408PubMedCrossRefGoogle Scholar
  22. Newman M (2003) The structure and function of complex networks. SIAM Rev 45: 167–256CrossRefGoogle Scholar
  23. Pakkenberg B, Gundersen H (1997) Neocortical neuron number in humans: effect of sex and age. J Comput Neurol 384: 312–320CrossRefGoogle Scholar
  24. Partzsch J, Schüffny R (2009) On the routing complexity of neural network models—Rent’s Rule revisited. In: ESANN, pp 595–600Google Scholar
  25. Schemmel J, Brüderle D, Grübl A, Hock M, Meier K, Millner S (2010) A wafer-scale neuromorphic hardware system for large-scale neural modeling. In: ISCAS, pp 1947–1950Google Scholar
  26. Schüz A, Chaimow D, Liewald D, Dortenmann M (2006) Quantitative aspects of corticocortical connections: a tracer study in the mouse. Cerebr Cortex 16: 1474–1486CrossRefGoogle Scholar
  27. Sporns O, Kötter R (2004) Motifs in brain networks. PLoS Biol 2(11): 1910–1918CrossRefGoogle Scholar
  28. Stepanyants A, Chklovskii D (2005) Neurogeometry and potential synaptic connectivity. Trends Neurosci 28(7): 387–394PubMedCrossRefGoogle Scholar
  29. Stroobandt D, Kurdahi F (1998) On the characterization of multi-point nets in electronic designs. In: 8th Great Lakes symposium on VLSI, pp 344–350Google Scholar
  30. Watts D, Strogatz S (1998) Collective dynamics of small-world networks. Nature 393: 440–442PubMedCrossRefGoogle Scholar
  31. Young M, Scannell J, Burns G (1995) In: Landes RG (ed) The analysis of cortical connectivity. Springer, New YorkGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Chair of Highly Parallel VLSI Systems and Neuromorphic Circuits, Institute of Circuits and Systems, Faculty of Electrical Engineering and Information TechnologyTechnische Universität DresdenDresdenGermany

Personalised recommendations