Dynamics of Evolving Feed-Forward Neural Networks and Their Topological Invariants

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9886)

Abstract

The evolution of a simulated feed-forward neural network with recurrent excitatory connections and inhibitory forward connections is studied within the framework of algebraic topology. The dynamics includes pruning and strengthening of the excitatory connections. The invariants that we define are based on the connectivity structure of the underlying graph and its directed clique complex. The computation of this complex and of its Euler characteristic are related with the dynamical evolution of the network. As the network evolves dynamically, its network topology changes because of the pruning and strengthening of the onnections and algebraic topological invariants can be computed at different time steps providing a description of the process. We observe that the initial values of the topological invariant computed on the network before it evolves can predict the intensity of the activity.

Keywords

Graph theory Network invariant Directed clique complex Recurrent neural dynamics Synfire chain Synaptic plasticity 

References

  1. 1.
    Abeles, M.: Local Cortical Circuits. An Electrophysiological Study. Studies of Brain Function, vol. 6. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  2. 2.
    Abeles, M.: Corticonics: Neural Circuits of the Cerebral Cortex. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  3. 3.
    Aviel, Y., Mehring, C., Abeles, M., Horn, D.: On embedding synfire chains in a balanced network. Neural Comput. 15(6), 1321–1340 (2003)CrossRefMATHGoogle Scholar
  4. 4.
    Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Csardi, G., Nepusz, T.: The igraph software package for complex network research. InterJ. Complex Syst. 1695 (2006). http://igraph.org
  6. 6.
    Dłotko, P., Hess, K., Levi, R., Nolte, M., Reimann, M., Scolamiero, M., Turner, K., Muller, E., Markram, H.: Topological analysis of the connectome of digital reconstructions of neural microcircuits. arXiv preprint arXiv:1601.01580 (2016)
  7. 7.
    Freeman, W.J.: Neural networks and chaos. J. Theor. Biol. 171, 13–18 (1994)CrossRefGoogle Scholar
  8. 8.
    Giusti, C., Ghrist, R., Bassett, D.S.: Two’s company, three (or more) is a simplex: algebraic-topological tools for understanding higher-order structure in neural data. arXiv preprint arXiv:1601.01704 (2016)
  9. 9.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  10. 10.
    Iglesias, J., Villa, A.E.P.: Effect of stimulus-driven pruning on the detection of spatiotemporal patterns of activity in large neural networks. Biosystems 89(1–3), 287–293 (2007)CrossRefGoogle Scholar
  11. 11.
    Kumar, A., Rotter, S., Aertsen, A.: Spiking activity propagation in neuronal networks: reconciling different perspectives on neural coding. Nat. Rev. Neurosci. 11(9), 615–627 (2010)CrossRefGoogle Scholar
  12. 12.
    Litvak, V., Sompolinsky, H., Segev, I., Abeles, M.: On the transmission of rate code in long feedforward networks with excitatoryinhibitory balance. J. Neurosci. 23(7), 3006–3015 (2003)Google Scholar
  13. 13.
    Masulli, P., Villa, A.E.P.: The topology of the directed clique complex as a network invariant. Springer Plus 5, 388 (2016)CrossRefGoogle Scholar
  14. 14.
    Petri, G., Expert, P., Turkheimer, F., Carhart-Harris, R., Nutt, D., Hellyer, P.J., Vaccarino, F.: Homological scaffolds of brain functional networks. J. R. Soc. Interf. 11(101), 20140873 (2014)CrossRefGoogle Scholar
  15. 15.
    Prut, Y., Vaadia, E., Bergman, H., Haalman, I., Slovin, H., Abeles, M.: Spatiotemporal structure of cortical activity: properties and behavioral relevance. J. Neurophysiol. 79(6), 2857–2874 (1998)Google Scholar
  16. 16.
    Tange, O.: GNU parallel - the command-line power tool: login. USENIX Mag. 36(1), 42–47 (2011)Google Scholar
  17. 17.
    Villa, A.E., Tetko, I.V., Hyland, B., Najem, A.: Spatiotemporal activity patterns of rat cortical neurons predict responses in a conditioned task. Proc. Nat. Acad. Sci. 96(3), 1106–1111 (1999)CrossRefGoogle Scholar
  18. 18.
    Waddington, A., Appleby, P.A., De Kamps, M., Cohen, N.: Triphasic spike-timing-dependent plasticity organizes networks to produce robust sequences of neural activity. Front. Comput. Neurosci. 6, 88 (2012)CrossRefGoogle Scholar
  19. 19.
    Zaytsev, Y.V., Morrison, A., Deger, M.: Reconstruction of recurrent synaptic connectivity of thousands of neurons from simulated spiking activity. J. Comput. Neurosci. 39(1), 77–103 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zheng, P., Triesch, J.: Robust development of synfire chains from multiple plasticity mechanisms. Front. Comput. Neurosci. 8, 66 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.NeuroHeuristic Research GroupUniversity of LausanneLausanneSwitzerland

Personalised recommendations