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Bootstrap Percolation in Living Neural Networks

Abstract

Recent experimental studies of living neural networks reveal that their global activation induced by electrical stimulation can be explained using the concept of bootstrap percolation on a directed random network. The experiment consists in activating externally an initial random fraction of the neurons and observe the process of firing until its equilibrium. The final portion of neurons that are active depends in a non linear way on the initial fraction. The main result of this paper is a theorem which enables us to find the final proportion of the fired neurons, in the asymptotic case, in the case of random directed graphs with given node degrees as the model for interacting network. This gives a rigorous mathematical proof of a phenomena observed by physicists in neural networks.

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References

  1. Aldous, D., Bandyopadhyay, A.: A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15, 1047–1110 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  2. Aldous, D., Lyons, R.: Processes on unimodular random networks. Electron. J. Probab. 12, 1454–1508 (2007)

    MATH  MathSciNet  Google Scholar 

  3. Amini, H.: Bootstrap percolation and diffusion in random graphs with given vertex degrees. Electron. J. Comb. 17, R25 (2010)

    MathSciNet  ADS  Google Scholar 

  4. Balogh, J., Peres, Y., Pete, G.: Bootstrap percolation on infinite trees and non-amenable groups. Comb. Probab. Comput. 15, 715–730 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  5. Balogh, J., Pittel, B.G.: Bootstrap percolation on the random regular graph. Random Structures & Algorithms 30, 257–286 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  6. Bertoin, J., Sidoravicius, V.: The structure of typical clusters in large sparse random configurations. J. Stat. Phys. 135, 87–105 (2009)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  7. Bollobás, B.: Random Graphs. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  8. Breskin, I., Soriano, J., Moses, E., Tlusty, T.: Percolation in living neural networks. Phys. Rev. Lett. 97, 188102 (2006)

    Article  ADS  Google Scholar 

  9. Cain, J., Wormald, N.: Encores on cores. Electron. J. Comb. 13, R81 (2006)

    MathSciNet  Google Scholar 

  10. Cohen, O., Kesselman, A., Martinez, M.R., Soriano, J., Moses, E., Tlusty, T.: Quorum percolation in living neural networks. Europhys. Lett. 89, 18008 (2010)

    Article  ADS  Google Scholar 

  11. Cooper, C., Frieze, A.M.: The size of the largest strongly connected component of a random digraph with a given degree sequence. Comb. Probab. Comput. 13, 319–337 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  12. Eckmann, J.P., Feinerman, O., Gruendlinger, L., Moses, E., Soriano, J., Tlusty, T.: The physics of living neural networks. Phys. Rep. 449, 54–76 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  13. van Enter, A.C.D.: Proof of Straley’s argument for bootstrap percolation. J. Stat. Phys. 48, 943–945 (1987)

    MATH  Article  ADS  Google Scholar 

  14. Fontes, L.R.G., Schonmann, R.H.: Bootstrap percolation on homogeneous trees has 2 phase transitions. J. Stat. Phys. 132, 839–861 (2008)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  15. Goltsev, A.V., Dorogovtsev, S.N., Mendes, J.F.F.: k-core (bootstrap) percolation on complex networks: critical phenomena and nonlocal effects. Phys. Rev. E 73, 056101 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  16. Holroyd, A.: The metastability threshold for modified bootstrap percolation in d dimensions. Electron. J. Probab. 11, 418–433 (2006)

    MathSciNet  Google Scholar 

  17. Janson, S.: The probability that a random multigraph is simple. Ann. Appl. Probab. 18, 205–225 (2008)

    Google Scholar 

  18. Molloy, M., Reed, B.: The size of the giant component of a random graph with a given degree sequence. Comb. Probab. Comput. 7, 295–305 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  19. Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118 (2001)

    Article  ADS  Google Scholar 

  20. Schwab, D.J., Bruinsma, R.F., Levine, A.J.: Rhythmogenic neuronal networks, pacemakers, and k-cores. Phys. Rev. Lett. (2008)

  21. Soriano, J., Martínez, M.R., Tlusty, T., Moses, E.: Development of input connections in neural cultures, Proc. Nat. Acad. Sci. (2008)

  22. Tlusty, T., Eckmann, J.-P.: Remarks on bootstrap percolation in metric networks. J. Phys. A, Math. Theor. 42, 205004 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  23. Watts, D.J.: A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. 99, 5766–5771 (2002)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  24. Wormald, N.: Differential equations for random processes and random graphs. Ann. Appl. Probab. 5, 1217–1235 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  25. Wormald, N.: The differential equation method for random graph processes and greedy algorithms. Lect. Approx. Random. Algorithms (1999)

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Correspondence to Hamed Amini.

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Amini, H. Bootstrap Percolation in Living Neural Networks. J Stat Phys 141, 459–475 (2010). https://doi.org/10.1007/s10955-010-0056-z

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  • DOI: https://doi.org/10.1007/s10955-010-0056-z

Keywords

  • Bootstrap percolation
  • Phase transition
  • Random graphs
  • Neural networks