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Fractal Analyses of Networks of Integrate-and-Fire Stochastic Spiking Neurons

  • Ariadne A. Costa
  • Mary Jean Amon
  • Olaf Sporns
  • Luis H. Favela
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)

Abstract

Although there is increasing evidence of criticality in the brain, the processes that guide neuronal networks to reach or maintain criticality remain unclear. The present research examines the role of neuronal gain plasticity in time-series of simulated neuronal networks composed of integrate-and-fire stochastic spiking neurons and the utility of fractal methods in assessing network criticality. Simulated time-series were derived from a network model of fully connected discrete-time stochastic excitable neurons. Monofractal and multifractal analyses were applied to neuronal gain time-series. Fractal scaling was greatest in networks with a mid-range of neuronal plasticity, versus extremely high or low levels of plasticity. Peak fractal scaling corresponded closely to additional indices of criticality, including average branching ratio. Networks exhibited multifractal structure, or multiple scaling relationships. Multifractal spectra around peak criticality exhibited elongated right tails, suggesting that the fractal structure is relatively insensitive to high-amplitude local fluctuations. Networks near critical states exhibited mid-range multifractal spectra width and tail length, which is consistent with the literature suggesting that networks poised at quasi-critical states must be stable enough to maintain organization but unstable enough to be adaptable. Lastly, fractal analyses may offer additional information about critical state dynamics of networks by indicating scales of influence as networks approach critical states.

Keywords

1/f scaling Self-organized criticality Fractal analysis Multifractal analysis Neuronal networks 

Notes

Acknowledgements

This article was produced as part of the activities of FAPESP Research, Innovation and Dissemination Center for Neuromathematics (grant #2013/07699-0, S.Paulo Research Foundation). AAC also thanks grants \(\#\)2016/00430-3 and \(\#\)2016/20945-8 São Paulo Research Foundation (FAPESP).

References

  1. 1.
    Chialvo, D.R.: Critical brain networks. Phys. A 340, 756–765 (2004)CrossRefGoogle Scholar
  2. 2.
    Beggs, J., Timme, N.: Being critical of criticality in the brain. Front Psychol. 3, 163 (2012)Google Scholar
  3. 3.
    Favela, L.H.: Radical embodied cognitive neuroscience: addressing "grand challenges" of the mind sciences. Front Hum Neurosci. 8, 796 (2014)CrossRefGoogle Scholar
  4. 4.
    Hesse, J., Gross, T.: Self-organized criticality as a fundamental property of neural systems. Front Syst. Neurosci. 8, 166 (2014)CrossRefGoogle Scholar
  5. 5.
    Poil, S.S., van Ooyen, A., Linkenkaer-Hansen, K.: Avalanche dynamics of human brain oscillations: relation to critical branching processes and temporal correlations. Hum. Brain Mapp 29, 770–777 (2008)CrossRefGoogle Scholar
  6. 6.
    Petermann, T., Thiagarajan, T.C., Lebedev, M.A., Nicolelis, M.A.L., Chialvo, D.R., Plenz, D.: Spontaneous cortical activity in awake monkeys composed of neuronal avalanches. Proc. Natl. Acad. Sci. 106, 15921–15926 (2009)ADSCrossRefGoogle Scholar
  7. 7.
    Hahn, G., Petermann, T., Havenith, M.N., Yu, S., Singer, W., Plenz, D., Nikolić, D.: Neuronal avalanches in spontaneous activity in vivo. J. Neurophysiol. 104, 3312–3322 (2010)CrossRefGoogle Scholar
  8. 8.
    Favela, L.H., Coey, C.A., Griff, E.R., Richardson, M.J.: Fractal analysis reveals subclasses of neurons and suggests an explanation of their spontaneous activity. Neurosci. Lett. 626, 54–58 (2016)CrossRefGoogle Scholar
  9. 9.
    Beggs, J.M., Plenz, D.: Neuronal avalanches in neocortical circuits. J. Neurosci. 23, 11167–11177 (2003)Google Scholar
  10. 10.
    Beggs, J.M., Plenz, D.: Neuronal avalanches are diverse and precise activity patterns that are stable for many hours in cortical slice cultures. J. Neurosci 24(22), 5216–29 (2004)CrossRefGoogle Scholar
  11. 11.
    de Arcangelis, L., Perrone-Capano, C., Herrmann, H.J.: Self-organized criticality model for brain plasticity. Phys. Rev. Lett. 96, 028107 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    Levina, A., Herrmann, J.M., Geisel, T.: Dynamical synapses causing self-organized criticality in neural networks. Nat. Phys. 3, 857–860 (2007)CrossRefGoogle Scholar
  13. 13.
    Costa, A.A., Copelli, M., Kinouchi, O.: Can dynamical synapses produce true self-organized criticality? J. Stat. Mech. Theory Exp. 2015, P06004 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Brochini, L., Costa, A.A., Abadi, M., Roque, A.C., Stolfi, J., Kinouchi, O.: Phase transitions and self-organized criticality in networks of stochastic spiking neurons. Sci. Rep. 6 (2016)Google Scholar
  15. 15.
    Kinouchi, O., Copelli, M.: Optimal dynamical range of excitable networks at criticality. Nat. Phys. 2, 348–351 (2006)CrossRefGoogle Scholar
  16. 16.
    Shew, W.L., Yang, H., Petermann, T., Roy, R., Plenz, D.: Neuronal avalanches imply maximum dynamic range in cortical networks at criticality. J. Neurosci. 29, 15595–15600 (2009)CrossRefGoogle Scholar
  17. 17.
    Beggs, J.M.: The criticality hypothesis: how local cortical networks might optimize information processing. Philos. Trans. R. Soc. A 366, 329–343 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Shew, W.L., Plenz, D.: The functional benefits of criticality in the cortex. Neuroscientist 19, 88–100 (2013). PMID: 22627091CrossRefGoogle Scholar
  19. 19.
    Massobrio, P., de Arcangelis, L., Pasquale, V., Jensen, H.J., Plenz, D.: Criticality as a signature of healthy neural systems. Front. Syst. Neurosci. 9, (2015)Google Scholar
  20. 20.
    Mandelbrot, B.B.: The fractal geometry of nature, Updated edn. W. H. Freeman and Company, New York (1982)MATHGoogle Scholar
  21. 21.
    Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987)ADSCrossRefGoogle Scholar
  22. 22.
    Watkins, N.W., Pruessner, G., Chapman, S.C., Crosby, N.B., Jensen, H.J.: 25 years of self-organized criticality: concepts and controversies. Space Sci Rev. 198, 3–44 (2016)ADSCrossRefGoogle Scholar
  23. 23.
    Tetzlaff, C., Okujeni, S., Egert, U., Wörgötter, F., Butz, M.: Self-organized criticality in developing neuronal networks. PLoS Comput. Biol. 6, e1001013 (2010)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Galves, A., Löcherbach, E.: Infinite systems of interacting chains with memory of variable length - a stochastic model for biological neural nets. J. Stat. Phys. 151, 896–921 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Costa, A.A., Brochini, L., Kinouchi, O.: Self-organized supercriticality and oscillations in networks of stochastic spiking neurons. Entropy. 19, 399 (2017)ADSCrossRefGoogle Scholar
  26. 26.
    Gerstner, W., van Hemmen, J.L.: Associative memory in a network of ‘spiking’ neurons. Netw. Comput. Neural 3, 139–164 (1992)CrossRefMATHGoogle Scholar
  27. 27.
    Gerstner, W., Kistler, W.M.: Spiking Neuron Models: Single Neurons, Populations. Plasticity. Cambridge University Press, Cambridge (2002)CrossRefMATHGoogle Scholar
  28. 28.
    Lapicque, L.: Recherches quantitatives sur l’excitation électrique des nerfs traitée comme une polarisation. J. Physiol. Pathol. Gen. 9, 620–635 (1907): Translation: Brunel, N., van Rossum, M.C.: Quantitative investigations of electrical nerve excitation treated as polarization. Biol. Cybern. 97, 341–349 (2007)Google Scholar
  29. 29.
    Larremore, D.B., Shew, W.L., Ott, E., Sorrentino, F., Restrepo, J.G.: Inhibition causes ceaseless dynamics in networks of excitable nodes. Phys. Rev. Lett. 112, 138103 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    Duarte, A., Ost, G.: A model for neural activity in the absence of external stimuli. Markov Process. Relat. Fields 22, 37–52 (2016)MathSciNetMATHGoogle Scholar
  31. 31.
    De Masi, A., Galves, A., Löcherbach, E., Presutti, E.: Hydrodynamic limit for interacting neurons. J. Stat. Phys. 158, 866–902 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Galves, A., Löcherbach, E.: Modeling networks of spiking neurons as interacting processes with memory of variable length. J. Soc. Franc. Stat. 157, 17–32 (2016)MathSciNetMATHGoogle Scholar
  33. 33.
    Kole, M.H., Stuart, G.J.: Signal processing in the axon initial segment. Neuron 73, 235–247 (2012)CrossRefGoogle Scholar
  34. 34.
    Campos, J.G.F., Costa, A.A., Copelli, M., Kinouchi, O.: Correlations induced by depressing synapses in critically self-organized networks with quenched dynamics. Phys. Rev. E 95, 042303 (2017)ADSCrossRefGoogle Scholar
  35. 35.
    Kello, C.T., Beltz, B.C., Holden, J.G., Van Orden, G.C.: The emergent coordination of cognitive function. J. Exp. Psychol. Gen 136, 551 (2007)CrossRefGoogle Scholar
  36. 36.
    Holden, J.G., Van Orden, G.C., Turvey, M.T.: Dispersion of response times reveals cognitive dynamics. Psychol. Rev 116, 318 (2009)CrossRefGoogle Scholar
  37. 37.
    Van Orden, G.C., Kloos, H., Wallot, S.: Living in the pink: Intentionality, wellbeing, and complexity. In: Philosophy of Complex Systems, Handbook of the philosophy of science, vol. 10. (2011)Google Scholar
  38. 38.
    Gilden, D.L.: Cognitive emissions of 1/f noise. Psychol. Rev 108, 33 (2001)CrossRefGoogle Scholar
  39. 39.
    Holden, J.G.: Gauging the fractal dimension of response times from cognitive tasks, pp. 267–318. A Webbook Tutorial, Contemporary Nonlinear Methods for Behavioral Scientists (2005)Google Scholar
  40. 40.
    Delignieres, D., Ramdani, S., Lemoine, L., Torre, K., Fortes, M., Ninot, G.: Fractal analyses for ‘short’ time-series: a re-assessment of classical methods. J. Math. Psychol 50, 525–544 (2006)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Ihlen, E.A.: Introduction to multifractal detrended fluctuation analysis in matlab. Front Psychol. 3 (2012)Google Scholar
  42. 42.
    Botcharova, M., Farmer, S.F., Berthouze, L.: Markers of criticality in phase synchronization. Front. Syst. Neurosci. 8 (2014)Google Scholar
  43. 43.
    Hardstone, R., Poil, S.S., Schiavone, G., Jansen, R., Nikulin, V.V., Mansvelder, H.D., Linkenkaer-Hansen, K.: Detrended fluctuation analysis: a scale-free view on neuronal oscillations. Front Psychol. 3 (2012)Google Scholar
  44. 44.
    Linkenkaer-Hansen, K., Nikouline, V.V., Palva, J.M., Ilmoniemi, R.J.: Long-range temporal correlations and scaling behavior in human brain oscillations. J. Neurosci. 21, 1370–1377 (2001)Google Scholar
  45. 45.
    Eke, A., Herman, P., Bassingthwaighte, J., Raymond, G., Percival, D., Cannon, M., Balla, I., Ikrényi, C.: Physiological time-series: distinguishing fractal noises from motions. Pflügers Arch. 439, 403–415 (2000)CrossRefGoogle Scholar
  46. 46.
    Delignières, D., Marmelat, V.: Theoretical and methodological issues in serial correlation analysis. In: Progress in Motor Control, pp. 127–148. Springer, Berlin (2013)Google Scholar
  47. 47.
    Haldeman, C., Beggs, J.M.: Critical branching captures activity in living neural networks and maximizes the number of metastable states. Phys. Rev. Lett. 94, 058101 (2005)ADSCrossRefGoogle Scholar
  48. 48.
    Timme, N.M., Marshall, N.J., Bennett, N., Ripp, M., Lautzenhiser, E., Beggs, J.M.: Criticality maximizes complexity in neural tissue. Front. Psychol. 7, (2016)Google Scholar
  49. 49.
    Wilting, J., Priesemann, V.: Branching into the unknown: inferring collective dynamical states from subsampled systems (2016). arXiv preprint arXiv:1608.07035
  50. 50.
    Lopes, R., Ayache, A.: Tenets, methods, and applications of multifractal analysis in neurosciences. In: The Fractal Geometry of the Brain, pp. 65–79. Springer, Berlin (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Ariadne A. Costa
    • 1
  • Mary Jean Amon
    • 1
  • Olaf Sporns
    • 1
  • Luis H. Favela
    • 2
  1. 1.Department of Psychological and Brain SciencesIndiana UniversityBloomingtonUSA
  2. 2.Department of Philosophy and Cognitive Sciences ProgramUniversity of Central FloridaOrlandoUSA

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