Modeling Functional Connectivity on Empirical and Randomized Structural Brain Networks

  • Şeyma Bayrak
  • Philipp HövelEmail author
  • Vesna Vuksanović
Original Research


This study combines modeling of neuronal activity and networks derived from neuroimaging data in order to investigate how the structural organization of the human brain affects the temporal dynamics of interacting brain areas. The dynamics of the neuronal activity is modeled with FitzHugh–Nagumo oscillators and the blood-oxygen-level-dependent (BOLD) time series is inferred via the Balloon–Windkessel hemodynamic model. The simulations are based on anatomical probability maps between considered brain regions of interest. These maps were derived from diffusion-weighted magnetic resonance imaging measurements. In addition, the length of the fiber tracks allows for inference of coupling delays due to finite signal propagation velocities. We aim to investigate (i) graph-theoretical properties of the network topology derived from neuroimaging data and (ii) how randomization of structural connections influences the dynamics of neuronal activity. The network characteristics of the structural connectivity data are compared to density-matched Erdős–Rényi random graphs. Furthermore, the neuronal and BOLD activity are modeled on both empirical and random (Erdős–Rényi type) graphs. The simulated temporal dynamics on both graphs are compared statistically to capture whether the spatial organization of these network affects the modeled time series. Results support previous findings that key topological network properties such as small-worldness of our neuroimaging data are distinguishable from random networks. We also show that simulated BOLD activity is affected by the underlying network topology and the strength of connections between the network nodes. The difference of the modeled temporal dynamics of brain networks from the dynamics on randomized graphs suggests that anatomical connections in the human brain together with dynamical self-organization are crucial for the temporal evolution of the resting-state activity.


Brain networks Functional and anatomical connectivity Hemodynamic model Resting state Time-delayed oscillations 



This study was assisted by BMBF (Grant No. 01Q1001B) in the framework of BCCN Berlin (Project B7). We would like to thank Yasser Iturria-Medina for sharing the DW-MRI data used in this work. Şeyma Bayrak acknowledges additionally the support by Jochen Braun.


  1. 1.
    Acebrón, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Modern Phys. 77(1), 137 (2005)CrossRefGoogle Scholar
  2. 2.
    Bassett, D., Bullmore, E.: Small-world brain networks. Neuroscientist 12(6), 512–523 (2006)CrossRefGoogle Scholar
  3. 3.
    Bassett, D.S., Bullmore, E., Verchinski, B.A., Mattay, V.S., Weinberger, D.R., Meyer-Lindenberg, A.: Hierarchical organization of human cortical networks in health and schizophrenia. J. Neurosci. 28(37), 9239–9248 (2008)CrossRefGoogle Scholar
  4. 4.
    Bassett, D.S., Meyer-Lindenberg, A., Achard, S., Duke, T., Bullmore, E.: Adaptive reconfiguration of fractal small-world human brain functional networks. Proc. Natl. Acad. Sci. USA 103(51), 19518–19523 (2006)CrossRefGoogle Scholar
  5. 5.
    Bhattacharyya, A.: On a measure of divergence between two statistical populations defined by their probability distributions. Bull. Calcutta Math. Soc. 35, 99–109 (1943)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Biswal, B., Yetkin, F.Z., Haughton, V.M., Hyde, J.S.: Functional connectivity in the motor cortex of resting human brain using echo-planar MRI. Magn. Reson. Med. 34(4), 537–541 (1995)CrossRefGoogle Scholar
  7. 7.
    Bogacki, P., Shampine, L.F.: A 3(2) pair of Runge–Kutta formulas. Appl. Math. Lett. 2(4), 321–325 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Braun, U., et al.: Dynamic brain network reconfiguration as a potential schizophrenia genetic risk mechanism modulated by NMDA receptor function. Proc. Natl. Acad. Sci. USA 113(44), 12568–12573 (2016)CrossRefGoogle Scholar
  9. 9.
    Breakspear, M., Heitmann, S., Daffertshofer, A.: Generative models of cortical oscillations: neurobiological implications of the Kuramoto model. Front. Hum. Neurosci. 4(190), 1–14 (2010)Google Scholar
  10. 10.
    Bressler, S.L., Menon, V.: Large-scale brain networks in cognition: emerging methods and principles. Trends Cogn. Sci. 14(6), 277–290 (2010)CrossRefGoogle Scholar
  11. 11.
    Bullmore, E.T., Bassett, D.S.: Brain graphs: graphical models of the human brain connectome. Ann. Rev. Clin. Psychol. 7, 113–140 (2011)CrossRefGoogle Scholar
  12. 12.
    Bullmore, E.T., Sporns, O.: Complex brain networks: graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci. 10(3), 186–198 (2009)CrossRefGoogle Scholar
  13. 13.
    Cabral, J., Hugues, E., Kringelbach, M.L., Deco, G.: Modeling the outcome of structural disconnection on resting-state functional connectivity. NeuroImage 62, 1342–1353 (2012)CrossRefGoogle Scholar
  14. 14.
    Cabral, J., Hugues, E., Sporns, O., Deco, G.: Role of local network oscillations in resting-state functional connectivity. NeuroImage 57(1), 130–139 (2011)CrossRefGoogle Scholar
  15. 15.
    Cabral, J., Kringelbach, M., Deco, G.: Functional graph alterations in schizophrenia: a result from a global anatomic decoupling? Pharmacopsychiatry 45(1), S57 (2012)Google Scholar
  16. 16.
    Cabral, J., Kringelbach, M.L., Deco, G.: Exploring the network dynamics underlying brain activity during rest. Prog. Neurobiol. 114, 102–131 (2014)CrossRefGoogle Scholar
  17. 17.
    Damoiseaux, J.S., Rombouts, S.A.R.B., Barkhof, F., Scheltens, P., Stam, C.J., Smith, S.M., Beckmann, C.F.: Consistent resting-state networks across healthy subjects. Proc. Natl. Acad. Sci. USA 103(37), 13848–13853 (2006)CrossRefGoogle Scholar
  18. 18.
    Deco, G., Jirsa, V.K.: Ongoing cortical activity at rest: criticality, multistability, and ghost attractors. J. Neurosci. 32(10), 3366–3375 (2012)CrossRefGoogle Scholar
  19. 19.
    Deco, G., Jirsa, V.K., McIntosh, A.R., Sporns, O., Kötter, R.: Key role of coupling, delay, and noise in resting brain fluctuations. Proc. Natl. Acad. Sci. USA 106(25), 10302–10307 (2009)CrossRefGoogle Scholar
  20. 20.
    Erdős, P., Rényi, A.: On random graphs. I. Publ. Math. 6, 290–297 (1959)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Fell, D.A., Wagner, A.: The small world of metabolism. Nat. Biotechnol. 18(11), 1121–1122 (2000)CrossRefGoogle Scholar
  22. 22.
    FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)CrossRefGoogle Scholar
  23. 23.
    Flunkert, V., Schöll, E.: Pydelay—a python tool for solving delay differential equations. arXiv:0911.1633 [nlin.CD] (2009)
  24. 24.
    Friston, K., Mechelli, A., Turner, R., Price, C.J.: Nonlinear responses in fMRI: the balloon model, Volterra kernels, and other hemodynamics. NeuroImage 12(4), 466–477 (2000)CrossRefGoogle Scholar
  25. 25.
    Friston, K.J., Harrison, L., Penny, W.: Dynamic causal modelling. NeuroImage 19(4), 1273–1302 (2003)CrossRefGoogle Scholar
  26. 26.
    Ghosh, A., Rho, Y., McIntosh, A.R., Kötter, R., Jirsa, V.K.: Cortical network dynamics with time delays reveals functional connectivity in the resting brain. Cogn. Neurodyn. 2(2), 115–120 (2008)CrossRefGoogle Scholar
  27. 27.
    Ghosh, A., Rho, Y., McIntosh, A.R., Kötter, R., Jirsa, V.K.: Noise during rest enables the exploration of the brain’s dynamic repertoire. PLoS Comput. Biol. 4(10), e1000196 (2008)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hagberg, A.A., Schult, D.A., Swart, P.J.: Exploring network structure, dynamics, and function using NetworkX. In: Proceedings of the 7th Python in Science Conference (SciPy2008), pp. 11–15 (2008)Google Scholar
  29. 29.
    Hagmann, P., Cammoun, L., Gigandet, X., Meuli, R., Honey, C., Wedeen, V., Sporns, O.: Mapping the structural core of human cerebral cortex. PLoS Biol. 6(7), 15 (2008)CrossRefGoogle Scholar
  30. 30.
    Honey, C.J., Kötter, R., Breakspear, M., Sporns, O.: Network structure of cerebral cortex shapes functional connectivity on multiple time scales. Proc. Natl. Acad. Sci. USA 104, 10240–10245 (2007)CrossRefGoogle Scholar
  31. 31.
    Hövel, P.: Control of Complex Nonlinear Systems with Delay. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  32. 32.
    Humphries, M.D., Gurney, K.: Network ’small-world-ness’: a quantitative method for determining canonical network equivalence. PLoS One 3(4), e0002051 (2008)CrossRefGoogle Scholar
  33. 33.
    Iturria-Medina, Y., Sotero, R.C., Canales-Rodríguez, E.J., Alemán-Gómez, Y., Melie-García, L.: Studying the human brain anatomical network via diffusion-weighted MRI and graph theory. NeuroImage 40(3), 1064–1076 (2008)CrossRefGoogle Scholar
  34. 34.
    Jirsa, V.K., McIntosh, A.R.: Handbook of Brain Connectivity. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  35. 35.
    Kretzschmar, M., Morris, M.: Measures of concurrency in networks and the spread of infectious disease. Math. Biosci. 133(2), 165–195 (1996)CrossRefzbMATHGoogle Scholar
  36. 36.
    Latora, V., Marchiori, M.: Efficient Behavior of small-world networks. Phys. Rev. Lett. 87, 198701 (2001)CrossRefGoogle Scholar
  37. 37.
    Lopes da Silva, F., Pijn, J., Velis, D., Nijssen, P.: Alpha rhythms: noise, dynamics and models. Int. J. Psychophysiol. 26(1–3), 237–249 (1997)CrossRefGoogle Scholar
  38. 38.
    Lynall, M.E., Bassett, D.S., Kerwin, R., McKenna, P.J., Kitzbichler, M., Muller, U., Bullmore, E.: Functional connectivity and brain networks in schizophrenia. J. Neurosci. 30(28), 9477–9487 (2010)CrossRefGoogle Scholar
  39. 39.
    Maslov, S., Sneppen, K.: Specificity and stability in topology of protein networks. Science 296, 910–913 (2002)CrossRefGoogle Scholar
  40. 40.
    May, R.M.: Will a large complex system be stable? Nature 238, 413–414 (1972)CrossRefGoogle Scholar
  41. 41.
    May, R.M., Lloyd, A.L.: Infection dynamics on scale-free networks. Phys. Rev. E 64(6), 066112-1–066112-4 (2001)Google Scholar
  42. 42.
    Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962)CrossRefGoogle Scholar
  43. 43.
    Newman, M.E., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64(2), 026118-1–026118-17 (2001)Google Scholar
  44. 44.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Newman, M.E.J.: Networks: An Introduction. Oxford University Press Inc, New York (2010)CrossRefzbMATHGoogle Scholar
  46. 46.
    Nunez, P.L.: Electric Fields of the Brain: The Neurophysics of EEG. Oxford University Press, New York (1998)Google Scholar
  47. 47.
    Nunez, P.L.: Toward a quantitative description of large-scale neocortical dynamic function and EEG. Behav. Brain Sci. 23(03), 371–398 (2000)CrossRefGoogle Scholar
  48. 48.
    Olde Dubbelink, K.T., et al.: Disrupted brain network topology in Parkinson’s disease: a longitudinal magnetoencephalography study. Brain 137, 197–207 (2014)CrossRefGoogle Scholar
  49. 49.
    Penn, A.A., Shatz, C.J.: Brain waves and brain wiring: the role of endogenous and sensory-driven neural activity in development. Pediatr. Res. 45, 447–458 (1999)CrossRefGoogle Scholar
  50. 50.
    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(7), 770–777 (2008)CrossRefGoogle Scholar
  51. 51.
    Ponten, S., Douw, L., Bartolomei, F., Reijneveld, J., Stam, C.: Indications for network regularization during absence seizures: weighted and unweighted graph theoretical analyses. Exp. Neurol. 217(1), 197–204 (2009)CrossRefGoogle Scholar
  52. 52.
    Rubinov, M., Sporns, O.: Complex network measures of brain connectivity: uses and interpretations. NeuroImage 52(3), 1059–1069 (2010)CrossRefGoogle Scholar
  53. 53.
    Rubinov, M., Sporns, O., van Leeuwen, C., Breakspear, M.: Symbiotic relationship between brain structure and dynamics. BMC Neurosci. 10(1), 55 (2009)CrossRefGoogle Scholar
  54. 54.
    Samu, D., Seth, A.K., Nowotny, T.: Influence of wiring cost on the large-scale architecture of human cortical connectivity. PLoS Comput. Biol. 10(4), e1003557 (2014)CrossRefGoogle Scholar
  55. 55.
    Seth, A.K., Chorley, P., Barnett, L.C.: Granger causality analysis of fMRI BOLD signals is invariant to hemodynamic convolution but not downsampling. NeuroImage 65, 540–555 (2013)CrossRefGoogle Scholar
  56. 56.
    Simpson, S.L., Hayasaka, S., Laurienti, P.J.: Exponential random graph modeling for complex brain networks. PLoS One 6(5), e20039 (2011)CrossRefGoogle Scholar
  57. 57.
    Simpson, S.L., Moussa, M.N., Laurienti, P.J.: An exponential random graph modeling approach to creating group-based representative whole-brain connectivity networks. NeuroImage 60(2), 1117–1126 (2012)CrossRefGoogle Scholar
  58. 58.
    Sporns, O., Chialvo, D.R., Kaiser, M., Hilgetag, C.C.: Organization, development and function of complex brain networks. Trends Cognit. Sci. 8(9), 418–425 (2004)CrossRefGoogle Scholar
  59. 59.
    Sporns, O., Zwi, J.D.: The small world of the cerebral cortex. Neuroinformatics 2(2), 145–162 (2004)CrossRefGoogle Scholar
  60. 60.
    Stam, C.J.: Modern network science of neurological disorders. Nat. Rev. Neurosci. 15(10), 683–95 (2014)CrossRefGoogle Scholar
  61. 61.
    Stam, C., De Haan, W., Daffertshofer, A., Jones, B., Manshanden, I., Van Walsum, A.V.C., Montez, T., Verbunt, J., De Munck, J., Van Dijk, B., et al.: Graph theoretical analysis of magnetoencephalographic functional connectivity in Alzheimer’s disease. Brain 132(1), 213–224 (2009)CrossRefGoogle Scholar
  62. 62.
    Stam, C.J., Reijneveld, J.C.: Graph theoretical analysis of complex networks in the brain. Nonlinear Biomed. Phys. 1(1), 3 (2007)CrossRefGoogle Scholar
  63. 63.
    Tzourio-Mazoyer, N., Landeau, B., Papathanassiou, D., Crivello, F., Etard, O., Delcroix, N., Mazoyer, B., Joliot, M.: Automated anatomical labeling of activations in SPM using a macroscopic anatomical parcellation of the MNI MRI single-subject brain. NeuroImage 15(1), 273–289 (2002)CrossRefGoogle Scholar
  64. 64.
    Vincent, J.L., Patel, G.H., Fox, M.D., Snyder, A.Z., Baker, J.T., van Essen, D.C., Zempel, J.M., Snyder, L.H., Corbetta, M., Raichle, M.E.: Intrinsic functional architecture in the anaesthetized monkey brain. Nature 447(7140), 83–86 (2007)CrossRefGoogle Scholar
  65. 65.
    Vuksanović, V., Hövel, P.: Functional connectivity of distant cortical regions: role of remote synchronization and symmetry in interactions. NeuroImage 97, 1–8 (2014)CrossRefGoogle Scholar
  66. 66.
    Vuksanović, V., Hövel, P.: Dynamic changes in network synchrony reveal resting-state functional networks. Chaos Interdiscip. J. Nonlinear Sci. 25(2), 023116-1–023116-9 (2015)Google Scholar
  67. 67.
    Vuksanović, V., Hövel, P.: Large-scale neural network model for functional networks of the human cortex. In: Pelster, A., Wunner, G. (eds.) Self-organization in complex systems: the past. present, and future of synergetics, pp. 345–352. Springer, Berlin (2016)Google Scholar
  68. 68.
    Vuksanović, V., Hövel, P.: Role of structural inhomogeneities in resting-state brain dynamics. Cognit. Neurodyn. 10(4), 361–365 (2016)Google Scholar
  69. 69.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393, 440–442 (1998)CrossRefGoogle Scholar
  70. 70.
    Wiener, N.: Cybernetics: Or, Control and Communication in the Animal and the Machine. MIT press, Cambridge (1961)CrossRefzbMATHGoogle Scholar
  71. 71.
    Zhang, L.I., Poo, M.M.: Electrical activity and development of neural circuits. Nat. Neurosc. 4, 1207–1214 (2001)CrossRefGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2017

Authors and Affiliations

  1. 1.Max-Planck Institute for Human Cognitive and Brain SciencesLeipzigGermany
  2. 2.Institute of Theoretical PhysicsTechnische Universität BerlinBerlinGermany
  3. 3.Bernstein Center for Computational Neuroscience BerlinHumboldt-Universität zu BerlinBerlinGermany
  4. 4.Aberdeen Biomedical Imaging CentreUniversity of AberdeenAberdeenUK

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