Skip to main content

Determining Network Structure from Data: Nonlinear Modeling Methods

  • Reference work entry
  • First Online:
  • 76 Accesses

Definition

In this entry we focus on inference of network structures from data. One possible approach to studying networks is to model the nodes, such as neurons, and generate networks and run simulations and observe the network behavior. This approach requires on a priori assumptions about the constituent parts; for instance, Hodgkin-Huxley neurons may be coupled and the resulting network behavior is investigated. The model behaviors can be compared to the measured neuronal signals through statistical analysis. However, this approach only provides indirect information about the network structure. An alternative approach to studying network structure is to use parametric, semiparametric, and nonparametric analyses of the observed signals and reconstruct the network connections. Such analyses are essential tools for systems in which network structure is not known, such as when anatomical connections are not known or information is not sufficient.

A particular challenge when inferring...

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   2,499.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  • Baccala LA, Sameshima K (2001) Partial directed coherence: a new concept in neural structure determination. Biol Cybern 84:463–474

    CAS  PubMed  Google Scholar 

  • Bandrivskyy A, Bernjak A, McClintock PVE, Stefanovska A (2004) Wavelet phase coherence analysis: application to skin temperature and blood flow. Cardiovasc Eng 4:89–93

    Google Scholar 

  • Boccaletti S, Kurths J, Osipov G, Valladares DL, Zhou CS (2002) The synchronization of chaotic systems. Phys Rep 366:1–101

    CAS  Google Scholar 

  • Dahlhaus R (2000) Graphical interaction models for multivariate time series. Metrika 51:157–172

    Google Scholar 

  • Ding M, Chen Y, Bressler SL (2006) Granger causality: basic theory and application to neuroscience. In: Schelter B, Winderhalder M, Timmer J (eds) Handbook of time series analysis. Wiley-VCH, Weinheim, pp 437–460

    Google Scholar 

  • Eckmann J-P, Oliffson Kamphorst S, Ruelle D (1987) Recurrence plots of dynamical systems. Europhys Lett 4:973–977

    Google Scholar 

  • Eichler M (2005) A graphical approach for evaluating effective connectivity in neural systems. Phil Trans R Soc B 360:953–967

    PubMed Central  PubMed  Google Scholar 

  • Eichler M (2007) Granger-causality and path diagrams for multivariate time series. J Econom 137:334–353

    Google Scholar 

  • Granger CWJ (1969) Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37:424–438

    Google Scholar 

  • Hellwig B, Haeussler S, Schelter B, Lauk M, Guschlbauer B, Timmer J, Luecking CH (2001) Tremor correlated cortical activity in essential tremor. Lancet 357:519–523

    CAS  PubMed  Google Scholar 

  • Hellwig B, Schelter B, Guschlbauer B, Timmer J, Luecking CH (2003) Dynamic synchronisation of central oscillators in essential tremor. Clin Neurophysiol 114:1462–1467

    CAS  PubMed  Google Scholar 

  • Kantz H, Schreiber T (1997) Nonlinear time series analysis. Cambridge University Press, Cambridge

    Google Scholar 

  • Kocarev L, Parlitz U (1996) Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys Rev Lett 76:1816–1819

    CAS  PubMed  Google Scholar 

  • Kralemann B, Pikovsky A, Rosenblum M (2011) Reconstructing phase dynamics of oscillator networks. Chaos 21:025104

    PubMed  Google Scholar 

  • Le van Quyen M, Martinerie J, Navarro V, Boon P, D’Have M, Adam C, Renault B, Varela F, Baulac M (2001) Anticipation of epileptic seizures from standard EEG recordings. Lancet 357:183–188

    Google Scholar 

  • Louis ED, Ford B, Wendt KJ, Cameron G (1998) Clinical characteristics of essential tremor: data from a community-based study. Mov Disord 13:803–808

    CAS  PubMed  Google Scholar 

  • Luetkepohl H (1993) Introduction to multiple time series analysis. Springer, New York

    Google Scholar 

  • Mardia K, Jupp P (2000) Directional statistics. Wiley, West Sussex

    Google Scholar 

  • Marwan N, Carmen Romano M, Thiel M, Kurths J (2007) Recurrence plots for the analysis of complex systems. Phys Rep 438:237–329

    Google Scholar 

  • Mormann F, Lehnertz K, David P, Elger CE (2000) Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients. Physica D 144:358–369

    Google Scholar 

  • Nawrath J, Romano MC, Thiel M, Kiss IZ, Wickramasinghe M, Timmer J, Kurths J, Schelter B (2010) Distinguishing direct and indirect interactions in oscillatory networks with multiple time scales. Phys Rev Lett 104:038701

    PubMed  Google Scholar 

  • Osipov GV, Hu B, Zhou C, Ivanchenko MV, Kurths J (2003) Three types of transitions to phase synchronization in chaotic oscillators. Phys Rev Lett 91:024101

    PubMed  Google Scholar 

  • Packard N, Crutchfield J, Farmer D, Shaw R (1980) Geometry from a time series. Phys Rev Lett 45:712

    Google Scholar 

  • Palus M, Stefanovska A (2003) Direction of coupling from phases of interacting oscillators: an information theoretic approach. Phys Rev E 67:055201

    Google Scholar 

  • Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64:821–824

    PubMed  Google Scholar 

  • Pikovsky A, Rosenblum M, Kurths J (2000) Phase synchronization in regular and chaotic systems. Int J Bifurc Chaos 10:2291–2305

    Google Scholar 

  • Pikovsky A, Rosenblum M, Kurths J (2001) Synchronization – a universal concept in nonlinear sciences. Cambridge University Press, Cambridge

    Google Scholar 

  • Poincare H (1890) Sur les equations de la dynamique et le probleme de trois corps. Acta Mathematica 13:1–270

    Google Scholar 

  • Roessler OE (1976) An equation for continuous chaos. Phys Lett A 57:397–398

    Google Scholar 

  • Romano MC, Thiel M, Kurths J, Kiss IZ, Hudson JL (2005) Detection of synchronization for non-phase-coherent and non-stationary data. Europhys Lett 71:466–472

    CAS  Google Scholar 

  • Rosenblum MG, Pikovsky AS, Kurths J (1996) Phase synchronization of chaotic oscillators. Phys Rev Lett 76:1804–1807

    CAS  PubMed  Google Scholar 

  • Rosenblum MG, Pikovsky AS, Kurths J (1997) From phase to lag synchronization in coupled chaotic oscillators. Phys Rev Lett 78:4193–4196

    CAS  Google Scholar 

  • Rosenblum MG, Pikovsky A, Kurths J, Schaefer C, Tass PA (2001) Phase synchronization: from theory to data analysis. In: Moss F, Gielen S (eds) Handbook of biological physics, vol 4, Neuroinformatics. Elsevier, Amsterdam, pp 279–321

    Google Scholar 

  • Runge J, Heitzig J, Marwan N, Kurths J (2012) Quantifying causal coupling strength: a lag-specific measure for multivariate time series related to transfer entropy. Phys Rev E 86:061121

    Google Scholar 

  • Sauer T, Yorke J, Casdagli M (1991) Embedology. J Stat Phys 65:579–616

    Google Scholar 

  • Schelter B, Winterhalder M, Timmer J (eds) (2006a) Handbook of time series analysis. Wiley-VCH, Berlin

    Google Scholar 

  • Schelter B, Winterhalder M, Dahlhaus R, Kurths J, Timmer J (2006b) Partial phase synchronization for multivariate synchronizing system. Phys Rev Lett 96:208103

    PubMed  Google Scholar 

  • Schelter B, Winterhalder M, Eichler M, Peifer M, Hellwig B, Guschlbauer B, Luecking CH, Dahlhaus R, Timmer J (2006c) Testing for directed influences among neural signals using partial directed coherence. J Neurosci Methods 152:210–219

    PubMed  Google Scholar 

  • Smirnov D, Schelter B, Winterhalder M, Timmer J (2007) Revealing direction of coupling between neuronal oscillators from time series: phase dynamics modeling versus partial directed coherence. Chaos 17:013111

    PubMed  Google Scholar 

  • Takens F (1981) Detecting strange attractors in turbulence. In: Rand DA, Young L-S (eds) Dynamical systems and turbulence (Warwick 1980), vol 898, Lecture notes in mathematics. Springer, Berlin, pp 366–381

    Google Scholar 

  • Tass PA, Rosenblum MG, Weule J, Kurths J, Pikovsky A, Volkmann J, Schnitzler A, Freund HJ (1998) Detection of n : m phase locking from noisy data: application to magnetoencephalography. Phys Rev Lett 81:3291–3295

    CAS  Google Scholar 

  • van der Pol B (1922) On oscillation-hysteresis in a simple triode generator. Phil Mag 43:700–719

    Google Scholar 

  • Volkmann J, Joliot M, Mogilner A, Ioannides AA, Lado F, Fazzini E, Ribary U, Llinas R (1996) Central motor loop oscillations in Parkinsonian resting tremor revealed by magnetoencephalography. Neurology 46:1359–1370

    CAS  PubMed  Google Scholar 

  • Wibral M, Wollstadt P, Meyer U, Pampu N, Priesemann V, Vicente R (2012) Revisiting Wiener‘s principle of causality – interaction-delay reconstruction using transfer entropy. In: Proceedings of the 34th annual international conference of the IEEE EMBS (EMBC 2012), San Diego

    Google Scholar 

  • Wiesenfeldt M, Parlitz U, Lauterborn W (2001) Mixed state analysis of multivariate time series. Int J Bifurc Chaos 11:2217–2226

    Google Scholar 

Further Reading

  • Hellwig B, Haeussler S, Lauk M, Koester B, Guschlbauer B, Kristeva-Feige R, Timmer J, Luecking CH (2000) Tremor-correlated cortical activity detected by electroencephalography. Electroencephalogr Clin Neurophysiol 111:806–809

    CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bjoern Schelter .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media New York

About this entry

Cite this entry

Schelter, B., Thiel, M. (2015). Determining Network Structure from Data: Nonlinear Modeling Methods. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6675-8_439

Download citation

Publish with us

Policies and ethics