Article | Published:

Role of graph architecture in controlling dynamical networks with applications to neural systems

Nature Physics volume 14, pages 9198 (2018) | Download Citation

Abstract

Networked systems display complex patterns of interactions between components. In physical networks, these interactions often occur along structural connections that link components in a hard-wired connection topology, supporting a variety of system-wide dynamical behaviours such as synchronization. Although descriptions of these behaviours are important, they are only a first step towards understanding and harnessing the relationship between network topology and system behaviour. Here, we use linear network control theory to derive accurate closed-form expressions that relate the connectivity of a subset of structural connections (those linking driver nodes to non-driver nodes) to the minimum energy required to control networked systems. To illustrate the utility of the mathematics, we apply this approach to high-resolution connectomes recently reconstructed from Drosophila, mouse, and human brains. We use these principles to suggest an advantage of the human brain in supporting diverse network dynamics with small energetic costs while remaining robust to perturbations, and to perform clinically accessible targeted manipulation of the brain’s control performance by removing single edges in the network. Generally, our results ground the expectation of a control system’s behaviour in its network architecture, and directly inspire new directions in network analysis and design via distributed control.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

References

  1. 1.

    Networks: An Introduction (Oxford Univ. Press, 2010).

  2. 2.

    The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003).

  3. 3.

    & Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).

  4. 4.

    The architecture of complexity. Proc. Am. Phil. Soc. 10, 467–482 (1962).

  5. 5.

    & Network neuroscience. Nat. Neurosci. 20, 353–364 (2017).

  6. 6.

    , , & Functional structure of cortical neuronal networks grown in vitro. Phys. Rev. E 75, 021915 (2007).

  7. 7.

    & Small-world brain networks revisited. Neuroscientist (2016).

  8. 8.

    & Modular brain networks. Annu. Rev. Psychol. 67, 613–640 (2016).

  9. 9.

    , & Network medicine: a network-based approach to human disease. Nat. Rev. Genet. 12, 56–68 (2011).

  10. 10.

    , & Distributed control in a mean-field cortical network model: implications for seizure suppression. Phys. Rev. E 86, 021920 (2012).

  11. 11.

    , , , & Virtual cortical resection reveals push-pull network control preceding seizure evolution. Neuron 91, 1170–1182 (2016).

  12. 12.

    et al. Intra-operative multi-site stimulation: expanding methodology for cortical brain mapping of language functions. PLoS ONE 12, e0180740 (2017).

  13. 13.

    & Optical techniques in optogenetics. J. Mod. Opt. 62, 949–970 (2015).

  14. 14.

    & Classifying and quantifying basins of attraction. Chaos 25, 083101 (2015).

  15. 15.

    , & Realistic control of network dynamics. Nat. Commun. 4, 1942 (2013).

  16. 16.

    , & Temporal metastates are associated with differential patterns of time-resolved connectivity, network topology, and attention. Proc. Natl Acad. Sci. USA 113, 9888–9891 (2016).

  17. 17.

    et al. Dynamic network centrality summarizes learning in the human brain. J. Complex Netw. 1, 83–92 (2013).

  18. 18.

    , , , & Cross-linked structure of network evolution. Chaos 24, 013112 (2014).

  19. 19.

    Mathematical description of linear dynamical systems. J. SIAM Control Ser. A 1, 152–192 (1963).

  20. 20.

    Structural controllability. IEEE Trans. Autom. Control 19, 201–208 (1974).

  21. 21.

    , & Controllability of complex networks. Nature 473, 167–173 (2011).

  22. 22.

    & Control profiles of complex networks. Science 343, 1373–1376 (2014).

  23. 23.

    et al. Controllability of structural brain networks. Nat. Commun. 6, 8414 (2015).

  24. 24.

    et al. Optimal trajectories of brain state transitions. Neuroimage 148, 305–317 (2017).

  25. 25.

    , , , & Optimally controlling the human connectome: the role of network topology. Sci. Rep. 6, 30770 (2016).

  26. 26.

    et al. Stimulation-based control of dynamic brain networks. PLoS Comput. Biol. 12, e1005076 (2016).

  27. 27.

    et al. A mesoscale connectome of the mouse brain. Nature 508, 207–214 (2014).

  28. 28.

    , , & Wiring cost and topological participation of the mouse brain connectome. Proc. Natl Acad. Sci. USA 112, 10032–10037 (2015).

  29. 29.

    et al. Connectomics-based analysis of information flow in the Drosophila brain. Curr. Biol. 25, 1249–1258 (2015).

  30. 30.

    Linear Systems (Prentice-Hall, 1980).

  31. 31.

    On how network architecture determines the dominant patterns of spontaneous neural activity. PLoS ONE 3, e2148 (2008).

  32. 32.

    et al. Predicting human resting-state functional connectivity from structural connectivity. Proc. Natl Acad. Sci. USA 106, 2035–2040 (2009).

  33. 33.

    , , , & Modeling the impact of lesions in the human brain. PLoS Comput. Biol. 5, e1000408 (2009).

  34. 34.

    , & Modification of central metabolic pathway in Escherichia coli to reduce acetate accumulation by heterologous expression of the Bacillus subtilis acetolactate synthase gene. Biotechnol. Bioeng. 44, 944–951 (1994).

  35. 35.

    & CRISPR-Cas systems for editing, regulating and targeting genomes. Nat. Biotechnol. 32, 347–355 (2014).

  36. 36.

    Networkcontrology. Chaos 25, 097621 (2015).

  37. 37.

    , & Controllability metrics, limitations and algorithms for complex networks. IEEE Trans. Control Netw. Syst. 1, 40–52 (2014).

  38. 38.

    et al. Emotion induction after direct intracerebral stimulations of human amygdala. Cereb. Cortex 17, 1307–1313 (2007).

  39. 39.

    , , & Sparse overlapping group lasso for integrative multi-omics analysis. J. Comput. Biol. 22, 73–84 (2015).

  40. 40.

    , , , & Sparse inverse covariance estimation with L0 penalty for network construction with omics data. J. Comput. Biol. 23, 192–202 (2016).

  41. 41.

    , & Hierarchical structure and the prediction of missing links in networks. Nature 453, 98–101 (2008).

  42. 42.

    & An information-theoretic model for link prediction in complex networks. Sci. Rep. 5, 13707 (2015).

  43. 43.

    et al. Harnessing plasticity for the treatment of neurosurgical disorders: an overview. World Neurosurg. 82, 648–659 (2014).

  44. 44.

    & Noninvasive brain stimulation in the treatment of aphasia: exploring interhemispheric relationships and their implications for neurorehabilitation. Restor. Neurol. Neurosci. 29, 375–394 (2011).

  45. 45.

    , & Emerging frontiers of neuroengineering: a network science of brain connectivity. Annu. Rev. Biomed. Eng. 19, 327–352 (2017).

  46. 46.

    et al. Neuromodulation for brain disorders: challenges and opportunities. IEEE Trans. Biomed. Eng. 60, 610–624 (2013).

  47. 47.

    , , & Neurobiologically realistic determinants of self-organized criticality in networks of spiking neurons. PLoS Comput. Biol. 7, e1002038 (2011).

  48. 48.

    et al. Adaptation to sensory input tunes visual cortex to criticality. Nat. Phys. 11, 659–663 (2015).

  49. 49.

    , , & Mapping the functional connectome in traumatic brain injury: what can graph metrics tell us? Neuroimage S1053–8119, 30694–30692 (2016).

  50. 50.

    et al. Altered wiring of the human structural connectome in adults with mild traumatic brain injury. J. Neurotrauma 34, 1035–1044 (2017).

  51. 51.

    Random Graphs (Academic, 1985).

  52. 52.

    , , & Resolving structural variability in network models and the brain. PLoS Comput. Biol. 10, e1003491 (2014).

  53. 53.

    , & Classification of weighted networks through mesoscale homological features. J. Complex Netw. 5, 245–273 (2017).

  54. 54.

    et al. Six networks on a universal neuromorphic computing substrate. Front Neurosci. 7, 11 (2013).

  55. 55.

    , , , & Topological and geometric measurements of force-chain structure. Phys. Rev. E 94, 032909 (2016).

Download references

Acknowledgements

J.Z.K. acknowledges support from National Institutes of Health T32-EB020087, PD: F. W. Wehrli, and the National Science Foundation Graduate Research Fellowship No. DGE-1321851. J.M.S. and D.S.B. acknowledge support from the John D. and Catherine T. MacArthur Foundation, the Alfred P. Sloan Foundation, the US Army Research Laboratory and the US Army Research Office through contract numbers W911NF-10-2-0022 and W911NF-14-1-0679, the National Institute of Health (2-R01-DC-009209-11, 1R01HD086888-01, R01-MH107235, R01-MH107703, R01MH109520, 1R01NS099348 R21-M MH-106799, and T32-EB020087), the Office of Naval Research, and the National Science Foundation (BCS-1441502, CAREER PHY-1554488, BCS-1631550, and CNS-1626008). A.E.K. and J.M.V. acknowledge support from the US Army Research Laboratory contract number W911NF-10-2-0022. F.P. acknowledges support from the National Science Foundation (BCS-1430280 and BCS 1631112). The content is solely the responsibility of the authors and does not necessarily represent the official views of any of the funding agencies.

Author information

Affiliations

  1. Department of Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

    • Jason Z. Kim
    • , Jonathan M. Soffer
    • , Jean M. Vettel
    •  & Danielle S. Bassett
  2. Department of Neuroscience, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

    • Ari E. Kahn
  3. US Army Research Laboratory, Aberdeen, Maryland 21001, USA

    • Ari E. Kahn
  4. Human Research & Engineering Directorate, US Army Research Laboratory, Aberdeen, Maryland 21001, USA

    • Jean M. Vettel
  5. Department of Psychological and Brain Sciences, University of California, Santa Barbara, California 93106, USA

    • Jean M. Vettel
  6. Department of Mechanical Engineering, University of California, Riverside, California 92521, USA

    • Fabio Pasqualetti
  7. Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

    • Danielle S. Bassett

Authors

  1. Search for Jason Z. Kim in:

  2. Search for Jonathan M. Soffer in:

  3. Search for Ari E. Kahn in:

  4. Search for Jean M. Vettel in:

  5. Search for Fabio Pasqualetti in:

  6. Search for Danielle S. Bassett in:

Contributions

J.Z.K., D.S.B. and F.P. wrote and revised the bulk of the manuscript. J.Z.K. developed the mathematical framework and analysed the data with feedback from F.P. and D.S.B. J.M.S. collected the human diffusion data, and A.E.K. processed the data to produce structural connectivity matrices with support from J.M.V.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Danielle S. Bassett.

Supplementary information

PDF files

  1. 1.

    Supplementary information

    Supplementary information

About this article

Publication history

Received

Accepted

Published

DOI

https://doi.org/10.1038/nphys4268

Further reading